tag:blogger.com,1999:blog-37414104180967168272021-05-09T10:53:41.582-07:00Unizor - Creative Mind through Art of MathematicsUnizor is a site where students can learn the high school math (and in the future some other subjects) in a thorough and rigorous way. It allows parents to enroll their children in educational programs and to control the learning process.Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.comBlogger469125tag:blogger.com,1999:blog-3741410418096716827.post-80440584448687747992021-05-09T10:53:00.001-07:002021-05-09T10:53:40.112-07:00Energy of Oscillation: UNIZOR.COM - Physics4Teens - Waves - Mechanical O...<iframe width="480" height="270" src="https://youtube.com/embed/L22abaR9mio" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Energy of Oscillation</u><br /><br/>Let's consider an object of mass <i><b>m</b></i> on an ideal spring of elasticity <i><b>k</b></i> in ideal conditions (no gravity, no friction, no air resistance etc.)<br/><br/>What happens from the energy viewpoint, when we stretch this spring by a distance <i><b>a</b></i> from its neutral position?<br/>Obviously, we supply it with some potential energy.<br/><br/>The object on a spring's free end will have this potential energy and, when we let a spring go, a spring will pull the object towards a neutral position, increasing its speed and, therefore, its kinetic energy.<br/><br/>The potential energy, meanwhile, is diminishing since the spring retracts towards its neutral position.<br/>At the moment of crossing the neutral position an object has no potential energy, all its energy is converted into kinetic energy.<br/><br/>When an object moves further, squeezing a spring, it slows down, while squeezing a spring further and further, loses it kinetic energy, but increases potential energy, since a spring is squeezed more and more.<br/><br/>At the extreme position of a squeezed spring all the energy is again potential. An object momentarily stops at this point, having no speed and, therefore, no kinetic energy.<br/><br/>Then the oscillation continues in the opposite direction with similar transformation of energy from potential to kinetic and then back to potential.<br/><br/>Of course, total amount of energy, potential plus kinetic, should remain constant because of the Law of Energy Conservation.<br/><br/>Let's calculate the potential energy we give to an object on a spring by initially stretching a spring by a distance <i><b>a</b></i> from its neutral position.<br/><br/>According to the Hooke's Law, stretching a spring by an infinitesimal distance from position <i><b>x</b></i> to position <i><b>x+</b>d<b>x</b></i> requires a force <i><b>F(x)</b></i> proportional to <i><b>x</b></i> with a coefficient of proportionality <i><b>k</b></i> that depends on the properties of a spring called elasticity.<br/><br/>On the distance <i>d<b>x</b></i> this force does some infinitesimal amount of work that is equal to<br/><i>d<b>W(x) = F(x)·</b>d<b>x = k·x·</b>d<b>x</b></i>.<br/>Integrating this infinitesimal amount of work on a segment from <i><b>x=0</b></i> to <i><b>x=a</b></i>, we will obtain the total amount of work <i><b>W(a)</b></i> we have to spend to stretch a spring by a distance <i><b>a</b></i> from its neutral position.<br/>This amount of work is the amount of potential energy <i><b>U(a)</b></i> we supply to an object on a stretched spring.<br/><i><b>U(a) = <font size=5>∫</font></b></i><sub>[0,a]</sub><i><b>k·x·</b>d<b>x = k·a²/2</b></i><br/><br/>This formula for potential energy is true for any displacement <i><b>a</b></i>. This displacement can be positive (stretching) or negative (squeezing), the potential energy is always positive or zero (for <i><b>a=0</b></i> at the neutral point).<br/><br/>Now we can easily find a speed of an object <i><b>v<sub><sub>0</sub></sub></b></i> when it crosses the neutral position. This is the object's maximum speed, since potential energy at this point is zero and all energy is kinetic, which is proportional to a square of its velocity. Its kinetic energy at this point must be equal to the above value <i><b>U(a)</b></i>. At the same time, if its speed is <i><b>v<sub><sub>0</sub></sub></b></i>, its kinetic energy is <i><b>E<sub>0</sub>=m·v<sub>0</sub>²/2</b></i>.<br/>Therefore,<br/><i><b>U(a) = m·v<sub><sub>0</sub></sub>²/2</b></i><br/><i><b>k·a²/2 = m·v<sub><sub>0</sub></sub>²/2</b></i><br/><i><b>|v<sub><sub>0</sub></sub>| = √<span style='text-decoration:overline'>k/m</span>·a = ω·a</b></i><br/>where <i><b>ω = √<span style='text-decoration:overline'>k/m</span></b></i> is the same parameter used in expressing the harmonic oscillations in a form <i><b>x(t)=a·cos(ωt)</b></i>.<br/><br/>In the above expression we used absolute value <i><b>|v<sub><sub>0</sub></sub>|</b></i> because this speed is positive when an object moves from a squeezed position to a stretched one and negative in an opposite direction.<br/><br/>Using this approach we can find an object's velocity <i><b>v<sub><sub>d</sub></sub></b></i> at any distance <i><b>d</b></i> from the neutral position.<br/>The potential energy of an object in this case is <i><b>U(d)=k·d²/2</b></i>.<br/>Its kinetic energy is <i><b>E(d)=m·v<sub><sub>d</sub></sub>²/2</b></i>.<br/>Since the total energy is supposed to be equal the potential energy at initial position <i><b>U(a)=k·a²/2</b></i>, the kinetic energy equals to<br/><i><b>E(d) = U(a) − U(d) =<br/>= k·a²/2 − k·d²/2</b></i><br/>From this we can find <i><b>v<sub><sub>d</sub></sub></b></i>:<br/><i><b>m·v<sub><sub>d</sub></sub>²/2 = k·a²/2 − k·d²/2</b></i><br/><i><b>|v<sub><sub>d</sub></sub>| = √<span style='text-decoration:overline'>(k/m)·(a²−d²)</span></b></i><br/><i><b>|v<sub><sub>d</sub></sub>| = ω·√<span style='text-decoration:overline'>a²−d²</span></b></i><br/><br/>The above formula for <i><b>|v<sub><sub>d</sub></sub>|</b></i> corresponds to speed being equal to zero at the extreme position of an object at distance <i><b>a</b></i> from the neutral point and it being maximum at a neutral point, where <i><b>d=0</b></i>.<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-18238907815526209012021-05-07T08:40:00.000-07:002021-05-07T08:40:05.341-07:00Rotational Oscillation: UNIZOR.COM - Physics4Teens - Waves - Mechanical ...<iframe width="480" height="270" src="https://youtube.com/embed/xVB97wm9il8" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Rotational Oscillation</u><br /><br/><i>Rotational oscillations</i> (also called <i>torsional oscillations</i>) can be observed in movements of a balance wheel inside hand watches. It rotates, that's why it's <i>rotational</i>, and it moves along the same trajectory back and forth, that's why it's <i>oscillation</i>.<br/><br/>Another example might be a weightless horizontal rod with two identical weights at its opposite ends hanging on a vertical steel wire attached to a rod's midpoint.<br/><img src='http://www.unizor.com/Pictures/RotationalOscillation.png' style='width:200px;height:200px;'><br/>If we wind up the horizontal rod, as shown on the picture, and let it go, it will create a tension in the twisted wire that will start untwisting, returning the rod into its original position, then winding in an opposite direction etc., thus oscillating rotationally.<br/><br/>Recall the concept of a <i>torque</i> for rotational movement<br/><i><b>τ = R·F</b></i><br/>In a simple case of a force acting perpendicularly to a radius (the only case we will consider) the above can be interpreted just as a multiplication. In a more general case, assuming both force and radius are vectors, the above represents a vector product of these vectors, making a torque also a vector.<br/><br/>While the tension force of a twisted steel wire <i><b>F<sub>1</sub></b></i> might be significant, it acts on a very small radius of a wire <i><b>r</b></i>, so the force <i><b>F<sub>2</sub></b></i>, acting on each of two weights on opposite sides of a rod of radius <i><b>R</b></i> and having the same torque <i><b>τ</b></i>, is proportionally weaker<br/><i><b>τ = r·F<sub>1</sub> = R·F<sub>2</sub></b></i><br/>from which follows<br/><i><b>F<sub>1</sub> <font size=4>/</font>F<sub>2</sub> = R <font size=4>/</font>r</b></i><br/>and<br/><i><b>F<sub>2</sub> = (r<font size=4>/</font>R)·F<sub>1</sub> = τ <font size=4>/</font>R</b></i><br/><br/>Dynamics of reciprocating (back and forth) movement are expressed in terms of <i>inertial mass <b>m</b></i>, <i>force <b>F</b></i> and <i>acceleration <b>a</b></i> by the Second Newton's Law<br/><i><b>F = m·a</b></i><br/><br/>In case of a rotational movement with a radius of rotation <i><b>R</b></i> the dynamics are expressed in terms of <i>moment of inertia <b>I=m·R²</b></i>, <i>torque <b>τ=R·F</b></i> and <i>angular acceleration <b>α=a/R</b></i> by the rotational equivalent of the Second Newton's Law<br/><i><b>τ = I·α</b></i><br/><br/>There is a rotational equivalent of a Hooke's Law. It relates a <i>torque <b>τ</b></i> and an <i>angular displacement <b>φ</b></i> from a neutral (untwisted) position<br/><i><b>τ = −k·φ</b></i><br/><br/>For rotational oscillations an <i>angular displacement <b>φ</b></i> is a function of time <i><b>φ(t)</b></i>. <i>Angular acceleration <b>α</b></i> is a second derivative of an <i>angular displacement <b>φ(t)</b></i>.<br/><br/>Therefore, we can equate the <i>torque</i> expressed according to the rotational equivalent of the Second Newton's Law to the one expressed according to the rotational equivalent of the Hooke's law, getting an equation<br/><i><b>I·α = −k·φ</b></i><br/>or<br/><i><b>I·φ"(t) = −k·φ(t)</b></i><br/>or<br/><i><b>φ"(t) = −(k<font size=4>/</font>I)·φ(t)</b></i><br/><br/>The differential equation above is of the same type as for an oscillations of a weight on a spring discussed in the previous lecture. The only difference is that, instead of a mass of an object <i><b>m</b></i> we use <i>moment of inertia <b>I=m·R²</b></i>.<br/><br/>For initial angular displacement (initial twist) of a steel wire <i><b>φ(0)=γ</b></i> and no initial angular speed (<i><b>φ'(0)=0</b></i>) the solution to this equation is<br/><i><b>φ(t) = γ·cos(√<span style='text-decoration:overline'>k/I</span>·t)</b></i><br/><br/>The rotational oscillations in our case have a period (the shortest time the object returns to its original position)<br/><i><b>T = 2π·√<span style='text-decoration:overline'>I/k</span></b></i><br/><br/>Frequency of rotational oscillations <i>f = 1/T</i>.<br/>Therefore, our rod with two weights makes<br/><i><b>f = 1/T = (1/2π)·√<span style='text-decoration:overline'>k/I</span></b></i><br/>oscillations per second.<br/><br/>Since <i><b>I=m·R²</b></i>, the period is greater (and the frequency is smaller) when objects are more massive and on a greater distance from a center of a rod, where the wire is attached.<br/><br/>Notice that a period and a frequency of these oscillations are not dependent on initial angle of turning the rod <i><b>φ(0)=γ</b></i>. This parameter <i><b>γ</b></i> defines only the amplitude of oscillations, but not their period and frequency.<br/><br/>This is an important factor used, for example, in watch making with a balance wheel oscillating based on its physical characteristics and an elasticity of a spiral spring.<br/>No matter how hard you wind a spring (or how weak it becomes after it worked for awhile), a balance wheel will maintain the same period and frequency of its oscillations.<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-66891543219280952602021-05-04T09:11:00.000-07:002021-05-04T09:11:37.330-07:00Periodic Movement: UNIZOR.COM - Physics4Teens - Waves - Mechanical Oscil...<iframe width="480" height="270" src="https://youtube.com/embed/whDkuLMjL9c" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Periodic Movement</u><br /><br/>Mechanics, as a subject, deals with movements of different objects. Among these movements there are those that we can call "repetitive". Examples of these repetitive movements, occurring during certain time segment, are rotation of a carousel, swinging of a pendulum, vibration of a musical tuning fork, etc. Here we are talking about certain time segment during which these movements are repetitive, because after some time these movements are changing, if left to themselves.<br/><br/>These repetitive movements might be of a kind when the repetitions are to a high degree exactly similar to each other (like in case of a pendulum) or some of the characteristics of the motion change in time (like in case of a tuning fork).<br/><br/>Repetitive movements that can be divided into equal time segments, during which the movements to a high precision repeat exactly each other, are called <i>periodic</i>.<br/>The time segments of such a repetitive movement are called <i>periods</i>.<br/><br/>If a position <i><b>P</b></i> of an object making periodic movement with a period <i><b>T</b></i> is defined by a set of Cartesian coordinates <i><b>P=(x</b>,<b>y</b>,<b>z)</b></i> as a vector function of time <i><b>P(t)</b></i>, the periodicity means that for any time moment <i><b>t</b></i><br/><i><b>P(t) = P(t+T)</b></i><br/>which is exactly the mathematical definition of a periodic function.<br/><br/>For example, a <i>period</i> of rotational movement of a carousel equals to a time it takes to make one circle. The period of a movement of a pendulum is the time it moves from left most position all the way to the right most and back to the left.<br/><br/>A case of a vibrating tuning fork is a bit more complex because gradually the vibrations, after being initiated, diminish with time. The <i>period</i> of vibration might be the same during this process, but the amplitude (deviation from a middle point) would diminish with time.<br/><br/>A special type of <i>periodic</i> movement is <i>oscillation</i>. It's characterized by a periodic movement of an object that repeats the same trajectory of movement in alternating directions, back and forth. For example, a pendulum, an object on a spring, a tuning fork, a buoy on a surface of water under ideal weather conditions etc.<br/><br/>In all those systems we can observe a specific middle point position from which an object can deviate in both directions. If put initially at this position, an object would remain there, unless some external force acts on it. This is a point of a <i>stable equilibrium</i>. Then, after some external force is applied, it will move along its trajectory back and forth, each time passing this <i>equilibrium</i> point.<br/><br/>From this point an object can move along a trajectory to some extreme position, then back through an <i>equilibrium</i> point to another extreme position, then back again, repeating a movement along the same trajectory in alternating directions.<br/><br/><i>Oscillation</i> is only possible if some external forces act on a moving object towards <i>stable equilibrium</i> point. Otherwise, it would never return to an <i>equilibrium</i>. These forces must depend on the position, not acting at the equilibrium point, acting in one direction in case an object deviated from an equilibrium to one side along its trajectory and acting in the opposite direction in case an object deviated to the other side along a trajectory.<br/><br/>A very important type of <i>oscillations</i> are so-called <i>harmonic oscillations</i>.<br/>An example of this type of a movement is an object on an initially stretched (or squeezed) spring with the only force acting on an object during its movement to be the spring's elasticity.<br/><br/>According to the Hooke's Law, the force of elasticity of a spring is proportional to its stretch or squeeze length and directed towards a neutral point of no stretch nor squeeze.<br/><br/>If a string is positioned along the X-axis on a Cartesian system of coordinates with one end fixed to some point with negative coordinate on this axis, while its neutral point at <i><b>x=0</b></i>, the position of an object attached to this spring and oscillating can be described as a function of time <i><b>x(t)</b></i> that satisfies two laws:<br/><br/>(1) the Second Newton's Law connecting the force of elasticity <i><b>F(t)</b></i> to the <b>mass</b> <i><b>m</b></i> and acceleration (second derivative of position)<br/><i><b>F(t) = m·x"(t) = m·</b>d²<b>x(t)/</b>d<b>x²</b></i><br/><br/>(2) the Hooke's Law connecting the force of elasticity <i><b>F</b></i> with a displacement of a free end of a spring from its neutral position<br/><i><b>F(t) = −k·x(t)</b></i><br/>(where <i><b>k</b></i> is a <b>coefficient of elasticity</b> that is a characteristic of a spring).<br/><br/>From these two equations we can exclude the force <i><b>F(t)</b></i> and get a simple differential equation that defines the position of an object at the free end of a spring <i><b>x(t)</b></i>.<br/><i><b>m·x"(t) = m·</b>d²<b>x(t)/</b>d<b>x² = −k·x(t)</b></i><br/>or<br/><i><b>x"(t) = −(k/m)·x(t)</b></i><br/><br/>Obviously, trigonometric functions <i>sin(t)</i> and <i>cos(t)</i> are good candidates for a solution to this equation since their second derivative looks like the original function with some coefficients<br/><i><b>sin"(t) = -sin(t)</b></i><br/><i><b>cos"(t) = -cos(t)</b></i><br/><br/>General solution to the above linear differential equation is<br/><i><b>x(t) = C<sub>1</sub>·cos(ωt) + C<sub>2</sub>·sin(ωt)</b></i><br/>where <i><b>ω</b></i> depends on coefficients of the differential equation and constants <i><b>C<sub>1</sub></b></i> and <i><b>C<sub>2</sub></b></i> depend on initial conditions (initial displacement of the object off the neutral position on a spring and its initial speed).<br/><br/>Then<br/><i><b>x'(t) = −C<sub>1</sub>·ω·sin(ωt) +<br/>+ C<sub>2</sub>·ω·cos(ωt)</b></i><br/><i><b>x"(t) = −C<sub>1</sub>·ω²·cos(ωt) −<br/>− C<sub>2</sub>·ω²·sin(ωt)</b></i><br/><br/>Since<br/><i><b>x"(t) = −(k/m)·x(t)</b></i><br/>we conclude that<br/><i><b>−(k/m)·x(t) = −C<sub>1</sub>·ω²·cos(ωt) −<br/>− C<sub>2</sub>·ω²·sin(ωt)</b></i><br/>or<br/><i><b>−(k/m)·[C<sub>1</sub>·cos(ωt) +<br/>+ C<sub>2</sub>·sin(ωt)] =<br/>−C<sub>1</sub>·ω²·cos(ωt) −<br/>− C<sub>2</sub>·ω²·sin(ωt)</b></i><br/>from which immediately follows<br/><i><b>ω = √<span style='text-decoration:overline'>k/m</span></b></i><br/><br/>Assume, initially we stretch a spring by a distance <i><b>a</b></i> from the neutral position (that is, <i><b>x(0)=a</b></i>) and let it go without any push (that is, <i><b>x'(0)=0</b></i>).<br/>From these initial conditions we can derive the values of constants <i><b>C<sub>1</sub></b></i> and <i><b>C<sub>2</sub></b></i><br/><i><b>a = x(0) =<br/>= C<sub>1</sub>·cos(0) + C<sub>2</sub>·sin(0) = C<sub>1</sub></b></i><br/><i><b>0 = x'(0) =<br/>= −C<sub>1</sub>·ω·sin(0) + C<sub>2</sub>·ω·cos(0) =<br/>= C<sub>2</sub>·ω</b></i><br/>from which immediately follows<br/><i><b>C<sub>1</sub> = a</b></i><br/><i><b>C<sub>2</sub> = 0</b></i><br/>and the solution for our differential equation with given initial conditions is<br/><i><b>x(t) = a·cos(√<span style='text-decoration:overline'>k/m</span>·t)</b></i><br/><br/>The oscillations described by the above function <i><b>x(t)</b></i> in its general form <i><b>x(t)=a·cos(ω·t)</b></i> are called <i>simple harmonic oscillations</b></i>.<br/><br/>Parameter <i><b>a</b></i> characterizes the <b>amplitude</b> of harmonic oscillations, while parameter <b>ω</b></i> represents the <b>angular speed</b> of oscillations.<br/>Function <i>cos(t)</i> is periodical with a period <i>T=2π</i>.<br/>Function <i>cos(ωt)</i> is also periodical with a period <i>T=2π/ω</i>.<br/>Indeed,<br/><i>cos(ω(t+T)) =<br/>= cos(ω(t+2π/ω)) =<br/>= cos(ωt+2π) =<br/>= cos(ωt)</i><br/>Therefore, the simple harmonic oscillations in our case have a period (the shortest time the object returns to its original position)<br/><i><b>T = 2π/ω = 2π·√<span style='text-decoration:overline'>m/k</span></b></i><br/><br/>If one full cycle the oscillation process makes in time <i>T</i>, we can find how many cycles it makes in a unit of time (1 sec) using a simple proportion<br/><i>1</i> cycle - <i>T</i> sec<br/><i>f</i> cycles - <i>1</i> sec<br/>Hence, <i>f = 1/T</i><br/>Therefore, the object on a spring we deal with makes<br/><i><b>f = 1/T = (1/2π)·√<span style='text-decoration:overline'>k/m</span></b></i><br/>oscillations per second.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-34789489862362617592021-04-26T08:36:00.000-07:002021-04-26T08:36:12.411-07:00Transistors: UNIZOR.COM - Physics4Teens - Electromagnetism - Semiconduct...<iframe width="480" height="270" src="https://youtube.com/embed/zrlI1bPPtpM" frameborder="0"></iframe>New lecture is released to UNIZOR.COM.<br/>It's about transistors and the principles of using semiconductors in amplifying an analogue signal or working as ON/OFF switch.<br/>UNIZOR.COM - Physics 4 Teens - Electromagnetism - Semiconductors in Electronics - Junction Transistors<br/><br/><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Bipolar Junction Transistors</u><br /><br/>Below is a schematic representation of a so-called <b>n-p-n bipolar junction transistor</b> - one of the most popular types.<br/><img src='http://www.unizor.com/Pictures/Transistor.png' style='width:200px;height:200px;'><br/>Functionally, it's playing a role similar to triodes discussed in one of the previous lectures. It can amplify a signal and it can work as On/Off switch.<br/><br/>To understand how it works, we will "build" it, step by step, gradually introducing new components.<br/><br/>As the first step in this process, imagine the device above without a narrow layer of <i>p-type</i> semiconductor in the middle and wiring connecting it to a battery (that is, without components marked in blue color).<br/><br/>Then this device would represent a solid <i>n-type</i> semiconductor built, let's assume, from silicon foundation with added atoms of phosphorus that have excess of one valence electron for each atom, which does not fit into a crystalline structure of silicon.<br/><br/>This <i>n-type</i> semiconductor is attached to a battery through two electrodes.<br/>Let's assume that the difference in electric potential (voltage) between electrodes is sufficient to attract extra valence electrons of phosphorus (held only by electrostatic attraction to a nucleus, but not participating in <i>covalent bonds</i>) toward a positive electrode, but it's not strong enough to rip off other valence electrons participating in both the <i>covalent bonds</i> and electrostatic attraction to a nucleus, as electrons of silicon atoms are.<br/><br/>When such voltage is present between the electrodes, there is a small current going through a semiconductor, as the extra valence electrons from phosphorus atoms will be attracted to a positive electrode, rendering these phosphorus atoms positively charged, while the negative electrode will compensate the loss of electrons in the body of a semiconductor.<br/>The current, obviously, depends on density of phosphorus atoms within silicon foundation and voltage applied.<br/><br/>The next step is to split the solid <i>n-type</i> semiconductor in two halves and put a layer of <i>p-type</i> semiconductor (let's assume its silicon with boron additive) in between two halves of <i>n-type</i> semiconductor (thin blue layer on the picture above represents this <i>p-type</i> semiconductor). Let's not connected this <i>p-type</i> semiconductor to a battery yet.<br/><br/>This will stop the current and here is why.<br/>As before, the positive electrode will attract extra valence electrons of phosphorus atoms from the <i>n-type</i> semiconductor (the right side on the picture above).<br/><br/>The other side of <i>n-type</i> semiconductor, connected to a negative electrode, will have its extra valence electrons of phosphorus penetrating the <i>n-p junction</i> into <i>p-type</i> semiconductor. Both random moving of these free electrons and repelling off the negative electrode are the factors in this process. Crossed the <i>n-p junction</i>, they will be captured by holes in the <i>covalent bonds</i> of <i>p-type</i> semiconductor.<br/><br/>After some time there will be a saturation of the <i>covalent bonds</i> in the <i>p-type</i> semiconductor and the flow of electrons from the <i>n-type</i> to <i>p-type</i> stops, as the barrier of extra electrons in the <i>p-type</i> layer would repel electrons from the <i>n-type</i> connected to a negative electrode.<br/><br/>At this time the <i>covalent bonds</i> of the <i>p-type</i> layer will be filled, which renders it to be negatively charged. Extra electrons around boron nuclei will be held only by <i>covalent bonds</i>, but not attracted to a nucleus.<br/>The <i>n-type</i> connected to a negative electrode will be neutral, as the electrons crossed to the <i>p-type</i> layer through <i>n-p junction</i> will be compensated from the negative electrode.<br/>The <i>n-type</i> connected to a positive electrode will be positively charged, as its free valence electrons, not participating in <i>covalent bonds</i>, will be consumed by a positive electrode.<br/><br/>Now we will introduce the last component of this device - connect the <i>p-type</i> layer to a positive electrode of another battery and the negative electrode of this battery connect to the <i>n-type</i> semiconductor that already has a connection to a negative electrode of another battery.<br/>Thus this <i>n-type</i> semiconductor will be connected to negative electrodes of both batteries.<br/><br/>What's important now is that the electrons, crossed into <i>p-type</i> layer through <i>n-p junction</i>, making it negatively charged and making a barrier on the migration of new electrons from the <i>n-type</i> semiconductor connected to a negative electrode, will be attracted by a positive electrode of a new battery. These electrons are held only by <i>covalent bonds</i> inside the <i>p-type</i> layer, not the electrostatic attraction to a nucleus. Positive electrode of a new battery will attract some of them from the place where they acted as a barrier at the <i>n-p junction</i>, the barrier weakens, and new electrons from the <i>n-type</i> semiconductor could penetrate the <i>n-p junction</i> barrier into <i>p-type</i> layer.<br/><br/>The process does not stop here. Since the barrier is weakened, the electrons from <i>n-type</i> semiconductor connected to a negative electrode continue migrating to <i>p-type</i> layer. That's why this <i>n-type</i> semiconductor is called <b>emitter</b>. Some of these electrons go further through <i>p-n junction</i> between a <i>p-type</i> layer and the other <i>n-type</i> semiconductor connected to a positive electrode, establishing an electric current between two initial electrodes. That's why that other <i>n-type</i> semiconductor is called <b>collector</b>. The semiconductor of <i>p-type</i>, making the layer between <b>emitter</b> and <b>collector</b> is called <b>base</b>.<br/><br/>By changing the voltages on two batteries involved and by changing the amounts of additives into <b>emitter</b>, <b>collector</b> and <b>base</b> we can control the current between <b>emitter</b> and <b>collector</b>.<br/><br/>Under some conditions the <b>n-p-n bipolar junction transistor</b>, whose principles of work are described above, can act as an amplifier of the signal between <b>emitter</b> and <b>base</b> into a stronger signal between <b>emitter</b> and <b>collector</b>.<br/><br/>Under some other conditions <b>n-p-n bipolar junction transistor</b> can act as the On/Off switch, opening or closing a circuit between <b>emitter</b> and <b>collector</b> by applying some voltage between <b>emitter</b> and <b>base</b>.<br/><br/>There are other ways to connect <b>emitter</b>, <b>collector</b>, <b>base</b> and batteries that we will not consider, as our purpose is to introduce a concept, rather than going into details of implementation. The development of contemporary transistors, their theoretical and technological aspects took a lot of efforts and time, so now we have a pretty advanced devices. But the principles of their work are still the same, those we demonstrated on the example presented above.<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-27796240657733820982021-04-22T07:49:00.000-07:002021-04-22T07:49:06.152-07:00p-n Junction Diodes: UNIZOR.COM - Physics4Teens - Electromagnetism - Sem...<iframe width="480" height="270" src="https://youtube.com/embed/p-cpRth8AXE" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>p-n Junction Diodes</u><br /><br/>Consider a semiconductor of <i>n-type</i>, like <b>silicon</b> (<i><b>Si</b></i>) with <b>phosphorus</b> (<i><b>P</b></i>) additive.<br/>Atoms of <i><b>P</b></i> with its five valence electrons are embedded in the crystalline structure of <i><b>Si</b></i> with four of these valence electrons falling nicely into a structure, making <i>covalent bonds</i> with four neighboring atoms of <i><b>Si</b></i>, while the fifth one cannot get a pair to establish a <i>covalent bond</i>, so it's rather loosely attached to its atom of <i><b>P</b></i> and might randomly travel around.<br/><img src='http://www.unizor.com/Pictures/SiliconPhosphorus.jpg' style='width:200px;height:200px;'><br/><br/>During this process of randomly fluctuating electrons within an <i>n-type</i> of a semiconductor the whole structure remains electrically neutral because these traveling negatively charged electrons are balanced by extra protons in atoms of phosphorus.<br/><br/>The number of these traveling electrons is not great, so, if we apply a low voltage to a semiconductor, we can observe some low conductivity, but increasing the voltage will show that the conductivity is greater. The mechanism of this conductivity is based on a combination of electrostatic forces and forces of covalent bonds that maintain a crystalline structure of a semiconductor.<br/><br/>Freely traveling excess electrons of an <i>n-type</i> semiconductor will be electrostatically attracted by a positive electrode that has deficiency of electrons. Then the body of a previously electrically neutral semiconductor becomes positively charged, since it lost some electrons. Consequently, electrons from a negative electrode will enter the semiconductor, will not be captured by covalent bonds and become free traveling ones, replacing those consumed by a positive electrode.<br/><br/>This process of electrons moving from a negative electrode through an <i>n-type</i> semiconductor to a positive electrode will continue, as long as the voltage between electrodes is maintained at sufficient level.<br/><br/>Now consider a <i>p-type</i> semiconductor, like <b>silicon</b> (<i><b>Si</b></i>) with <b>boron</b> (<i><b>B</b></i>) additive.<br/>Atoms of <i><b>B</b></i>, embedded in the crystalline structure of <i><b>Si</b></i>, have their three valence electrons bonded to valence electrons of three surrounding atoms of <i><b>Si</b></i>, while one valence electron of the fourth neighboring atom of <i><b>Si</b></i> cannot get a pair to establish a covalent bond with atom of <i><b>B</b></i>, thus a hole in the crystalline structure is formed.<br/><img src='http://www.unizor.com/Pictures/SiliconBoron.jpg' style='width:200px;height:200px;'><br/>Any neighboring atom of <i><b>Si</b></i>, if properly excited, can have its valence electron to jump from its own atom to fill the gap. That leaves a hole in another spot of a crystalline structure. Then some other electron from yet another atom can fill the new hole, opening a hole in yet another area. So, holes in the crystalline structure are randomly traveling inside a <i>p-type</i> of semiconductor.<br/><br/>This process of randomly moving holes in the crystalline structure of <i>p-type</i> semiconductor is analogous to the above described random moving of electrons in the <i>n-type</i> semiconductor.<br/><br/>It also assures that, if a sufficient voltage is applied to this type of semiconductor, the conductivity can be observed. The mechanism of this conductivity is based on a combination of electrostatic forces and forces of covalent bonds that maintain a crystalline structure of a semiconductor.<br/><br/>Consider holes in the crystalline structure caused by a deficiency of one valence electron of atoms of <i><b>B</b></i>. There are electrons randomly traveling from a negative electrode into a semiconductor and back. Some electrons that happened to be in the vicinity of a hole will be captured by the forces of covalent bonds and fill the holes in the crystalline structure.<br/>Then the semiconductor will become negatively charged, that is will have excess of electrons. The excess electrons will be attracted by a positive electrode and new holes will be created in the crystalline structure.<br/><br/>This process of electrons moving from a negative electrode through a <i>p-type</i> semiconductor to a positive electrode and, correspondingly, holes moving in the opposite direction will continue, as long as the voltage between electrodes is maintained at sufficient level.<br/><br/>Next topic is to analyze the behavior of electrons and holes at the border or <i>junction</i> between the <i>n-type</i> semiconductor and the <i>p-type</i> one. This border area is called <i>p-n junction</i>.<br/><br/>Initially electrically neutral, a semiconductor on the <i>n-type</i> side of a junction has electrons that do not fit in its crystalline structure and not involved in the covalent bonding.<br/>On the <i>p-type</i> side of a junction, also initially electrically neutral, there are holes in the crystalline structure of a semiconductor waiting to capture some electron into a covalent bonding.<br/><br/>The first consideration is related to activity related to the forces of covalent bonds. Randomly traveling electrons from the <i>n-type</i> semiconductor can jump through a junction into the <i>p-type</i> semiconductor and be captured by covalent bonds to restore missing electrons of a crystalline structure of a semiconductor in the area adjacent to the junction.<br/><br/>That, in turn, creates a layer of negative charge near the junction on the <i>p-type</i> side and a positively charged layer near the junction on the <i>n-type</i> side.<br/><br/>Excess electrons kept by covalent bonds near the junction on the <i>p-type</i> side act as an electrostatic barrier to prevent new electrons from the <i>n-type</i> side to jump the junction.<br/>Some kind of equilibrium is developed between electrons on the <i>n-type</i> side not involved in the crystalline structure and ready to jump the junction and electrons already jumped to the <i>p-type</i> side.<br/><br/>The covalent bonds are restored on the <i>p-type</i> side and there are no electrons not linked through covalent bonds on the <i>n-type</i> side of the junction. The price for this proper crystalline structure on both sides of a junction is that there is an excess electrons on the <i>p-type</i> side and deficiency of electrons on the <i>n-type</i> side.<br/><br/>Let's connect the <i>p-type</i> side of a junction to a positive electrode and the <i>n-type</i> to a negative electrode of a battery.<br/>Excess electrons from the <i>p-type</i> will be attracted to a positive electrode and the barrier of electrons on the <i>p-type</i> side that prevented new electrons from the <i>n-type</i> to jump the junction will no longer be there.<br/><br/>This will cause new electrons from the <i>n-type</i> side to be able to jump the junction into <i>p-type</i>. In parallel, electrons from the negative electrode will be attracted to a positively charged <i>n-type</i> side, compensating electrons that have already jumped to the <i>p-type</i> side.<br/><br/>As we see, there will be an electric current through a <i>p-n junction</i>, as long as the voltage is maintained.<br/><br/>Now let's connect the electrodes in an opposite way. Positive to <i>n-type</i> side and negative to the <i>p-type</i>.<br/>Electrons that do not fit a crystalline structure on the <i>n-type</i> will be attracted to a positive electrode and will no longer participate in the process.<br/>New electrons from the negative electrode will penetrate the body of the <i>p-type</i> side filling all the covalent bonds and reinforcing the electrons' barrier near a junction to completely stop all flow of electrons from the <i>n-type</i> side.<br/><br/>Now the crystalline structure on both sides of a <i>p-n junction</i> are complete and no new electrons are traveling in any direction. There is no current through a <i>p-n junction</i>.<br/><br/>As we see, the electrons can go through a <i>p-n junction</i> only if <i>n-type</i> side is connected to a negative electrode and <i>p-type</i> side is connected to a positive electrode. So, <i>p-n junction</i> acts like a <b>diode</b>. This configuration of <i>p-type</i> and <i>n-type</i> semiconductors joined together and acting as a <b>diode</b> is called <i><b>p-n junction diode</b></i>.<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-44667435535879681362021-04-17T10:08:00.003-07:002021-04-17T10:10:41.315-07:00Covalent Bonds: UNIZOR.COM - Physics4Teens - Electromagnetism - Theory o...<iframe width="480" height="270" src="https://youtube.com/embed/_sTtoZbE3Io" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Covalent Bonds</u><br /><br/>Now we know that the theory of semiconductors is built upon introducing some impurity into crystalline structure of such a material as <i>silicon (Si)</i> or <i>germanium (Ge)</i>.<br/>Let's consider this crystalline structure in more details to understand its underlying principles.<br/><br/>According to the orbital model of an atom, electrons are rotating around a nucleus on certain orbits, each orbit representing certain energy level possessed by all electrons on it. There are certain principles related to quantum mechanics and wave theory of elementary particles that prescribe electrons to rotate only on certain distinct radiuses around a nucleus. The reason for this we will discuss when addressing the atomic structure in more details in a corresponding part of this course.<br/><br/>On every distinct radius the rotating electrons experience the attraction of the protons in a nucleus and repelling of other electrons rotating on the same orbit (on the same radius from a nucleus). Obviously, for every distinct radius of an orbit there is a maximum number of electrons that can rotate around a nucleus without pushing each other so strongly that one of them must jump out of the orbit to, most likely, the orbit of a larger radius or even outside of the atom.<br/><br/>Calculations based on principles of quantum mechanics show that this maximum number for the first (closest to a nucleus) orbit is 2, for the next - 8, next - 18 etc. The formula <i><b>N=2n²</b></i> where <i><b>n</b></i> is the orbit number and <i><b>N</b></i> is the maximum number of electrons on this orbit describes this in many (but not all) cases.<br/><br/>The atom of every element has certain number of protons in a nucleus and the same number of electrons rotating on different orbits. These orbits are packed with electrons not to exceed their maximum capacity.<br/><br/>In a simplified model of an atom of carbon <i><b>C<sup>12</sup></b></i> with 6 protons and 6 neutrons in a nucleus there are 6 electrons on two orbits: 2 on the closest to a nucleus and 4 on the next one.<br/><img src='http://www.unizor.com/Pictures/Atom_C.jpg' style='width:200px;height:230px;'><br/><br/>In a simplified model of an atom of silicon <i><b>Si<sup>28</sup></b></i> with 14 protons and 14 neutrons in a nucleus there are 14 electrons on three orbits: 2 on the closest to a nucleus, 8 on the next one and 4 on the outer most orbit.<br/><br/><img src='http://www.unizor.com/Pictures/Atom_Si.jpg' style='width:200px;height:200px;'><br/><br/>In a simplified model of an atom of titanium <i><b>Ti<sup>48</sup></b></i> with 22 protons and 26 neutrons in a nucleus there are 22 electrons on four orbits with, correspondingly, 2, 8, 10 and 2 electrons on them.<br/><img src='http://www.unizor.com/Pictures/Atom_Ti.jpg' style='width:200px;height:220px;'><br/><br/>In a simplified model of an atom of radon <i><b>Rn<sup>222</sup></b></i> with 86 protons and 136 neutrons in a nucleus there are 86 electrons on five orbits with, correspondingly, 2, 8, 18, 32, 18 and 8 electrons on them.<br/><img src='http://www.unizor.com/Pictures/Atom_Rn.jpg' style='width:200px;height:230px;'><br/><br/>Electrons populating the outer most orbit are called <i>valence</i> electrons. While in theory belonging to particular atoms, they are very active in their relationships with other atoms and their <i>valence</i> electrons.<br/><br/>In fact, valence electrons from different atoms might bond together, thus two atoms have a shared pair of electrons. The reason to this is a tendency to fill up the orbit of valence electrons. The magic number of valence electrons is 2 for atoms with only one orbit (there are only two elements with a single orbit of electrons - hydrogen and helium) or 8 for atoms with two or more orbits, so two or more atoms might come into bonding by sharing the valence electrons, thus filling all outer orbits up to a magic number of electrons (shared and not shared).<br/><br/>Here is an example of a molecule of methane that consists of one atom of carbon <i><b>C<sup>12</sup></b></i> (two orbits with 2 electrons on an inner orbit, not shown on a picture, and 4 valence electrons) and four atoms of hydrogen <i><b>H<sup>1</sup></b></i> (single orbit with 1 electron on it).<br/><img src='http://www.unizor.com/Pictures/Covalent_CH4.png' style='width:200px;height:240px;'><br/>As we see, the orbit with valence electrons of an atom of carbon has now 8 electrons, 4 its own and 4 shared with atoms of hydrogen, thus filling a magic number of valence electrons. At the same time each atom of hydrogen has 2 electrons (also a magic number for an atom of hydrogen), 1 its own and one shared with an atom of carbon.<br/><br/>These <i>covalent bonds</i> between atoms are the basis for certain different atoms to combine into molecules or the same atoms to form a crystalline structure.<br/><br/>Consider now a crystalline structure of a <i>silicon</i>, our main subject of discussion.<br/>Each atom of silicon contains 4 valence electrons on its outer (third) orbit. In three-dimensional world these electrons are positioned at the vertices of a tetrahedron with a nucleus of an atom at the center of this tetrahedron.<br/>Four valence electrons of one atom of silicon pair with valence electrons of neighboring atoms to form a complicated three-dimensional crystalline structure that in two-dimensional representation looks as follows.<br/><img src='http://www.unizor.com/Pictures/Silicon_2D.png' style='width:200px;height:200px;'><br/>Four neighboring silicon atoms contribute to a center one their valence electrons to share (one from each neighbor), thus the center atom has a magic number of electrons on its outer orbit. Each valence electron of each atom is shared with a neighboring atom and, therefore, every atom has 8 electrons (including both its own 4 electrons and 4 from its four neighbors, one from each). This makes a strong crystalline structure of material.<br/><br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-27927577250336271032021-04-08T14:04:00.000-07:002021-04-08T14:04:27.939-07:00Theory of Semiconductors: UNIZOR.COM - Physics4Teens - Electromagnetism ...<iframe width="480" height="270" src="https://youtube.com/embed/k69pBz0XN1g" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Theory of Semiconductors</u><br /><br/><i>Conductivity</i> is the quality of a material to conduct electricity. Since electricity is carried by electrons, it's important how "free" electrons are in the material and how easy it is for negatively charged electrons to fly away from attraction of a positively charged nucleus of an atom.<br/><br/>Electrons can circle around a nucleus on certain stationary orbits (more precisely, narrow <i>bands</i> of orbits). So, to fly away from a nucleus, electron needs certain amount of energy to jump over a gap from one orbit to a higher one and, eventually, to fly away. If the distance between bands of an atom is minimal, electrons will jump over the gap between them easier and, potentially, the conductivity of this type of a material is better. If this gap is large, conductivity will be lower or none at all.<br/><br/>It happens that most metals have a very small gap between the bands of electron orbits, so electrons can relatively free jump from one band to a higher one and, eventually, fly away from their nuclei to other nuclei, thus making enough free electrons to facilitate the electric current. That's why metals are good <i>conductors</i> of electricity.<br/><br/>On another hand, diamonds have atoms with large gap between the bands of electron orbits, which makes it very difficult for electrons to escape the attraction of their nuclei. It's not impossible, but needs a lot of energy to excite them, enabling a long jump from one band to the next. This makes diamond and other elements with large gap between the bands <i>insulators</i>.<br/><br/>Finally, there is an "in-between" kind of elements with gaps between bands of electron orbits larger than in metals, but still not large enough, so relatively small energy supplied to electrons (heat, light etc.) will make them conductors, while without this extra energy they are insulators. These elements are <i>semiconductors</i>.<br/><br/>Let's start with the most important example of a semiconductor - silicon (<i><b>Si</b></i>).<br/>Silicon is a metalloid, it has crystalline structure and its atoms have four valence electrons on the outer most orbit, next down orbit contains eight electrons and the inner most - two electrons.<br/><br/>Silicon is one of the most frequently occurring elements on Earth and, together with oxygen, forms molecules of silicon dioxide <i><b>SiO<sub>2</sub></b></i> - main component of sand.<br/><br/>All silicon atoms are neutral since the number of electrons in each one and the number of protons in each nucleus are the same and equal to 14.<br/>Atoms are connected by their covalent bonds of valence electrons on the outer orbits into a lattice-like crystalline structure.<br/><br/>Picture below represents the crystalline structure of silicon<br/><img src='http://www.unizor.com/Pictures/SiliconCrystal.jpg' style='width:200px;height:200px;'><br/>For the purposes of this lecture we will represent inner structure of silicon atoms with only outer orbit of each atom with four valence electrons, as electrons on the inner orbits do not participate in the process of generating electricity from light.<br/><img src='http://www.unizor.com/Pictures/SiliconStructure.jpg' style='width:200px;height:200px;'><br/>The covalent bonds are quite strong, they do not easily release electrons. As a result, under normal conditions silicon is practically a dielectric.<br/>However, if we excite the electrons sufficiently enough to break the covalent bonds, some electrons will move. For example, if we increase the temperature of a piece of silicon or put it under a bright sun light and measure its electrical resistance, we will observe the resistance diminishing.<br/>That's why silicon and similar elements are called <b>semiconductors</b>.<br/><br/>While excited electrons of any semiconductor decrease its electrical resistance, they don't produce electromotive force because the material as a whole remains electrically neutral.<br/>To build a semiconductor that conducts electricity we will introduce two kinds of "impurities" into crystalline lattice of silicon.<br/>One is an element with five valence electrons on the outer most orbit, like phosphorus (<i><b>P</b></i>). When it's embedded into a crystalline lattice of silicon, one electron on that outer orbit of phosphorus would remain not attached to any neighboring atom through a covalent bond.<br/><img src='http://www.unizor.com/Pictures/SiliconPhosphorus.jpg' style='width:200px;height:200px;'><br/>This creates the possibility for this electron to start traveling, replacing other electrons and pushing them out, which are, in turn, push out others etc. Basically, we create as many freelance electrons as many atoms of phosphorus we add to silicon base.<br/>The whole material is still neutral, but it has certain number of freelance electrons and the same number of stationary positive ions - those nuclei of phosphorus that lost electrons to freelancers.<br/>Silicon with such addition is called <b><i>n-type</i></b> (letter <b>n</b> for <i>negative</i>).<br/><br/>Another type of "impurity" that we will add to silicon is an element with three valence electrons on the outer orbit, like boron (<i><b>B</b></i>).<br/>When atoms of boron are embedded into a crystalline lattice of silicon, the overall structure of this combination looks like this<br/><img src='http://www.unizor.com/Pictures/SiliconBoron.jpg' style='width:200px;height:200px;'><br/>In this case the lattice has a deficiency of an electron that is traditionally called a "hole". Existence of a "hole" opens the opportunity for neighboring valence electrons to fill it, thus creating a "hole" in another spot. These "holes" behave like positively charged particles traveling inside silicon with added boron inasmuch as negatively charged electrons travel in silicon with added phosphorus.<br/>Silicon with such addition of boron is called <b><i>p-type</i></b> (letter <b>p</b> for <i>positive</i>).<br/><br/>Now imagine two types of "impure" silicon, <b><i>n-type</i></b> and <b><i>p-type</i></b>, contacting each other. In practice, it's two flat pieces (like very thin squares) on top of each other.<br/>Let's examine what happens in a thin layer of border between these different types of material.<br/><br/>Initially, both pieces of material are electrically neutral with <i>n-type</i> having free traveling electrons and equal number of stationary positive nuclei of phosphorus inside a crystalline structure and with <i>p-type</i> having traveling "holes" and equal number of stationary positive nuclei of boron inside a crystalline structure.<br/><br/>As soon as contact between these two types of material is established, certain exchange between electrons of the <i>n-type</i> material and "holes" of the <i>p-type</i> takes place in the border region called <b>p-n junction</b>. Electrons and "holes" in this border region combine, thus reconstituting the lattice.<br/>This process is called <b>recombination</b>.<br/><br/>The consequence of this process of recombination is that <i>n-type</i> material near the border loses electrons, thus becoming positively charged, while the <i>p-type</i> gains the electrons, that is loses "holes", thus becoming negatively charged.<br/><br/>Eventually, the diffusion between <i>n-type</i> and <i>p-type</i> materials stops because negative charge of the <i>p-type</i> sufficiently repels electrons from the <i>n-type</i>. There will be some equilibrium between both parts.<br/><br/>These qualities of <i>n-type</i> and <i>p-type</i> semiconductors, as well as the processes occurring in a thin layer between these two types of semiconductors allow to use them in electronics. <i>Transistors</i> are electronic devices made of semiconductors and functionally equivalent to electronic devices described in the previous section and made using vacuum tubes.<br/>The details of usage semiconductors in electronics will be presented in the next lectures.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-41657396247936790592021-03-20T07:47:00.002-07:002021-03-20T07:47:41.995-07:00Flip-Flop (Latch): UNIZOR.COM - Physics4Teens - Electromagnetism - Elect...<iframe width="480" height="270" src="https://youtube.com/embed/SA7zFQmxwZM" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Implementation of Electronic<br/>Flip Flop (Latch)</u><br/><br/>In the beginning of computer era one of the first problem was to be able to store information. And the first step was to design a circuit that can store one bit of information, 1 or 0.<br/><br/>A digital <b>flip-flop</b> or <b>latch</b> is such a one bit storage, an electronic device that<br/>(a) has two states (1 and 0);<br/>(b) can be set to one of these two states by a proper impulse;<br/>(c) retains the state after an impulse is gone.<br/><br/>The main component of a design we will discuss in this lecture is a combination of a binary logical operation <b><i>OR</i></b> and a unary logical operation <b><i>NOT</i></b>, both of which were discussed in previous lectures.<br/><br/>Let's recall the implementation of the logical <b><i>OR</i></b> circuit that involves diodes. <img src='http://www.unizor.com/Pictures/Logic_OR.png' style='width:200px;height:200px;'><br/><br/>The implementation of the <b><i>NOT</i></b> logical operation that involves triodes was suggested as follows <img src='http://www.unizor.com/Pictures/Logic_NOT_triode.png' style='width:200px;height:270px;'><br/><br/>Combining them sequentially, we first implement <b><i>OR</i></b> and then <b><i>NOT</i></b> of the result of <i><b>OR</b></i>, thus implementing the binary logical operation <b><i>NOR</i></b> as follows<br/><b><i>NOT(A OR B) = ¬(A|B)</i></b><br/><img src='http://www.unizor.com/Pictures/Logic_NOR_gate.png' style='width:200px;height:140px;'><br/><br/>The logical <i><b>NOR</b></i> operation is denoted by the following symbol<br/><img src='http://www.unizor.com/Pictures/NOR_symbol.jpg' style='width:200px;height:100px;'><br/>The table below denotes the results of <i><b>NOR</b></i> operation for all pairs of input values of arguments<br/><i><b>¬(0|0) = ¬0 = 1</b></i><br/><i><b>¬(0|1) = ¬1 = 0</b></i><br/><i><b>¬(1|0) = ¬1 = 0</b></i><br/><i><b>¬(1|1) = ¬1 = 0</b></i><br/><br/>Using the <i><b>NOR</b></i> gate, we will implement a circuit that can serve as a one bit memory cell.<br/><br/>Consider the following schema of a circuit called <b>SR flip-flop</b> or <b>SR latch</b>.<br/><img src='http://www.unizor.com/Pictures/SR_FlipFlop.png' style='width:200px;height:200px;'><br/>Here terminals <i><b>R</b></i> (Reset) and <i><b>S</b></i> (Set) are controlling terminals, using which we store information.<br/><br/>Information is stored in a pair of terminals <i><b>Q</b></i> and its opposite <i><b>NOT(Q)=¬Q</b></i> in a way that either <i><b>Q=1</b></i> (and <i><b>¬Q=0</b></i>) or <i><b>Q=0</b></i> (and <i><b>¬Q=1</b></i>).<br/><br/>These two states are stable in a sense that, after we send a positive impulse at terminal <i><b>S</b></i>, while <i><b>R=0</b></i>, the <i><b>Q</b></i> will be equal to <i><b>1</b></i> irrespective of its prior value, the <i><b>¬Q</b></i> will be equal to <i><b>0</b></i> irrespective of its prior value, and they both will retain the same values <i><b>Q=1</b></i> and <i><b>¬Q=0</b></i> after the impulse at <i><b>S</b></i> is gone (that is when <i><b>S=0</b></i>).<br/><br/>Similarly, after we send a positive impulse at terminal <i><b>R</b></i>, while <i><b>S=0</b></i>, the <i><b>Q</b></i> will be equal to <i><b>0</b></i> irrespective of its prior value, the <i><b>¬Q</b></i> will be equal to <i><b>1</b></i> irrespective of its prior value, and they both will retain the same values <i><b>Q=0</b></i> and <i><b>¬Q=1</b></i> after the impulse at <i><b>R</b></i> is gone (that is when <i><b>R=0</b></i>).<br/><br/>Let's examine the work of this SR latch in detail.<br/><br/><br/><i>Set <b>Q=1</b> and <b>¬Q=0</b></i><br/><br/>Assume, we set the voltage at terminal <i><b>S</b></i> to a positive value, keeping terminal <i><b>R</b></i> at zero potential. It means in logical sense that <i><b>S=1</b></i> and <i><b>R=0</b></i>.<br/>The <i><b>NOR gate</b></i> connected to terminal <i><b>S</b></i>, irrespective of its second connection to it, will produce a signal <i><b>0</b></i> as an output. Indeed, the <i><b>OR</b></i> operation between <i><b>S=1</b></i> and any other signal produces <i><b>1</b></i> and <i><b>NOT(1)=0</b></i>.<br/><br/>This immediately sets <i><b>¬Q=0</b></i> and send zero potential to the <i><b>NOR gate</b></i> connected to terminal <i><b>R</b></i> that has zero potential. So, both inputs to <i><b>NOR gate</b></i> connected to <i><b>R</b></i> terminal are at zero potential. The logical <i><b>OR</b></i> of two zeroes is zero. Subsequent <i><b>NOT</b></i> converts the signal to <i><b>1</b></i> that sets terminal <i><b>Q=1</b></i>.<br/>So, raising the potential at terminal <i><b>S</b></i> to a positive value, keeping terminal <i><b>R</b></i> at zero potential switches terminals <i><b>Q</b></i> and <i><b>¬Q</b></i> to, correspondingly, values <i><b>1</b></i> and <i><b>0</b></i>.<br/><br/>Let's see what happens when the positive impulse at terminal <i><b>S</b></i> is gone, so <i><b>S=0</b></i>, while terminal <i><b>R</b></i> is still at zero potential, terminal <i><b>Q=1</b></i> and <i><b>¬Q=0</b></i>.<br/><br/>The <i><b>NOR gate</b></i> connected to terminal <i><b>S</b></i> has one input from terminal <i><b>S=0</b></i>, while the other input is from terminal <i><b>Q</b></i> that was set to <i><b>1</b></i> on a previous step. The output from this <i><b>NOR gate</b></i> will be zero, since the logical <i><b>OR</b></i> between <i><b>0</b></i> and <i><b>1</b></i> gives <i><b>1</b></i>, and subsequent logical <i><b>NOT</b></i> converts it to <i><b>0</b></i>.<br/><br/>This sets <i><b>¬Q=0</b></i>, but this terminal already had this value, so <i><b>¬Q</b></i> does not change and remains at <i><b>0</b></i>.<br/><br/>From there the value <i><b>¬Q=0</b></i> goes to the input of the <i><b>NOR gate</b></i> connected to terminal <i><b>R</b></i> that still has zero potential. Both input to this <i><b>NOR gate</b></i> are zero, so the output is 1, which sets <i><b>Q=1</b></i>, which it already has. So <i><b>Q</b></i> does not change and remains at <i><b>1</b></i>.<br/><br/>As we see, the positive impulse on terminal <i><b>S</b></i> sets terminals <i><b>Q</b></i> and <i><b>¬Q</b></i> to stable values <i><b>1</b></i> and <i><b>0</b></i> correspondingly, which they retain after the impulse is gone.<br/>In other words, the positive impulse on terminal <i><b>S</b></i> sets a <b>flip-flop</b> to a stable state <i><b>Q=1</b></i> and <i><b>¬Q=0</b></i>.<br/><br/><br/><i>Reset <b>Q=0</b> and <b>¬Q=1</b></i><br/><br/>Notice how symmetrical is the circuit of our flip-flop.<br/>Sending a positive impulse to terminal <i><b>R</b></i>, while keeping zero potential on terminal <i><b>S</b></i> is fully analogous to a previous case, except now the flip-flop will be in a stable state <i><b>Q=0</b></i> and <i><b>¬Q=1</b></i>.<br/><br/><br/><i><b>Conclusion</b></i><br/><br/><b>The flip-flop presented above can serve as a memory bit</b>, since we can set its value to <i><b>1</b></i> (<i><b>Q=1</b></i> and <i><b>¬Q=0</b></i>) by sending a positive impulse to terminal <i><b>S</b></i> or to <i><b>0</b></i> (<i><b>Q=0</b></i> and <i><b>¬Q=1</b></i>) by sending a positive impulse to terminal <i><b>R</b></i>.<br/><br/>An important detail about this design is that we should never have a situation when both <i><b>S=1</b></i> and <i><b>R=1</b></i>. The reason is, the state of a flip-flop in this cases is unpredictable, it will constantly change the values on its output terminals. So, this situation of both input terminals being equal to <i><b>1</b></i> must be avoided.<br/>Allowed are only positive impulse (that is, an electric potential jumps to some positive value, after which it goes down to zero) on terminal <i><b>S</b></i>, keeping <i><b>R=0</b></i>, which sets the value of a flip-flop to <i><b>1</b></i> (that is, <i><b>Q=1</b></i> and <i><b>¬Q=0</b></i>) or positive impulse on terminal <i><b>R</b></i>, keeping <i><b>S=0</b></i>, which sets the value of a flip-flop to <i><b>0</b></i> (that is, <i><b>Q=0</b></i> and <i><b>¬Q=1</b></i>). Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-6200482466127651102021-03-08T08:20:00.002-08:002021-03-08T08:20:43.179-08:00Logical NOT: UNIZOR.COM - Physics4Teens - Electromagnetism - Electronics<iframe width="480" height="270" src="https://youtube.com/embed/Jn-yC6AxFkU" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Logical NOT Implementation</u><br/><br/>Unary logical operation NOT (symbol ¬) is characterized by two simple rules:<br/><i><b>¬(TRUE) = ¬1 = 0 = FALSE</b></i><br/><i><b>¬(FALSE) = ¬0 = 1 = TRUE</b></i><br/><br/>Let's implement the logical NOT operation in an electronic circuit using a plain switch.<br/>Consider the following schema.<br/><img src='http://www.unizor.com/Pictures/Logic_NOT_switch.png' style='width:200px;height:270px;'><br/>When ON/OFF switch is in OFF position, there is no current in a circuit and the potential at the output point will be positive as on the corresponding battery terminal.<br/><br/>If we change the switch to ON position, there will be an electric current flow between the ground and the positive battery terminal, the potential at the output point will be the same as that of the grounding, that is zero.<br/><br/>This circuit converts ON position of the switch into zero potential at the output, while OFF position of the switch is converted into positive potential at the output.<br/><br/>With this in mind, all we have to change in this circuit to make it purely electronic is to implement the switch electronically.<br/>We can use a triode for this purpose with grounded cathode.<br/><br/>As we know, the positive potential on the grid of a triode enhances the attraction of an anode for an electron cloud around a grounded but heated cathode, thereby opening the movement of electrons from cathode to anode. Since cathode is grounded, all electrons that moved to an anode will be compensated from the ground, and there will be a steady flow of electrons from cathode to anode, which is equivalent to a switch to be in ON position.<br/><br/>At the same time, grounded (zero potential) grid does not help the electrons of the grounded cathode to reach the positive anode, there will be very weak, if any, electric current between anode and cathode, which with proper parameters of the components of a circuit can be brought practically to zero, which is equivalent to a switch to be in OFF position.<br/><br/>This brings us to the following circuit that implements the logical NOT operation (inverter).<br/><img src='http://www.unizor.com/Pictures/Logic_NOT_triode.png' style='width:200px;height:270px;'><br/>This is a very primitive implementation, only for educational purposes. The real inverters are more complex and usually implemented using transistors, which we will discuss later in the course.<br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-28375444313355219902021-03-08T08:19:00.002-08:002021-03-08T08:19:55.086-08:00Logical XOR: UNIZOR.COM - Physics4Teens - Electromagnetism - Electronics<iframe width="480" height="270" src="https://youtube.com/embed/u-vJVwvohO4" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Logical XOR Implementation</u><br/><br/>Let's analyze how diodes can be used to implement operation XOR (or addition by modulo 2) of mathematical (and computer) logic.<br/><br/>The standard is that TRUE or 1 is represented by some positive potential on a wire relative to the ground, while FALSE or 0 is represented by zero potential relative to the ground.<br/><br/>Exclusive OR (<i><b>XOR</b></i>) operation is represented by a symbol of addition in a circle <b>⊕</b>.<br/>The rules are:<br/><i><b>0 ⊕ 0 = 0<br/>0 ⊕ 1 = 1<br/>1 ⊕ 0 = 1<br/>1 ⊕ 1 = 0 </b></i><br/><br/>Now let's design a schema that represents a logical operation <i>XOR</i> of two logical values.<br/><br/><img src='http://www.unizor.com/Pictures/Logic_XOR.png' style='width:200px;height:170px;'><br/><br/>There are four variations of input values <i><b>A</b></i> and <i><b>B</b></i>:<br/><i><b>A=0, B=0</b></i><br/><i><b>A=0, B=1</b></i><br/><i><b>A=1, B=0</b></i><br/><i><b>A=1, B=1</b></i><br/><br/>For each of these cases we will determine the output value at point <i><b>C</b></i>.<br/><br/>1. If <i><b>A=0</b></i> and <i><b>B=0</b></i>, that is the potential is zero at both points relative to the ground and points <i><b>D</b></i> and <i><b>F</b></i> are at zero potential as well. In theory, electrons from hot cathodes in diodes at segments <i><b>ED</b></i> and <i><b>EF</b></i> can move to corresponding anodes only if these anodes are positively charged. But this is not the case since the potentials at points <i><b>D</b></i> and <i><b>F</b></i> are zero. Therefore, no electrons escape point <i><b>E</b></i>. Point <i><b>G</b></i> is connected only to anodes and, therefore, no electrons escape this point either. So, at point <i><b>C</b></i> the potential is zero, that is <i><b>C=0</b></i><br/><br/>2. Consider a case of <i><b>A=0</b></i> and <i><b>B=1</b></i>, that is the potential is zero at points <i><b>A</b></i> and <i><b>D</b></i>, but is positive (deficiency of electrons) at points <i><b>B</b></i> and <i><b>F</b></i> relative to the ground.<br/>A diode on a segment <i><b>EF</b></i> has its anode positively charged from point <i><b>F</b></i>. Therefore, electrons in the cloud produced by thermionic emission on its cathode will be attracted to positively charged anode, <u>thus creating a deficiency of electrons (positive potential) on this cathode</u> and at points <i><b>E</b></i> and <i><b>C</b></i>. Point <i><b>C</b></i> becomes positively charged, that is <i><b>C=1</b></i>.<br/>Deficiency of electrons at point <i><b>E</b></i> cannot be compensated through a segment <i><b>DE</b></i> because a diode on this segment is working in an opposite direction. So, deficiency of electrons will be compensated from point <i><b>G</b></i> from point <i><b>G</b></i>, which becomes positively charged and attracts electrons from point <i><b>D</b></i> through a diode at segment <i><b>DG</b></i>. These new electrons, coming from point <i><b>A</b></i> to point <i><b>D</b></i>, through thermionic emission to point <i><b>G</b></i> and to point <i><b>E</b></i>, will be immediately dispatched to a thermionic emission cloud and attracted by positively charged anode of a diode on <i><b>EF</b></i> segment, thus maintaining a constant flow of electrons from point <i><b>G</b></i> (neutral) to point <i><b>E</b></i> (positive).<br/>This process of electrons moving from zero potential at point <i><b>A</b></i> to point <i><b>D</b></i>, to thermionic emission cloud on a cathode of <i><b>DG</b></i> segment, to point <i><b>G</b></i>, to point <i><b>E</b></i> will continue as long as point <i><b>B</b></i> has positive potential.<br/>Therefore, the cathode of a diode on a segment <i><b>EF</b></i> will always be positively charged, and so is a wire at points <i><b>E</b></i> and <i><b>C</b></i> connected to it. That means that <i><b>C=1</b></i>.<br/><br/>3. If <i><b>A=1</b></i> (positively charged, that is deficiency of electrons) and <i><b>B=0</b></i> (neutral), situation is analogous to a previous one, except all the electrons will flow from point <i><b>B</b></i> to point <i><b>F</b></i>, to cathode of a diode on a segment <i><b>FG</b></i> to point <i><b>G</b></i>, to point <i><b>E</b></i>, to thermionic cloud of a cathode on segment <i><b>ED</b></i> to positively charged points <i><b>D</b></i> and <i><b>A</b></i>. Since there is a deficiency of electrons at point <i><b>E</b></i>, the potential at point <i><b>C</b></i> will be positive, that is <i><b>C=1</b></i>.<br/><br/>4. Case <i><b>A=1</b></i> and <i><b>B=1</b></i> (both deficient of electrons). There is no source of electrons to compensate this deficiency. If point <i><b>E</b></i> was connected to such a source, there would be a flow of electrons from it through segments <i><b>ED</b></i> and <i><b>EF</b></i>, but this is not the case. No electrons are moving through point <i><b>G</b></i> either. Therefore, potential at points <i><b>E</b></i>, <i><b>G</b></i> and <i><b>C</b></i> is zero, that is <i><b>C=0</b></i><br/><br/>As we see, if any one input potentials at points <i><b>A</b></i> and <i><b>B</b></i> equal to 1, the output <i><b>C=1</b></i> as well. If both <i><b>A=0</b></i> and <i><b>B=0</b></i>, the potential at <i><b>C</b></i> is zero, that is <i><b>C=0</b></i>. If both <i><b>A=1</b></i> and <i><b>B=1</b></i>, the potential at <i><b>C</b></i> is also zero, that is <i><b>C=0</b></i>.<br/><br/>That fully corresponds to the rules of logical XOR operation.<br/><br/>Therefore, the circuit above is a proper electronic implementation of logical operation of <i>exclusive OR</i> (or addition by modulo 2).<br/><br/>Obviously, the real implementation requires some concrete values for voltage, resistance, characteristics of diodes etc. that we will not address here, as our purpose is to introduce only the principle behind this implementation.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-87132552575952068002021-03-08T08:18:00.002-08:002021-03-08T08:18:49.903-08:00Logical AND: UNIZOR.COM - Physics4Teens - Electromagnetism - Electronics<iframe width="480" height="270" src="https://youtube.com/embed/IqRc9_iFgTA" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Logical AND Implementation</u><br/><br/>Let's analyze how diodes can be used to implement operation AND of mathematical (and computer) logic.<br/><br/>The standard is that TRUE or 1 is represented by some positive potential on a wire relative to ground, while FALSE or 0 is represented by zero potential relative to ground.<br/><br/>Conjunction <b>AND</b> is represented by a symbol of multiplication <b>·</b> or an ampersand <b>&</b>.<br/>The rules are:<br/><i><b>0 & 0 = 0<br/>0 & 1 = 0<br/>1 & 0 = 0<br/>1 & 1 = 1 </b></i><br/><br/>Now let's design a schema that represents a logical operation <i>AND</i> of two logical values.<br/><br/><img src='http://www.unizor.com/Pictures/Logic_AND.png' style='width:200px;height:200px;'><br/>Here points <i><b>A</b></i> and <i><b>B</b></i> are on the cathode sides of corresponding diodes, while both anodes are connected to a positive potential wire from a battery.<br/><br/>If both <i><b>A</b></i> and <i><b>B</b></i> represent the value of 1 (that is, have positive potential and deficiency of electrons), there is no difference in potentials between anodes and corresponding cathodes, no thermionic clouds and the potential at point <i><b>C</b></i> will be positive, that is <i><b>C=1</b></i> in this case.<br/><br/>If the potential at point <i><b>A</b></i> or at point <i><b>B</b></i>, or at both of these points is maintained at 0, electrons from thermionic cloud will fly to a corresponding anode and, further, neutralize any positive potential (deficiency of electrons) that might exist at point <i><b>C</b></i>.<br/>Therefore, in this case the logical value at point <i><b>C</b></i> will be equal to 0.<br/><br/>This is exactly the same as the definition of logical operation <i>AND</i> between two boolean variables <i><b>A</b></i> and <i><b>B</b></i>.<br/><br/>Therefore, the circuit above is a proper electronic implementation of logical operation of <i>conjunction</i>.<br/><br/>Obviously, the real implementation requires some concrete values for voltage, resistance, characteristics of diodes etc. that we will not address here, as our purpose is to introduce only the principle behind this implementation.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-22353586712736735672021-03-08T08:17:00.001-08:002021-03-08T08:17:54.237-08:00Logical OR: UNIZOR.COM - Physics4Teens - Electromagnetism - Electronics<iframe width="480" height="270" src="https://youtube.com/embed/f73TCODZaPg" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Logical OR Implementation</u><br/><br/>Let's analyze how diodes can be used to implement operation OR of mathematical (and computer) logic.<br/><br/>The standard is that TRUE or 1 is represented by some positive potential on a wire relative to the ground, while FALSE or 0 is represented by zero potential relative to the ground.<br/><br/>Disjunction <b>OR</b> is represented by a symbol of addition <b>+</b> or a vertical bar <b>|</b>.<br/>The rules are:<br/><i><b>0 | 0 = 0<br/>0 | 1 = 1<br/>1 | 0 = 1<br/>1 | 1 = 1 </b></i><br/><br/>Now let's design a schema that represents a logical operation <i>OR</i> of two logical values.<br/><br/>Assume, the wires <i><b>A</b></i> and <i><b>B</b></i>, connected to corresponding anodes of two diodes, represent some logical values, TRUTH (1) or FALSE (0) and find the value represented by wire <i><b>C</b></i> on the following picture<br/><img src='http://www.unizor.com/Pictures/Logic_OR.png' style='width:200px;height:200px;'><br/><br/>There are four variations of values <i><b>A</b></i> and <i><b>B</b></i>:<br/><i><b>A=0, B=0</b></i><br/><i><b>A=0, B=1</b></i><br/><i><b>A=1, B=0</b></i><br/><i><b>A=1, B=1</b></i><br/><br/>For each of these cases we will determine the value at point <i><b>C</b></i>.<br/><br/>1. If <i><b>A=0</b></i> and <i><b>B=0</b></i>, that is the potential is zero at both points relative to ground, there is no electric current anywhere. Even with cathode on any of the diodes heated, there is no current since there is no positive charge on anode to attract the electrons in the cloud produced by thermionic emission. So, at point <i><b>C</b></i> the potential is zero, that is <i><b>C=0</b></i><br/><br/>2. Consider a case of <i><b>A=0</b></i> and <i><b>B=1</b></i>, that is the potential is zero at point <i><b>A</b></i>, but is positive at point <i><b>B</b></i> relative to the ground.<br/>The electrons in the cloud produced by thermionic emission on cathode opposite to point <i><b>B</b></i> will be attracted to positively charged anode at point <i><b>B</b></i>, <u>thus creating a deficiency of electrons (positive potential) on this cathode</u>.<br/>New electrons from the neutral ground will compensate the loss of electrons on this positively charged cathode, but will be immediately dispatched to a thermionic emission cloud and attracted by positively charged anode at point <i><b>B</b></i>, thus maintaining a constant flow of electrons from the neutral ground through positively charged cathode to point <i><b>B</b></i>.<br/>This process of electrons moving from the ground to positively charged cathode opposite to point <i><b>B</b></i>, to thermionic emission cloud and to point <i><b>B</b></i> will continue as long as point <i><b>B</b></i> has positive potential.<br/>Therefore, the cathode opposite to point <i><b>B</b></i> will always be positively charged, and so is a wire at point <i><b>C</b></i> connected to it. That means that <i><b>C=1</b></i>.<br/><br/>3. If <i><b>A=1</b></i> and <i><b>B=0</b></i>, situation is analogous to a previous one, except all the electrons will flow to point <i><b>A</b></i>. The potential at point <i><b>C</b></i> will be positive, that is <i><b>C=1</b></i>.<br/><br/>4. Case <i><b>A=1</b></i> and <i><b>B=1</b></i> is not much different. Now the electrons will flow from the ground to both points <i><b>A</b></i> and <i><b>B</b></i>. The point <i><b>C</b></i> will be positively charged, that is <i><b>C=1</b></i>.<br/><br/>As we see, if any one or both input potentials at points <i><b>A</b></i> and <i><b>B</b></i> equal to 1, the output <i><b>C=1</b></i> as well. Only if both <i><b>A=0</b></i> and <i><b>B=0</b></i>, the potential a <i><b>C</b></i> is zero, that is <i><b>C=0</b></i>.<br/><br/>That fully corresponds to the rules of logical OR operation.<br/><br/>Therefore, the circuit above is a proper electronic implementation of logical operation of <i>disjunction</i>.<br/><br/>Obviously, the real implementation requires some concrete values for voltage, resistance, characteristics of diodes etc. that we will not address here, as our purpose is to introduce only the principle behind this implementation.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-11186093920281983392021-03-05T09:29:00.000-08:002021-03-05T09:29:53.344-08:00Electronic Logic: UNIZOR.COM - Physics4Teens - Electromagnetism - Electr...<iframe width="480" height="270" src="https://youtube.com/embed/nQRJzbdY4T0" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Electronic Logic</u><br/><br/>Let's analyze how diodes can be used to implement operations of mathematical (and computer) logic.<br/>These are operations on variables that can have only two possible values - TRUE (in many cases represented by number 1) and FALSE (represented by 0).<br/><br/><i>TRUE and FALSE in Electronics</i><br/><br/>First of all, we have to agree on what represents logical TRUE and FALSE or 1 and 0 in electrical circuits.<br/>The standard is that TRUE or 1 is represented by some positive potential on a wire relative to ground, while FALSE or 0 is represented by zero potential relative to ground.<br/><br/>Consider the following circuit<br/><img src='http://www.unizor.com/Pictures/Logic_1_0.png' style='width:200px;height:120px;'><br/>According to our rules about what represents logical TRUE (1) and FALSE (0), the logical value at point <i><b>A</b></i> is FALSE (0) and at point <i><b>B</b></i> it is TRUE (1).<br/><br/><br/><i>Binary logical operations</i><br/><br/>Disjunction <b>OR</b> (sometimes represented by symbol of addition <b>+</b> or vertical bar <b>|</b>).<br/>The rules are: <i><b>0|0=0, 0|1=1, 1|0=1, 1|1=1</b></i><br/><br/>Conjunction <b>AND</b> (might be represented by symbol of multiplication <b>·</b> or ampersand <b>&</b>),<br/>The rules are: <i><b>0&0=0, 0&1=0, 1&0=0, 1&1=1</b></i><br/><br/>Exclusive disjunction <b>XOR</b> (might be represented by symbol <b>⊕</b>).<br/>The rules are: <i><b>0</i>⊕<i>0=0, 0</i>⊕<i>1=1, 1</i>⊕<i>0=1, 1</i>⊕<i>1=0</b></i><br/><br/><br/><i>Unary logical operation</i><br/><br/>Negation <b>NOT</b> (sometimes represented by exclamation sign <b>!</b> or symbol <b>¬</b>).<br/>The rules are: <i><b>¬0=1, ¬1=0</b></i><br/><br/><br/><i>Implementation</i><br/><br/>Implementation of binary logical operations in electronics involves developing a schema with two input contacts, <i><b>A</b></i> and <i><b>B</b></i>, and one output contact <i><b>C</b></i>. On two input contacts there might be four different combinations of electric potential:<br/><i><b>A=0, B=0</b></i><br/><i><b>A=0, B=1</b></i><br/><i><b>A=1, B=0</b></i><br/><i><b>A=1, B=1</b></i><br/>For each of these combinations the value of electric potential at output point <i><b>C</b></i> should correspond to the rules of the corresponding binary logical operation implemented in this schema.<br/><br/>Implementation of unary logical operations in electronics involves developing a schema with one input contacts <i><b>A</b></i> and one output contact <i><b>B</b></i>. On input contact <i><b>A</b></i> there might be two different electric potentials:<br/><i><b>A=0</b></i><br/><i><b>A=1</b></i><br/>For each of these combinations the value of electric potential at output point <i><b>B</b></i> should correspond to the rules of the corresponding unary logical operation implemented in this schema.<br/><br/>Examples of implementation of certain logical operations in electronics are presented in subsequent lectures.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-20125326111081761712021-03-02T09:23:00.000-08:002021-03-02T09:23:05.830-08:00Electronics - Triode: UNIZOR.COM - Physics4Teens - Electromagnetism - Usage<iframe width="480" height="270" src="https://youtube.com/embed/CTH-CbwVdVs" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Electronics: Triode</u><br/><br/><i>Triodes</i> are another class of relatively simple electronic components that are present in practically all electronic devices.<br/>The main purpose of a <i>triode</i> is to amplify the electric current in addition to rectifying it.<br/><br/>Placing a negatively charged grid between a source of an electron cloud (<b>cathode</b>) and a plate receiving electrons produced by this <i>thermionic emission</i> (<b>anode</b>) allows to regulate the flow of electrons from cathode to anode.<br/><br/>The greater negative charge of an electrostatic field of a grid is - the more electrons will be repelled back to cathode and will not reach anode.<br/>The electrically neutral grid causes a triode to act as a diode since the grid is no longer repels electrons, so they are attracted and fly to anode without any resistance.<br/><br/>There is a direct correlation between the <i>voltage</i> of a grid relatively to cathode and the <i>amperage</i> of the current reaching the anode.<br/><br/>Below is a schematic representation of a vacuum tube triode with a heated filament separated from a cathode, so the battery which heats the filament does not interfere with current from cathode to anode.<br/><img src='http://www.unizor.com/Pictures/TriodeTube.png' style='width:200px;height:200px;'><br/><br/>The following schema represents the basic principle of amplifying a signal using a triode.<br/><img src='http://www.unizor.com/Pictures/TriodeAmplifier.png' style='width:200px;height:130px;'><br/>The DC power supply causes a flow of electrons from an electron cloud around cathode, caused by thermionic emission, to anode plate.<br/>Input voltage is applied to a grid inside a triode, that affects the flow of electrons from cathode to anode.<br/><br/>Varying the base voltage of DC power supply, we can regulate the strength of the output voltage relatively to input voltage.<br/>Output voltage changes synchronously with the input voltage with the same frequency but different amplitude. This enables amplifying the input signal.<br/><br/>Vacuum tube triodes are rarely used nowadays. Semiconductors and integrated circuits are the main technological base. Basic functionality of triodes is implemented in semiconductors called <i>transistors</i>. These will be discussed in a separate chapter of this course dedicated to semiconductors.<br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-83531053116837609652021-02-22T09:32:00.001-08:002021-02-22T09:32:56.779-08:00<i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Relativity - Transformation of Space-Time Coordinates</u><br/>(notes to item #3 of Einstein's "Electrodynamics")<br /><br />The following is an example of how a system of linear equations can be used to derive formulas of special theory of relativity. Albert Einstein has derived these formulas in his "Electrodynamics" in a more physical, more intuitive way. The following is pure mathematics and, as such, causes much less problems in understanding.<br/><br/>Assumptions:<br />1. Assume we have two systems of coordinates, one stationary with coordinates {X,T} (assuming for simplicity all the movements will occur in one space dimension along X-axis and one time dimension) and another with coordinates {x,t} moving along the X-axis with constant speed V.<br />2. Assume that at time T=0 systems coincide (i.e. X=0, t=0 and x=0).<br />3. Assume that the speed of something, as measured in both stationary and moving systems is the same and equal to C regardless of the direction of the movement (that "something" is the light in vacuum, but it's physical characteristics are unimportant)<br />4. Assume further that we are looking for linear orthogonal (i.e. preserving the distance between points and angles between vectors) transformation of coordinates from (X,T) to (x,t) that satisfies the above criteria. What would this transformation be?<br /><br />Linear transformation from {X,T} system to {x,t} system should look like this:<br /><b><i>x = pX + qT<br />t = rX + sT<br /></i></b>where p, q, r and s are 4 unknown coefficients of transformation, which we are going to determine by constructing a system of 4 linear equations with them.<br /><br />We should not add any constants into above transformations since {X=0,T=0} should transform into {x=0,t=0}.<br /><br/>A. Notice that the property of orthogonality is needed to preserve geometry (i.e. no deformation) and, therefore, to preserve the form of all physical equations of motion. As it is well known, orthogonal transformations have determinant of the matrix of coefficients equal to 1, i.e. <br /><b><i>ps − qr = 1</i></b>. An unfamiliar with this property student can study this subject separately (we at Unizor plan to include this into corresponding topic on vectors).<br />The above is the first equation to determine unknown coefficients. It's not linear, it is rather quadratic, but the rest of the equations will be linear and that's why we included this particular problem in the topic dedicated to linear systems.<br />We need three more equations to determine all the unknown coefficients.<br /><br />B. Since moving system moves along X-axis with speed V, its beginning of coordinate (point x=0) must at the moment of time T be on a distance VT from the beginning of coordinates of a stationary system. Hence, if X=VT, x=0 for any T. From this and the first transformation equation x = pX + qT we derive:<br /><b><i>0 = pVT + qT</i></b> or<br/><b><i>0 = (pV + q)T</i></b>.<br /><br/>Since this equality is true for any T, <br /><b><i>pV + q = 0</i></b> <br />and, unconditionally, <b><i>q = −pV</i></b>.<br />This is the second equation for our unknown coefficients.<br /><br />C. Since the speed of light C is the same in both systems {X,T} and {x,t}, an equation of its motion in the stationary system must be X = CT and in the moving system x = Ct. Therefore, if X=CT, then x=Ct. Put X=CT into equations of transformation of coordinates. We get x = pCT + qT, t = rCT + sT. Substitute these expressions into x=Ct: <br /><b><i>pCT + qT = rC<sup>2</sup>T + sCT</i></b>.<br />Reduce by T, <br /><b><i>pC + q = rC<sup>2</sup> + sC</i></b>.<br />This is the third equation for unknown coefficients.<br /><br />D. Repeat the logic of a previous paragraph for the light moving in the opposite direction with a speed −C. We get, if X = −CT, then x = −Ct. Therefore, x = −pCT + qT, t = −rCT + sT and (since x = −Ct) <br /><b><i>−pCT + qT = rC<sup>2</sup>T − sCT</i></b>. <br />Reduce by T, <br /><b><i>−pC + q = rC<sup>2</sup> − sC</i></b>.<br />This the fourth equation for unknown coefficients.<br /><br />So, this is the system of equations for 4 unknown coefficients of transformation <b><i>p, q, r, s</i></b>:<br/>(a) <b><i>ps − qr = 1</i></b><br/>(b) <b><i>q = −pV</i></b><br />(c) <b><i>pC + q = rC<sup>2</sup> + sC</i></b><br />(d) <b><i>−pC + q = rC<sup>2</sup> − sC</i></b><br /><br/>It's not exactly linear, but it has sufficient number of linear equations (all but one) to solve it using the known methodology. Let's solve this system of equations by combining the methods of substitution and elimination. We will express unknown variables <b><i>q, r, s</i></b> in terms of <b><i>p</i></b> using equations (b), (c) and (d). Then we will substitute them into (a) to get an equation for <b><i>p</i></b>. Solving it will allow to evaluate all other unknowns.<br/><br/>E. From (c) and (d), adding and subtracting these equations, we get:<br /><b><i>2q = 2rC<sup>2</sup></i></b>, therefore <b><i>q = rC<sup>2</sup></i></b><br /><b><i>2pC = 2sC</i></b>, therefore <b><i>p = s</i></b><br /><br />Since from (b) q = −pV,<br /><b><i>−pV = rC<sup>2</sup></i></b> and <b><i>r = −pV/C<sup>2</sup></i></b>.<br /><br />Now all coefficients of a transformation are expressed in terms of one unknown coefficient <b><i>p</i></b>. To get the value of <b><i>p</i></b>, use the first equation (a).<br /><br />F. Substituting q, r and s, expressed in terms of p, into an equation (a) <b><i>ps − qr = 1</i></b>, we get:<br /><b><i>p<sup>2</sup> − (−pV)·(−pV)/C<sup>2</sup> = 1</i></b>, therefore <br /><b><i>p<sup>2</sup>·(1 − V<sup>2</sup>/C<sup>2</sup>) = 1</i></b> and <br /><b><i>p = 1/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span></i></b>.<br /><br />From this all other coefficients of a transformation matrix are derived:<br /><b><i>q = −V/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span><br />r = −(V/C<sup>2</sup>)/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span><br />s = 1/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span><br /></i></b><br />G. The final transformation matrix looks exactly like in Einstein's article on electrodynamics, but seems to be much simpler to arrive at and the derivation is strictly mathematical.<br /><b><i>x = (1/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span>)·X − (V/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span>)·T<br />t = ((−V/C<sup>2</sup>)/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span>)·X + (1/√</b><span style='text-decoration:overline;'><b>1−V<sup>2</sup>/C<sup>2</sup> </span>)·T<br /></i></b><br />Traditionally, factor V/C is replaced with Greek letter β, which results in formulas:<br /><b><i>x = (1/√</b><span style='text-decoration:overline;'><b>1−β<sup>2</sup> </span>)·X + (−V/√</b><span style='text-decoration:overline;'><b>1−β<sup>2</sup> </span>)·T<br />t = ((−V/C<sup>2</sup>)/√</b><span style='text-decoration:overline;'><b>1−β<sup>2</sup> </span>)·X + (1/√</b><span style='text-decoration:overline;'><b>1−β<sup>2</sup> </span>)·T<br /></i></b><br />One more simplification is usually done by introducing Lorentz factor <b><i>γ</i></b> equals to <b><i>1/√</b><span style='text-decoration:overline;'><b>1−β<sup>2</sup> </span></i></b>:<br /><b><i>x = γX − γVT<br/> = γ(X − VT)</i></b><br /><b><i>t = −γVX/C<sup>2</sup> + γT<br/> = γ(T − VX/C<sup>2</sup>)</i></b><br /><br />The final form of transformation of coordinates in the Special Theory of Relativity is:<br><b><i>x = (X − VT)/√</b><span style='text-decoration:overline;'><b>1−(V/C)<sup>2</sup> </span></i></b><br /><b><i>t = (T − VX/C<sup>2</sup>)/√</b><span style='text-decoration:overline;'><b>1−(V/C)<sup>2</sup> </span></i></b><br /><br />Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-70108092140186369022021-02-08T08:39:00.002-08:002021-02-08T08:39:41.602-08:00Electronics - Diode: UNIZOR.COM - Physics4Teens - Electromagnetism - Usage<iframe width="480" height="270" src="https://youtube.com/embed/wEuL4rA7R_8" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Electronics: Diode</u><br/><br/>To the category of <b>electronic</b> devices we relate all devices that use electricity and whose primary purpose is not just to heat or to do mechanical work.<br/>Radio, television, computers, all kinds of non-mechanical switches and regulators, phones, hardware that runs Internet and many others devices belong to this category of electronic devices.<br/><br/>Obviously, we cannot talk about how all these devices work. Instead, we will spend some time to understand the basic components that are used in these devices.<br/><br/><i>Diodes</i> are one of the simplest electronic components that are present in practically all electronic devices.<br/>The main purpose of a <i>diode</i> is to let the electric current only in one direction. This process of allowing electric current to go only in one direction is called <i>rectifying</i> the current.<br/><br/>The origin of diodes lies in an observations made by scientists and engineers (Fleming, Edison and others) at the end of 19th century. Two wires that did not touch each other were placed inside a vacuum tube, one connected to a positive pole of a battery (<i>anode</i>) and another connected to a negative one (<i>cathode</i>). At normal temperature there was no electric current between them, because they did not touch each other. But, if the negative wire was heated, some electric current between these wires was observed, while heating the positive wire did not cause any current in the circuit.<br/><br/>The explanation of this phenomenon is simple. Heating increases the activity of elementary particles inside a negatively charged metal of a wire that has an excess of electrons. With this increased activity certain electrons escape from the surface of the metal and form some kind of electron cloud. This is called <i>thermionic emission</i>.<br/>In the presence of a positively charged wire some electrons from a cloud are attracted to a positive wire, thus forming a current. New electrons from a negative poll of a battery replace the escaped electrons, maintaining a fresh supply of new electrons, which enables a steady current.<br/><br/>Obviously, if a positive wire with its deficiency of electrons is heated, even if some electrons do escape because of high temperature, they will be repelled by a negative wire and attracted back to a positive one. No current would be formed.<br/><br/>Below is a schematic representation of a vacuum tube diode with ampere meter in a circuit showing the existence of electric current.<br/><img src='http://www.unizor.com/Pictures/DiodeTube.png' style='width:200px;height:200px;'><br/><br/>The symbol for a diode in electronic schemas is<br/><img src='http://www.unizor.com/Pictures/DiodeSymbol.png' style='width:200px;height:100px;'><br/><br/>The primary usage of diodes is to <i>rectify</i> the alternating current, where they allow the current to go only in one direction, thus converting AC to DC.<br/>They are also used in signal isolation, filtering and mixing.<br/>Vacuum tube diodes are used now only in high capacity <i>rectifiers</i> with semiconductors based diodes used in all the electronic devices we usually deal with.<br/><br/>Let's analyze the process of rectifying AC using diodes.<br/>As we know, the current in a regular AC circuit is sinusoidal, changing the direction and the value.<br/>If a diode is included into a circuit in sequence, the current in one direction will go through, while it will be prohibited to go in the opposite direction.<br/>This causes the alternating current to change from a regular sinusoidal wave-like behavior into irregular direct current.<br/><img src='http://www.unizor.com/Pictures/DiodeRectifying.jpg' style='width:200px;height:200px;'><br/><br/>The irregularity of the current in a circuit can be improved by using a <i>bridge rectifier</i> built from 4 diodes as follows<br/><img src='http://www.unizor.com/Pictures/BridgeDiodes.png' style='width:200px;height:170px;'><br/><br/>If the AC generator produces positive charge at point A and negative at point B, the flow of electrons is in the direction BEFNMDCA. The electric current is defined as going against the flow of electrons in the opposite direction ACDMNFEB.<br/><br/>If the AC generator produces positive charge at point B and negative at point A, the flow of electrons is in the direction ACFNMDEB. The electric current is defined as going against the flow of electrons in the opposite direction BEDMNFCA.<br/><br/>As you see, the direction of a current in both cases is from point M to point N, regardless of polarity of a generator contacts.<br/><br/>The AC current rectified by diodes that form a bridge is better than rectified by a single diode, but still is quite irregular, comparing with DC current from a battery.<br/>Additional improvements can be achieved by splitting a current into two separate lines and putting a capacitor on one of them to change the phase of the oscillations and then combining signals together.<br/>Here is how the combination works.<br/><img src='http://www.unizor.com/Pictures/TwoRectifiers.png' style='width:200px;height:250px;'><br/><br/>The combined <nobr><i><b><font color=blue>Signal 1+2</font></b></i></nobr> still has some irregularity, but is much more stable than each of its components <nobr><i><b><font color=green>Signal 1</font></b></i></nobr> or <nobr><i><b><font color=brown>Signal 2</font></b></i></nobr>.<br/><br/>In general, by combining currents, shifted by phase relative to each other, helps improving the stability of the flow of electrons. Real life rectifiers are build on this principle.<br/><br/>Vacuum tube diodes have been largely replaced by semiconductors, but the principle of their work is very similar.<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-30428960586444014992021-02-04T08:01:00.002-08:002021-02-04T08:01:48.270-08:00Electric Devices: UNIZOR.COM - Physics4Teens - Electromagnetism<iframe width="480" height="270" src="https://youtube.com/embed/LSr04WyOwWM" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Electric Devices</u><br/><br/>In this chapter we will discuss different usages of electricity by grouping them according to certain criteria.<br/><br/>Two major groups to differentiate are <b>electric</b> and <b>electronic</b> devices.<br/>To the category of <b>electric</b> devices we relate devices that use electricity for two primary purposes - to produce mechanical work, like rotation, and to produce heat, like in electric stove, including producing light by means of heating, like in incandescent lamps.<br/><br/>Regardless of such a simple definition of this category, the number of devices in this group is enormous, and these devices are the first ones invented to make our lives easier. Examples of these electric devices are a subject of this lecture.<br/><br/>To the category of <b>electronic</b> devices we relate all other devices that use electricity and whose primary purpose is not just to heat or to do mechanical work. They will be discussed in the next lecture.<br/><br/><br/><i>Mechanical Work</i><br/><br/>The easiest and most common example of the usage of electricity to produce mechanical work is an electric motor. This is a device that converts electricity into rotation.<br/><br/>For alternating current numerous one-phase and three-phase motors are used all around us.<br/>The primary motion they produce is rotation, which in some cases is converted into other forms of motion.<br/><br/>They pump water, rotate fans, work in refrigerators, rotate wheels of electric trains, lift elevator cabins, drill for oil and gas, operate machinery at manufacturing plants, move construction cranes.<br/><br/>Direct current in most cases comes from batteries and is used in direct current electric motors, like the one that starts the car engine, rotates a hard disk in a computer, rotates the battery powered electric drill etc.<br/><br/>Electric wall clock is another example of using electricity to move the wheels of a clock. Some electric clocks work off alternating current, some off direct one.<br/><br/>Washing machine has an electric pump to deal with water pumped in and out and another motor to rotate the drum.<br/><br/>Just as an illustration, let's calculate the technical characteristics of a motor that should supply water to a building, where I live in.<br/><br/>The water is pumped to the roof tank, then it flows down to all apartments.<br/><br/>We have <nobr><i>12 floors</i></nobr>, each about <nobr><i>3 meters</i></nobr> high, <nobr><i>200 apartments</i></nobr>, each apartment needs about <nobr><i>100 liters</i></nobr> of water during <nobr><i>3 hours</i></nobr> in the morning.<br/>So, the pump should pump <nobr><i>200·100=20,000 liters</i></nobr> of water to the height <nobr><i>12·3=36 meters</i></nobr> during <nobr><i>3 hours</i></nobr> time.<br/><br/>This allows us to calculate work <i>W</i> performed during this time and power <i>P</i> of the pump needed to perform this work.<br/><br/>Each liter of water has a mass of <nobr><i>1 kg</i></nobr> and, therefore, the weight of <nobr><i>9.8 N</i></nobr>.<br/><i><b>W = 9.8·20,000·36 =<br/>= 7,056,000 J</b> (joules)</i><br/><br/>Since the time to do this work is <nobr><i>T = 3 hours</i></nobr> and each hour has <nobr><i>3,600 seconds</i></nobr>, the power of the motor is<br/><i><b>P=W/T=7,056,000/(3·3,600)≅<br/>≅ 653 J/sec</b> (watts)</i><br/><br/>Usually, we need some excess of power to prevent shortage during some extra work requirements and to account for losses of power in the motor itself due to friction and heating, so a motor of about <nobr><i>1000 watt (1 kilowatt)</i></nobr> should suffice, if we allow it to work without interruption.<br/><br/>In practice, the motor should start and stop periodically, depending on the level of water in the tank, so it has to pump faster than water is consumed and we need a more powerful motor, say <nobr><i>1.5 KW</i></nobr>.<br/>With voltage to such a motor at the level of <nobr><i>220V</i></nobr> the current flowing through this motor is<br/><i><b>I = 1500W/220V ≅ 6.8A</b></i><br/><br/>In addition, considering that things break and we need an uninterrupted water supply, we need the same pump with the motor of the same power to be ready to automatically pick up the load in case the main pump breaks.<br/>That makes a design a bit more complicated with two pumps working in parallel, alternating their work and, in case one breaks, another working alone. This requires some electronic switching mechanism.<br/><br/><br/><i>Heat</i><br/><br/>Electric heater and incandescent lamp represent this group of electric devices. They warm and light up our homes.<br/><br/>We use electric stove to prepare our food.<br/><br/>Electric hair drying fan is an example of a combined mechanical (to push the air) and heating (to heat up the air) electric device.<br/><br/>Drying machine uses electricity to rotate a drum, produce heat and blow it into the drum using a fan.<br/><br/>For illustration, let's do some calculations related to incandescent lamps.<br/>Consider a lamp with marked power consumption of <nobr><i>P=100W</i></nobr> and voltage <nobr><i>U=120V</i></nobr>.<br/>Here we are talking about alternating current and, therefore, all characteristics are <i>effective</i>.<br/><br/>The effective electric current running through it is<br/><i><b>I = 100W/120V ≅ 0.8333A</b></i><br/>The resistance of the spiral in this lamp is<br/><i><b>R = 120V/0.8333A = 144Ω</b></i><br/><br/>Obviously, we can check that<br/><i><b>P = U²/R = I²·R</b></i><br/>With given voltage in the circuit, the power consumed by an incandescent lamp will be more when the resistance is less. That's why a lamp consuming <i>100W</i> has a thicker spiral (with less resistance) than a lamp consuming <i>40W</i> for the same voltage.<br/><br/>Another interesting example of using electricity to produce heat is welding. This is a process when an electric arc between two electrodes is formed and used to melt metal.<br/><br/>There are many different types of welding machines. An important characteristic is the electric current going through the arc formed between electrodes. Usually it's in hundreds of amperes, like <i>500A-1000A</i> with voltage in the range <i>30V-60V</i>.<br/>That makes the power consumption of a welding machine to be somewhere from <i>15KW</i> to <i>60KW</i>, which is a lot, comparing to a power of about <i>1.5KW</i> needed for a water pump described above.<br/><br/>These characteristics fluctuate as the welding process goes, they depend on the length of an electric arc and materials used as electrodes.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-37577092315479179512021-01-29T06:48:00.000-08:002021-01-29T06:48:41.195-08:00Electricity to Consumers: UNIZOR.COM - Physics4Teens - Electromagnetism ...<iframe style="background-image:url(https://i.ytimg.com/vi/yFqUetqW4Vs/hqdefault.jpg)" width="480" height="270" src="https://youtube.com/embed/yFqUetqW4Vs" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Distribution to Consumers</u><br/><br/>When we described the <b>grid</b> in the previous lecture, we concentrated on main principles of its work - synchronization of the generators of electric energy with the grid. It might make a wrong impression that the grid is one gigantic wire that have certain voltage, frequency and phase, and all the generators must adhere to this standard, when connected to it.<br/>The real situation is more complex.<br/><br/>Considering the grid covers large distances, we must maintain the high voltage in it to avoid waste of energy to heat (hundreds of thousands of volts). Most consumers, however, need relatively lower voltage (no more than a few hundred volts).<br/><br/>To connect each consumer to a grid through a powerful and very expensive transformer is impractical. Instead, we combine a group of geographically close consumers into it's own grid, connected at one point to a main ultra high voltage grid through a transformer that lowers the voltage, and then we connect consumers to this second level grid.<br/><br/>For example, the whole city can form this secondary grid with lower voltage than in the main ultra high voltage grid. Thus formed, this secondary grid can go to each street with lower voltage better suitable for consumption. And, because this grid is relatively localized, there will be no big loss of energy to distribute electricity at a lower voltage.<br/><br/>Making the picture even more complex, we can arrange third level grid lowering the voltage for each building in the city from street level of the secondary grid to a lower voltage building level that goes to each apartment.<br/><br/>Yet another complication can be introduced by connecting other generators to a grid. Since our grid now consists of many grids connected via transformers from main ultra high voltage to second level with lower voltage in the city and third level for each building, we can introduce generators into each grid, and the only requirement for this is to make sure that each generator conforms to a corresponding grid's voltage, frequency and phase.<br/><br/>For example, a building's management decided to put solar panels on the roof. They generate electricity for a building and, if there is excess of energy, it goes to a grid for general consumption. That, probably, is the third level grid that serves this building, it has it's own characteristics, and the output of solar panels must conform to these characteristics.<br/><br/>Similarly, a city decided to build a power plant working on burning the garbage. This power plant produces electricity that should go to a grid that serves the whole city, which we call the second level grid. The voltage produced by this plant is higher than the one in any building level grid but not as high as in main ultra high voltage grid that supplies the whole city.<br/><br/>The overall picture of distribution of electricity consists of different grids of different voltage, connected through transformers, having producers and consumers in each grid. Each consumer of electricity should have its parameters to be the same as the grid it's connected to (voltage and frequency). Each producer of electricity should be synchronized with the grid it's connected to in voltage, frequency and phase.<br/><br/>On the picture below we have schematically displayed the generation of electricity at <i>13,000V</i>, transformers that increase the voltage to <i>600,000V</i> to transmit along the long distances, transformers that decrease the voltage to <i>7,200V</i> at the entrance to a city that supplies this electricity to buildings and, finally, transformers that decrease the voltage to <i>240V</i> before entering buildings.<br/>Inside the buildings this voltage is distributed to individual apartments.<br/><img src='http://www.unizor.com/Pictures/GridA2Z.png' style='width:200px;height:170px;'><br/>In practice there are more devices participating in the grid, stabilizing the voltage, sensing the abnormal conditions, controlling different functions of the grid, protecting the grid against disastrous conditions, attacks or human errors etc.<br/><br/>The grid is constantly changing as new sources of energy come on line, new consumers are attached to a grid, new more efficient maintenance devices are introduced Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-88132076282570838052021-01-24T08:52:00.003-08:002021-01-24T08:52:45.433-08:00Electric Grid: UNIZOR.COM - Physics4Teens - Electromagnetism - Distribution<iframe width="480" height="270" src="https://youtube.com/embed/R_jVITpgmZM" frameborder="0"></iframe><br/><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>The Grid</u><br/><br/>The <i><b>grid</b></i> is an extremely important solution to most problems with electric power interruption due to different technical issues at power plants that generate electricity.<br/><br/>Consider a simple analogy of distributing water to apartments in a large building from a tank on a roof.<br/>If you have one tank, and it needs repair or cleaning, the water to all apartments must be shut off while the work performed.<br/>If, however, you have two water tanks on the roof both connected to a common distribution pipe, from which the water is flowing to all apartments, we can just close the connection of the tank to be cleaned to a distribution pipe, leaving another tank operational and water supply to apartments uninterrupted.<br/><br/>This same principle is used in combining many generating electricity power plants to a common distribution wiring, thus assuring uninterrupted power supply. This system of interconnected power plants forms a <i><b>grid</b></i> that feeds those consumers of electricity connected to it, assuring their uninterrupted work.<br/><br/>Obviously, with electricity it's much more complex than with water supply.<br/><br/>Let's enumerate problem we have to resolve when connecting different electric generators to a common power distribution system.<br/><br/><i>Direct Current</i><br/><br/>Let's connect two batteries generating direct current parallel to each other and power up a lamp.<br/><img src='http://www.unizor.com/Pictures/ParallelBatteries.png' style='width:200px;height:265px;'><br/>The proper connection requires the same <i>polarity</i> (positive pole of one battery is connected to positive pole of another, negative - to negative) and the same <i>voltage</i> generated by these batteries. Only then there will be no electric current between the batteries, only from a battery to a device consuming the electricity (a lamp in this case).<br/>These two conditions, similar <i>polarity</i> and equal <i>voltage</i>, are necessary and sufficient conditions to successfully connect two batteries in parallel. The overall energy capacity of these two batteries will be twice as big. They will last twice as long on the same load (one lamp) or they can double the load (have two lamps parallel to each other) and serve the same time as one battery on a single load.<br/><br/><br/><i>Alternating Current</i><br/><br/>Analogously, two generators of alternating current (AC) must have the same output voltage, if connected in parallel. Otherwise, there will be an unnecessary electric current between them, which diminishes their usefulness.<br/><br/>But for AC generators there are more characteristic parameters than just a voltage. Voltage varies as a sinusoid with time and is characterized by <i>amplitude</i>, <i>frequency</i> and <i>phase</i>.<br/><br/>All three parameters must be the same for a proper parallel connection of two generators. Their output voltages, as functions of time, must coincide exactly to each other. And this is a big challenge to build a grid with many different generators, each contributing their part in overall power supply.<br/><img src='http://www.unizor.com/Pictures/ParallelGenerators.png' style='width:200px;height:170px;'><br/><br/>The above considerations dictate strict restrictions on how any new source of electricity, like a new power plant or a new solar panel are hooked to a <i><b>grid</b></i>.<br/><br/>First of all, the output of a generator at the point of connection to a grid must be alternating. Solar panels, for example, produce direct current, so, before connecting to a grid, the DC electricity must be converted to AC. Special devices called <i>inverters</i> provide this type of conversion, assuring the frequency of voltage oscillation to be that of the grid.<br/><br/>Then, depending on a point of connection to a grid, the output voltage must be equalized with that of the grid at the connection point. This can be done with proper <i><b>transformers</b></i>.<br/><br/>Finally, we have to adjust the phase to synchronize the output of the generator with the phase of a grid. This can be achieve by adjusting the speed of a generator's rotor while monitoring the phase difference on a special sensor until proper <i><b>synchronization</b></i> is achieved.<br><br/>Overall, the connection to a grid is a sophisticated process that requires special devices, tools, instrumentation and care.<br/>There are many controls that monitor, adjust and maintain the regime of work of a grid. In many ways it's automated, computer controlled and reliable. However, human errors do happen and some of them result in significant distortions of power supply to large areas and affecting a lot of people. As an example, the 2003 blackout in Ohio resulted in power loss across Eastern United States and even some areas in Canada.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-54438069816225472312021-01-22T08:16:00.000-08:002021-01-22T08:16:37.062-08:00Electricity In Transit: UNIZOR.COM - Physics4Teens - Electromagnetism - ...<iframe width="480" height="270" src="https://youtube.com/embed/6PiSthNGN1Q" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Electricity in Transit</u><br/><br/>Let's discuss how electricity is delivered from the power plants to consumers.<br/><br/>The only way to deliver the electricity from the place it's generated to a place it's used is via electric wires.<br/>Since the distance between the power plant and a consumer can be substantial, may be even hundreds of kilometers, the problem of losses of electric energy in transit because of wire resistance is extremely important.<br/><br/>Examine a simple electric circuit consisting of a source of electricity (generator) and a consumer (like an electric motor).<br/>In theory, we have four places where electric energy is spent:<br/>(a) inside a generator due to internal resistance,<br/>(b) inside a wire from a generator to a motor due to wire resistance,<br/>(c) inside a motor due to useful work the electricity does and internal resistance,<br/>(d) inside a wire from a motor back to a generator due to wire resistance.<br/><img src='http://www.unizor.com/Pictures/ElectricityTransit.png' style='width:200px;height:60px;'><br/><br/>Obviously, only in a motor the energy is spent with some useful purpose, in other places the energy is spent just because it's unavoidable, that is wasted.<br/><br/>Amount of energy wasted to heat per unit of time due to wire resistance <i><b>W</b><sub>wire</sub></i> depends on the current running through wire <i><b>I</b><sub>wire</sub></i> and the wire resistance <i><b>R</b><sub>wire</sub></i> according to a formula<br/><i><b>W</b><sub>wire</sub><b> = I<sup>2</sup></b><sub>wire</sub><b>·R</b><sub>wire</sub></i><br/><br/>For practical example, let's calculate the amount of energy wasted in some long piece of copper wire.<br/>The resistance of a wire depends on <i>resistivity</i> of material it's made of <i>ρ</i>, is proportional to the length of a wire <i>L</i> and is inversely proportional to its cross-section area <i>A</i><br/><i><b>R</b><sub>wire</sub><b> = ρ·L<font size=4>/</font>A</b></i><br/><br/>For copper the resistivity is approximately<br/><i>ρ≅1.70·10<sup>−8</sup> Ω·m</i>.<br/>Assume, the combined wire length to and from a consumer of electricity is<br/><i>L = 1 km = 1000 m</i><br/>and its diameter is<br/><i>D = 2 mm = 2·10<sup>−3</sup> m</i>,<br/>which gives the area of its cross-section<br/><i>A = π·D²/4 ≅ 3.14·10<sup>−6</sup> m²</i><br/>Then the resistance of this peace of wire is<br/><i>R<sub>wire</sub> ≅ 5.4 Ω</i><br/><br/>For example, we are supposed to run an electric motor working at voltage<br/><i>U<sub>mot</sub>=220 volt</i><br/>and delivering power of<br/><i>W<sub>mot</sub>=2.2 kilowatt</i>.<br/><br/>Then the current it requires is<br/><i>I<sub>mot</sub> = W<sub>mot</sub>/U<sub>mot</sub> = 10 A</i><br/><br/>The current <i>I<sub>mot</sub>=10 A</i> must go through the wire<br/><i>I<sub>wire</sub> = I<sub>mot</sub> = 10 A</i><br/>Then the amount of energy wasted to heat in the copper wire of resistance <i>R<sub>wire</sub>=5.4 Ω</i> per unit of time (a second) is<br/><i>W<sub>wire</sub> = I²<sub>wire</sub>·R<sub>wire</sub> = 540 W</i><br/><br/>For a price of about $0.1 per kilowatt this amounts to about $0.054 per second. For 24 hours uninterrupted work the financial waste amounts to $4,665, and that is every day of operation of one motor, which is absolutely unacceptable.<br/><br/>Reducing the resistance of a wire by making it thicker or using multiple parallel wires has its practical limitations because of cost of wires. Therefore, our solution to reduce the energy wasted to heat due to wire resistance, while staying within reasonable limits with the cost of a wire, must be related to reducing the current <i>I</b><sub>wire</sub></i> running through a wire without reduction of power that is supposed to be delivered to consumers of electricity.<br/><br/>This can be accomplished by using transformers.<br/>Immediately after generation, the alternating current is directed to a <i>transformer</i> that increases the voltage and proportionally decreases the amperage.<br/><br/>At the output of this transformer the voltage reaches thousands of volts - from low voltage of 1000V to ultra high voltage above 800,000V, depending on the length of wires from generators to consumers.<br/>This high voltage electricity is delivered to consumers, where another transformers reduce the voltage to standard needed to run all their different devices. <br/><br/>Consumers of electricity get the voltage required to run their equipment, but the current running in the long wires between generators and consumers is low, thus reducing waste of electric energy.<br/><br/>Consider an example above with a motor that needs <i>W<sub>mot</sub>=2.2 kilowatt</i> of electricity at voltage <i>U<sub>mot</sub>=220 volt</i> and, therefore, requires <i>I<sub>mot</sub>=10 A</i> electric current.<br/>If, instead of transmitting electricity with these parameters, we increase the voltage by a <i>transformer</i> before sending it to long wires to, say, <i>2200V</i>, thus proportionally reducing the amperage by the same factor, our amperage will be<br/><i>A = W/U = 2200/2200 = 1A</i><br/>Reducing the amperage from <i>10A</i> to </i>1A</i> reduces the energy waste by a factor of <b>100</b> because the heat formula depends on a square of amperage.<br/><br/>The distribution of electricity, therefore, should include transformers that increase the voltage before sending electricity along long wires and decrease it wherever it's needed for usage by consumers.<br/>With this modification the picture that corresponds to practical aspects of distribution of electricity looks like this<br/><img src='http://www.unizor.com/Pictures/ElectricityTransformed.png' style='width:200px;height:80px;'><br/><br/>To increase the electrical systems' reliability, improve the energy balancing and make sure of uninterrupted power supply, the sources of electrical energy (electric power plants and other installations producing electric energy) are combined into a network called the <i><b>grid</b></i>.<br/>The principles of this networking are a subject of the next lecture.<br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-65267088255934775682021-01-10T09:25:00.000-08:002021-01-10T09:25:15.934-08:00Electricity at Power Plants: UNIZOR.COM - Physics4Teens - Electromagneti...<iframe width="480" height="270" src="https://youtube.com/embed/6R9a4AX-qrc" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Electricity at Power Plants</u><br /><br/>Addressing distribution of electricity, we will primarily discuss the way electricity, produced at the power plants, is delivered to consumers.<br/>We will concentrate on the process of distribution of electricity generated from the kinetic energy of rotating turbines, as the most quantitatively significant source of electricity.<br/><br/>Three stages of this distribution are<br/>(a) at the power plant<br/>(b) in transit<br/>(c) at consumers.<br/><br/>This lecture is about what's going on at the power plant that produces the electricity from the kinetic energy of rotating turbines.<br/><br/>Turbines at electric power plants are rotated because of a flow of steam or water, or wind. Turbines are acting as <i>rotors</i> in the electric power generator, while the electricity is produced in <i>stators</i> based on the principles of electromagnetic induction.<br/><br/><br/><i>Hydroelectric Stations</i><br/><br/>Let's estimate theoretically an amount of energy that a hydroelectric station can produce.<br/><img src='http://www.unizor.com/Pictures/HydroelectricPlant.jpg' style='width:200px;height:160px;'><br/><br/>First important component in generating electricity at a hydroelectric station is falling water. We can have falling water by building a dam on a river, like Hoover Dam on Colorado river or use natural difference in the level of water of the waterfalls, like at Niagara falls.<br/><br/>Having the water at two levels, we should direct the flow from top to bottom onto turbines through pipes. Amount of electricity that can be generated obviously depends on the amount of potential energy water at the top has relatively to the bottom level. As the water falls through the pipes onto turbine, its potential energy is converted into kinetic energy of moving water. This is how much energy we can use to generate electricity. It depends on the amount of water flowing through pipes and the height difference between the top and the bottom levels.<br/><br/>Let's assume that the amount of water falling down through pipes from top to bottom level is <nobr><i><b>M</b> (kg/sec)</i></nobr> and the difference in height from top to bottom is <nobr><i><b>H</b> (m)</i>.</nobr><br/>That means that the amount of energy falling water is losing per unit of time is<br/><i><b>P</b><sub>water</sub><b> = M·g·H</b> (J/sec </i>or<i> W)</i><br/><br/>Then we have to solve a purely technical problem to convert this energy into rotational energy of turbines.<br/><img src='http://www.unizor.com/Pictures/HydroTurbine.jpg' style='width:200px;height:135px;'><br/><br/>Different designs of turbines have been used and tested during a long time of using hydroelectric power. Contemporary turbines are pretty efficient in this process of conversion, but still far less than 100% effective. Losses of energy always exist, and we need some coefficient of efficiency of a turbine to get exact amount of rotational energy produced by falling water.<br/>Let's assume that <i><b>k</b></i> is such a coefficient. It has a value from 0 (absolutely ineffective conversion) to 1 (full energy amount of falling water is converted into rotational energy of a turbine). Then the amount of rotational energy produced by turbines per unit of time is<br/><i><b>P</b><sub>turbine</sub><b> = k·M·g·H</b></i><br/><br/>Next step is to convert rotational energy of turbines into electric energy.<br/>This is done by generators, which we discussed in previous lectures.<br/><img src='http://www.unizor.com/Pictures/RotorStator.jpg' style='width:200px;height:100px;'><br/><br/>The contemporary generators are pretty effective with norm being above 90%, so we can assume that the coefficient of effectiveness <i><b>k</b></i> introduced above encompasses both effectiveness of converting energy of falling water into rotation of turbines and conversion of rotation into electric energy.<br/>So, overall energy produced by a hydroelectric power station per unit of time (that is, the power produced) is<br/><i><b>P = k·M·g·H</b></i><br/>The hydroelectric power stations can be very large and can produce a lot of electric energy. The most powerful electric power stations are hydroelectric. The problem is, there are not too many rivers suitable for building hydroelectric power stations and an environmental effect of building a hydroelectric power station can be significant.<br/><br/>At the same time, the hydroelectric power stations are pretty efficient, the coefficient <i><b>k</b></i> in the formula above can be above 0.8, which means that about 80% of the power of water falling on turbines is effectively converted into electric power.<br/><br/><br/><i>Coal Burning Stations</i><br/><br/>Almost a third of electricity generated in the world is produced by fossil fuel burning power stations.<br/>Let's examine the coal burning power station.<br/><img src='http://www.unizor.com/Pictures/CoalBurningElectricPlant.jpg' style='width:200px;height:135px;'><br/><br/>The main steps of producing electricity by burning fossil fuel are<br/>(a) burning fossil fuel to boil water, converting chemical energy of burning fuel into kinetic energy of produced steam,<br/>(b) converting kinetic energy of steam into rotation of turbines,<br/>(c) converting rotational energy of turbines into electricity by generators.<br/><br/>Coal is a major source of fossil fuel with natural gas and oil following.<br/>Convenience of putting a coal burning electric power station anywhere should be weighed against environmental impact of such a plant.<br/><br/>Producing energy from burning coal is not a very efficient way to extract chemical energy. Significant portion of the energy produced by burning coal is wasted on each step and the overall efficiency of such a power station is about 40%. Most of the energy losses occur during the first stage of generating electricity - burning coal to boil water and produce steam.<br/>Some efficiency can be achieved by pulverizing coal to powder. However, the main product of burning fossil fuel - carbon dioxide <i><b>CO<sub>2</sub></b></i> - produces some unavoidable negative environmental effect.<br/><br/><br/><i>Nuclear Power Plants</i><br/><br/>The difference between a nuclear power plant and coal burning one is at the first stage to boil the water. While at coal burning plants the source of heat to boil water is burning coal, at the nuclear power plant the source of heat is energy released by breaking nuclei of heavy elements, like Uranium or Plutonium, into lighter components using neutrons.<br/><img src='http://www.unizor.com/Pictures/NuclearFission.jpg' style='width:200px;height:120px;'><br/><br/>The main steps of producing electricity in a nuclear power plant are<br/>(a) bombarding the enriched radioactive material (Uranium, Plutonium or other) with neutrons causing the nuclei of this material to break, releasing certain amount of heat to boil water getting steam,<br/>(b) converting kinetic energy of steam into rotation of turbines,<br/>(c) converting rotational energy of turbines into electricity by generators.<br/><br/>Efficiency of nuclear power plants is quite limited inasmuch as in coal burning power stations and is about 40%. That is, about 40% of the energy generated by heat is converted into electricity.<br/><br/>Of interest is a process of nuclear fission that produces the heat. Here is a simplified model of this process.<br/><br/>When a nucleus of Uranium-235 (92 protons + 143 neutrons) is bombarded with a neutron, it temporarily accepts this neutron inside, becoming Uranium-236 (92 protons + 144 neutrons).<br/>This isotope is not stable and a nucleus breaks into different parts. This is a complex process and parts might be different.<br/><br/>A typical scenario might be as follows.<br/>Broken parts are Barium (56 protons + 83 neutrons), Krypton (36 protons + 58 neutrons) and 3 neutrons are released to bombard other nuclei of Uranium 235, causing a <b>chain reaction</b>.<br/><br/>The combined mass of all parts is less than the mass of initial components. The remaining mass of an unstable nucleus of Uranium-236 is converted into radiation (heat, gamma-rays). The heat is used to boil the water, converting it into high energy steam to rotate the turbines.<br/><br/>The corresponding equations describing nuclear fission is:<br/><i><b><sup>1</sup>n<sub>0</sub> + <sup>235</sup>U<sub>92</sub> → <sup>236</sup>U<sub>92</sub> →<br/>→ </b>(fission)<b> →<br/>→ <sup>139</sup>Ba<sub>56</sub> + <sup>94</sup>Kr<sub>36</sub> + 3<sup>1</sup>n<sub>0</sub> + γ</b></i><br/><br/>In reality the process is much more complex because the broken parts of a nucleus might be different, themselves not stable and further emitting elementary particles.<br/>The process must be controlled by reducing the number of neutrons flying in all the directions after fission to prevent a nuclear explosion.<br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-74465376178827381492020-12-27T07:19:00.000-08:002020-12-27T07:19:09.335-08:00Electricity from Solar Energy: UNIZOR.COM - Physics4Teens - Electromagne...<iframe width="480" height="270" src="https://youtube.com/embed/AIcuexrQQO8" frameborder="0"></iframe><br/><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Solar Energy → Electricity</u><br /><br/>Solar panels that produce electricity are not physically or chemically changed during this process. So, the only source of energy is the light, and the energy it carries is transformed by a solar panel into electricity, that is into movement of electrons.<br/>In this lecture we will analyze how the energy of light forces the electrons to move.<br/>The complete theory behind this process is based on principles of quantum mechanics and we will not be able to dive deep into this area, but a schematic description of the process will be presented.<br/><br/>Commercial solar panel are much more complex that might seem from the following explanation, but their engineering is not a subject of this lecture, which is only about the principle of generation of electricity from sun light.<br/><br/>Let's start with the most important component of a solar cell - silicon (<i><b>Si</b></i>).<br/>Silicon is a metalloid, it has crystalline structure and its atoms have four valence electrons on the outer most orbit, next down orbit contains eight electrons and the inner most - two electrons.<br/><br/>Silicon is one of the most frequently occurring elements on Earth and, together with oxygen, forms molecules of silicon dioxide <i><b>SiO<sub>2</sub></b></i> - main component of sand.<br/><br/>All silicon atoms are neutral since the number of electrons in each one and the number of protons in each nucleus are the same and equal to 14.<br/>Atoms are connected by their covalent bonds of valence electrons on the outer orbits into a lattice-like crystalline structure.<br/><br/>Picture below represents the crystalline structure of silicon<br/><img src='http://www.unizor.com/Pictures/SiliconCrystal.jpg' style='width:200px;height:200px;'><br/>For the purposes of this lecture we will represent inner structure of silicon atoms with only outer orbit of each atom with four valence electrons, as electrons on the inner orbits do not participate in the process of generating electricity from light.<br/><img src='http://www.unizor.com/Pictures/SiliconStructure.jpg' style='width:200px;height:200px;'><br/>The covalent bonds are quite strong, they do not easily release electrons. As a result, under normal conditions silicon is practically a dielectric.<br/>However, if we excite the electrons sufficiently enough to break the covalent bonds, some electrons will move. For example, if we increase the temperature of a piece of silicon or put it under a bright sun light and measure its electrical resistance, we will observe the resistance diminishing.<br/>That's why silicon and similar elements are called <b>semiconductors</b>.<br/><br/>While excited electrons of any semiconductor decrease its electrical resistance, they don't produce electromotive force because the material as a whole remains electrically neutral.<br/>To build a solar cell that produces electricity under sun light we will introduce two kinds of "impurities" into crystalline lattice of silicon.<br/>One is an element with five valence electrons on the outer most orbit, like phosphorus (<i><b>P</b></i>). When it's embedded into a crystalline lattice of silicon, one electron on that outer orbit of phosphorus would remain not attached to any neighboring atom through a covalent bond.<br/><img src='http://www.unizor.com/Pictures/SiliconPhosphorus.jpg' style='width:200px;height:200px;'><br/>This creates the possibility for this electron to start traveling, replacing other electrons and pushing them out, which are, in turn, push out others etc. Basically, we create as many freelance electrons as many atoms of phosphorus we add to silicon base.<br/>The whole material is still neutral, but it has certain number of freelance electrons and the same number of stationary positive ions - those nuclei of phosphorus that lost electrons to freelancers.<br/>Silicon with such addition is called <b>n-type</b> (letter <b>n</b> for <i>negative</i>).<br/><br/>Another type of "impurity" that we will add to silicon is an element with three valence electrons on the outer orbit, like boron (<i><b>B</b></i>).<br/>When atoms of boron are embedded into a crystalline lattice of silicon, the overall structure of this combination looks like this<br/><img src='http://www.unizor.com/Pictures/SiliconBoron.jpg' style='width:200px;height:200px;'><br/>In this case the lattice has a deficiency of an electron that is traditionally called a "hole". Existence of a "hole" opens the opportunity for neighboring valence electrons to fill it, thus creating a "hole" in another spot. These "holes" behave like positively charged particles traveling inside silicon with added boron inasmuch as negatively charged electrons travel in silicon with added phosphorus.<br/>Silicon with such addition of boron is called <b>p-type</b> (letter <b>p</b> for <i>positive</i>).<br/><br/>Now imagine two types of "impure" silicon, <b>n-type</b> and <b>p-type</b>, contacting each other. In practice, it's two flat pieces (like very thin squares) on top of each other.<br/>Let's examine what happens in a thin layer of border between these different types of material.<br/><br/>Initially, both pieces of material are electrically neutral with n-type having free traveling electrons and equal number of stationary positive nuclei of phosphorus inside a crystalline structure and with p-type having traveling "holes" and equal number of stationary negative nuclei of silicon inside a crystalline structure.<br/><br/>As soon as contact between these two types of material is established, certain exchange between electrons of the n-type material and "holes" of the p-type takes place in the border region called <b>p-n junction</b>. Electrons and "holes" in this border region combine, thus reconstituting the lattice.<br/>This process is called <b>recombination</b>.<br/><br/>The consequence of this process of recombination is that n-type material near the border loses electrons, thus becoming positively charged, while the p-type gains the electrons, that is loses "holes", thus becoming negatively charged.<br/><br/>Eventually, the diffusion between n-type and p-type materials stops because negative charge of the p-type sufficiently repels electrons from the n-type. There will be some equilibrium between both parts.<br/><br/>If we introduce heat or bright sun light to electrons of n-type part, the diffusion will be longer and greater charge will be accumulated on both sides of the material - positive on the n-type and negative on the p-type.<br/><br/>Now the key point is to connect n-type side to p-type through some kind of electrical connection and extra electrons from the p_type will go to positively charged n-type, thus creating an electric current. This current is weak because only the diffusion between n-type and p-type in the border region is a contributing factor, but it's still the electric current.<br/><br/>Obviously, the more excited electrons in the n-type material are - the stronger diffusion inside the p-n junction is and the stronger current is produced. That's why, if sun light is used to excite the electrons, the more light falls on the n-type side - the stronger current is produced.<br/><br/>A pair of n-type and p-type materials form a <i>cell</i>. Since we are talking about using sun light to excite electrons, these cells are called <i>solar cells</i> or <i>photovoltaic cells</i>. Cells can be connected in a series increasing the produced electromotive force (voltage) or they can be connected in parallel to increase the electric current (amperage). <i>Solar panels</i> are sets of <i>solar cells</i> connected in series to increase the generated electromotive force (voltage).<br/><br/>Materials used to produce commercial solar cells can be different, not necessarily silicon with phosphorus or boron additions, but the main principle of using p-n junction between superconductors is the same for all.<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-75461795306853161912020-12-20T09:47:00.000-08:002020-12-20T09:47:45.156-08:00Chemical Energy to Electricity: UNIZOR.COM - Physics4Teens - Electromagn...<iframe width="480" height="270" src="https://youtube.com/embed/Mai5bN-U4H8" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Chemical Energy → Electric Energy</u><br /><br/>To generate energy, the source components participating in the process should have more energy than the final components of the process. In case of generation of any form of energy from chemical energy we deal with chemical reaction, during which the inter-atomic energy of source components should exceed the inter-atomic energy of products of chemical reaction.<br/><br/>The simple form of such process is burning that produces heat. For example, burning methane <i><b>CH<sub>4</sub></b></i> in atmosphere that contains oxygen <i><b>O<sub>2</sub></b></i> is a chemical reaction described by<br/><i><b>CH<sub>4</sub> + 2O<sub>2</sub> = CO<sub>2</sub> + 2H<sub>2</sub>O</b></i><br/>The inter-atomic energy of one molecule of methane and two molecules of oxygen must be greater than inter-atomic energy of a molecule of carbon dioxide and two molecules of water, otherwise there will not be any energy released as heat.<br/><br/>To generate electric energy from chemical we need the same type of inequality: the inter-atomic energy of primary components of the chemical reaction must be greater than inter-atomic energy of the resulting components.<br/>The way how this excess of energy represented depends on the chemical reaction. In the process of burning the excess of energy is in a form of heat, in case of the generating of electricity the excess of energy is in a form of electric current that has certain voltage and amperage and, therefore, is a carrier of energy.<br/><br/>Transformation of chemical energy into electricity is typically occurring in <b>batteries</b>.<br/>There are different types of batteries, and in this lecture we will mention a few with some details of how they work.<br/><br/>Consider a <b>lead-acid battery</b> used in many cars.<br/>In a simplified way it has three major components: solid <i>anode</i> (negative electrode) made of <i>lead <b>Pb</b></i>, solid <i>cathode</i> (positive electrode) made of <i>lead dioxide <b>PbO<sub>2</sub></b></i> and liquid <i>electrolyte</i> in-between them containing <i>sulfuric acid <b>H<sub>2</sub>SO<sub>4</sub></b></i> diluted in <i>water <b>H<sub>2</sub>O</b></i>.<br/><img src='http://www.unizor.com/Pictures/LeadAcidBattery.png' style='width:200px;height:190px;'><br/>To understand why electricity is generated by this device, let's look inside the atomic structure of its components and analyze what happens when there is a load (like a lamp) connected to its terminals.<br/><br/>Recall the classical planetary model of an atom.<br/>There are protons and neutrons forming its nucleus and electrons circulating around this nucleus on different orbits.<br/>Those electrons on outer most orbits are less attached to a host nucleus and many of them can fly around, potentially getting attached by other host nuclei.<br/>This creates certain amount of "free" positively (with deficit of electrons) charged <i>ions</i> called <b>cations</b> and negatively (with access of electrons) charged <i>ions</i> called <b>anions</b>.<br/><br/>Analogously, molecules are also not completely stable and can lose atoms, if inter-atomic forces are not strong enough.<br/>Applied to molecular structure of any substance, typical contents not only contains electrically neutral molecules, but also molecules with certain missing or extra atoms or electrons - <i>ions</i>.<br/><br/>Consider sulfuric acid inside a battery pictured above.<br/>Structural composition of atoms of each molecule of this acid can be represented as<br/><img src='http://www.unizor.com/Pictures/SulfuricAcid.jpg' style='width:200px;height:100px;'><br/>Many of electrically neutral molecules of sulfuric acid <b>H<sub>2</sub>SO<sub>4</sub></b> are partially breaking losing a positive ions (nuclei with a single proton) of hydrogen but retaining their electrons, which can be represented as<br/><i><b>H<sub>2</sub>SO<sub>4</sub> → 2H<sup> +</sup> + SO<sub>4</sub><sup>2−</sup></b></i><br/>This can be explained by the fact that hydrogen nucleus contains only one proton, this is the lightest and electrically weakest nucleus and, being so light and volatile, it easily breaks its inter-atomic links with the main molecule of sulfuric acid. The resulting <i>cations <b>H<sup> +</sup></b></i> and <i>anions <b>SO<sub>4</sub><sup>2−</sup></b></i> are actually responsible for burning and corrosion caused by sulfuric acid. In batteries, however, these ions are responsible for producing electricity.<br/>Here is how.<br/><br/>The negative <i>anode</i> of a lead-acid battery is made of lead (chemical symbol <i><b>Pb</b></i>), which, when going into chemical reactions, exhibits either 2 or 4 links to other atoms in the molecules.<br/>The negative ion (<i>anion</i>) <i><b>SO<sub>4</sub><sup>2−</sup></b></i>, produced during the breaking of the molecules of sulfuric acid, chemically reacts with led producing <i>lead sulfate</i>, and releasing two electrons left after ions of hydrogen broke free from the molecule of sulfuric acid as follows:<br/><i><b>Pb+SO<sub>4</sub><sup>2−</sup> → PbSO<sub>4</sub> + 2e<sup>−</sup></b></i><br/>Structural composition of atoms of each molecule of <i>lead sulfate</i> can be represented as<br/><img src='http://www.unizor.com/Pictures/LeadSulfate.png' style='width:200px;height:130px;'><br/><br/>Keep an eye on two electrons <i><b>2e<sup>−</b></i> produced at <i>anode</i>. They are produced on the surface of a lead <i>anode</i> and accumulated inside it up to a certain concentration that gives certain negative charge to an <i>anode</i>. These electrons will be the ones that produce the electric current, when some load (like a lamp) is connected to terminals of a battery.<br/><br/>Meanwhile, at the <i>cathode</i> terminal of a led-acid battery another reaction goes on between its main component <i>lead dioxide <b>PbO<sub>2</sub></b></i>, having the following structure<br/><img src='http://www.unizor.com/Pictures/LeadDioxide.png' style='width:200px;height:90px;'><br/>and electrolyte that still contains unused positive ions of hydrogen <i><b>2H<sup>+</sup></b></i> from the reaction on an <i>anode</i> and another pair of ions, positive <i><b>2H<sup>+</sup></b></i> and negative <i><b>SO<sub>4</sub><sup>2−</sup></b></i> of the sulfuric acid.<br/><br/>Now two reactions simultaneously go on at the surface of a <i>cathode</i>.<br/>First of all, lead dioxide connects with negative ion of sulfuric acid, producing lead sulfate and releasing two negative ions of oxygen:<br/><i><b>PbO<sub>2</sub> + SO<sub>4</sub><sup>2−</sup> →<br/>→ PbSO<sub>4</sub> + 2O<sup>−</sup></b></i><br/>Secondly, remaining two positive ions of hydrogen from a molecule of sulfuric acid participating in the above reaction and two positive ions of hydrogen from the reaction on an <i>anode</i> meet two negative ions of oxygen from the above reaction, forming two molecules of water <i><b>H<sub>2</sub>O</b></i> with two electrons still missing:<br/><i><b>4H<sup>+</sup> + 2O<sup>−</sup> → 2H<sub>2</sub>O<sup>2+</sup></b></i><br/><br/>Concentration of these molecules of water missing two electrons creates a positive charge on the <i>cathode</i> terminal of a battery. When this concentration reaches certain level, positive ions of hydrogen cannot reach a <i>cathode</i>, ions of hydrogen are not readily breaking off the molecules of sulfuric acid and reaction stops at certain level of positive charge, unless we connect <i>anode</i> and <i>cathode</i> through some external electric load to allow accumulated on an <i>anode</i> electrons compensate missing electrons on a <i>cathode</i>.<br/><br/>So, we have a shortage of two electrons on a <i>cathode</i> terminal of a battery to form electrically neutral molecules of lead dioxide and water, but these two electrons can travel through an outside load between the terminals of a battery from its <i>anode</i>.<br/>The overall reaction on a <i>cathode</i> looks like<br/><i><b>PbO<sub>2</sub> + 4H<sup>+</sup> + SO<sub>4</sub><sup>2−</sup> + 2e<sup>−</sup> →<br/>→ PbSO<sub>4</sub> + 2H<sub>2</sub>O</b></i><br/><br/>As a result of these chemical reactions electrons released by a reaction on an <i>anode</i> travel to a <i>cathode</i> through external load, thus creating an <b>electric current</b>.</br>In an absence of an external load reactions on terminals of a battery continue until some limit of concentration of negative charge on an <i>anode</i> and positive charge on a <i>cathode</i> is reached, after which the reactions stops as further ionization is prevented by accumulated charges.<br/><br/>This is how chemical energy (inter-molecular links) is converted into electrical energy in a lead-acid battery.<br/><br/>The above described lead-acid battery is capable to work in reverse, to accumulate chemical energy, if external electromotive force is applied to its terminals. In this case all reactions go in reverse, that's how car battery is charged by alternator, when a car is in motion.<br/>The overall picture of the lead-acid battery to discharge and charge is as follows<br/><img src='http://www.unizor.com/Pictures/LeadAcidProcess.png' style='width:200px;height:300px;'><br/><br/>Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-5405065304407990502020-12-07T08:00:00.000-08:002020-12-07T08:00:27.890-08:00Electricity from Kinetic Energy: UNIZOR.COM - Physics4Teens - Electromag...<iframe width="480" height="270" src="https://www.youtube.com/embed/9h04wQ904J8" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>Kinetic Energy → Electric Energy</u><br /><br/>The main principle used in converting kinetic energy into electric is the principle of electromagnetic induction.<br/>Recall the Faraday's Law that defines the induced EMF as being proportional to a rate of changing the magnetic flux<br/><i><b>EMF = −</b>d<b>Φ/</b>d<b>t</b></i><br/><br/>We can achieve a variable magnetic flux by either rotating the wire frame in the permanent magnetic field or rotating the magnetic field inside the wire frame. The corresponding designs were discussed earlier in this course.<br/><br/>In this lecture we will talk about how we can make a rotation of a rotor in the electric generator.<br/><br/>The simplest form of generating a rotational movement is if we already have some mechanical movement, so all we need is to transform one form of motion (usually, along some trajectory) into a rotational one.<br/><br/>As the first example of such purely mechanical device, consider a propeller.<br/><img src='http://www.unizor.com/Pictures/Propeller.jpg' style='width:200px;height:150px;'><br/>It can be used to convert the flow of water in <b>hydroelectric plants</b> or the flow of wind through a <b>wind turbine</b> into a rotation. Once we have a rotational motion of the rotor in an electric generator, the rest goes along the previously described way of generation of electricity according to the laws of induction.<br/>It can be a generation of alternating current, including three-phase one, or direct current. These were discussed in details in previous lectures.<br/><br>In other cases we do not have already available motion that we can transform into a rotation, but we can artificially create one, using some other form of energy.<br/>The common process <b>thermal power station</b> is to generate a flow of steam by heating the water or a flow of some kind of combustion gases. This can be done by using the burners that burn coal, oil or natural gas.<br/>Another way of heating is to use nuclear energy to heat the water by controlling the chain reaction inside the radioactive core of a nuclear reactor.<br/>In some cases the solar energy is used to heat the water to produce a flow of steam.<br/>Rarely used types of heat are geothermal and ocean thermal sources.<br/><br/>In all these cases some kind of <b>turbines</b> are used to convert the flow of moving substance (water, air, gases, steam) into a rotation.<br/>Steam turbines are just a more sophisticated type of propeller (or rather coaxial propellers), allowing to extract as much as possible energy from the steam flow.<br/><img src='http://www.unizor.com/Pictures/Turbine.jpg' style='width:200px;height:150px;'><br/><br/>Another form of generating electricity from heat and kinetic energy is internal combustion engines. The work of such engine results in a reciprocating motion of a piston, converted, using a connecting rod and a crankshaft, into rotation of a rotor of an electric generator that generates electricity.<br/><img src='http://www.unizor.com/Pictures/Crankshaft.jpg' style='width:200px;height:150px;'><br/><br/>In all the above cases the electricity is produced from kinetic energy of some substance, which is either readily available in nature (like water flowing along a river) or produced as a result of some process (like heating the water to produce a flow of steam).<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0tag:blogger.com,1999:blog-3741410418096716827.post-48975085707000515982020-11-22T09:12:00.002-08:002020-11-22T09:12:57.764-08:00AC Power: UNIZOR.COM - Physics4Teens - Electromagnetism - AC Ohm's Law<iframe width="480" height="270" src="https://www.youtube.com/embed/Una1qjDReO8" frameborder="0"></iframe><i>Notes to a video lecture on http://www.unizor.com</i><br/><br/><u>AC Power</u><br /><br/><b>Power</b> is a rate of work performed by some source of energy (like electric current) per unit of time. More precisely, it's a derivative of work performed by a source of energy, as a function of time, by time.<br/><br/>As we know from the properties of a <i>direct electrical current</i>, its power is<br/><i><b>P = U·I = I²R = U²/R</b></i><br/>where <i><b>U</b></i> is the voltage around a resistor of resistance <i><b>R</b></i> and <i><b>I</b></i> is the electric current going through this resistor.<br/>All values in the above expression are constant for <i>direct current</i>.<br/><br/><i>Alternating current</i> presents a problem of having the voltage and the current to be variable and dependent not only on the resistors, but also on the presence of inductors and capacitors in the circuit.<br/><br/>As described in the previous lecture, the alternating current in the circuit of a resistor, an inductor and a capacitor connected in a series to a generator of sinusoidal EMF equals to<br/><i><b>I(t) = (E<sub>0</sub> <font size=4>/</font>Z)·sin(ωt+φ) =<br/>= I<sub>0</sub>·sin(ωt+φ)</b></i><br/>where impedance <i><b>Z=√<span style='text-decoration:overline'>(X<sub><font size=1>C</font></sub>−X<sub><font size=1>L</font></sub>)²+R²</span></b></i>,<br/><i><b>tan(φ)=(X<sub><font size=1>C</font></sub>−X<sub><font size=1>L</font></sub>)<font size=4>/</font>R</b></i>.<br/><br/>Having expressions for generated EMF <i><b>E(t)=E<sub>0</sub>·sin(ωt)</b></i> and electric current in a circuit <i><b>I(t)=I<sub>0</sub>·sin(ωt+φ)</b></i>, we can find an instantaneous power<br/><i><b>P(t) = E(t)·I(t) =<br/>= E<sub>0</sub>·I<sub>0</sub>·sin(ωt)·sin(ωt+φ)</b></i><br/><br/>When people talk about voltage or amperage in the AC circuit, they understand that these characteristics are variable and, to be more practical, they use <i>effective voltage <nobr><b>E<sub>eff</sub> = E<sub>0</sub> <font size=4>/</font>√<span style='text-decoration:overline'>2</span></b></i></nobr> and <i>effective amperage <nobr><b>I<sub>eff</sub> = I<sub>0</sub> <font size=4>/</font>√<span style='text-decoration:overline'>2</span></b></i></nobr> of the electric current. The usage of these characteristics allows to calculate the power consumed by a resistor-only circuit during a period [<i><b>0,T</b></i>] of time (for <i><b>T</b></i> significantly greater than one oscillation of a current) without integrating a variable function <i><b>P(t)=E(t)·I(t)</b></i> on interval [<i><b>0,T</b></i>], but just performing a multiplication of constants:<br/><i><b>W<sub>[0,T]</sub> = E<sub>eff</sub> · I<sub>eff</sub> · T</b></i><br/><br/>Adding an inductor and a capacitor brings some complication because of a phase difference between EMF and a current. To express the power consumed by an AC circuit that includes a resistor, an inductor and a capacitor in terms of effective voltage and effective amperage, let's find the work performed by an electric current during a period of oscillation in terms of <i><b>E<sub>eff</sub></b></i> and <i><b>I<sub>eff</sub></b></i> and divide it by this period. The result would be an average power consumed by a circuit per time of one oscillation that we will call the <b>effective power</b> of a circuit.<br/><br/>The period of one oscillation with angular speed <i><b>ω</b></i> is <i><b>T=2π/ω</b></i>.<br/>The instantaneous power consumption by a circuit is<br/><i><b>P(t) = E(t)·I(t) =<br/>= E<sub>0</sub>·sin(ωt)·I<sub>0</sub>·sin(ωt+φ)</b></i><br/>The energy consumed by a circuit during one period of oscillation <i><b>T=2π/ω</b></i> equals to<br/><i><b>W<sub>[0,T]</sub> = <font size=5>∫</font><sub>[0,T]</sub>P(t)·</b>d<b>t</b></i><br/>where<br/><i><b>P(t)=E<sub>0</sub>·sin(ωt)·I<sub>0</sub>·sin(ωt+φ)</b></i><br/><br/>We can simplify the product of two trigonometric functions to make it easier to integrate:<br/><i><b>sin(x)·sin(y) =<br/>= (1/2)·</b></i>[<i><b>cos(x−y)−cos(x+y)</b></i>]<br/>Using this for <i><b>x=ωt</b></i> and <i><b>y=ωt+φ</b></i>, we obtain<br/><i><b>sin(ωt)·sin(ωt+φ) =<br/>= (1/2)·</b></i>[<i><b>cos(φ)−cos(2ωt+φ)</b></i>]<br/><br/>To find the power consumption during one period of oscillation <i><b>T</b></i>, we have to calculate the following integral<br/><i><b>W<sub>[0,T]</sub> = <font size=5>∫</font><sub>[0,T]</sub>P(t)·</b>d<b>t</b></i><br/>where period <i><b>T=2π/ω</b></i> and<br/><i><b>P(t) = E<sub>0</sub>·I<sub>0</sub>·<br/>·(1/2)·</b></i>[<i><b>cos(φ)−cos(2ωt+φ)</b></i>]<br/><br/>This integral can be expressed as a difference of two integrals<br/><i><b><font size=5>∫</font><sub>[0,T]</sub>E<sub>0</sub>·I<sub>0</sub>·(1/2)·cos(φ)·</b>d<b>t</b></i><br/>which, considering <i><b>cos(φ)</b></i> is a constant for a given circuit, is equal to<br/><i><b>E<sub>0</sub>·I<sub>0</sub>·(1/2)·cos(φ)·T =<br/>= E<sub>0</sub>·I<sub>0</sub>·(1/2)·cos(φ)·2π/ω</b></i><br/>and<br/><i><b><font size=5>∫</font><sub>[0,T]</sub>E<sub>0</sub>·I<sub>0</sub>·(1/2)·cos(2ωt+φ)·</b>d<b>t</b></i><br/>which is equal to zero because integral of a periodical function <i><b>cos(x)</b></i> over any argument interval that equals to one or more periods equals to zero.<br/>The same can be proven analytically<br/><i><b><font size=5>∫</font><sub>[0,T]</sub>cos(2ωt+φ)·</b>d<b>t =<br/>= sin(2ωt+φ)/(2ω)<font size=5>|</font><sub>[0,T]</sub> =<br/>= sin(2ω·2π/ω+φ)/(2ω) −<br/>− sin(φ)/(2ω) =<br/>= </b></i>[<b><i>sin(4π+φ)−sin(φ)</b></i>]<b><i>/(2ω) = 0</b></i><br/><br/>Hence, the energy consumed by a circuit during one period of oscillation equals to<br/><i><b>W<sub>[0,T]</sub> = E<sub>0</sub>·I<sub>0</sub>·(1/2)·cos(φ)·2π/ω</b></i><br/>The average power consumption, that is the average rate of consuming energy that we will call <b>effective power</b>, equals to this amount of energy divided by time, during which it was consumed - one period of oscillation <i><b>T=2π/ω</b></i>:<br/><i><b>P<sub>eff</sub> = W<sub>[0,T]</sub><fonr size=4>/</font>T = <br/>= E<sub>0</sub>·I<sub>0</sub>·(1/2)·cos(φ)</b></i><br/>Since <nobr><i><b>E<sub>eff</sub> = E<sub>0</sub> <font size=5>/</font>√<span style='text-decoration:overline'>2</span></b></i></nobr> and <i><nobr><b>I<sub>eff</sub> = I<sub>0</sub> <font size=5>/</font>√<span style='text-decoration:overline'>2</span></b></i></nobr>, the last expression for power equals to<br/><i><b>P<sub>eff</sub> = E<sub>eff</sub>·I<sub>eff</sub>·cos(φ)</b></i><br/>where a phase shift <i><b>φ</b></i> depends on resistance and reactances of a circuit as follows<br/><i><b>tan(φ) = (X<sub>C</sub> − X<sub>L</sub>) <font size=5>/</font>R</b></i><br/><i><b>X<sub>C</sub> = 1/(ωC)</b></i> - capacitive reactance,<br/><i><b>X<sub>L</sub> = ωL</b></i> - inductive reactance,<br/><i><b>R</b></i> - resistance.<br/>The above formula is derived for RLC-circuit that contains a resistor or resistance <i><b>R</b></i>, a capacitor of capacitance <i><b>C</b></i> and an inductor of inductance <i><b>L</b></i><br/>Let's analyze different circuits and their effective power consumption rate.<br/><br/><i>R-Circuit</i><br/>R-circuit contains only a resistor. Therefore, both reactances <i><b>X<sub>C</sub></b></i> and <i><b>X<sub>L</sub></b></i> are zero and phase shift <i><b>φ</b></i> is zero as well. Since <i><b>cos(0)=1</b></i>, the effective power for this R-circuit is<br/><i><b>P<sub>eff</sub> = E<sub>eff</sub>·I<sub>eff</sub></b></i><br/>which fully corresponds to a power for a circuit with a direct current running through it.<br/><br/><i>RC-Circuit</i><br/>RC-circuit contains a resistor and a capacitor in a series. Reactance <i><b>X<sub>L</sub></b></i> is zero.<br/><i><b>P<sub>eff</sub> = E<sub>eff</sub>·I<sub>eff</sub>·cos(φ)</b></i><br/>where <i><b>tan(φ) = X<sub>C</sub> <font size=5>/</font>R</b></i><br/><br/><i>RL-Circuit</i><br/>RC-circuit contains a resistor and an inductor in a series. Reactance <i><b>X<sub>C</sub></b></i> is zero.<br/><i><b>P<sub>eff</sub> = E<sub>eff</sub>·I<sub>eff</sub>·cos(φ)</b></i><br/>where <i><b>tan(φ) = −X<sub>L</sub> <font size=5>/</font>R</b></i><br/>The negative values of <i><b>tan(φ)</b></i> is not important since function <i><b>cos(φ)</b></i> is even and <i><b>cos(φ)=cos(−φ)</b></i>.<br/><br/>Interestingly, if our circuit contains a resistor, but a capacitor and an inductor are in <i>resonance</i>, that is <i><b>X<sub>C</sub>=X<sub>L</sub></b></i>, the phase shift will be equal to zero, as if only a resistor is present in a circuit.<br/><br/><i>L-, C- and LC-Circuits</i><br/>If no resistor is present in the circuit (assuming the resistance of wiring is zero), the denominator in the expression<br/><i><b>tan(φ) = (X<sub>C</sub> − X<sub>L</sub>) <font size=5>/</font>R</b></i><br/>is equal to zero.<br/>Therefore, the phase shift is <i>φ=π/2=90°</i>, <i><b>cos(π/2)=0</b></i> and the power consumption is zero. So, only resistors contribute to a power consumption. Inductors and capacitors are not consuming any energy, they only shift the current phase relatively to a generated EMF. And, if an inductor and a capacitor are in <i>resonance</i>, there is no phase shift, they neutralize each other.<br/><br/> Unizorhttp://www.blogger.com/profile/06592791874048701921noreply@blogger.com0