Thursday, January 27, 2022

Parabolic Reflector: UNIZOR.COM - Physics4Teens - Waves - Properties of ...

Notes to a video lecture on http://www.unizor.com

Parabolic Reflector

Consider now a more complicated case of a curved reflection surface.
Any smooth surface can be considered as an infinite set of infinitesimally small flat pieces with each piece reflecting light in a direction that can be determined by a plane tangential to a surface at that point.
So, to determine the reflected light at some point of a surface we can just replace a surface with a tangential plane at that point and use the Laws of Reflection presented in the previous lecture.

Let's illustrate this on a concrete example of a paraboloid as a surface reflecting the light. Paraboloid is a surface obtained by rotating a parabola z=k·x² in the XZ-plane around the Z-axis.

As a result of this rotation, the three-dimensional formula for a paraboloid is
z = k·(x² + y²)

We will examine how vertically going down rays of light are reflected by this surface.

Assume, a light ray falls down parallel to the Z-axis of a paraboloid within XZ-plane at distance a from this axis and hits paraboloid at point B (vertical blue line on the picture below).

After the reflection off the surface of paraboloid, which we will analyze as if reflected off the tangential plane to paraboloid at point B, the reflected ray of light crosses the Z-axis of this paraboloid at point C (black line BC on the picture below).
The reflected ray of light should cross the Z-axis at some point C from the considerations of rotational symmetry of the paraboloid.

We will analyze this using a two-dimensional cut along the plane going through a point B on the surface of paraboloid, where the light ray hits its surface and the vertical axis of this paraboloid with Z-axis coinciding with the axis of paraboloid

The light blue colored line represents the light going down at a distance a=OA from the Z-axis. It hits a point B on a parabola z=k·x² (red curved line) and the solid black line represents the reflected ray of light that hits the Z-axis at point C.
The green line is tangential to a parabola at point B and should be used to determine the direction of the reflected light by establishing a normal to a parabola line (a purple line perpendicular to a green tangential line) and using the law of reflection about equality between an incidence angle θi and the reflection angle θr.

Our task is to determine a distance OC from the origin of coordinates to point C, where the reflected ray of light intersects the Z-axis of this parabola.

The analysis of this task, going from what is to be found back to what's known, is:
1. Find OC as the difference between AB (known to be the value of z=k·x² at x=a, that is k·a²) and an unknown BD.
2. To find BD, we will use the formula
BD = CD·cot(∠CBD),
where CD=OA=a
3. Angle ∠CBD is the difference between π and angle ∠θi+∠θr, that is
(since θir)
∠CBD = π−2θ
4. Since ∠BEA=∠θ and BE is a tangential to our parabola z=k·x², tangent of ∠BEA equals to a derivative of z=k·x² at point x=a, from which follows:
tan(∠θ) = 2k·a

Based on this analysis, we derive the following:
(a) tan(∠BEA) =
= tan(∠θ) = 2k·a

(b) tan(∠CBD) =
= tan(π−2θ) = −tan(2θ) =
= −2tan(θ)/(1−tan²(θ)) =
= 4k·a/(4k²·a²−1)

(c) cot(∠CBD) =
= 1/tan(∠CBD) =
= (4k²·a²−1)/(4k·a) =
= k·a − 1/(4k·a)

(d) BD = a·cot(∠CBD) =
= k·a² −1/(4k)

(e) OC = AB − BD =
= k·a² − (k·a² −1/(4k)) =
= 1/(4k)


So, as we see, the reflected ray of light will intersect the Z-axis at point C at a distance OC=1/(4k) from the bottom of a paraboloid.

What's remarkable about this result is that the location of point C does not depend on the value of parameter a - the distance of the incident light from the Z-axis.

So, any vertically directed ray of light will be reflected by a paraboloid towards the same point on its axis - its focal point - located at distance f=1/(4k) from the bottom, where parameter k defines the "steepness" of a paraboloid.

Using the parabolic mirror, we can "gather" the sun rays into a focal point and boil the water positioned there to use the steam to generate electricity.

If the source of light is positioned at the focal point of a parabolic mirror, all its emitted light will be directed in one direction parallel to the axis of a paraboloid. That's the principle of work of a projector.

The dish-like parabolic antenna, directed towards a stationary satellite broadcasting some radio signals, collects all the radio waves falling into it, reflecting all these signals towards its focal point, where a radio receiver is located. This allows to catch even a relatively weak radio signal.

When we don't hear a distant sound, we make a sort of a "dish" with our hand, directing the reflected sound towards the ear to hear better.

All the above examples and many others are the usages of a principle of focusing the waves by parabolic (or almost parabolic) reflectors.

Tuesday, January 25, 2022

Reflection: UNIZOR.COM - Physics4Teens - Waves - Properties of Light

Notes to a video lecture on http://www.unizor.com

Reflection

Reflection and refraction are effects of changing the direction of light propagation after the light hits some surface or, more precisely, when light reaches the border between two different media, "old", where it's coming from, and "new", which the light hits on its path.

Reflection happens when light returns back to the "old" medium after hitting its border with a "new" medium and continues to propagate there in a different direction, while refraction is the penetration of the light inside the "new" medium, where it continues to propagate in, generally speaking, different direction and different speed, as compared to the original direction and speed.

In this lecture we will address the effect of reflection.

Before addressing the Laws of Reflection, let's accept as an intuitively understood axiom, the Fermat's Principle of the Least Time of light propagation. This principle, proposed by French mathematician Pierre Fermat in 1662, states that the light travels from its source to some point along such a trajectory that the travel time is the least among all possible trajectories.

In particular, if the environment the light travels through is uniform (like vacuum or glass of a uniform consistency), the light travels along a straight line, because a straight line is the shortest distance between any two points, which results in the least travel time for light that travels with a constant speed.
It means that, if the source of light S emits light in all directions in a uniform environment, at some observing point A we see only the ray that travels along a straight line SA.

Reflection is easily understood from the viewpoint of the corpuscular theory of light, which might be a factor in dominance of this theory, when scientists first attempted to understand the nature of light.
Indeed, reflected light behaves exactly like billiard balls hitting the border of a billiard table.

Many experiments have shown that the direction of the reflected ray of light is determined by its initial direction before it hits the reflecting border between two media and the geometry of this border.

Consider the simplest case of a border between two media being an ideal plane that reflects all the light coming on it, like a mirror.
Let's examine how the light is reflected by this mirror from the viewpoint of the Fermat's Principle of the Least Time.

Let point S be a source of light. Choose one particular ray emitted by it at a certain angle to a plane of a mirror (this is an angle between a line of a ray and a plane of a mirror, which is measured as an angle between this line and its projection on the plane).
This ray is reflected by a mirror. Let point A be any point on the reflected ray.

Before hitting a mirror the ray travels within a uniform environment along a straight line. After the reflection light also travels to point A in a uniform environment along another straight line.
Our task is to determine a point R, where the light hits a reflecting plane before traveling to point A.

Since both segments the light travels (SR before hitting a mirror and RA from a reflecting mirror to point A) are in the same environment, where the speed of light is the same, the Principle of the Least Time will be satisfied if the whole distance from the source S to a reflection point R and to point A is minimal among all possible trajectories.

Consider now a purely geometric problem. Given two points in space S and A on the same side of a plane α, find a point R on plane α such that the sum of the lengths of two segments SR and RA is minimal.

The following picture represents a solution:
Find a point A' symmetrical to point A relatively to a given reflecting plane α by dropping a perpendicular to plane α from point A and choose on this perpendicular point A' on the opposite side of a plane such that AB=BA', where B is intersection point of this perpendicular with plane α.

Next, connect points S and A' by a straight line. Point R is an intersection of line SA' with plane α. From equality of right triangles ΔARB and ΔA'RB, that follows from the equality of their catheti, follows equality of hypotenuses RA and RA'.
The point R is the point where reflection occurs and the sum of distances SR and RA is the least among all other reflection points on plane α.

Indeed, consider any other point R' as the reflection point. It's obvious that R'A=R'A' (analogously to why RA=RA', as proved above) and, therefore,
SR'+R'A=SR'+R'A'
is greater than
SR+RA=SR+RA'=SA'
because SR+RA' is a straight line, while SR'+R'A' is not.
So, any other ray, not coinciding with AR, will not hit point A because the trajectory from point S to a different reflection point R' and then to point A will be longer than straight line SA'.

The following easily provable statements are direct consequences of the method of construction of the reflection point R.

(a) Plane of light rays β that contains initial ray of light SR and reflected ray of light RA is perpendicular to a reflecting plane α because it contains the point A' that lies on a continuation of line SR and it goes through a perpendicular to α line AA'.

(b) Projection S' of the source of light S onto reflection plane α also lies in the plane β because line SS' is parallel to AA' that belongs to plane β and point S is on that plane as well.

(c) Perpendicular RR' from a reflection point R to reflecting plane α (normal to plane α at the point of reflection) also lies in the plane β because line RR' is parallel to AA' that belongs to plane β and point R is on that plane as well.

(d) Points S', R and B lie on the same straight line - the line of intersection of two planes α and β; from this follows that ∠SRS' equals to ∠A'RB as vertical within plane β.

(e) ∠A'RB equals to ∠ARB from the equality of triangles ΔARB and ΔA'RB within plane β.

(f) ∠SRS' equals to ARB, as follows from the two previous statements.

(g) Complementary to the two equal angles of the previous statement, incidence angle ∠SRR' (between an incident ray and a normal to a reflecting plane at the reflection point) and reflection angle ∠ARR' (between a reflected ray and a normal to a reflecting plane at the reflection point) also are equal to each other.

The last statement about equality of an incidence angle and a reflection angle is very important.
Now, using the properties described above, we can formulate the Laws of Reflection as consequences of the Fermat's Principle of the Least Time.

1. Three lines, an incident ray, a normal to a reflection plane at a point of reflection and a reflected ray, lie in the same plane.

2. An incidence angle equals to a reflection angle.

3. Incident and reflected rays lie on different sides relatively to a normal at a point of reflection.




Let's support our derivation of the above Laws of Reflection, based on the Principle of the Least Time, with more physical considerations from the viewpoint of the corpuscular theory that states that the ray of light is a set of particles flying in the same direction with certain constant speed along a straight line.

Consider a frame of reference with XY-plane being the reflecting plane and a light particle flying with constant linear speed from some point in the second quadrant of the XZ-plane towards the origin of coordinates along a straight line, so its Y-coordinate and Y-component of its speed are always zero.
Then the above picture represents the trajectory in the XZ-plane.

Assume that a ray of light originated at time t=0 at a distance D from the incidence point (from the origin of coordinates) and flies toward it along a straight line at an angle of incidence θi with constant speed c.
The ray will reach a point of incidence at the time moment T=D/c, at which point its coordinates will be {x(T)=0;y(T)=0;z(T)=0}.

At the incidence point the velocity vector of a light particle will be
Vi(t)={c·sin(θi);0;−c·cos(θi)}.

Assuming the ideally elastic reflection, the X-component of the particle's velocity will be unchanged because it's parallel to the reflective XY-plane, Y-component will remain at zero, while Z-component after the contact with reflecting XY-plane will be inversed by an ideal reflection.
Therefore, the velocity vector of a light particle after the reflection will be
Vr(t)={c·sin(θi);0;c·cos(θi)}.

After the reflection the light will go along the trajectory that coincides with its velocity vector.

Since Y-component of the velocity vector was, is and will always be zero, the reflected ray from the reflection point (the origin of coordinates) will continue its motion within the same XZ-plane it came from. So, the incident ray, normal to a reflecting XY-plane (that is, Z-axis) and reflected ray lie within XZ-plane, which supports the above mentioned first law of reflection.

If the angle of reflection is θr, the vector of velocity is
Vr(t)={c·sin(θr);0;c·cos(θr)}.
Therefore, we have two expressions for the same vector of velocity after the reflection, and they must be equal to each other:
{c·sin(θi);0;c·cos(θi)} =
=
{c·sin(θr);0;c·cos(θr)}
Obviously, if
sin(θi) = sin(θr) and
cos(θi) = cos(θr),
angles θi and θr are equal to each other.
This supports the second law of reflection about equality of the incidence and reflection angles.

Since before the reflection X-coordinate of a light particle is negative and it becomes positive after the reflection, while Y-coordinate is always zero and Z-coordinate is always non-negative, incident ray lies in the second quadrant of the XZ-plane, while reflected ray lies in the first quadrant.
This supports the third law of reflection.

Saturday, January 22, 2022

Basics of Light: UNIZOR.COM - Physics4Teens - Waves - Properties of Light

Notes to a video lecture on http://www.unizor.com

Basic Characteristics of Light

What is Light?

When we talk about light, we mean electromagnetic waves (oscillations of the electromagnetic field) that our eyes can detect.
Not all the oscillations of electromagnetic field are sensed by our eyes, but only within a visible spectrum of frequencies. This spectrum of frequencies of visible light varies for different people, but, in general, it's usually defined as from flow = 4·1014Hz to fhigh = 8·1014Hz.

With the speed of light in vacuum approximately c = 300,000,000 m/sec, using the formula for the wavelength λ=c/f, we can approximate the low and high wavelengths for visible light:
λlow = c/fhigh = 750·10−9(m) =
= 750(nm)

λhigh = c/flow = 375·10−9(m) =
= 375(nm)



Colors

Traditionally, we divide the visible spectrum of light based on the difference in how we sense it in terms of different colors.
Though different people see colors slightly differently, here is the division by colors, as is traditionally defined, as a function of the wavelength in nanometers:

As you see from the picture above, the color becomes almost black, when we approach high and low boundaries of visible spectrum, that is the light becomes almost invisible for the eyes, though younger people usually have more sensitive eyes and see a slightly broader spectrum of light.
Invisible for an eye light with wavelength of less than 375 nm is called ultraviolet.
Invisible for an eye light with wavelength of greater than 750 nm is called infrared.


Speed

The speed of light mentioned above as 300,000,000 m/sec is an approximation. The exact speed depends on the substance where the light propagates.
In vacuum it's the fastest.

In vacuum it's exactly 299,792,458 m/sec. We emphasize the exactness of this speed because in SI system of units meter is defined through a speed of light, as the length traveled by light in vacuum during the time
T = 1 /299,792,458 sec.
Speed of light in water is slower than in vacuum by, approximately, 1.33 times and equals to 2.25·108m/sec. Obviously, it depends on the chemical composition of water.
Analogously, speed of light is different in all translucent substances, but always slower than in vacuum.
According to the Theory of Relativity by Albert Einstein, the speed of light in vacuum is the fastest speed possible to achieve.


Source

There are many different sources of light.

Chemical reaction can produce a visible light. For example, coal or wood burning is a chemical reaction between carbon in the coal or wood and oxygen in the air, producing carbon dioxide and energy in a form of heat and visible light.

Electric current can be a source of light, when a sufficiently strong flow of electrons passes through a conductor, producing heat and light.
This can be observed in the incandescent lamps.

Nuclear reactions of fission and fusion, occurring within stars, including our Sun, produces visible light.

Luminescence is a general term that encompasses close in their nature but slightly different sources of light:
fluorescence,
phosphorescence,
chemiluminescence.
They all involve absorption of light energy in some form and its emission as a visible light of different wavelengths immediately after absorption or at a later time.

Recently new way of producing light is light emitting diodes (LED).


Theories

There have been many theories of light, each one explaining this or that property of light. Discovery of each new property of light was the cause to re-evaluate the concept of light and, in most cases, developing a new theory.

Particle or corpuscular theory of light was developed, primarily, by Pierre Gassendi, Isaac Newton and other scientists. According to this theory, light consists of particles (corpuscles) emitted by the source and flying in all directions.
The corpuscular theory explained many properties of light, but had problems explaining certain observable phenomenons, like interference. Eventually, this theory was rejected by scientists.

Wave theory explained quite well such property as interference, but required a medium for wave propagation - aether. Many scientists contributed to this theory, including Hooke, Huygens and others. Numerous experiments, however, contradicted the concept of aether and, eventually, this theory was rejected as well.

Electromagnetic theory of light became the dominant because of work by Faraday, Maxwell and Hertz. According to this theory, light is the oscillations of electromagnetic field with variable electric and magnetic components causing each other. This theory is the foundation of contemporary usage of radio waves, including TV, cell phones, remote controls etc.

Quantum theory complemented the electromagnetic theory and provided a better explanation of certain corpuscular properties of light, like photoelectric effect. Works of Planck, Einstein and other physicists were essential to developing the quantum theory of light, which is now considered as the current model.

Thursday, December 23, 2021

FM Equation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

FM Equation

Our plan is to come up with a mathematical representation of a frequency modulated oscillations of carrier waves that incorporate given sound waves.

Sound waves are changes in the air pressure and can be represented by some function of time m(t) (m stands for modulation, as we want this signal to modulate the frequency of sinusoidal carrier signal).

Practically, m(t) is the changes in the current of some circuit that contains a microphone, that converts the air pressure changes into synchronous changes in the electric current.

Let's discuss the concept of angular frequency of the carrier signal in more mathematical terms.
Frequency f, if constant, is the number of periods of oscillations per unit of time (second).
It's often expressed in radians per second ω=2πf and unmodulated oscillations of an electric current in the carrier LC circuit that produces base oscillations would be
I(t) = A·cos(ω·t)

This sinusoidal function of time I(t) that represents oscillations of electric current can be viewed as an X-coordinate of a point on a circle of radius A (amplitude of oscillations) with a center at the origin of coordinates, that rotates along a circle counterclockwise with constant angular speed ω radians per second, assuming that at t=0 it is located on the X-axis at point (A,0).

So, angular frequency in more mechanical terms is angular speed.

From now on we will consider the above presented rotation of a point as mathematical representation of sinusoidal oscillations.

Another important characteristic of an oscillation is its phase φ(t).
The definition of a phase is an angle a rotating point has rotated to during the time of rotation from its start up to a current moment t, which can be expressed as a time-dependent function φ(t).

From this definition immediately follows the analogy between kinematics terms distance, speed and concepts phase, angular frequency used in radio electronics.

This analogy is complete in a sense that the relationship between a phase φ(t) and angular frequency ω(t) is similar to that between a distance S(t) covered by a moving object and its instantaneous speed V(t).

The instantaneous speed at moment in time t in, generally, a non-uniform movement, as a function of time, is a derivative of a distance covered by a moving object from the start of movement up to a position at time t, as a function of time:
V(t) = dS/dt = S'(t)

Similarly, angular frequency (sometimes called instantaneous angular frequency or simply instantaneous frequency) is the first derivative of a phase (angle of rotation) as a function of time:
ω(t) = dφ/dt = φ'(t)

Knowing speed V(t) of a moving object at every moment of time from start to t, we can restore the distance S(t) covered by this object as a function of time
S(t) = [0,t]V(τ)·dτ

Similarly, we can restore the phase φ(t) (that is, an angle a point has rotated from the start of its rotation to a moment in time t), knowing the instantaneous angular frequency ω(t) at each moment of time.
φ(t) = [0,t]ω(τ)·dτ

Consider the main equation of oscillations of an electric current in the LC circuit of a carrier without any sound modulation
I(t) = A·cos(ω0·t)

The argument to a function cos() is a product of a constant angular frequency (speed) ω0 by time, which is an angular distance of rotation or, using terminology introduced above, a phase of the rotation at time t
φ(t) = ω0·t

Therefore, our representation of carrier signal can be expressed in a more general form, suitable even for non-uniform rotation:
I(t) = A·cos(φ(t))

In a non-uniform rotation with variable instantaneous angular frequency ω(t) we can always derive this frequency from the phase:
ω(t) = φ'(t)

If we want to combine the carrier signal I(t) with some frequency modulating signal m(t) in such a way that the resulting variable instantaneous frequency ωmod(t) of a modulated signal reflected the modulation, we need to satisfy the following equation:
ωmod(t) = ω0 + m(t)
where
ω0 is the carrier own unmodulated constant frequency determined by it main LC circuit,
m(t) is a modulating add-on to reflect the sound waves to be transmitted.

We can even vary the degree by which the modulating affects the output signal by adding a modulating index λ as a factor to a modulator m(t):
ωmod(t) = ω0 + λ·m(t)

Knowing the target instantaneous frequency ωmod(t) and the above expression of a phase in terms of this frequency
φ(t) = [0,t]ω(τ)·dτ
we can express the modulated phase as
φmod(t) = [0,t]ωmod(τ)·dτ

This modulated phase will be an argument to a modulated signal of a carrier
Imod(t) = A·cos[φmod(t)] =
= A·cos
[[0,t]ωmod(τ)·dτ] =
= A·cos
[[0,t]0+λ·m(τ))·dτ] =
= A·cos
[ω0·t+λ·[0,t]m(τ)·dτ]

The role of modulation index λ in this formula is to define how significantly base carrier frequency should change with a change in sound waves frequency.

Below is a picture of a rather complicated air pressure oscillations (red) and the resulting modulated signal (purple) that represents the carrier frequency modulation by this sound together with equations that represent all the components of this frequency modulation

(you can click the right mouse button and open this picture in another tab for better view)

Tuesday, December 21, 2021

Frequency Modulation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

Frequency Modulation

Unfortunately, amplitude modulation (AM), with its carrier frequency in the range from 540kHz to 1,600kHz, does not fit to transmit good quality (Hi-Fi) sound, especially on higher pitch notes, as was explained in the lecture "Amplitude Modulation" of this chapter.
The maximum audio frequency that is practical to transmit through AM radio is about 4.4kHz, while the sensitivity of the human ear goes up to 20kHz.
In addition, this type of modulation is too sensitive to radio noise.

The necessity to accommodate high quality transmission of sound waves caused new ideas and designs solutions.
A very useful invention was frequency modulation (FM).

First of all, frequency modulation is implemented within a carrier frequency range from 88mHz to 108mHz, which by itself helps to transmit a higher pitch audio signal of up to 15kHz.

Secondly, the basic principle of FM transmission is to represent sound waves, which are just changes in the air pressure, by a carrier signal's deviations from the base frequency assigned to it. Higher air pressure is represented by higher frequency of the carrier signal, lower air pressure is represented by a lower carrier frequency.

Amplitude of the carrier signal frequency remains always the same, only its frequency is fluctuating in synch with changes of the air pressure that carries a sound, as converted by a microphone into electric current.

All the deviations from the base frequency of the carrier signal must be within certain range assigned for each base frequency, so different radio stations, having different base frequencies of signal transmission, do not step over each other.

In the US the FM transmission uses the frequencies from 88mHz to 108mHz divided into 100 channels of 0.2mHz wide. That means, the frequency deviation from the base frequency of each FM transmitter should not exceed 0.1mHz up or down.

A picture below shows how the frequency of the carrier's signal (blue line) is changing in response to the most simple sound waves that represent a single note of a constant intensity (red line). Mathematically, the red line is a graph of an air pressure at some point in space as a function of time - a sinusoid for a single note of constant intensity.

(you can click the right mouse button and open this picture in another tab for better view)

What remains to discuss is a physical implementation of the frequency modulation.
This implementation involves changing the frequency of the transmitting oscillations of an electric current in the LC circuit connected to an antenna in synch with sound waves input from a microphone.

This can be accomplished by adding a variable capacitor into a circuit that is controlled by a current that represents the sound waves on the output of a microphone.
Changing the capacitance C in the LC circuit will effectively change the angular frequency ω=1/√L·C.

There are many different ways to put together such a device, we will leave this to special courses on radio electronics.

Finally, as an example of a more practical case of complicated sound waves that result in superposition of many different sources of sound, here is a picture of the sound waves (red) and modulated carrier signal (purple) with its frequency in synch with the sound waves.

(you can click the right mouse button and open this picture in another tab for better view)

As you see, higher intensity of sound waves (higher pressure of air) is represented by more frequent oscillations of the carrier signal and lower intensity of sound waves is represented by lower frequency of the carrier signal.

This picture intentionally presents the carrier signal of, generally, lower range of frequencies in order to visualize the individual oscillations.
In practice that range of oscillations of the carrier signal should be significantly higher to better represent each curve of sound waves.

Saturday, December 18, 2021

AM Equation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

AM Equation

Let's approach amplitude modulation (AM) from a mathematical standpoint.

High frequency oscillations of the electric current in the LC circuit with an inductor of inductance L and a capacitor of capacitance C can be described by a function
I(t) = Ac·cos(ωc·t)
where
I(t) is the circuit's own oscillation of the current,
Ac is the amplitude of these oscillations,
ωc = 1/√L·C is an angular frequency of these oscillations.

Simple air pressure oscillations resulting from a sound can be expressed by a similar function P(t) = As·cos(ωs·t)
where
P(t) is the oscillation of the air pressure around a source of sound,
As is the amplitude of these air pressure oscillations,
ωs is an angular frequency of these oscillations.

Note that our design requires ωc to be substantially greater than ωs.
Also note that real sound is a combination (superposition) of different overtones with different amplitudes and different angular frequencies and phases, like
P(t) = As1·cos(ωs1·(t+φs1)) +
+ As2·cos(ωs2·(t+φs2)) +
+ As3·cos(ωs3·(t+φs3)) + ...


Amplitude modulation alters the amplitude of the carrier's oscillations by changing it in synch with sound oscillations.
The simplest way to achieve it mathematically is to incorporate the sound waves into an amplitude of carrier's waves:
Im(t) = [Ac+P(t)]·cos(ωc·t)

Here is an example of this type of modulation of a carrier's signal.

Assume, the carrier's high frequency oscillations have an amplitude Ac=4 and angular frequency ωc=20, which can be described by an equation
I(t) = 4·cos(20t)

(you can click the right mouse button and open this picture in another tab for better view)

The sound makes air pressure oscillations that combine two different tones, one with an amplitude As1=2 and angular frequency ωs1=1 and another with an amplitude As2=1 and angular frequency ωs2=2, which can be described by the following equation
P(t) = 2·cos(t) + cos(2t)

Then the modulated signal can be described by an equation
Im(t) =
=
[4+2·cos(t)+cos(2t)]·cos(20t)
that graphically looks like this:


Just as a demonstration of the importance of having a high carrier frequency of a signal to properly represent a sound, here is what the transmitted signal would look like if the LC circuit of a carrier has a frequency comparable to a frequency of sound.

Assume, the carrier's oscillations have an amplitude Ac=4 and angular frequency ωc=3, which can be described by an equation
I(t) = 4·cos(3t)
The sound waves are assumed to be as above
P(t) = 2·cos(t) + cos(2t)
Then the modulated signal can be described by an equation
Im(t) =
=
[4+2·cos(t)+cos(2t)]·cos(3t)
The following graph represents both the sound wave (purple) and modulated signal (blue):

As you see, the representation of sound waves by a modulated signal is far from exact. This lower frequency of amplitude modulation cannot be used for transmitting sound.

Thursday, December 16, 2021

Amplitude Modulation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

Amplitude Modulation

Now we have some basic understanding of how a radio signals (oscillations of electromagnetic field) are transmitted and received.
But, by themselves, these oscillations do not carry any useful information, which can be voice, image, data or any other type of information.
The oscillations of an electromagnetic field are only carriers of the information and their frequency represents the carrying frequency of the radio communication.

Transmitting and receiving information using the carrying radio waves requires certain additional design that is called modulation.
This lecture is about basic principles of amplitude modulation (AM) to transmit sound waves (oscillations of the air or other medium).

Electromagnetic oscillations are synchronous waves of electric and magnetic field forces perpendicular to each other (see "Field Waves" of this topic "Waves") that can be represented graphically as follows

(you can click the right mouse button and open this picture in another tab for better view)

Since these two kinds of forces act synchronously (in phase), we will represent the intensity of the electromagnetic field as a simple two-dimensional sinusoidal graph

At this point we'd like to state that the frequency of the carrier waves represented above should be relatively high as compared to frequency of the sound waves to accomplish successful sound transmission. The reason for this will be obvious after we explain what amplitude modulation is.

Our first task is to convert sound into oscillations of electric current. This is accomplished by a microphone or any other sound capturing device.
Notice that the oscillations of the electric current must represent the sound in its frequency and amplitude. Moreover, real sound is usually a combination of different frequencies of oscillation of air or other medium with different amplitudes, and the corresponding oscillations of electric current should reflect all this multitude of different parameters.

Considering we have accomplished producing an electric current oscillations in some circuit that reflects the sound. Here is a graphical representation of these oscillations:

The next task is to transmit these oscillations.
Experiments show that these oscillations are of too low frequency and not well propagated using any kind of an antenna. Its the high frequency oscillations that needed to do it with some sort of success.

Here comes the amplitude modulation. This method is using the high frequency of the carrier waves produced by an electronic RLC oscillator (red on a picture below), superimposing on its oscillations the changes in amplitude in synch with oscillations of the sound frequency (green on a picture below).
The result is the carrier oscillations modulated by amplitude by sound waves (blue on a picture below):


It should be obvious now why the carrier frequency of radio waves oscillations must be significantly higher than the sound waves frequency. If this difference is not significant, it would be impossible to distinguish the signal of the carrier from oscillations of sound waves.

Sound waves frequencies are those in the range from 20Hz to 20,000Hz (1Hz is 1 period of oscillations per second, which is, actually, sec−1), while the range of carrier frequencies for amplitude modulation (AM range) is from 540,000Hz to 1,600,000Hz (that is from 540kHz to 1,600kHz). Such a difference in ranges assures that the sound waves reflected in the amplitude of carrier waves are distinguishable from the carrier waves themselves.

For example, WCBS radio station in New York broadcasts at the carrier's frequency
f=880kHz=0.88·106sec−1
At the same time, the sound frequency of High A note is 440Hz. That means that for each period of this sound wave the carrier makes 2,000 oscillations.

The higher the sound's pitch (which is, higher frequency of sound waves) - the less difference there is between its frequency and the frequency of the carrier radio waves. With the highest frequency that human ear can hear, which is about 20,000Hz, the WCBS radio waves with frequency of 880kHz are only 440 times more frequent. Generally, it's still sufficient to select the sound waves from the carrier's but some small details of sound waves might be lost. That's why AM radio cannot carry really high fidelity sound, but for regular speech it works fine.

Incidentally, considering the speed of propagation of radio waves is, approximately,
c=0.3·109m/sec,
the WCBS radio waves have the length of
λ = c/f ≅ 341 m

These relatively long waves allow AM radio signals to go around large objects and even be reflected from the ionosphere. As a result, AM radio can be received at a significantly more distant from a transmitter location than waves of a shorter length.

What remains to discuss is a physical implementation of the amplitude modulation.
This implementation involves reducing the amplitude of the oscillations of an electric current in the LC circuit connected to an antenna.

This can be accomplished by adding a variable resistor into a circuit that is controlled by a current that represents the sound waves on the output of a microphone.
There are many different ways to put together such a device, we will leave this to special courses on radio electronics.