AC Power: UNIZOR.COM - Physics4Teens - Electromagnetism - AC Ohm's Law

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

AC Power

Power 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.

As we know from the properties of a direct electrical current, its power is
P = U·I = I²R = U²/R
where U is the voltage around a resistor of resistance R and I is the electric current going through this resistor.
All values in the above expression are constant for direct current.

Alternating current 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.

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
I(t) = (E₀ /Z)·sin(ωt+φ) =
= I₀·sin(ωt+φ)
where impedance Z=√(X_C−X_L)²+R²,
tan(φ)=(X_C−X_L)/R.

Having expressions for generated EMF E(t)=E₀·sin(ωt) and electric current in a circuit I(t)=I₀·sin(ωt+φ), we can find an instantaneous power
P(t) = E(t)·I(t) =
= E₀·I₀·sin(ωt)·sin(ωt+φ)

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 effective voltage E_eff = E₀ /√2 and effective amperage I_eff = I₀ /√2 of the electric current. The usage of these characteristics allows to calculate the power consumed by a resistor-only circuit during a period [0,T] of time (for T significantly greater than one oscillation of a current) without integrating a variable function P(t)=E(t)·I(t) on interval [0,T], but just performing a multiplication of constants:
W_[0,T] = E_eff · I_eff · T

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 E_eff and I_eff 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 effective power of a circuit.

The period of one oscillation with angular speed ω is T=2π/ω.
The instantaneous power consumption by a circuit is
P(t) = E(t)·I(t) =
= E₀·sin(ωt)·I₀·sin(ωt+φ)
The energy consumed by a circuit during one period of oscillation T=2π/ω equals to
W_[0,T] = ∫_[0,T]P(t)·dt
where
P(t)=E₀·sin(ωt)·I₀·sin(ωt+φ)

We can simplify the product of two trigonometric functions to make it easier to integrate:
sin(x)·sin(y) =
= (1/2)·[cos(x−y)−cos(x+y)]
Using this for x=ωt and y=ωt+φ, we obtain
sin(ωt)·sin(ωt+φ) =
= (1/2)·[cos(φ)−cos(2ωt+φ)]

To find the power consumption during one period of oscillation T, we have to calculate the following integral
W_[0,T] = ∫_[0,T]P(t)·dt
where period T=2π/ω and
P(t) = E₀·I₀·
·(1/2)·[cos(φ)−cos(2ωt+φ)]

This integral can be expressed as a difference of two integrals
∫_[0,T]E₀·I₀·(1/2)·cos(φ)·dt
which, considering cos(φ) is a constant for a given circuit, is equal to
E₀·I₀·(1/2)·cos(φ)·T =
= E₀·I₀·(1/2)·cos(φ)·2π/ω
and
∫_[0,T]E₀·I₀·(1/2)·cos(2ωt+φ)·dt
which is equal to zero because integral of a periodical function cos(x) over any argument interval that equals to one or more periods equals to zero.
The same can be proven analytically
∫_[0,T]cos(2ωt+φ)·dt =
= sin(2ωt+φ)/(2ω)|_[0,T] =
= sin(2ω·2π/ω+φ)/(2ω) −
− sin(φ)/(2ω) =
= [sin(4π+φ)−sin(φ)]/(2ω) = 0

Hence, the energy consumed by a circuit during one period of oscillation equals to
W_[0,T] = E₀·I₀·(1/2)·cos(φ)·2π/ω
The average power consumption, that is the average rate of consuming energy that we will call effective power, equals to this amount of energy divided by time, during which it was consumed - one period of oscillation T=2π/ω:
P_eff = W_[0,T]/T =
= E₀·I₀·(1/2)·cos(φ)
Since E_eff = E₀ /√2 and I_eff = I₀ /√2, the last expression for power equals to
P_eff = E_eff·I_eff·cos(φ)
where a phase shift φ depends on resistance and reactances of a circuit as follows
tan(φ) = (X_C − X_L) /R
X_C = 1/(ωC) - capacitive reactance,
X_L = ωL - inductive reactance,
R - resistance.
The above formula is derived for RLC-circuit that contains a resistor or resistance R, a capacitor of capacitance C and an inductor of inductance L
Let's analyze different circuits and their effective power consumption rate.

R-Circuit
R-circuit contains only a resistor. Therefore, both reactances X_C and X_L are zero and phase shift φ is zero as well. Since cos(0)=1, the effective power for this R-circuit is
P_eff = E_eff·I_eff
which fully corresponds to a power for a circuit with a direct current running through it.

RC-Circuit
RC-circuit contains a resistor and a capacitor in a series. Reactance X_L is zero.
P_eff = E_eff·I_eff·cos(φ)
where tan(φ) = X_C /R

RL-Circuit
RC-circuit contains a resistor and an inductor in a series. Reactance X_C is zero.
P_eff = E_eff·I_eff·cos(φ)
where tan(φ) = −X_L /R
The negative values of tan(φ) is not important since function cos(φ) is even and cos(φ)=cos(−φ).

Interestingly, if our circuit contains a resistor, but a capacitor and an inductor are in resonance, that is X_C=X_L, the phase shift will be equal to zero, as if only a resistor is present in a circuit.

L-, C- and LC-Circuits
If no resistor is present in the circuit (assuming the resistance of wiring is zero), the denominator in the expression
tan(φ) = (X_C − X_L) /R
is equal to zero.
Therefore, the phase shift is φ=π/2=90°, cos(π/2)=0 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 resonance, there is no phase shift, they neutralize each other.

RLC Circuit Ohm Law: UNIZOR.COM - Physics4Teens - Electromagnetism - AC ...

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

RLC Circuit Ohm's Law

This lecture combines the material of the previous two ones dedicated to the Ohm's Law in alternating current circuits.
The first one was analyzing a circuit with a resistor and a capacitor.
The second one analyzed a circuit with a resistor and an inductor.
This lecture analyzes a circuit with all three elements - resistor, inductor and capacitor.

Consider a closed AC circuit with a generator with no internal resistance producing a sinusoidal electromotive force (EMF or voltage)
E(t)=E₀·sin(ωt)(volt)
where E₀ is a peak EMF produced by an AC generator,
ω is angular speed of EMF oscillations and
t is time.

In this circuit we have a resistor with resistance R, an inductor with inductance L and a capacitor with capacitance C connected in series with EMF generator. This is RLC-circuit.
Since the circuit is closed, the electric current I(t) going through a circuit is the same for all components of a circuit.

Assume that the voltage drop on a resistor (caused by its resistance R) is V_R(t), the voltage drop on an inductor (caused by its inductive reactance X_L=ω·L) is V_L(t) and the voltage drop on a capacitor (caused by its capacitive reactance X_C=1/(ω·C)) is V_L(t).

Since a resistor, an inductor and a capacitor are connected in a series, the sum of these voltage drops should be equal to a generated EMF E(t):
E(t) = V_R(t) + V_L(t) + V_C(t)

As explained in the previous lectures the Ohm's Law localized around a resistor states that
I(t) = V_R(t)/R or
V_R(t) = I(t)·R

The inductance of an inductor L, voltage drop on this inductor caused by a self-induction effect V_L(t), electromagnetic flux going through an inductor Φ(t) and electric current going through it I(t), as explained in the lecture "AC Inductors" of the chapter "Electromagnetism - Alternating Current Induction", are in a relationship
V_L = dΦ/dt = L·dI(t)/dt

The capacity of a capacitor C, voltage drop on this capacitor V_C(t) and electric charge accumulated on its plates Q_C(t), according to a definition of a capacitance C, are in a relationship
C = Q_C(t)/V_C(t) or
V_C(t) = Q_C(t)/C

We have expressed both voltage drops V_R(t) and V_L(t) in terms of an electric current I(t):
V_R(t) = I(t)·R
V_L(t) = L·I'(t)
Since an electric charge Q_C(t) is also involved to express the voltage drop on a capacitor, we will use this electric charge as a main variable, using the definition of an electric current as a rate of change of electric charge
I(t) = Q_C'(t) and
I'(t) = Q_C"(t)

Now all three voltage drops can be expressed as functions of Q_C(t) and, equating their sun to a generated EMF E(t), we can the following differential equation
E₀·sin(ωt) = Q_C'(t)·R +
+ L·Q_C"(t) + Q_C(t)/C =
= L·Q_C"(t) + R·Q_C'(t) +
+ (1/C)·Q_C(t)

As we know, capacitive and inductive reactance are functionally equivalent to resistance, and they all have the same unit of measurement - ohm. Therefore, it's convenient, instead of capacitance C and inductance L, to use corresponding reactance X_C=1/(ω·C) and X_L=ω·L.
So, we will substitute
C = 1/(ω·X_C)
L = X_L/ω

Substitute Q_C(t)=y(t) for brevity. Then our equation looks like
(X_L/ω)·y"(t) + R·y'(t) +
+ ω·X_C·y(t) = E₀·sin(ωt)

Solving this equation for y(t)=Q_C(t) and differentiating it by time t, we will obtain the expression for an electric current I(t) in this circuit as a function of all given parameters and time.

First of all, let's find a particular solution of this differential equation.
The form of the right side of this equation prompts to look for a solution in trigonometric form
y(t) = F·sin(ωt) + G·cos(ωt)
Then
y'(t) =
= ω·[F·cos(ωt)−G·sin(ωt)]
y"(t) =
= −ω²·[F·sin(ωt)+G·cos(ωt)] =
= −ω²·y(t)

Using this trigonometric representation of potential solution y(t), the left side of our equation is
(X_L/ω)·y"(t) + R·y'(t) +
+ ω·X_C·y(t) =
= −(X_L/ω)·ω²·y(t) + R·y'(t) +
+ ω·X_C·y(t) =
= −ω·X_L·y(t) + R·y'(t) +
+ ω·X_C·y(t) =
= R·y'(t) + ω·(X_C−X_L)·y(t) =
= R·ω·[F·cos(ωt)−G·sin(ωt)] +
+ ω·(X_C−X_L)·[F·sin(ωt)+
+G·cos(ωt)] =
= ω·[(X_C−X_L)·F−R·G]·sin(ωt)+
+ω·[R·F+(X_C−X_L)·G]·cos(ωt)

Since this is supposed to be equal to E₀·sin(ωt), we have the following system of two linear equations with two variables F and G
ω·[(X_C−X_L)·F−R·G] = E₀
ω·[R·F+(X_C−X_L)·G] = 0
or
(X_C−X_L)·F−R·G = E₀/ω
R·F+(X_C−X_L)·G = 0

Determinant of the matrix that defines this system is
(X_C−X_L)²+R².
It's always positive and usually is denoted as
Z² = (X_C−X_L)²+R²
The value Z is called the impedance of a circuit and, as we will see, plays the role of a resistance for an entire RLC-circuit.

This system of two linear equations with two variables has a solution:
F = (E₀/ω)·(X_C−X_L)/Z²
G = −(E₀/ω)·R/Z²

Consider two expressions that participate in the above solutions F and G:
(X_C−X_L)/Z² and R/Z²
Since Z² = (X_C−X_L)²+R², we can always find an angle φ such that
(X_C−X_L)/Z = sin(φ) and
R/Z = cos(φ)

Using this, we express the solutions to the above system as
F = (E₀/(ω·Z))·sin(φ)
G = −(E₀/(ω·Z))·cos(φ)

Now the solution of the differential equation for y(t)=Q_C(t) that we were looking for in a format
y(t) = F·sin(ωt) + G·cos(ωt)
looks like
y(t)=(E₀/(ω·Z))·sin(φ)sin(ωt) −
− (E₀/(ω·Z))·cos(φ)cos(ωt) =
= −(E₀/(ω·Z))·cos(ωt+φ)

As we noted before, differentiation of this function gives the electric current in the circuit I(t):
I(t) = y'(t) =
= (E₀ /(ω·Z))·ω·sin(ωt+φ) =
= (E₀ /Z)·sin(ωt+φ) =
= I₀·sin(ωt+φ)
where I₀=E₀ /Z is an equivalent of the Ohm's Law for an RLC-circuit.

Similar relationship exists between effective voltage and effective current
I_eff = I₀ /√2 =
= E₀ /(√2·Z) = E_eff /Z

Here impedance Z, defined by resistance R, inductive reactance X_L and capacitive reactance X_C as
Z = √(X_C−X_L)²+R²
plays a role of a resistance in the RLC-circuit.

There is a phase shift φ of the electric current oscillations relative to EMF. It is also defined by the same characteristics of an RLC-circuit:
(X_C−X_L)/Z = sin(φ)
R/Z = cos(φ)
Hence
tan(φ) = (X_C−X_L)/R
This makes phase shift positive or negative depending on the values of capacitive and inductive reactance.
If X_C is greater than X_L, the phase shift is positive, if the reverse is true, the phase shift is negative.
If the values of reactance are the same, that is X_C=X_L, there is no phase shift. From definition of reactance, it happens when
1/(ω·C) = ω·L or
1/(L·C) = ω² or
L·C = 1/ω²
This relationship between inductance, capacitance and angular speed of EMF oscillation is call resonance.

Expressions for electric current
I(t) = (E₀ /Z)·sin(ωt+φ) =
= I₀·sin(ωt+φ)
in RLC-circuit, where impedance Z=√(X_C−X_L)²+R² and tan(φ)=(X_C−X_L)/R, correspond to analogous formulas for R-, RC- and RL-circuits presented in the previous lectures. All it takes is to set the appropriate value of capacitance or inductance to zero.

Ohm's Law for RL_Circuit: UNIZOR.COM - Physics4Teens - Electromagnetism ...

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

RL Circuit Ohm's Law

Consider a closed AC circuit with a generator with no internal resistance producing a sinusoidal electromotive force (EMF or voltage)
E(t)=E₀·sin(ωt)(volt)
where E₀ is a peak EMF produced by an AC generator,
ω is angular speed of EMF oscillations and
t is time.
In this circuit we have a resistor with resistance R and an inductor with inductance L connected in series with EMF generator. This is RL-circuit.
Since the circuit is closed, the electric current I(t) going through a circuit is the same for all components of a circuit.

Assume that the voltage drop on a resistor caused by its resistance is V_R(t) and the voltage drop on an inductor caused by self-induction effect is V_L(t). Since a resistor and an inductor are connected in a series, the sum of these voltage drops should be equal to a generated EMF E(t):
E(t) = V_R(t) + V_L(t)

As explained in the previous lecture for an R-circuit, the Ohm's Law localized around a resistor states that
I(t) = V_R(t)/R or
V_R(t) = I(t)·R

The inductance of an inductor L, voltage drop on this inductor caused by a self-induction effect V_L(t), electromagnetic flux going through an inductor Φ(t) and electric current going through it I(t), as explained in the lecture "AC Inductors" of the chapter "Electromagnetism - Alternating Current Induction", are in a relationship
V_L = dΦ/dt = L·dI(t)/dt

We have expressed both voltage drops V_R(t) and V_L(t) in terms of an electric current I(t):
V_R(t) = I(t)·R
V_L(t) = L·I'(t)

Now we can substitute them into a formula for their sum being equal to a generated EMF
E(t) = V_R(t) + V_L(t) =
= I(t)·R + I'(t)·L

Since E(t)=E₀·sin(ωt) we should solve the following differential equation to obtain function I(t)
E₀·sin(ωt) = I(t)·R + I'(t)·L

Let's divide this equation by L and use simplified notation for brevity:
y(t) = I(t)
a = R/L
b = E₀/L
Then our equation looks simpler
y'(t)+a·y(t) = b·sin(ωt)

This exact differential equation was solved in the previous lecture dedicated to RC-circuit, notes for that lecture contain detail analysis of its solution
y(t) = −b·cos(ωt+ψ)/√(a²+ω²) + K
where phase shift ψ=arctan(a/ω) and K is a constant that can be determined by initial condition.

Using original notation,
I(t) = −(E₀/L)·cos(ωt+ψ)/
/√(R/L)²+ω² + K =
= −E₀·cos(ωt+ψ)/
/√R²+(ω·L)² + K =
= −E₀·cos(ωt+ψ)/√R²+X_L² + K
where X_L=ω·L is inductive reactance of an inductor - a characteristic of an inductor functionally equivalent to a resistance for a resistor and measured in the same units - ohm and
tan(ψ) = a/ω = R/(L·ω) = R/X_L

Let's apply some Trigonometry to simplify the above formula.
−cos(ωt+ψ) =
= −sin((π/2)−ωt−ψ) =
= −sin((π/2)−ψ−ωt) =
= sin(ωt−((π/2)−ψ)) =
= sin(ωt−φ)
where φ=(π/2)−ψ and, therefore, tan(φ)=1/tan(ψ)=X_L/R.

Using phase shift φ, the equation for the current I(t) looks like
I(t) = E₀·sin(ωt−φ)/√R²+X_L² + K
where tan(φ)=X_L/R.

The value of a constant K can be defined by initial conditions. Since we don't really know these conditions (like what is the value of a current at some moment in time), traditionally this constant is assigned the value of zero, motivating it by the fact that, if the generated EMF is oscillating between its minimum and maximum of the same magnitude with different signs, the current also will be "symmetrical" relative to zero level, which requires the value K=0

The final version of the current in this RL-circuit is
I(t) = E₀·sin(ωt−φ)/√R²+X_L² =
= I₀·sin(ωt−φ)
where I₀ = E₀/√R²+X_L²

The last issue is to analyze the effective current in this RL-circuit. Since for sinusoidal oscillations the effective current is by √2 less than peak amperage, the effective current is
I_eff = I₀ /√2 =
= E₀ /(√X_L²+R²·√2) =
= E_eff /√X_L²+R²
This is the Ohm's Law for effective voltage and amperage in RC-circuit.

It's important to notice that in the absence of a resistor (that is, R=0) the formula for I(t) transforms into
I(t) = I₀·sin(ωt−φ)
where peak amperage I₀=E₀/X_L and phase shift φ=arctan(∞)=π/2, which corresponds to results obtained in a lecture "AC Inductors" of the "Alternating Current Induction" chapter of this course dedicated to only an inductor in the AC circuit, taking the current as given and deriving the EMF.

Ohm's Law for R- and RC-Circuits: UNIZOR.COM - Physics4Teens - Electromagnetism - AC O...

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

R and RC Circuit Ohm's Law

In this chapter we will examine different aspects of the Ohm's Law as they occur in different alternating current (AC) circuits.
Four different types of AC circuits will be considered in this and subsequent lectures:
(a) R-circuit that contains only a resistor;
(b) RC-circuit that contains a resistor and a capacitor;
(c) RL-circuit that contains a resistor and an inductor;
(d) RLC-circuit that contains a resistor, an inductor and a capacitor.
The first two are subject to this lecture.


R-circuit

Consider a closed AC circuit with a generator with no internal resistance producing a sinusoidal electromotive force (EMF or voltage)
E(t)=E₀·sin(ωt)(volt)
where E₀ is a peak EMF produced by an AC generator,
ω is angular speed of EMF oscillations
and t is time.
In this circuit there is only a resistor with resistance R(ohm).

During any infinitesimal time interval the electric current going through a circuit depends only on the generated EMF and a resistance of a circuit components. Since the only resistive component is a resistance R, the electric current at any moment of time will obey the Ohm's Law for direct current.

Therefore the electric current I(t) in this R-circuit is
I(t) = E(t)/R = E₀·sin(ωt)/R =
= (E₀/R)·sin(ωt) = I₀·sin(ωt)
where I₀ = E₀/R is a peak current.
Now we can express the Ohm's Law for peak voltage and peak amperage in the R-circuit as
I₀ = E₀ /R

Since in many cases we are interested in effective voltage and effective electric current, and knowing that
E_eff = E₀ /√2 and I_eff = I₀ /√2,
the Ohm's Law for effective voltage and effective amperage in the R-circuit can be expressed in a form
I_eff = E_eff /R


RC-circuit

Let's add a capacitor with capacity C (farad) to R-circuit, connecting it in series with a resistor. This is RC-circuit.
As we know, alternating current goes through a capacitor, so we have a closed circuit of resistor R and capacitor C connected in a series with EMF generator that produces sinusoidal voltage E(t)=E₀·sin(ωt).
The electric current I(t) going through a circuit is the same for all components of a circuit.

Assume that the voltage drop on a resistor is V_R(t) and the voltage drop on a capacitor is V_C(t). Since a resistor and a capacitor are connected in a series, the sum of these voltage drops should be equal to a generated EMF E(t):
E(t) = V_R(t) + V_C(t)

As explained above for an R-circuit, the Ohm's Law localized around a resistor states that
I(t) = V_R(t)/R or
V_R(t) = I(t)·R

The capacity of a capacitor C, voltage drop on this capacitor V_C(t) and electric charge accumulated on its plates Q_C(t), according to a definition of a capacity, are in a relationship
C = Q_C(t)/V_C(t) or
V_C(t) = Q_C(t)/C

At the same time, by definition of the electric current, the electric current going through a capacitor is just a rate of change of electrical charge on its plates, that is, a derivative of an accumulated electric charge by time:
I(t) = Q'_C(t)

This is the same electric current that goes through a resistor, since the circuit is closed. Therefore, we can substitute this expression of an electric current into a formula for a voltage drop on a resistor
V_R(t) = Q'_C(t)·R

We have expressed both voltage drops V_R(t) and V_C(t) in terms of an electric charge Q_C(t) accumulated on the plates of a capacitor:
V_R(t) = Q'_C(t)·R
V_C(t) = Q_C(t)/C

Now we can substitute them into a formula for their sum being equal to a generated EMF
E(t) = V_R(t) + V_C(t) =
= Q'_C(t)·R + Q_C(t)/C

Since E(t)=E₀·sin(ωt) we should solve the following differential equation to obtain function Q_C(t)
E₀·sin(ωt) =
= Q'_C(t)·R + Q_C(t)/C

Let's divide this equation by R and use simplified notation for brevity:
y(t) = Q_C(t)
a = 1/(R·C)
b = E₀/R
Then our equation looks simpler
y'(t)+a·y(t) = b·sin(ωt)

Let's discuss how to solve (integrate) this differential equation.
Recall the formula for a derivative of a product of two functions
(x(t)·y(t))' =
= x(t)·y'(t) + x'(t)·y(t)

Let's find such function x(t) that, if we use it as a multiplier of our differential equation, the left side will look like a complete derivative of x(t)·y(t).

Multiplied by this function x(t), our equation looks like this:
x(t)·y'(t)+a·x(t)·y(t) =
= b·x(t)·sin(ωt)
Let's find function x(t) such that a·x(t) is x'(t). Then the left side of the equation will be equal to (x(t)·y(t))'.

Obvious choice for such function is e^a·t. It's an intelligent guess, but it can be determined analytically.
Indeed,
a·x(t) = x'(t)
a·x(t) = dx/dt
a·dt = dx/x(t)
d(a·t) = dln(x(t))
a·t = ln(x(t)) + K₁
x(t) = K₂·e^a·t
Here K₁ and K₂ are any constants, so let's use K₂=1.
Then x(t)=e^a·t.

Now the equation takes the following form
e^a·t·y'(t)+a·e^a·t·y(t) =
= b·e^a·t·sin(ωt)
which is equivalent to
[e^a·t·y(t)]' = b·e^a·t·sin(ωt)

The above equation should be integrated to get e^a·t·y(t) and, finally y(t).

The indefinite integral of the left side is e^a·t·y(t).

The indefinite integral of the right side can be found straight forward using Euler formula
cos(φ)+i·sin(φ) = e^i·φ
and following from it expressions for trigonometric functions
cos(φ) = (e^i·φ+e^−i·φ)/2
sin(φ) = (e^i·φ−e^−i·φ)/(2i)

The result of integration of the right side (without multiplier b) is
∫e^a·t·sin(ωt)·dt =
= [e^a·t/(a²+ω²)]·
·[a·sin(ωt)−ω·cos(ωt)] + K₃
(K₃ is any constant)

Therefore,
e^a·t·y(t) = b·[e^a·t/(a²+ω²)]·
·[a·sin(ωt)−ω·cos(ωt)] + K₃

Reducing by e^a·t both sides, the expression for y(t)=Q_C(t) looks like
y(t) = [b/(a²+ω²)]·
·[a·sin(ωt)−ω·cos(ωt)] + K₄

Consider two constants a/√(a²+ω²) and ω/√(a²+ω²). It is convenient to represent them as sin(φ) and cos(φ) correspondingly for some angle φ=arctan(a/ω).
Then
a·sin(ωt)−ω·cos(ωt) =
= √(a²+ω²)·
·[sin(φ)·sin(ωt)−cos(φ)·cos(ωt)]

The last expression equals to
−√(a²+ω²)·cos(ωt+φ)
and our equation for y(t) looks like
y(t) = −b·cos(ωt+φ)/√(a²+ω²) +
+ K₄

Since y(t) is the electric charge Q_C(t) accumulated on the plates of a capacitor, its derivative is an electric current in the circuit:
I(t) = b·ω·sin(ωt+φ)/√(a²+ω²)

Let's restore this equation to original constants
a = 1/(R·C)
b = E₀/R
The factor at sin(ωt+φ) in the equation above then is
b·ω/√(a²+ω²) =
= E₀·ω/(R·√1/(R·C)²+ω²) =
= E₀/(√1/(ωC)²+R²) =
= E₀/(√X_C²+R²)
where X_C=1/(ωC) was defined in one of the previous lectures as reactance of a capacitor (or capacitive reactance).

The angle φ=arctan(a/ω) in terms of original constants is
φ=arctan(1/(R·C·ω)) =
= arctan (X_C / R)
This angle is called phase shift of the current from the voltage.

Now we have a formula for an electric current in the RC-cirucit that connects generated EMF, resistance of a resistor and capacity of a capacitor
I(t) = E₀·sin(ωt+φ)/√X_C²+R² =
= I₀·sin(ωt+φ)
where I₀=E₀/√X_C²+R² is the peak amperage of the electric current.
This is the Ohm's Law for RC-circuit.

The expression √X_C²+R² plays in this case the same role as a resistance in case of direct current circuits.

It is important that there is a phase shift φ of the oscillations of an electric current relatively to oscillation of generated EMF.

The last issue is to analyze the effective current in this RC-circuit. Since for sinusoidal oscillations the effective current is by √2 less than peak amperage, the effective current is
I_eff = I₀ /√2 =
= E₀ /(√X_C²+R²·√2) =
= E_eff /√X_C²+R²
This is the Ohm's Law for effective voltage and amperage in RC-circuit.

It's important to notice that in the absence of a resistor (that is, R=0) the formula for I(t) transforms into
I(t) = I₀·sin(ωt+φ)
where peak amperage I₀=E₀/X_C and phase shift φ=arctan(∞)=π/2, which corresponds to results obtained in a lecture "AC Capacitors" of the "Alternating Current Induction" chapter of this course dedicated to only a capacitor in the AC circuit.

Problems on LC Circuits: UNIZOR.COM - Physics4Teens - Electromagnetism -...

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

Problems on LC Circuit

Problem A

Consider a circuit that contains an AC generator, an inductor of inductance L and a capacitor of capacity C in a series.


Generated EMF has frequency f=50Hz and effective voltage E_eff=220V (volt).
The effective AC current is I_eff=5A (ampere).
The capacitance of a capacitor is C=10μF (microfarad).
Find inductance L (henry) of an inductor?

Solution

Let's start with an expression for the AC current in the LC circuit in terms of generated EMF and reactance of the capacitor and inductor.

EDF generated by a source of electricity
E(t) = E₀·sin(ωt)
where
E₀ is a peak voltage on the terminals of a generator,
ω=2πf is an angular velocity of a rotor in radians per second, where f is a frequency of generated EMF in number of cycles per second.

Alternating electric current in the circuit
I(t) = I₀·cos(ωt)
where
I₀ = E₀/(X_C−X_L) is peak amperage,
X_C = 1/(ω·C) is capacitive reactance,
X_L = ω·L is inductive reactance

Effective voltage and effective amperage are less than, correspondingly, peak voltage and peak amperage by √2.
Therefore, from the expression for AC current follows
I(t) = I₀·cos(ωt) =
= E₀·cos(ωt)/(X_C−X_L)
I_eff = I₀/√2 =
= E₀/[√2(X_C−X_L)] =
= E_eff/(X_C−X_L)

The unknown in this expression is X_L that can be found:
X_C−X_L = E_eff/I_eff
X_L = X_C−E_eff/I_eff

Since X_C=1/(ω·C) and X_L=ω·L L·ω = 1/(ω·C) − E_eff/I_eff
L = 1/(ω²·C) − E_eff/(ω·I_eff)

Substituting
ω=2πf=2π·50Hz=314(1/sec)
C=10μF=10⁻⁵F
E_eff=220V
I_eff=5A
into above expression for L, we obtain (rounded to 0.001)
L= 1/(314²·10⁻⁵)−220/(314·5)=
= 0.874H (henry)


Problem B

Consider a circuit that contains a source of a noisy electrical signal, which is a combination of many sinusoidal waves with different frequencies, amplitudes and phase shifts, a resistor with resistance R and a capacitor with capacitance C connected parallel to a resistor.


The signal (electrical current) coming through a resistor will have higher frequencies weakened by a presence of a capacitor.
Explain why.

Explanation

Reactance of a capacitor X_C, which functionally similar to a resistance of a resistor, is inversely proportional to a frequency of the voltage on its ends
X_C = 1/(ω·C) = 1/(2πf·C),
where
ω=2πf is the angular velocity of oscillations,
f is a frequency of oscillations,
C is the capacitance of a capacitor.
Therefore, the greater frequency - the less reactance of a capacitor.
Since reactance of a capacitor is functionally similar to a resistance of a resistor, this circuit is similar to a circuit with two parallel resistors with one of them having lower resistance with higher frequencies of the oscillations of the electrical current.

Using this analogy, we see that higher frequency oscillations of the electric current going through a capacitor will meet less resistance than lower frequency oscillations. Therefore, higher frequency oscillations of the current will go easier through a capacitor, thus having less impact on a resistor.

More precisely, the current going through parallel resistors is inversely proportional to their resistance (see lecture "DC Ohm's Law" in this part of the course). This is true not only for constant direct current produced by constant EMF E, but also at any moment t, when generated EMF oscillates (even irregularly) as a function of time E(t).
Therefore, a capacitor will lower reactance will have higher current going through it, thus weakening the higher frequency oscillations of the current going through a resistor.

Problem C

Consider a circuit that contains a source of a noisy electrical signal, which is a combination of many sinusoidal waves with different frequencies, amplitudes and phase shifts, a resistor with resistance R and an inductor with inductance L connected parallel to a resistor.


The signal (electrical current) coming through a resistor will have lower frequencies weakened by a presence of a inductor.
Explain why.

Explanation

Reactance of an inductor X_L, which functionally similar to a resistance of a resistor, is proportional to a frequency of the voltage on its ends
X_L = ω·L = 2πf·L,
where
ω=2πf is the angular velocity of oscillations,
f is a frequency of oscillations,
L is the inductance of an inductor.
Therefore, the lower frequency - the less reactance of an inductor.
Since reactance of an inductor is functionally similar to a resistance of a resistor, this circuit is similar to a circuit with two parallel resistors with one of them having lower resistance with lower frequencies of the oscillations of the electrical current.

Using this analogy, we see that lower frequency oscillations of the electric current going through an inductor will meet less resistance than higher frequency oscillations. Therefore, lower frequency oscillations of the current will go easier through an inductor, thus having less impact on a resistor.

More precisely, the current going through parallel resistors is inversely proportional to their resistance (see lecture "DC Ohm's Law" in this part of the course). This is true not only for constant direct current produced by constant EMF E, but also at any moment t, when generated EMF oscillates (even irregularly) as a function of time E(t).
Therefore, an inductor will lower reactance will have higher current going through it, thus weakening the lower frequency oscillations of the current going through a resistor.

SUMMARY
Capacitors and inductors can be used to filter certain "useful" frequencies from the noisy signals.

Series LC Circuit: UNIZOR.COM - Physics4Teens - Electromagnetism - Alter...

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



Series LC Circuit



Consider a circuit that contains an AC generator, an inductor of inductance L and a capacitor of capacity C in a series.




The current I_L(t) going through an inductor is the same as the current I_C(t) going through a capacitor. So, we will use an expression I(t) for both.



The electromotive force (EMF) generated by an AC generator depends only
on its own properties and can be described as a function of time t

E(t) = E₀·sin(ωt)

where

E₀ is a peak voltage on the terminals of a generator,

ω is an angular velocity of a rotor in radians per second.



Inductance L of an inductor and capacity C of a capacitor produce voltage drops V_L(t) and V_C(t) correspondingly.



As we know, the voltage drop on an inductor is causes by self-induction
and depends on the rate of change (that is, the first derivative by
time) of a magnetic flux Φ(t) going through it

V_L(t) = dΦ(t)/dt

Magnetic flux, in turn, depends on a current going through a wire of an inductor I(t) and the inductor's inductance L

Φ(t) = L·I(t)

Therefore, the voltage drop on an inductor equals to

V_L(t) = L·dI(t)/dt



As we know, the amount of electricity Q(t) accumulated in a capacitor is proportional to voltage V_C(t) applied to its plates (that is, voltage drop on a capacitor). The constant capacity of a capacitor C
is the coefficient of proportionality (see lecture "Electric Fields" -
"Capacitors" in this course) that depends on a type of a capacitor

C = Q(t)/V_C(t)

Therefore,

Q(t)=C·V_C(t)



Knowing the amount of electricity Q(t) accumulated in a capacitor as a function of time t, we can determine the electric current I(t) in a circuit, which is a rate of change (that is, the first derivative by time) of the amount of electricity

I(t) = dQ(t)/dt = C·dV_C(t)/dt



The sum of voltage drops on an inductor and a capacitor is supposed to be equal to EMF produced by an AC generator E(t)=E₀·sin(ωt), which is the final equation in our system:

V_L(t) = L·dI(t)/dt

I(t) = C·dV_C(t)/dt

E₀·sin(ωt) = V_L(t) + V_C(t)



To solve this system of three equations, including two differential ones, let's resolve the third equation for V_C(t) and substitute it in the second one.

V_C(t) = E₀·sin(ωt) − V_L(t)

I(t)=C·d[E₀·sin(ωt)−V_L(t)]/dt



In the last equation we can differentiate each component and, using symbol ' for a derivative, obtain

I(t)=CωE₀·cos(ωt)−C·V'_L(t)



Together with the first equation from the original system of three
equations above we have reduced the system to two equations (again, we
use symbol ' for brevity to signify differentiation)

V_L(t) = L·I'(t)

I(t)=CωE₀·cos(ωt)−C·V'_L(t)



Substituting V_L(t) from the first of these equations into the second, we obtain one equation for I(t), which happens to be a differential equation of the second order (we will use symbol " to signify a second derivative of I(t) for brevity)

I(t)=CωE₀·cos(ωt)−CL·I"(t)

or in a more traditional for differential equation form

a·I"(t) + b·I(t) = E₀·cos(ωt)

where

a = L/ω

b = 1/(Cω)



Without getting too deep into a theory of differential equations, notice that the one and only known function in this equation that depends on time t is cos(ωt). It's second derivative also contains cos(ωt). So, if I(t) is proportional to cos(ωt), its second derivative I"(t) will also be proportional to cos(ωt) and we can find the coefficient of proportionality to satisfy the equation.



Let's try to find such coefficient K that function I(t)=K·cos(ωt) satisfies our equation.

I'(t) = −ωK·sin(ωt)

I"(t) = −ω²K·cos(ωt)



Now our differential equation for I(t) is

−a·ω²K·cos(ωt) + b·K·cos(ωt) = E₀·cos(ωt)



From this we can easily find a coefficient K

K = E₀/(b−aω²)



Since a=L/ω and b=1/(Cω)

K = E₀/{[1/(Cω)] − Lω}



In the previous lectures we have introduced concepts of capacitive reactance X_C=1/(Cω) and inductive reactance X_L=Lω. Using these variables, the expression for coefficient K is
K = E₀/(X_C−X_L)

Therefore,

I(t) = E₀·cos(ωt)/(X_C−X_L)

or

I(t) = I₀·cos(ωt)

where

I₀ = E₀/(X_C−X_L)



The last equation brings us to a concept of a reactance of the LC circuit

X_C−X_L

that is similar to resistance of regular resistors.

Using a concept of reactance, the last equation resembles the Ohm's Law.



Let's determine the voltage drops on a capacitor V_C(t) and an inductor V_L(t) using the expression for the current I(t).



Since Q(t)=C·V_C(t) and I(t)=Q'(t), we can find V_C(t) by integrating I(t)/C.

V_C(t) = ∫_[0,t]I(t)·dt/C =

= I₀·sin(ωt)/(C·ω) =

= X_C·E₀·sin(ωt)/(X_C−X_L)



V_L(t) = L·I'(t) =

= −L·E₀·sin(ωt)·ω/(X_C−X_L) =

= −X_L·E₀·sin(ωt)/(X_C−X_L)



Let's check that the sum of voltage drops in the circuit V_L(t) and V_C(t) is equal to the original EMF generated by a source of electricity.

Indeed,

V_L(t) + V_C(t) =

=(X_C−X_L)·E₀·sin(ωt)/(X_C−X_L)=

= E₀·sin(ωt)



Summary



EDF generated by a source of electricity

E(t) = E₀·sin(ωt)

where

E₀ is a peak voltage on the terminals of a generator,

ω is an angular velocity of a rotor in radians per second.



Alternating electric current in the circuit

I(t) = I₀·cos(ωt)

where

I₀ = E₀/(X_C−X_L)

X_C = 1/(ω·C)

X_L = ω·L



Voltage drop on a capacitor

V_C(t) = X_C·E₀·sin(ωt)/(X_C−X_L)



Voltage drop on an inductor

V_L(t) = −X_L·E₀·sin(ωt)/(X_C−X_L)



Phase Shift



Notice that cos(x)=sin(x+π/2). Graph of function y=sin(x+π/2) is shifted to the left by π/2 relative to graph of y=sin(x).

Therefore, oscillations of the current I(t) in the LC circuit are ahead of oscillation of the EMF generated by a source of electricity E(t) by a phase shift of π/2.



Oscillations of the voltage drop on a capacitor V_C(t) in the LC circuit are synchronized (in phase) with generated EMF.



Notice that −sin(x)=sin(x+π). Graph of function y=sin(x+π) is shifted to the left by π relative to graph of y=sin(x).

Therefore, oscillations of the voltage drop on an inductor V_L(t) in the LC circuit are ahead of oscillation of the EMF generated by a source of electricity E(t) by π (this is called in antiphase).

AC Inductors: UNIZOR.COM - Physics4Teens - Electromagnetism

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



Alternating Current and Inductors



For the purpose of this lecture it's important to be familiar with the concept of a self-induction explained in "Electromagnetism - Self-Induction" chapter of this course.



In this lecture we will discuss the AC circuit that contains an inductor - a wire wound in a reel or a solenoid, thus making multiple loops, schematically presented on the following picture.





Both direct and alternating current go through an inductor, but, while
direct current goes with very little resistance through a wire, whether
it's in a shape of a loop or not, alternating current meets some
additional resistance when this wire is wound into a loop.



Consider the following experiment.





Here the AC circuit includes a lamp, an inductor in a shape of a solenoid and an iron rod fit to be inserted into a solenoid.

While the rod is not inside a solenoid, the lamp lights with normal
intensity. But let's gradually insert an iron core into a solenoid. As
the core goes deeper into a solenoid, the lamp produces less and less
light, as if some kind of resistance is increasing in the circuit.

This experiment demonstrates that inductors in the AC circuit produce
effect similar to resistors, and the more "inductive" the inductor - the
more resistance can be observed in a circuit.

The theory behind this is explained in this lecture.



The cause of this resistance is self-induction. This concept was
explained earlier in this course and its essence is that variable
magnetic field flux, going through a wire loop, creates electromotive
force (EMF) directed against the original EMF that drives electric
current through a loop.



Any current that goes along a wire creates a magnetic field around this
wire. Since the current in our wire loop is alternating, the magnetic
field that goes through this loop is variable. According to the
Faraday's Law, the variable magnetic field going through a wire loop
generates EMF equal to a rate of change of the magnetic field flux and
directed opposite to the EMF that drives the current through a wire,
thus resisting it.



Magnetic flux Φ(t) going through inductor, as a function of time t, is proportional to an electric current I(t) going through its wire

Φ(t) = L·I(t)

where L is a coefficient of proportionality that depends
on physical properties of the inductor (number of loop in a reel, type
of its core etc.) called inductance of the inductor.



If the current is alternating as

I(t) = I_max·sin(ωt)

the flux will be

Φ(t) = L·I_max·sin(ωt)



According to Faraday's Law, self-induction EMF E_i
is equal in magnitude to a rate of change of magnetic flux and opposite
in sign (see chapter "Electromagnetism - "Self-Induction" in this
course)

E_i(t) = −dΦ/dt =

= −L·dI(t)/dt =

= −L·ω·I_max·cos(ωt) =

= −L·ω·I_max·sin(ωt+π/2) =

= −E_imax·sin(ωt+π/2)

where

E_imax = L·ω·I_max



The unit of measurement of inductance is henry (H) with 1H being an inductance of an inductor that generates 1V electromotive force, if the rate of change of current is 1A/sec.

That is,

henry = volt·sec/ampere = ohm·sec



An expression X_L=L·ω in the above formula for E_i is called inductive reactance. It plays the same role for an inductor as resistance for resistors.

The units of the inductive reactance is Ohm (Ω) because

henry/sec = ohm·sec/sec = ohm.



Using this concept of inductive reactance X_L of an inductor, the time dependent induced EMF is

E_i(t) = −X_L·I_max·sin(ωt+π/2) = −E_imax·sin(ωt+π/2)

and

E_imax = X_L·I_max,

which for inductors in AC circuit is an analogue of the Ohm's Law for resistors.



What's most important in the formula

E_i(t) = −E_imax·sin(ωt+π/2)

and the most important property of an inductor in an AC circuit is
that, while the electric current in a circuit oscillates with angular
speed ω, the voltage drop on an inductor oscillates with the same angular speed ω as the current, but its period is shifted in time by π/2 relative to the current.