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.