Inductor (L), Capacitor (C), Resistor (R)

Circuit (LCR) Notes

by Bernd A. Berg

Department of Physics Florida State University

Tallahassee, FL 32306, USA. (Version March 6, 2011)

Copyright \begin{matrix}c \\ ◯ \end{matrix} by the author.

Chapter 1

1.1 LCR circuit

We consider a resistor (R), a capacitor (C) and an inductor (L) in series, see figure 1.1. Charge conservation implies

I_{R} = I_{C} = I_{L} = I  (1.1)

where I_{R}, I_{C} and I_{L} are the currents at the resistor, capacitor and inductor, respectively.

Figure 1.1: Series LCR circuit with an ac generator.

The voltages are related to the currents by the following equations

V_{R} = R I and V_{C} = \frac{Q}{C} with \frac{dQ}{dt} = I  (1.2)

where Q(t) is the charge on the capacitor. As inductor we consider an idealized solenoid (i.e., we neglect boundary effects which distinguish a real solenoid from an infinitely long one). The induced (back) electromotive force (voltage over the inductor) is the derivative of the magnetic flux

ϵ_{back} = V_{L} = \frac{1}{c} \frac{dΦ_{m}}{dt} = L \frac{dI}{dt}  (1.3)

where the constant L > 0 is called self inductance of the coil. In accordance with Lenz’s law the induced electromotive ϵ_{back} will work against the electromotive force ϵ_{emf} which drives the circuit. We use complex calculus and the physical quantities are given by the real parts. For the total voltage we have

ϵ_{emf} = ϵ_{max} e^{−iωt} = V = V_{L} + V_{R} + V_{C}  (1.4)

where ω is the driving frequency. We replace in this relation the voltages by their equations listed above and eliminate the current I in favor of the time derivative of the charge of the capacitor and to arrive at the second order differential equation (−iω t is Jackson convention)

L \frac{d^{2}Q}{dt^{2}} + R \frac{dQ}{dt} + \frac{Q}{C} = ϵ_{max} e^{−iωt} .  (1.5)

The general solution of this equation is obtained as superposition of a special solution of the inhomogenous equation and the general solution of the homogeneous equation, Q = Q_{h} +Q_{i}. A special solution of the inhomogeneous equation is

Q_{i} = Q_{0} e^{−iωt}  (1.6)

where

(−ω^{2} L − i ω R + \frac{1}{C} ) Q_{0} = ϵ_{max} ⇒ Q_{0} = \frac{ϵ_{max}}{−ω^{2} L − i ω R + 1/C} .  (1.7)

The general solution of the homogeneous equation follows from the ansatz Q_{h} ∼ e^{iαt}, which implies the algebraic equation

−α^{2} L + i α R + C^{−1} = 0 .

The solution is

α = i \frac{R}{2L} ± ω_{h} with ω_{h} = \sqrt{\frac{1}{LC} − \frac{R^{2}}{4L^{2}}}  (1.8)

and we arrive at the general solution of the homogeneous equation

Q_{h} = A_{1} e^{−tR/(2L)} e^{+iω_{h} t} + A_{2} e^{−tR/(2L)} e^{−iω_{h} t} .  (1.9)

Due to the exponential damping the homogeneous solution dies out for large times and we are left with the inhomogeneous solution Q = Q_{i} (1.6). The homogeneous solution has to be used to get the initial values right, e.g. for turning the circuit on or off. In the large time limit the essential features of the LCR circuit are those of the inhomogeneous solution.

For the inhomogeneous solution (1.6) Q_{0} is a complex constant. To separate the amplitude (magnitude) of Q_{0} from its phase we write

Q_{0} = Q_{max} e^{−iδ} with Q_{max} = |Q_{0}| = \frac{ϵ_{max}}{\sqrt{(1/C − ω^{2} L)^{2} + ω^{2} R^{2}}} .  (1.10)

Following standard conventions, we rewrite the expression for Q_{max} as

Q_{max} = \frac{ϵ_{max}}{ω \sqrt{[1/(ω C) − ω L]^{2} + R^{2}}} = \frac{ϵ_{max}}{ω \sqrt{(X_{C} − X_{L})^{2} + R^{2}}} = \frac{ϵ_{max}}{ω Z} .  (1.11)

Here

X_{C} = \frac{1}{ω C} is called capacitive reactance,  (1.12)
X_{L} = ω L is called inductive reactance,  (1.13)

and the difference X_{L} − X_{C} is called total reactance. The quantity

Z = \sqrt{(X_{C} − X_{L})^{2} + R^{2}} is called impedance.  (1.14)

Using this notation the original equation (1.7) for Q_{0} becomes

Q_{0} = \frac{ϵ_{max}}{ω (X_{C} − X_{L} − iR)} = \frac{ϵ_{max} (X_{C} − X_{L} + i R)}{ω Z^{2}} .  (1.15)

It follows from Q_{0} = Q_{max} cos(δ) − i Q_{max} sin(δ) that the phase δ is given by^{1}

tan δ = −\frac{sin(δ)}{cos(δ)} = −\frac{Im Q_{0}}{Re Q_{0}} = \frac{R}{X_{L} − X_{C}} .  (1.16)

From equation (1.2) we see that V_{C} is in phase with the charge Q

V_{C} = \frac{Q_{max}}{C} e^{−iδ} e^{−iωt}  (1.17)

and the phase of the current I is shifted by π/2 with respect to V_{C}:

I = \frac{dQ}{dt} = −i ω Q_{max} e^{−iδ} e^{−iωt} = ω Q_{max} e^{−iδ−iπ/2} e^{−iωt}
= I_{max} e^{−iδ−iπ/2} e^{−iωt}  (1.18)

where we used −i = exp(−iπ/2). Note also that I_{max} = ω Q_{max} together with equation (1.10) implies

I_{max} = \frac{ϵ_{max}}{Z} = \frac{ϵ_{max}}{\sqrt{[1/(ω C) − ω L]^{2} + R^{2}}} .  (1.19)

1Note that in Tipler, Physics 1995 edition, the phase δ is defined with respect to the current instead of the charge, what leads to the equation δ = (X_{L} − X_{C})/R.

The voltage over the resistor is in phase with the current

V_{R} = \frac{I_{max}}{R} e^{−iδ−iπ/2} e^{−iωt}  (1.20)

whereas the voltage over the inductor is shifted by another π/2 phase factor:

V_{L} = L \frac{dI}{dt} = ω L I_{max} e^{−iδt−iπ} e^{iωt} .  (1.21)

The current amplitude (1.19) takes its maximum at the resonance frequency

ω_{0} = \frac{1}{\sqrt{LC}} .  (1.22)

Note that Q_{max}(ω_{0}) = ϵ_{max}/(ω_{0}R) due to equation (1.11). For R ≠ 0 the maximum of Q_{max} is shifted by an amount of order (R/L)^{2} away from ω_{0}. We find it by differentiation of Q_{max} with respect to ω^{2}:

−2 L (C^{−1} − ω_{max}^{2} L) + R^{2} = 0 ⟹ ω_{max} = \sqrt{\frac{1}{LC} − \frac{R^{2}}{2 L^{2}}} .  (1.23)

Note, that this ω_{max} value is lower than the frequency ω_{h} of the damped homogeneous solution (1.9), while the resonance frequency ω_{0} is larger.