Circuit analysis with the Laplace transform

The methods of the earlier lessons each handle one kind of problem: time-domain differential equations for transients (Lessons 1 and 2), phasors for sinusoidal steady state (Lessons 3 to 8), and frequency response for filters (Lesson 9). The Laplace transform unifies them. It turns differential equations into algebraic equations, includes initial conditions automatically, gives the complete response (transient and steady state) to any input, and leads to the transfer function, whose poles and zeros describe a circuit's behavior in both time and frequency. This lesson introduces the transform and its properties, circuit elements in the s-domain, partial-fraction inversion, transfer functions, and stability. ELE-250 develops the transform further for signals and systems.

The Laplace transform

The (one-sided) Laplace transform of a function \(f(t)\) defined for \(t \ge 0\) is

where \(s = \sigma + j\omega\) is the complex frequency. Starting the integral at \(0^-\) includes any impulse at \(t = 0\) and makes the initial conditions those just before switching, which are known. In practice, transforms are taken from a table:

The properties that make the transform useful for circuits are:

• Linearity: \(\mathcal{L}\{af + bg\} = aF + bG\).
• Differentiation: \(\mathcal{L}\{df/dt\} = sF(s) - f(0^-)\). Differentiation becomes multiplication by \(s\), with the initial value appearing automatically.
• Integration: \(\mathcal{L}\{\int_0^t f\,d\tau\} = F(s)/s\).
• Time shift: \(\mathcal{L}\{f(t - T)u(t - T)\} = e^{-sT}F(s)\), for delayed inputs such as pulses.
• Initial and final values: \(f(0^+) = \lim_{s\to\infty} sF(s)\) and, if the final value exists, \(f(\infty) = \lim_{s\to 0} sF(s)\).

Circuit elements in the s-domain

Transforming each element's equation gives an s-domain model, in which initial conditions appear as sources:

• Resistor: \(V = RI\).
• Inductor: \(V = sLI - L\,i(0^-)\): an impedance \(sL\) in series with a voltage source \(L\,i(0^-)\) (or in parallel with a current source \(i(0^-)/s\)).
• Capacitor: \(V = I/(sC) + v(0^-)/s\): an impedance \(1/(sC)\) in series with a voltage source \(v(0^-)/s\) (or in parallel with a current source \(C\,v(0^-)\)).

With these models, a circuit is analyzed exactly as in the phasor domain, with impedances \(R\), \(sL\) and \(1/(sC)\), using nodal, mesh, Thévenin or any other method. Setting \(s = j\omega\) recovers the phasor impedances, which shows that phasor analysis is a special case.

Inverting by partial fractions

For circuits, the transformed response is a ratio of polynomials in \(s\). It is inverted by expanding into partial fractions, each of which appears in the table:

The \(p_k\) are the poles (roots of the denominator) and the \(r_k\) the residues, found by \(r_k = [(s - p_k)F(s)]_{s = p_k}\). Complex poles come in conjugate pairs with conjugate residues, and each pair combines into a damped sinusoid. Repeated poles give terms in \(t e^{pt}\).

The same mechanics apply to the second-order example of Lesson 2:

import numpy as np
from scipy import signal

# Series RLC (R = 4, L = 1 mH, C = 10 uF) with v_C(0) = 6 V, i(0) = 0 and a 10 V step at t = 0.
# s-domain: (10/s - 6/s) = I(s) (R + sL + 1/(sC))  ->  V_C(s) = 6/s + I(s)/(sC)
R, L, C = 4.0, 1e-3, 10e-6
# V_C(s) = [10 w0^2 + 6 s^2 + 6 (R/L) s] / [s (s^2 + (R/L) s + w0^2)]
w02 = 1 / (L * C)
num = [6.0, 6.0 * R / L, 10.0 * w02]
den = np.polymul([1.0, 0.0], [1.0, R / L, w02])
r, p, k = signal.residue(num, den)
for ri, pi in zip(r, p):
    print(f"pole {pi.real:9.1f} {pi.imag:+10.1f}j   residue {ri.real:7.3f} {ri.imag:+7.3f}j")

# Initial and final value theorems
print("initial value v_C(0+) = lim s V(s) as s -> inf =", num[0] / den[0])
print("final value  v_C(inf) = lim s V(s) as s -> 0  =", num[-1] / w02)

# Time response from the residues, checked against scipy
t = np.linspace(0, 2e-3, 2001)
v = np.real(sum(ri * np.exp(pi * t) for ri, pi in zip(r, p)))
_, v_check = signal.impulse((num, den), T=t)
print(f"v_C at 0.1, 0.3, 1.0 ms: {np.interp([1e-4, 3e-4, 1e-3], t, v).round(2)} V")
print(f"max difference from scipy.signal.impulse: {np.max(np.abs(v - v_check)):.1e} V")

pole       0.0       +0.0j   residue  10.000  +0.000j
pole   -2000.0    +9798.0j   residue  -2.000  +0.408j
pole   -2000.0    -9798.0j   residue  -2.000  -0.408j
initial value v_C(0+) = lim s V(s) as s -> inf = 6.0
final value  v_C(inf) = lim s V(s) as s -> 0  = 10.0
v_C at 0.1, 0.3, 1.0 ms: [ 7.62 12.06 10.54] V
max difference from scipy.signal.impulse: 8.3e-14 V

The pole at \(s = 0\) with residue 10 is the final value (the forced response); the complex pair at \(-2000 \pm j9798\) gives the damped ringing (the natural response). The initial-value theorem confirms that \(v_C(0^+) = 6\) V, the capacitor's initial charge, and the final-value theorem that it settles at 10 V.

Transfer functions

With all initial conditions zero, the ratio of the transformed output to the transformed input is the transfer function:

The roots of the numerator are the zeros and those of the denominator the poles. The transfer function contains everything about the circuit's input-output behavior:

• The impulse response is \(h(t) = \mathcal{L}^{-1}\{H(s)\}\); the response to any input is \(Y(s) = H(s)X(s)\), or the convolution \(y = h * x\) in time.
• The frequency response is \(H(j\omega)\), evaluated on the imaginary axis (Lesson 9).
• The poles determine the natural response: each pole contributes a term \(e^{p t}\).

Poles and stability

A circuit is stable if its natural response decays, which requires every pole to have a negative real part, in the left half of the s-plane. Poles on the imaginary axis give sustained oscillation (marginal stability), and poles in the right half-plane give responses that grow until something limits them.

Passive RLC circuits are always stable, because resistance only removes energy. Circuits with active elements, such as op-amps, transistors and dependent sources, can supply energy and become unstable. This is how oscillators are designed (deliberately placing poles on the imaginary axis) and how amplifiers and control loops fail (accidentally placing poles in the right half-plane).

import numpy as np
from scipy import signal

# H(s) = V_C / V_in for a series RLC: w0^2 / (s^2 + (R/L) s + w0^2), L = 1 mH, C = 10 uF
L, C = 1e-3, 10e-6
w0 = 1 / np.sqrt(L * C)
for R in (40.0, 20.0, 4.0, 0.0, -2.0):
    poles = np.roots([1.0, R / L, w0 ** 2])
    if np.all(poles.real < 0):
        kind = "stable"
    elif np.all(poles.real <= 0):
        kind = "marginal: sustained oscillation"
    else:
        kind = "unstable: growing oscillation"
    zeta = R / (2 * L) / w0
    p = ", ".join(f"{q.real + 0.0:8.0f}{q.imag:+8.0f}j" for q in poles)
    print(f"R = {R:5.1f} ohm  zeta = {zeta:5.2f}  poles {p}  {kind}")

# Frequency response from the transfer function for R = 4 ohm
sys = signal.TransferFunction([w0 ** 2], [1.0, 4.0 / L, w0 ** 2])
w, mag, phase = signal.bode(sys, w=[1e3, 1e4, 1e5])
for wi, m, ph in zip(w, mag, phase):
    print(f"w = {wi:8.0f} rad/s  |H| = {m:6.1f} dB  phase = {ph:7.1f} deg")

R =  40.0 ohm  zeta =  2.00  poles   -37321      +0j,    -2679      +0j  stable
R =  20.0 ohm  zeta =  1.00  poles   -10000      +0j,   -10000      +0j  stable
R =   4.0 ohm  zeta =  0.20  poles    -2000   +9798j,    -2000   -9798j  stable
R =   0.0 ohm  zeta =  0.00  poles        0  +10000j,        0  -10000j  marginal: sustained oscillation
R =  -2.0 ohm  zeta = -0.10  poles     1000   +9950j,     1000   -9950j  unstable: growing oscillation
w =     1000 rad/s  |H| =    0.1 dB  phase =    -2.3 deg
w =    10000 rad/s  |H| =    8.0 dB  phase =   -90.0 deg
w =   100000 rad/s  |H| =  -39.9 dB  phase =  -177.7 deg

The sequence shows the poles moving from two real values (overdamped), to a repeated pole (critically damped), to a complex pair (underdamped), onto the imaginary axis (R = 0, a lossless LC oscillator) and into the right half-plane when the resistance becomes negative, as it effectively does in the active circuit of an oscillator.

Choosing a method

With several tools available, the choice depends on the question:

• DC or sinusoidal steady state only: DC analysis or phasors.
• First-order transients with step inputs: the inspection method of Lesson 1 is fastest.
• Second-order or higher transients, arbitrary inputs, or initial conditions with several elements: Laplace.
• Frequency response and filter design: transfer functions and Bode plots.
• Non-linear circuits or large networks: numerical simulation (Lesson 12).