Laplace-Transform Circuit Analysis

Introduction

Transient circuit analysis becomes difficult when a network contains stored energy, switching, discontinuous sources, or non-sinusoidal inputs. The unilateral Laplace transform converts linear differential equations valid for t ≥ 0 into algebraic equations in the complex-frequency variable s. Initial capacitor voltages and inductor currents enter these equations explicitly, making the method especially useful for switched circuits.

Laplace-domain analysis also provides a unified framework for transfer functions, poles, zeros, impulse responses, convolution, stability, and steady-state prediction. It applies to linear time-invariant lumped circuits whose component values do not vary with time.

Learning Objectives

After studying this chapter, students should be able to:

• apply unilateral Laplace transforms to circuit differential equations;
• represent resistors, inductors, and capacitors in the s domain;
• include initial capacitor voltage and inductor current correctly;
• obtain complete responses to step, impulse, pulse, and arbitrary inputs;
• derive transfer functions and identify poles and zeros;
• compute convolution responses;
• apply initial- and final-value theorems under their proper conditions;
• relate pole locations to natural modes and BIBO stability; and
• validate analytical results using physical constraints and simulation.

1. The Unilateral Laplace Transform

For a causal analysis beginning at t = 0, the unilateral Laplace transform of f(t) is

Here, s = σ + jω is complex frequency in s⁻¹, σ controls exponential growth or decay, and ω is angular frequency in rad/s. The lower limit 0^- allows switching events and impulses at the origin to be represented.

The integral exists in a region of convergence determined by the growth of f(t). Ordinary circuit transients constructed from decaying exponentials satisfy this requirement in an appropriate right half-plane.

1.1 Transform pairs

The time-shift relation assumes T > 0 and that F(s) is the transform of the unshifted causal function.

1.2 Differentiation and initial conditions

The principal advantage of the unilateral transform is its treatment of derivatives:

More generally,

These formulas assume that the required derivatives are piecewise continuous and of exponential order, with impulses handled consistently as distributions.

2. Circuit Elements in the Laplace Domain

Voltage and current reference directions must obey the passive sign convention: current enters the terminal marked positive. Initial values are evaluated immediately before switching.

2.1 Resistor

For a resistance R in ohms,

A resistor stores no energy and introduces no initial-condition term.

2.2 Inductor

For inductance L in henries,

Therefore,

The term -Li_L(0^-) accounts for initial magnetic energy. An inductor current cannot change instantaneously unless an impulse voltage is applied:

for finite switching voltages.

2.3 Capacitor

For capacitance C in farads,

Thus,

Equivalently,

A capacitor voltage cannot change instantaneously unless an impulse current is applied:

for finite switching currents.

Equivalent-source signs depend on the selected polarity and current direction. Direct transformation of the element equation is safer than memorizing a diagram.

3. A Systematic Analysis Procedure

A reliable Laplace-domain workflow is:

1. Draw the circuit for t ≥ 0 after all switches have moved.
2. Determine v_C(0^-) and i_L(0^-) from the pre-switch circuit.
3. Apply continuity where no impulses occur.
4. Transform independent sources and element equations.
5. Write KCL, KVL, nodal, or mesh equations in the s domain.
6. Solve algebraically for the desired transform.
7. Simplify into a form suitable for partial fractions or known transform pairs.
8. Invert the transform and state the valid time interval.
9. Check initial values, final values, dimensions, and physical plausibility.

Initial conditions are independent excitations. Setting them to zero produces the zero-state response; suppressing external sources while retaining them produces the zero-input response. Their sum is the complete response.

4. First-Order Circuits with Initial Energy

4.1 Explicit RC discharge topology

Consider a capacitor initially charged to V₀. At t = 0, it is connected directly across a resistor R, forming a closed resistor–capacitor discharge loop with no independent source. The resistor and capacitor share the same two nodes.

KCL gives

Transforming,

so

The inverse transform is

The time constant τ = RC has units of seconds. The capacitor energy, W_C = Cv²/2, is dissipated in the resistor as t → ∞.

Numerical example. Let R = 2 kΩ, C = 50 μF, and V₀ = 5 V. Then

At t = 0.200 s,

4.2 Driven RL circuit

For a series R–L circuit excited by v_s(t) with initial current I₀, KVL yields

Hence,

For a step source V_su(t),

This expression displays the forced value V_s/R and the natural mode with time constant L/R.

5. Second-Order RLC Circuits

Consider a series resistor, inductor, and capacitor driven by v_s(t), with output v_C(t). Let the loop current enter the capacitor's positive terminal.

Figure: Series RLC circuit with capacitor voltage as the output.

Because i(t)=C dv_C/dt, KVL produces

Let V₀=v_C(0^-) and I₀=i(0^-). Since v_C'(0^-)=I₀/C, transformation gives

Therefore,

This equation separates the external input from the two forms of initial energy. Dividing the characteristic polynomial by LC gives

where

The natural frequency ω_n is in rad/s, while ζ is dimensionless.

• ζ > 1: overdamped, two distinct real poles.
• ζ = 1: critically damped, one repeated real pole.
• 0 < ζ < 1: underdamped, complex-conjugate poles.
• ζ = 0: ideal undamped LC oscillation.

6. Transfer Functions, Poles, and Zeros

For a linear time-invariant circuit with all initial conditions set to zero, the transfer function is

The zero-initial-condition requirement is essential. A response containing initial-energy terms cannot generally be represented as H(s)X(s) alone.

For a rational transfer function,

where z_k are zeros, p_r are poles, and K is a gain constant. Poles determine natural modes; zeros shape how inputs and internal modes appear at the selected output.

A physically realizable lumped passive network normally has a proper transfer function, meaning m ≤ n. An improper function implies ideal differentiation or another nonphysical high-frequency behavior unless interpreted within a broader model.

Exact pole–zero cancellation in algebra may hide an unstable internal mode. Consequently, terminal BIBO behavior and internal stability should be distinguished in feedback and high-order systems. Component tolerances also make exact cancellation unreliable.

7. Inverse Transforms and Partial Fractions

If

with distinct poles, write

For a simple pole p_k, its residue is

Repeated poles produce terms such as t^ke^pt, while complex-conjugate poles combine into real damped sinusoids. Partial-fraction calculations should be performed only after common factors have been examined and any physically justified cancellation identified.

8. Impulse Response and Convolution

The impulse response is the output of a zero-state LTI circuit to a unit impulse:

For an arbitrary causal input x(t), the zero-state output is

The integration variable τ is a dummy time variable. Convolution is valid for LTI zero-state behavior when the integral exists.

Pulse response of an RC low-pass circuit

For an RC low-pass circuit with output across the capacitor,

Let the input be a pulse of amplitude A and duration T:

By superposition of two shifted step responses,

and

The capacitor charges during the pulse and discharges after it ends. This example illustrates how convolution handles an input of finite duration without deriving a new differential equation for each interval.

9. Initial- and Final-Value Theorems

9.1 Initial-value theorem

If f(t) contains no impulse at t=0 and its transform satisfies the theorem's regularity conditions, then

For the RC discharge,

which agrees with capacitor-voltage continuity.

The theorem must not be used blindly for signals containing impulses. For example, the inverse transform of a nonzero constant includes an impulse, for which an ordinary finite value at 0^+ is not defined.

9.2 Final-value theorem

If every pole of sF(s) lies strictly in the open left half-plane after valid simplification, then

The theorem fails for sustained oscillation, unbounded growth, and signals with right-half-plane poles. For example, F(s)=1/(s²+ω²) represents a sinusoidal response with no final value, and sF(s) has imaginary-axis poles.

The theorem predicts a final value only when the time-domain limit actually exists; it is not merely an algebraic shortcut.

10. Pole Locations and Stability

For a causal finite-dimensional rational transfer function with no unstable hidden modes, BIBO stability requires every transfer-function pole to lie strictly in the open left half of the s plane. Equivalently, the impulse response must be absolutely integrable:

Figure: Pole locations in the complex-frequency plane. Poles on the imaginary axis require separate analysis and are not BIBO stable for ordinary rational causal systems.

Passive RLC circuits with positive R, L, and C cannot generate increasing natural energy, although ideal lossless LC circuits can sustain oscillations.

11. Applied Case: Step Response of a Series RLC Measurement Network

Background

This illustrative case concerns a laboratory sensor interface modeled as a series RLC circuit. The measured output is the capacitor voltage. A rapid source change is applied to assess overshoot and settling behavior.

Problem

Let

• R = 1.00 Ω,
• L = 10.0 mH = 0.0100 H,
• C = 1.00 mF = 0.00100 F,
• v_s(t)=10u(t) V, and
• initial capacitor voltage and inductor current both equal zero.

Find the poles and capacitor-voltage response.

Analysis

With zero initial conditions,

Dividing by LC=10^-5 s² gives

The pole discriminant is

Therefore,

The undamped natural frequency, damping ratio, and damped natural frequency are

For V_s(s)=10/s,

The inverse transform is

Validation

The initial-value theorem gives

The final-value theorem is valid because the poles of sV_C(s) are in the open left half-plane:

The initial slope is zero because the inductor current starts at zero and dv_C/dt=i/C. The response is underdamped and exhibits overshoot, but its exponential envelope decays because both poles have real part -50 s^-1.

Outcome and Learning Points

The corrected characteristic polynomial has discriminant -390000, producing poles -50 ± j312.25 s^-1. Small damping leads to oscillatory transient behavior even though the circuit is stable. The case demonstrates why numerical pole calculations, initial-condition checks, and final-value checks should all be performed independently.

12. Practical Limitations and Professional Practice

Laplace models rely on idealized linear and time-invariant components. Real inductors have winding resistance and parasitic capacitance; capacitors have equivalent series resistance, leakage, and voltage dependence. Switches have finite transition times, and sources have limited bandwidth. High-frequency predictions may therefore differ from measurements.

Stored energy also creates safety concerns. Capacitors may retain hazardous voltage after a source is removed, while inductive interruption may generate damaging overvoltage. Professional designs should include discharge paths, current limiting, insulation, protective switching, and component ratings appropriate to the applicable safety standards.

Computer algebra, SPICE simulation, numerical inverse transforms, and state-space software can accelerate analysis. AI-assisted tools may propose models or decompositions, but their output must be checked against KCL, KVL, units, continuity conditions, pole locations, passivity, and measured data. Exact symbolic results based on inaccurate component models are not physically reliable.

Best practice is to preserve parameterized expressions until late in the analysis, state reference directions clearly, verify numerical roots independently, and compare calculated responses with limiting cases and energy constraints.

Chapter Summary

The unilateral Laplace transform converts circuit differential equations into algebraic equations while retaining initial capacitor voltages and inductor currents. Resistors retain impedance R, inductors contribute sL and an initial-current term, and capacitors contribute 1/(sC) and an initial-voltage term.

Transfer functions describe zero-state input–output behavior. Their poles govern natural modes, while zeros shape transmission to a specified output. Arbitrary causal inputs may be handled through multiplication in the transform domain or convolution in time. Initial- and final-value theorems provide efficient checks but are valid only under stated conditions. For ordinary causal rational systems, open-left-half-plane poles imply decaying modes and BIBO stability; right-half-plane or imaginary-axis poles prevent BIBO stability. Sound engineering practice combines analytical derivation, physical interpretation, numerical verification, and realistic component modeling.

Key Terms

Unilateral Laplace transform: Transform over t ≥ 0 that incorporates pre-switch initial conditions.

Complex frequency: Variable s=σ+jω used in Laplace-domain analysis.

Initial condition: Stored-energy variable evaluated immediately before or after switching.

Zero-input response: Response caused only by initial stored energy.

Zero-state response: Response caused by external inputs when all initial energy is zero.

Complete response: Sum of zero-input and zero-state responses.

Transfer function: Zero-initial-condition ratio Y(s)/X(s) for an LTI system.

Characteristic equation: Denominator equation whose roots determine natural modes.

Pole: Value of s at which a transfer function becomes unbounded.

Zero: Value of s at which a transfer-function numerator vanishes.

Natural response: Response associated with a circuit's characteristic modes.

Forced response: Response attributable to an applied external input.

Impulse response: Zero-state output produced by a unit impulse.

Convolution: Time-domain integral that combines an input with an impulse response.

Residue: Coefficient associated with a simple pole in a partial-fraction expansion.

Initial-value theorem: Method for obtaining f(0^+) from the high-frequency limit of sF(s), when applicable.

Final-value theorem: Method for obtaining f(∞) from the low-frequency limit of sF(s), when stability conditions hold.

BIBO stability: Property that every bounded input produces a bounded output.

Damping ratio: Dimensionless quantity ζ characterizing second-order transient decay.

Review Questions

Short Answer Questions

1. Why is the unilateral Laplace transform especially suitable for switched circuits?
2. State the transformed voltage–current equation of an inductor with nonzero initial current.
3. Why must initial conditions be set to zero when defining a transfer function?
4. What physical behavior is associated with a complex-conjugate pole pair in the left half-plane?
5. Under what pole condition may the final-value theorem be applied?

Long Answer Questions

1. Derive the complete current response of a series RL circuit driven by a DC step with nonzero initial current.
2. Starting from KVL, derive the capacitor-voltage transform for a series RLC circuit with arbitrary source voltage, initial capacitor voltage, and initial inductor current.
3. Explain the relationship among a transfer function, impulse response, convolution, poles, and zeros.
4. Compare overdamped, critically damped, underdamped, and undamped second-order responses using pole locations.
5. Discuss the limitations of ideal Laplace-domain circuit models and describe appropriate professional validation practices.

Analytical / Application Questions

1. A 20 μF capacitor charged to 12 V is connected directly across a 5 kΩ resistor at t=0. Determine its voltage and resistor current for t≥0, and verify the initial and final values.
2. For H(s)=400/(s²+12s+400), determine the poles, natural frequency, damping ratio, stability, and qualitative step-response shape.
3. An RC low-pass circuit has R=10 kΩ and C=10 μF. Determine its response to a 4 V pulse lasting 0.20 s, including expressions before and after the pulse ends.

Multiple Choice Questions with Answers

Question 1

What is the unilateral Laplace transform of df(t)/dt?

A. sF(s)-f(0^-)

B. sF(s)+f(0^-)

C. F(s)/s-f(0^-)

D. s²F(s)-f(0^-)

Correct Answer: A. sF(s)-f(0^-)

Question 2

Under the passive sign convention, which equation represents a capacitor with initial voltage V₀?

A. I_C(s)=sCV_C(s)-CV₀

B. I_C(s)=CV_C(s)/s-CV₀

C. V_C(s)=sCI_C(s)-V₀

D. I_C(s)=V_C(s)/(sC)+V₀/s

Correct Answer: A. I_C(s)=sCV_C(s)-CV₀

Question 3

A 2 kΩ resistor is connected directly across a 50 μF capacitor initially charged to 5 V. What is the capacitor voltage after 0.20 s?

A. 0.6767 V

B. 1.839 V

C. 3.033 V

D. 4.524 V

Correct Answer: A. 0.6767 V

Question 4

For the polynomial s²+100s+100000, what are the poles?

A. -50 ± j312.25 s^-1

B. -100 ± j300.00 s^-1

C. 50 ± j312.25 s^-1

D. -50 ± j273.86 s^-1

Correct Answer: A. -50 ± j312.25 s^-1

Question 5

Which condition is required for applying the final-value theorem to F(s)?

A. All poles of sF(s) must lie strictly in the open left half-plane after valid simplification.

B. All poles of F(s) must lie on the imaginary axis.

C. The numerator degree must exceed the denominator degree.

D. The signal must contain a sustained sinusoid.

Correct Answer: A. All poles of sF(s) must lie strictly in the open left half-plane after valid simplification.

Question 6

A circuit transfer function is defined as H(s)=Y(s)/X(s) under which condition?

A. All independent sources are zero.

B. All initial conditions are zero.

C. The output is in sinusoidal steady state.

D. The circuit contains no reactive components.

Correct Answer: B. All initial conditions are zero.

Question 7

For a causal finite-dimensional rational transfer function with no hidden unstable modes, which pole set gives BIBO stability?

A. Poles strictly in the open left half-plane

B. Simple poles anywhere on the imaginary axis

C. At least one pole in the right half-plane

D. Repeated poles at the origin

Correct Answer: A. Poles strictly in the open left half-plane

Question 8

An RC low-pass circuit has H(s)=1/(1+sRC). What is its impulse response?

A. e^-t/(RC)u(t)

B. (1/RC)e^-t/(RC)u(t)

C. RCe^-RCtu(t)

D. (1-e^-t/(RC))u(t)

Correct Answer: B. (1/RC)e^-t/(RC)u(t)

Question 9

For a series RLC circuit with capacitor output, what is the zero-initial-condition transfer function V_C(s)/V_s(s)?

A. 1/(LCs²+RCs+1)

B. sL/(LCs²+RCs+1)

C. RCs/(LCs²+RCs+1)

D. (LCs²+RCs+1)^-1s

Correct Answer: A. 1/(LCs²+RCs+1)

Question 10

A pole pair is located at -50 ± j312.25 s^-1. Which description is correct?

A. Undamped and BIBO unstable

B. Overdamped with two real modes

C. Underdamped with an exponentially decaying oscillation

D. Critically damped with a repeated pole

Correct Answer: C. Underdamped with an exponentially decaying oscillation