Transient and Transform-Domain Analysis

Introduction

Electrical networks move from one operating condition to another whenever a source changes, a switch operates, a fault occurs, or stored electromagnetic energy is redistributed. The resulting time-varying response is called a transient. Although a transient may last only microseconds in a high-frequency converter or several seconds in an electromechanical system, it can determine insulation stress, protection performance, signal integrity and system stability.

Direct solution of circuit differential equations is possible for simple networks. Transform-domain methods, particularly the unilateral Laplace transform, provide a more systematic procedure: differential equations become algebraic equations, initial energy is included explicitly, and poles reveal natural modes and stability. For advanced network analysis, these methods connect classical circuit theory with state-space analysis, system identification, electromagnetic-transient simulation and network synthesis.

Learning Objectives

After studying this chapter, the reader should be able to:

• formulate transient responses using initial conditions and continuity laws;
• distinguish natural, forced, zero-input and zero-state responses;
• derive first- and second-order network differential equations;
• represent capacitors and inductors correctly in the unilateral Laplace domain;
• calculate responses using transfer functions, poles, residues and convolution;
• apply initial- and final-value theorems under their validity conditions;
• interpret damping, stability and pole–zero structure;
• use state-transition methods for higher-order networks; and
• evaluate practical switching transients, modelling limitations and protection measures.

Stored Energy, Switching and Continuity

A resistor stores no energy. Ideal capacitors and inductors store electric and magnetic energy, respectively:

Here, C is capacitance in farads, L is inductance in henries, v_C is capacitor voltage in volts, and i_L is inductor current in amperes. Both expressions have units of joules.

The constitutive relations are

For finite current, capacitor voltage is continuous across a switching instant t₀:

For finite voltage, inductor current is continuous:

These statements do not prohibit jumps under impulsive excitation. A capacitor-voltage jump requires an impulsive current. Integrating its constitutive relation across the switching instant gives

Thus, an impulse carrying charge q_imp causes \Delta v_C=q_imp/C. Conversely, an inductor-current jump requires an impulsive voltage:

The voltage impulse has volt-second area equal to the change in flux linkage L\Delta i_L. These impulse relationships are important in ideal switching models, though real circuits distribute the transition through parasitic inductance, capacitance and finite switching time.

Figure: Correct continuity and impulse relationships for ideal capacitors and inductors.

Components of a Transient Response

For a linear time-invariant network, the complete response can be decomposed in two useful ways:

The natural response x_n is determined by the network’s natural modes. The forced response x_f is associated with the applied excitation. The zero-input response x_zi is caused by initial stored energy with independent inputs set to zero, while the zero-state response x_zs is caused by inputs when all initial conditions are zero.

Natural and forced responses need not always correspond term by term to zero-input and zero-state responses. For example, a zero-state step response contains both transient natural-mode terms and a steady forced term.

A reliable switching-analysis procedure is:

1. analyse the circuit for t<0;
2. determine v_C(0^-) and i_L(0^-);
3. apply continuity unless an appropriate impulse is present;
4. construct the post-switch circuit for t>0;
5. find the differential equation or transform-domain model; and
6. verify the initial value, final value, units and physical plausibility.

First-Order Networks

A first-order network has one independent energy-storage state. Its standard differential equation is

where \tau>0 is the time constant in seconds, K is the static gain, and u(t) is the input. If the input becomes constant after t=0, the response is

For a series RC circuit observed across the capacitor, \tau=RC. For a series RL circuit observed through its current, \tau=L/R_eq, where R_eq is the resistance seen by the inductor after independent voltage sources are short-circuited and independent current sources are opened. Dependent sources must remain active; a test source may be needed to determine R_eq.

Worked Example: RC Energization

A 10 V source is connected at t=0 to an initially uncharged 100\ \mu F capacitor through 2\ k\Omega. Then

The capacitor voltage and current are

At t=0.2\ s, v_C=10(1-e^-1)\approx6.32\ V. The initial current is 5 mA and decays toward zero.

Second-Order RLC Networks

Consider a series RLC circuit driven by v_s(t), with capacitor voltage as the response. Kirchhoff’s voltage law gives

Since i(t)=C\,dv_C/dt, substitution yields

Dividing the entire equation by LC gives the normalized equation

Define the undamped natural frequency and damping factor as

Both have units of rad/s, with radians dimensionless. The characteristic roots are

For the underdamped case,

and

Worked Example: Critically Damped Step Response

Let R=4\ \Omega, L=1\ H, C=0.25\ F, and let an 8 V step be applied with zero initial energy. Here,

The circuit is critically damped. Its capacitor-voltage transfer function is

For V_s(s)=8/s,

Partial-fraction inversion gives

The response begins at zero with zero initial slope and approaches 8 V without overshoot.

The Unilateral Laplace Transform

For causal switching problems, the unilateral Laplace transform is

Its derivative properties explicitly incorporate initial conditions:

The bilateral transform, integrated from -\infty to \infty, is more suitable for two-sided signals and requires specification of a region of convergence. For causal rational signals, the region of convergence lies to the right of the rightmost pole.

Transform-Domain Element Models

For a resistor,

For a capacitor with v_C(0^-)=V₀,

or equivalently,

Thus, a capacitor can be represented by impedance 1/(sC) together with an initial-condition source. For an inductor with i_L(0^-)=I₀,

These models must be used with consistent reference polarities. Simply replacing C by 1/(sC) or L by sL while ignoring nonzero initial conditions produces an incorrect result.

Network Functions, Poles and Zeros

With zero initial conditions, a network or transfer function is

where X(s) and Y(s) are chosen input and output transforms. Zeros are roots of N(s), while poles are roots of D(s). Poles determine natural modes; zeros shape transmission and can suppress particular response components.

A transfer function is defined under zero initial conditions because otherwise the output includes contributions not caused by the specified input. Internal stability cannot always be inferred from a reduced input–output expression: exact pole–zero cancellation may hide an unstable internal mode.

It is incorrect to assert that every passive network function must be proper. For example, the driving-point impedance of a passive ideal inductor is

which is an improper rational function. Driving-point impedances and admittances of passive networks are instead constrained by positive-realness, under the standard assumptions of real, rational, lumped, passive one-port networks. Ordinary voltage or current transfer functions in finite-dimensional causal models are often proper for physically selected variables, but properness depends on the input–output definition.

For a rational function with distinct poles,

The inverse transform is x(t)=\sum K_ke^p_ktu(t). Repeated poles require higher-order partial-fraction terms. Complex poles of real networks occur in conjugate pairs.

Figure: Pole locations and continuous-time natural-mode behaviour.

For an asymptotically stable finite-dimensional continuous-time system, every internal natural mode must have a pole strictly in the open left half-plane. Simple imaginary-axis poles may produce bounded persistent oscillation in an ideal model, but repeated imaginary-axis poles cause unbounded growth and are unstable.

Convolution and Impulse Response

For a causal linear time-invariant system with zero initial conditions, the impulse response is

The zero-state response to input x(t) is the convolution

The convolution theorem states that multiplication in the transform domain corresponds to convolution in time:

Convolution is especially useful for arbitrary waveforms, sampled excitation records and Green-function interpretations. It assumes linear time invariance and zero-state conditions unless the zero-input response is added separately.

Initial- and Final-Value Theorems

The initial-value theorem is

provided the limit exists and the signal does not contain an impulse at the origin that invalidates the ordinary finite initial value.

The final-value theorem is

provided all poles of sX(s) lie strictly in the open left half-plane, with no poles on or to the right of the imaginary axis. A removable cancellation must be handled carefully because hidden unstable modes may still matter physically.

For example,

has

The theorem is valid because the only pole of sX(s)=5/(s+2) is at s=-2. Partial fractions confirm x(t)=2.5(1-e^-2t)u(t).

State-Space Transient Analysis

Large networks are often modelled using capacitor voltages and inductor currents as state variables. A linear time-invariant state model is

The complete state response is

The matrix exponential e^{\mathbf{A}t} is the state-transition matrix. Its eigenvalues are the system’s internal natural frequencies. When the realization is minimal, the transfer matrix is

State-space analysis accommodates multiple inputs and outputs, controlled sources, numerical integration and topology changes. Descriptor models of the form \mathbf{E}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u} are often preferable for networks containing algebraic constraints or ideal elements.

Frequency-Domain Connection

If the imaginary axis lies within the region of convergence, the frequency response is obtained from

Sinusoidal phasor analysis describes steady-state behaviour after stable transients have decayed; it does not by itself determine switching transients or incorporate arbitrary initial energy. Resonance peaks, bandwidth and phase shift are nevertheless direct manifestations of pole and zero locations.

The Fourier transform may fail to exist in the ordinary sense for growing or non-decaying signals even when a Laplace transform exists for an appropriate region of convergence. The Laplace transform is therefore the broader tool for causal transient analysis.

Illustrative Case: Relay-Coil Energization and Release Protection

Background

An industrial controller energizes a 24 V DC relay coil modelled by R=12\ \Omega and L=0.6\ H. The relay picks up when coil current reaches 1.2 A. This is an illustrative engineering case; actual relay characteristics include magnetic saturation, armature motion and temperature dependence.

Problem

Determine the pickup delay, stored magnetic energy at steady state, and the consequence of opening the switch without a suppression path.

Analysis

The electrical time constant and steady current are

With zero initial current,

At pickup, i=1.2\ A, so

The idealized electrical pickup threshold is reached after approximately 45.8 ms. Mechanical travel adds further delay.

At steady state, the stored magnetic energy is

If the current path is interrupted instantaneously, the inductor attempts to maintain 2 A. Because v_L=L\,di/dt, a very rapid decrease in current requires a large negative coil voltage. Real voltage is limited by switch capacitance, arcing, avalanche conduction or insulation breakdown rather than becoming infinite.

Approach and Solution

A flyback diode provides a low-voltage recirculation path and reduces semiconductor stress, but it produces relatively slow current decay. In the ideal-diode approximation,

A suitably rated transient-voltage suppressor permits a higher controlled reverse voltage, accelerating release while keeping the switch voltage within its safe operating area. Component selection must account for repetitive energy, temperature, tolerances and the supply topology.

Outcome

Transform-domain analysis predicts pickup timing and explains why suppression is necessary. The preferred suppressor depends on the trade-off between electromagnetic compatibility, contact release speed, semiconductor rating and energy dissipation.

Learning Points

• Inductor current remains continuous at ordinary switching instants.
• Stored energy, not merely steady current, determines suppression duty.
• An ideal diode protects effectively but may delay relay release.
• Real validation requires nonlinear device models and measured parasitics.

Modelling Challenges and Contemporary Practice

Ideal lumped models become inaccurate when interconnections are electrically long, switching edges excite distributed modes, magnetic cores saturate, or component values depend strongly on temperature and frequency. Additional difficulties include uncertain initial conditions, switch bounce, nonlinear semiconductor capacitance, numerical stiffness and hidden parasitic resonances.

Professional best practice includes:

• preserving reference directions and initial-condition signs;
• checking continuity, energy and limiting behaviour;
• retaining dependent sources when calculating equivalent resistance;
• testing final-value theorem conditions rather than applying it mechanically;
• avoiding unjustified pole–zero cancellation;
• validating reduced-order models over the intended frequency range; and
• comparing analytical predictions with simulation and measurement.

Modern electromagnetic-transient programs use companion models, numerical integration and sparse matrix methods. Vector fitting can derive rational models from measured frequency data, while passivity enforcement prevents fitted passive devices from generating nonphysical energy. Model-order reduction enables large interconnect and power-network models to be simulated efficiently. Machine-learning surrogate models can accelerate parameter sweeps, but they should be constrained or checked for causality, stability, passivity and out-of-distribution error.

Safety, Governance and Regulatory Considerations

Transient calculations influence insulation coordination, protective-device selection, electromagnetic compatibility and personnel safety. Engineers must use applicable IEC, IEEE, national wiring and product-safety requirements for the relevant equipment class rather than treating an idealized calculation as certification evidence. Assumptions, parameter sources, software versions and model limitations should be documented. Deliberately omitting adverse switching conditions or presenting unvalidated simulations as measured results is both professionally and ethically unacceptable.

Chapter Summary

Transient analysis determines how electrical networks redistribute stored energy after switching or excitation changes. Capacitor voltage and inductor current are continuous under finite excitation; capacitor-voltage jumps require impulsive current, whereas inductor-current jumps require impulsive voltage. First- and second-order responses are governed by time constants, natural frequencies and damping.

The unilateral Laplace transform converts differential equations into algebraic relations while retaining initial conditions. Network functions, poles, zeros, residues and convolution provide complementary interpretations of dynamic behaviour. Initial- and final-value theorems are useful only when their applicability conditions are satisfied. State-space methods extend the analysis to higher-order and multi-input systems. Practical transient engineering must include parasitics, nonlinearities, passivity, safety requirements and validation against physical evidence.

Key Terms

Transient: Time-varying response during movement between operating conditions.

Natural response: Response associated with a network’s internal modes.

Forced response: Response associated with an external excitation.

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

Zero-state response: Response caused by input with zero initial energy.

Time constant: Parameter governing exponential decay or growth in a first-order mode.

Damping factor: Exponential decay rate \alpha of a second-order response.

Natural frequency: Frequency of an unforced network mode.

Laplace transform: Integral transform mapping a time-domain signal into the complex-frequency domain.

Region of convergence: Set of complex values for which a transform integral converges.

Transfer function: Zero-initial-condition ratio of output transform to input transform.

Pole: Complex frequency at which a network function becomes unbounded.

Zero: Complex frequency at which a network function becomes zero.

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

Convolution: Integral operation combining an input with an impulse response.

State variable: Minimal dynamic variable used to describe stored-energy evolution.

State-transition matrix: Matrix exponential that propagates an initial state through time.

Positive-real function: Function satisfying conditions associated with passive driving-point immittance.

Review Questions

Short Answer Questions

1. State the continuity conditions for ideal capacitor voltage and inductor current.
2. Distinguish the natural response from the zero-input response.
3. Why must initial-condition sources be included in transform-domain reactive-element models?
4. State the pole condition required by the final-value theorem.
5. Explain why the impedance Z(s)=sL disproves the claim that every passive network function is proper.

Long Answer Questions

1. Derive the complete response of a first-order circuit to a constant post-switch excitation.
2. Derive the series RLC capacitor-voltage equation and classify its damping regimes.
3. Develop the unilateral Laplace-domain models of a capacitor and an inductor with nonzero initial conditions.
4. Explain how poles, zeros, regions of convergence and hidden pole–zero cancellations affect network interpretation.
5. Compare differential-equation, Laplace-transform, convolution and state-space methods for transient analysis.

Analytical / Application Questions

1. A 48 V source energizes a series RL circuit with R=16\ \Omega, L=0.8\ H and zero initial current. Determine the current at 25 ms and the time required to reach 95% of its final value.
2. For R=6\ \Omega, L=1\ H and C=0.04\ F, calculate \alpha, \omega₀, the damping classification and the characteristic roots.
3. A network response is X(s)=(s+4)/[s(s-1)(s+3)]. Determine whether the final-value theorem is applicable, justify the conclusion from the poles of sX(s), and describe the expected long-term behaviour.

Multiple Choice Questions with Answers

Question 1

Which excitation is required to produce an instantaneous jump in the voltage of an ideal capacitor?

A. An impulsive current

B. A finite constant current

C. An impulsive voltage

D. A finite constant voltage

Correct Answer: A. An impulsive current

Question 2

For a series RLC circuit with capacitor voltage as output, which equation results after dividing LC\,d²v_C/dt²+RC\,dv_C/dt+v_C=v_s by LC?

A. v_C''+(R/C)v_C'+(1/L)v_C=v_s/L

B. v_C''+(R/L)v_C'+v_C/(LC)=v_s/(LC)

C. v_C''+(L/R)v_C'+LCv_C=v_s

D. v_C''+RCv_C'+LCv_C=LCv_s

Correct Answer: B. v_C''+(R/L)v_C'+v_C/(LC)=v_s/(LC)

Question 3

For a capacitor with v_C(0^-)=V₀, which unilateral Laplace-domain relation is correct?

A. I_C(s)=sCV_C(s)+CV₀

B. I_C(s)=V_C(s)/(sC)

C. I_C(s)=C[sV_C(s)-V₀]

D. V_C(s)=sCI_C(s)-V₀

Correct Answer: C. I_C(s)=C[sV_C(s)-V₀]

Question 4

For X(s)=5/[s(s+2)], what is the final value of x(t)?

A. 1

B. 2

C. 2.5

D. 5

Correct Answer: C. 2.5

Question 5

A series RLC network is underdamped when which condition holds?

A. \alpha>\omega₀

B. \alpha=\omega₀

C. 0<\alpha<\omega₀

D. \alpha<0

Correct Answer: C. 0<\alpha<\omega₀

Question 6

A 2\ k\Omega resistor charges a 100\ \mu F capacitor from a 10 V source. What is the time constant?

A. 0.02 s

B. 0.2 s

C. 2 s

D. 20 s

Correct Answer: B. 0.2 s

Question 7

Which passive driving-point function is improper?

A. Z(s)=R

B. Z(s)=1/(sC)

C. Z(s)=sL

D. Z(s)=R/(1+sRC)

Correct Answer: C. Z(s)=sL

Question 8

For the critically damped example with R=4\ \Omega, L=1\ H, C=0.25\ F and an 8 V step, what is v_C(t)?

A. 8(1-e^-2t)\ V

B. 8[1-(1+2t)e^-2t]\ V

C. 8e^-2t\ V

D. 8(1+2t)e^-2t\ V

Correct Answer: B. 8[1-(1+2t)e^-2t]\ V

Question 9

A 24 V source energizes a 12\ \Omega, 0.6\ H relay coil from zero current. Approximately when does the current reach 1.2 A?

A. 20.0 ms

B. 34.7 ms

C. 45.8 ms

D. 91.6 ms

Correct Answer: C. 45.8 ms

Question 10

When is the continuous-time final-value theorem valid for a rational response X(s)?

A. Whenever X(s) is proper

B. Whenever all poles of X(s) are real

C. When all poles of sX(s) lie strictly in the open left half-plane

D. When X(s) has at least one pole at the origin

Correct Answer: C. When all poles of sX(s) lie strictly in the open left half-plane