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Qwestrum Engineering360 · Electrical & Electronics · Network Analysis

Transient Response

Transient response describes how currents and voltages evolve from an initial to a final steady state after a switching event, governed by the time constant τ for first-order circuits and by damping for second-order RLC circuits.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Overdamped, critically damped, underdamped from discriminant
  • Initial capacitor voltage and inductor current continuity
  • Step response vs impulse response via Laplace transform

Topic details

Introduction

Any question with a switch closing at t = 0 is a transient problem. The universal first-order form x(t) = x(∞) + [x(0⁺) − x(∞)] e^(−t/τ) solves almost every RC/RL numerical if you can find three things: the initial value, the final value, and the time constant.

Scope in B.Tech and GATE syllabus

Second-order RLC circuits need the characteristic equation. The discriminant (R/L)² − 4/(LC) decides whether the response is overdamped (two real roots), critically damped (repeated root), or underdamped (complex roots giving decaying oscillation).

Key relations & formulas

RC:τ=RC;vC(t)=V(1−e(−tτ))RC: \tau = R C; v_{C}(t) = V(1 - e^(-\frac{t}{\tau}))
(charging)

Formulas (Indian textbook notation)

  • RL:τ=LR;i(t)=(VR)(1−e(−tτ))RL: \tau = \frac{L}{R}; i(t) = (\frac{V}{R})(1 - e^(-\frac{t}{\tau}))
RLC:s2+(RL)s+1(LC)=0RLC: s^{2} + (\frac{R}{L})s + \frac{1}{(LC)} = 0
(characteristic equation)

Notation and sign conventions

Relation 1 —
RC:τ=RC;vCRC: \tau = R C; v_{C}
RC:τ=RC;vC(t)=V(1−e(−tτ))RC: \tau = R C; v_{C}(t) = V(1 - e^(-\frac{t}{\tau}))
(charging)
Write this relation with symbols exactly as in Network Analysis — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
RL:τ=LR;iRL: \tau = \frac{L}{R}; i

Formulas (Indian textbook notation)

  • RL:τ=LR;i(t)=(VR)(1−e(−tτ))RL: \tau = \frac{L}{R}; i(t) = (\frac{V}{R})(1 - e^(-\frac{t}{\tau}))
Write this relation with symbols exactly as in Network Analysis — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
RLC:s2+RLC: s^{2} +
RLC:s2+(RL)s+1(LC)=0RLC: s^{2} + (\frac{R}{L})s + \frac{1}{(LC)} = 0
(characteristic equation)
Write this relation with symbols exactly as in Network Analysis — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Use continuity: capacitor voltage and inductor current cannot change instantaneously, so v_C(0⁺) = v_C(0⁻) and i_L(0⁺) = i_L(0⁻). At t = 0⁺ treat a capacitor with known voltage as a voltage source and an inductor with known current as a current source.

Governing relations in practice

At t → ∞ (DC steady state) the capacitor is an open circuit and the inductor a short circuit. This gives the final values directly.

Design and analysis considerations

The time constant is τ = R_th C or L/R_th, where R_th is the Thevenin resistance seen by the storage element with sources deactivated. After ≈ 5τ the transient is essentially complete (>99%).

Assumptions and validity limits

State assumptions explicitly before using any relation for transient response — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Network Analysis viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Network Analysis papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to transient response.
4. Use equation 1:
RC:τ=RC;vCRC: \tau = R C; v_{C}
.
5. Use equation 2:
RL:τ=LR;iRL: \tau = \frac{L}{R}; i
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Transient Response appears in all electrical engineering circuits. In Indian electrical curricula this topic is tested because it connects theory to DC/AC circuit analysis.
GATE and semester exams often combine transient response with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use transient response?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using the source resistance instead of the Thevenin resistance seen by C or L for τ
• Assuming v_C or i_L jumps at switching (violates continuity)
• Treating capacitor as short / inductor as open at final DC state (it is the opposite)
• Dropping the natural-response term and keeping only the forced response

Quick revision checklist

Before attempting transient response problems, confirm you can:
1. Overdamped, critically damped, underdamped from discriminant
2. Initial capacitor voltage and inductor current continuity
3. Step response vs impulse response via Laplace transform
Revise the solved examples in Network Analysis — Nagrath & Kothari and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

RC charging time constant

Problem

A 10 µF capacitor, initially uncharged, is switched onto a 100 V source through 5 kΩ. Find τ, the voltage after 30 ms, and the initial charging current.

Solution

τ = RC = 5000 × 10×10⁻⁶ = 0.05 s = 50 ms.
v_C(t) = 100(1 − e^(−t/τ)). At t = 30 ms: t/τ = 0.03/0.05 = 0.6.
v_C = 100(1 − e^(−0.6)) = 100(1 − 0.549) = 45.1 V.
Initial current i(0⁺) = V/R = 100/5000 = 20 mA (capacitor acts as short at t = 0⁺).

Conceptual check — Transient Response

Problem

In a Network Analysis semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of transient response." What should a complete answer include?

Exams & GATE

Nagrath & Kothari Ch. 5 — find τ and final value from circuit.

📖 Standard books (India)

  • Network Analysis — Nagrath & Kothari

    Read: Syllabus unit

    KCL, KVL, theorems, and three-phase circuits

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