Unit 03: Dynamic Circuit Time-Domain Analysis

Fundamentals of Circuit Analysis (2026) · Chapter 7

Transient Process First-Order Circuits Three-Element Method Step Response Impulse Response Second-Order Circuits

Knowledge Structure Overview

Dynamic Circuit Time-Domain Analysis
└─ 7. Dynamic Circuit Time-Domain Analysis
    ├─ 7.1 Transient Process and Switching Rules
    │   ├─ 7.1.1 Basic Concepts of Dynamic Circuits
    │   └─ 7.1.2 Switching Rules
    ├─ 7.2 First-Order Circuit Zero-Input Response
    │   ├─ 7.2.1 First-Order RC Circuit Zero-Input Response
    │   └─ 7.2.2 First-Order RL Circuit Zero-Input Response
    ├─ 7.3 First-Order Circuit Zero-State Response
    │   ├─ 7.3.1 First-Order RC Circuit Zero-State Response
    │   └─ 7.3.2 First-Order RL Circuit Zero-State Response
    ├─ 7.4 First-Order Circuit Full Response and Three-Element Method
    │   ├─ 7.4.1 Full Response of First-Order Circuits
    │   └─ 7.4.2 Three-Element Method for First-Order Circuits
    ├─ 7.5 Step Function and Step Response
    │   ├─ 7.5.1 Step Function
    │   └─ 7.5.2 Step Response
    ├─ 7.6 Impulse Function and Impulse Response
    │   ├─ 7.6.1 Impulse Function
    │   └─ 7.6.2 Impulse Response
    └─ 7.7 Second-Order Circuit Zero-Input Response

Chapter 7

7.1 Transient Process and Switching Rules

7.1.1 Basic Concepts of Dynamic Circuits

Dynamic Circuit: A circuit containing energy-storage elements such as capacitors or inductors
Transient Process (Transition Process): The intermediate process when a circuit transitions from one steady state to another
Cause: Energy in energy-storage elements cannot change instantaneously (energy accumulation and release require time)
Steady State: The state where all quantities in the circuit have reached stability (DC steady state or AC steady state)
Switching: A sudden change in circuit structure or parameters (switch operation, parameter change, etc.)

7.1.2 Switching Rules

Core Theorem Switching Rules

Statement: At the switching instant ($t = 0$), capacitor voltage and inductor current cannot change instantaneously

$$u_C(0^+) = u_C(0^-) \qquad i_L(0^+) = i_L(0^-)$$

Physical Basis: Energy cannot change instantaneously ($W_C = \frac{1}{2}Cu_C^2$, $W_L = \frac{1}{2}Li_L^2$)

Corollaries:

Capacitor current can change instantaneously
Inductor voltage can change instantaneously
Resistor voltage and current can both change instantaneously

Procedure Steps for Determining Initial Values

Find $u_C(0^-)$ and $i_L(0^-)$
at $t = 0^-$
(steady-state analysis)

Apply switching rules to get
$u_C(0^+)$ and $i_L(0^+)$

Draw the $t = 0^+$
equivalent circuit (capacitor replaced by voltage source, inductor by current source)

Find initial values
of other quantities

Chapter 7

7.2 First-Order Circuit Zero-Input Response

RC Circuit First-Order RC Circuit Zero-Input Response

Circuit Structure: A charged capacitor C discharging through a resistor R

$$RC\frac{du_C}{dt} + u_C = 0$$ $$u_C(t) = u_C(0^+)e^{-t/\tau}$$

Time Constant: $\tau = RC$ (unit: seconds)
Capacitor voltage decays exponentially to zero
Larger $\tau$ means slower decay
After $3\tau \sim 5\tau$, the transient process can be considered essentially complete

$$i(t) = -\frac{u_C(0^+)}{R}e^{-t/\tau}$$

Discharge Current: Maximum at the start, zero at the end

RL Circuit First-Order RL Circuit Zero-Input Response

Circuit Structure: An energized inductor L releasing energy through a resistor R

$$L\frac{di_L}{dt} + Ri_L = 0$$ $$i_L(t) = i_L(0^+)e^{-t/\tau}$$

Time Constant: $\tau = L/R$ (unit: seconds)
Inductor current decays exponentially to zero
Inductor voltage $u_L(t) = -Ri_L(0^+)e^{-t/\tau}$ (changes instantaneously)

Chapter 7

7.3 First-Order Circuit Zero-State Response

RC Circuit First-Order RC Circuit Zero-State Response

Circuit Structure: An uncharged capacitor C connected to a DC voltage source $U_s$

$$RC\frac{du_C}{dt} + u_C = U_s$$ $$u_C(t) = U_s(1 - e^{-t/\tau}), \quad \tau = RC$$

Capacitor voltage rises exponentially from zero to $U_s$
Charging current $i(t) = \frac{U_s}{R}e^{-t/\tau}$, maximum at start, zero at end
Capacitor acts as an open circuit (DC steady state)

RL Circuit First-Order RL Circuit Zero-State Response

Circuit Structure: An inductor L with zero current connected to a DC voltage source $U_s$ (in series with resistor R)

$$L\frac{di_L}{dt} + Ri_L = U_s$$ $$i_L(t) = \frac{U_s}{R}(1 - e^{-t/\tau}), \quad \tau = L/R$$

Inductor current rises exponentially from zero to $U_s/R$
Inductor voltage $u_L(t) = U_s e^{-t/\tau}$, maximum at start, zero at end
Inductor acts as a short circuit (DC steady state)

Chapter 7

7.4 Full Response and Three-Element Method

7.4.1 Full Response of First-Order Circuits

Definition Full Response

Definition: The sum of the zero-input response and the zero-state response

$$\text{Full Response} = \text{Zero-Input Response} + \text{Zero-State Response}$$

Second Decomposition: Steady-state component + Transient component

$$\text{Full Response} = \text{Steady-State Component} + \text{Transient Component}$$

RC Circuit Full Response:

$$u_C(t) = U_s + [u_C(0^+) - U_s]e^{-t/\tau}$$

$U_s$: Steady-state component (final value)
$[u_C(0^+) - U_s]e^{-t/\tau}$: Transient component (decaying term)
Physical Meaning: The full response reflects the complete change process of the circuit from its initial state to steady state

7.4.2 Three-Element Method for First-Order Circuits

Key Method Three-Element Method

Three Elements:

Initial Value $f(0^+)$: The value immediately after switching
Steady-State Value $f(\infty)$: The steady-state value as $t \to \infty$
Time Constant $\tau$: Determines the speed of the transient process

General Formula:

$$f(t) = f(\infty) + [f(0^+) - f(\infty)]e^{-t/\tau}$$

Applicable Range: Full response of any first-order linear circuit

Applies to: first-order linear circuits $\tau = R_{eq}C$ or $\tau = L/R_{eq}$

Procedure Three-Element Method Solution Steps

Find $f(0^+)$
using switching rules and
the $t=0^+$ equivalent circuit

Find $f(\infty)$
open-circuit capacitor, short-circuit inductor
solve for DC steady state

Find $\tau$
set independent sources to zero,
find $R_{eq}$

Substitute into formula
$f(t) = f(\infty) + [f(0^+) - f(\infty)]e^{-t/\tau}$

Chapter 7

7.5 Step Function and Step Response

7.5.1 Step Function

$$\varepsilon(t) = \begin{cases} 0, & t < 0 \\ 1, & t > 0 \end{cases}$$

Delayed Step: $\varepsilon(t - t_0)$ transitions at $t = t_0$
Amplitude Step: $K\varepsilon(t)$ with amplitude K
Physical Meaning: Represents a switch closing (or opening) at $t=0$

7.5.2 Step Response

Definition: The zero-state response of a first-order circuit under a step input

RC Circuit Step Response

$u_C(t) = U_s(1-e^{-t/\tau})\varepsilon(t)$

RL Circuit Step Response

$i_L(t) = \frac{U_s}{R}(1-e^{-t/\tau})\varepsilon(t)$

Significance: The step response is the basis for analyzing more complex inputs

Chapter 7

7.6 Impulse Function and Impulse Response

7.6.1 Impulse Function

$$\delta(t) = \begin{cases} 0, & t \neq 0 \\ \infty, & t = 0 \end{cases}, \quad \int_{-\infty}^{+\infty}\delta(t)dt = 1$$

Sifting Property: $\int_{-\infty}^{+\infty}f(t)\delta(t)dt = f(0)$
Physical Meaning: Represents an extremely large force acting over an extremely short time (e.g., hammer blow, lightning strike)
Relationship with Step Function: $\delta(t) = d\varepsilon(t)/dt$

7.6.2 Impulse Response

Definition: The zero-state response of a circuit under a unit impulse function $\delta(t)$

Solution Methods:

Method 1: Direct solution of the differential equation
Method 2: Differentiate the step response (for first-order circuits)

$$h(t) = \frac{1}{RC}e^{-t/RC}\varepsilon(t)$$

RC Circuit Impulse Response

Impulse response = derivative of step response Fourier transform pair with frequency response

Chapter 7

7.7 Second-Order Circuit Zero-Input Response

Overview Second-Order Circuit Basic Equation

Circuit Structure: Contains two energy-storage elements (e.g., RLC series or parallel circuit)

Differential Equation: Second-order constant-coefficient linear differential equation (RLC series as example)

$$LC\frac{d^2u_C}{dt^2} + RC\frac{du_C}{dt} + u_C = 0$$

Characteristic Equation:

$$s^2 + \frac{R}{L}s + \frac{1}{LC} = 0$$

Characteristic Roots:

$$s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}$$

$\alpha = R/(2L)$: Damping coefficient
$\omega_0 = 1/\sqrt{LC}$: Natural angular frequency

Four Response Types

Overdamped

α > ω₀

Two unequal real roots

Non-oscillatory decay

Slowest decay

Critically Damped

α = ω₀

Two equal real roots

Non-oscillatory decay

Fastest without oscillation

Underdamped

α < ω₀

Complex conjugate roots

Oscillatory decay

$\omega_d = \sqrt{\omega_0^2 - \alpha^2}$

Damped oscillation frequency

Undamped

α = 0

Pure imaginary roots

Sustained oscillation

Ideal case

Summary

RC vs RL First-Order Circuit Comparison

RC Zero-Input

$\tau = RC$

$u_C(t) = u_C(0^+)e^{-t/\tau}$

Capacitor discharge

Exponential decay to zero

RL Zero-Input

$\tau = L/R$

$i_L(t) = i_L(0^+)e^{-t/\tau}$

Inductor energy release

Exponential decay to zero

RC Zero-State

$\tau = RC$

$u_C(t) = U_s(1-e^{-t/\tau})$

Capacitor charging

Exponential rise to $U_s$

RL Zero-State

$\tau = L/R$

$i_L(t) = \frac{U_s}{R}(1-e^{-t/\tau})$

Inductor current buildup

Exponential rise to $U_s/R$