Unit 02: Sinusoidal Steady-State Analysis

Fundamentals of Circuit Analysis (2026) · Chapters 5-6

Sinusoidal Steady-State Phasor Method Power Resonance Coupled Inductors Transformers

Knowledge Structure Overview

Sinusoidal Steady-State Analysis
├─ 5. Sinusoidal Steady-State Circuit Analysis
│   ├─ 5.1 Basic Concepts of Sinusoidal Quantities
│   │   ├─ 5.1.1 Frequency and Period
│   │   ├─ 5.1.2 Amplitude and RMS Value
│   │   └─ 5.1.3 Initial Phase
│   ├─ 5.2 Phasor Representation of Sinusoidal Quantities
│   │   ├─ 5.2.1 Complex Numbers and Operations
│   │   ├─ 5.2.2 Concept of Phasors
│   │   ├─ 5.2.3 Phasor Diagrams
│   │   └─ 5.2.4 Operational Properties of Phasors
│   ├─ 5.3 Phasor Form of Two Constraints
│   │   ├─ 5.3.1 Phasor Form of Kirchhoff's Laws
│   │   └─ 5.3.2 Phasor Form of Element V-I Relations
│   ├─ 5.4 Impedance of Passive One-Port Networks
│   │   ├─ 5.4.1 Concept of Impedance
│   │   ├─ 5.4.2 Series and Parallel Impedance
│   │   └─ 5.4.3 Phasor Model of Circuits
│   ├─ 5.5 Phasor Method Analysis of Sinusoidal Steady-State Circuits
│   │   ├─ 5.5.1 Analysis of Simple AC Circuits
│   │   └─ 5.5.2 Mesh and Node Methods
│   ├─ 5.6 Power in Sinusoidal AC Circuits
│   │   ├─ 5.6.1 ~ 5.6.5 Four Types of Power
│   │   ├─ 5.6.6 Power Factor Correction
│   │   └─ 5.6.7 Maximum Power Transfer
│   ├─ 5.7 Circuit Frequency Response
│   │   ├─ 5.7.1 Network Functions and Frequency Characteristics
│   │   ├─ 5.7.2 RC Low-Pass Filter Circuit
│   │   └─ 5.7.3 RC High-Pass Filter Circuit
│   └─ 5.8 Resonance Circuits
│       ├─ 5.8.1 Series Resonance
│       └─ 5.8.2 Parallel Resonance
└─ 6. Coupled Inductors and Ideal Transformers
    ├─ 6.1 Coupled Inductors
    │   ├─ 6.1.1 Mutual Inductance
    │   ├─ 6.1.2 V-I Relations of Coupled Inductors
    │   ├─ 6.1.3 Dot Convention
    │   └─ 6.1.4 Equivalent Circuit Models
    ├─ 6.2 Analysis of Sinusoidal Steady-State Circuits with Coupled Inductors
    │   ├─ 6.2.1 Direct Equation Method
    │   └─ 6.2.2 Decoupling Equivalent Analysis
    ├─ 6.3 Air-Core Transformers
    └─ 6.4 Ideal Transformers
        ├─ 6.4.1 Main Characteristics
        └─ 6.4.2 Circuit Analysis with Ideal Transformers

Chapter 5

5.1 Basic Concepts of Sinusoidal Quantities

5.1.1 Frequency and Period

Period T: The time required for a sinusoidal quantity to complete one cycle of periodic change, unit: second (s)
Frequency f: The number of periodic cycles completed per unit time, $f = 1/T$, unit: Hertz (Hz)
Angular Frequency ω: The rate of phase change of a sinusoidal quantity, $\omega = 2\pi f = 2\pi/T$, unit: radians/second (rad/s)
Power Frequency: The standard frequency of power systems in China is 50Hz, i.e., $\omega = 100\pi \approx 314$ rad/s

5.1.2 Amplitude and RMS Value

Instantaneous Value: $i(t) = I_m \sin(\omega t + \varphi)$
Amplitude (Maximum Value) $I_m$: The maximum value of the sinusoidal quantity
RMS Value (Effective Value): The DC value that produces the same thermal effect as the sinusoidal quantity $$I = \frac{I_m}{\sqrt{2}} \approx 0.707 I_m$$
Significance of RMS: AC meter readings are typically RMS values; rated values of electrical equipment are also expressed in RMS

5.1.3 Initial Phase

Phase: $\omega t + \varphi$, represents the state of a sinusoidal quantity at a given instant
Initial Phase $\varphi$: The phase angle at $t=0$, $|\varphi| \leq \pi$
Physical Meaning: Reflects the state of the sinusoidal quantity at the timing origin
Phase Difference: The difference in initial phases between two sinusoidal quantities of the same frequency, $\varphi = \varphi_1 - \varphi_2$
- $\varphi > 0$: u leads i (or i lags u)
- $\varphi < 0$: u lags i (or i leads u)
- $\varphi = 0$: u and i are in phase
- $\varphi = \pi$: u and i are in opposite phase

Chapter 5

5.2 Phasor Representation of Sinusoidal Quantities

5.2.1 Complex Numbers and Operations

Complex Number Representations:
- Rectangular form: $A = a + jb$ ($j^2 = -1$)
- Exponential form: $A = |A|e^{j\theta}$
- Polar form: $A = |A|\angle\theta$
Basic Operations:
- Addition/Subtraction: Add/subtract real parts and imaginary parts separately
- Multiplication/Division: Multiply/divide magnitudes, add/subtract angles
- Complex conjugate: $\bar{A} = a - jb = |A|\angle(-\theta)$

5.2.2 Concept of Phasors

Definition: Using complex numbers to represent the amplitude (or RMS value) and initial phase of a sinusoidal quantity
Amplitude Phasor: $\dot{I}_m = I_m e^{j\varphi} = I_m\angle\varphi$
RMS Phasor: $\dot{I} = I e^{j\varphi} = I\angle\varphi$ (more commonly used in engineering)
Correspondence: $i(t) = I_m \sin(\omega t + \varphi) \leftrightarrow \dot{I}_m = I_m\angle\varphi$
Note: Phasors can only represent sinusoidal quantities of the same frequency; sinusoidal quantities of different frequencies cannot be represented by the same phasor

5.2.3 Phasor Diagrams

Definition: A diagram representing phasors as directed line segments on the complex plane
Drawing Method: Starting from the origin, length represents RMS value (or amplitude), angle represents initial phase
Applications: Intuitively show phase relationships between sinusoidal quantities; assist in phasor addition and subtraction

5.2.4 Operational Properties of Phasors

Time-domain differentiation corresponds to phasor multiplication: $\frac{d}{dt} \rightarrow j\omega$
Time-domain integration corresponds to phasor division: $\int dt \rightarrow \frac{1}{j\omega}$
Linear operations on same-frequency sinusoidal quantities: Can be performed directly using phasors

Chapter 5

5.3 Phasor Form of Two Constraints

5.3.1 Phasor Form of Kirchhoff's Laws

Phasor form of KCL: $\sum \dot{I} = 0$ (the sum of phasor currents entering a node is zero)
Phasor form of KVL: $\sum \dot{U} = 0$ (the sum of phasor voltages around a loop is zero)
Note: The phasor forms of KCL and KVL are identical in form to the DC circuit versions, but the operands are complex numbers

5.3.2 Phasor Form of Element V-I Relations

Resistor R

$\dot{U}_R = R\dot{I}_R$

Voltage and current are in phase

Inductor L

$\dot{U}_L = j\omega L \cdot \dot{I}_L$

Voltage leads current by 90°

Inductive reactance $X_L = \omega L$

Capacitor C

$\dot{U}_C = \frac{1}{j\omega C}\dot{I}_C$

Voltage lags current by 90°

Capacitive reactance $X_C = \frac{1}{\omega C}$

Chapter 5

5.4 Impedance of Passive One-Port Networks

Impedance 5.4.1 Concept of Impedance

$$Z = \frac{\dot{U}}{\dot{I}} = R + jX = |Z|\angle\varphi_z$$

Resistance part (real): $R = |Z|\cos\varphi_z$
Reactance part (imaginary): $X = |Z|\sin\varphi_z$
Impedance magnitude: $|Z| = \sqrt{R^2 + X^2}$
Impedance angle: $\varphi_z = \arctan(X/R)$ (i.e., the angle by which voltage leads current)

Admittance: $Y = 1/Z = G + jB$

5.4.2 Series and Parallel Impedance

Series: $Z_{eq} = Z_1 + Z_2 + \cdots + Z_n$
Parallel: $Y_{eq} = Y_1 + Y_2 + \cdots + Y_n$ (i.e., $\frac{1}{Z_{eq}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}$)
Voltage Divider Formula: $\dot{U}_k = \frac{Z_k}{Z_1+Z_2+\cdots+Z_n} \cdot \dot{U}$
Current Divider Formula: $\dot{I}_k = \frac{Y_k}{Y_{eq}} \cdot \dot{I}$ (i.e., $\dot{I}_k = \frac{Z_{eq}}{Z_k} \cdot \dot{I}$)

5.4.3 Phasor Model of Circuits

Construction Method: Replace each circuit element with its phasor model
- Resistor → $R$
- Inductor → $j\omega L$
- Capacitor → $1/(j\omega C)$
- Voltage source → $\dot{U}_s$, Current source → $\dot{I}_s$
Characteristic: All quantities in the phasor model are complex numbers; analysis methods are the same as for DC resistive circuits

Chapter 5

5.5 Phasor Method Analysis of Sinusoidal Steady-State Circuits

5.5.1 Analysis of Simple AC Circuits

Convert the time-domain circuit
to a phasor model

Simplify the circuit using
impedance series/parallel

Write equations using
KCL, KVL, and Ohm's law

Solve for phasors and convert
back to time domain

5.5.2 Mesh and Node Methods in AC Circuits

Mesh Method: Same form as DC circuits, except impedance is complex $$Z_{11}\dot{I}_{m1} + Z_{12}\dot{I}_{m2} + \cdots = \dot{U}_{s11}$$
Node Method: Same form as DC circuits, except admittance is complex $$Y_{11}\dot{U}_{n1} + Y_{12}\dot{U}_{n2} + \cdots = \dot{I}_{s11}$$

Chapter 5

5.6 Power in Sinusoidal AC Circuits

5.6.1 Instantaneous Power

Definition: $p(t) = u(t) \cdot i(t)$
Expression: $p(t) = UI\cos\varphi_z - UI\cos(2\omega t + \varphi_u + \varphi_i)$
- Constant component (average power) + double-frequency oscillating component

Chapter 5

5.6 Four Types of Power at a Glance

Active Power

Average Power P

The power actually consumed by the circuit, unit: Watt (W)

$P = UI\cos\varphi_z$

$\lambda = \cos\varphi_z$ Reflects electrical energy utilization

Reactive Power

Reactive Power Q

Reflects the scale of energy exchange between electric and magnetic fields, unit: volt-ampere reactive (var)

$Q = UI\sin\varphi_z$

$Q > 0$: Inductive $Q < 0$: Capacitive

Apparent Power

Apparent Power S

The product of RMS voltage and RMS current, unit: volt-ampere (VA)

$S = UI = \sqrt{P^2 + Q^2}$

Equipment rated capacity

Complex Power

Complex Power $\tilde{S}$

Unified representation of active and reactive power; complex power is conserved

$\tilde{S} = \dot{U}\dot{I}^* = P + jQ$

$\sum \tilde{S} = 0$

Chapter 5

5.6.6 Power Factor Correction

Power Factor Shunt Capacitor Compensation

Significance: Reduces line current, decreases transmission losses, and improves equipment utilization

Method: Connect appropriate capacitors in parallel (compensate for reactive power of inductive loads)

$$C = \frac{P(\tan\varphi_1 - \tan\varphi_2)}{\omega U^2}$$

$\varphi_1$: Power factor angle before compensation
$\varphi_2$: Target power factor angle after compensation
$P$: Load active power
$U$: RMS source voltage

Chapter 5

5.6.7 Maximum Power Transfer

Max Power Conjugate Matching Condition

$$Z_L = Z_s^* \quad \Rightarrow \quad R_L = R_s, \; X_L = -X_s$$ $$P_{max} = \frac{|U_s|^2}{8R_s}$$

Condition: Load impedance equals the complex conjugate of the source internal impedance (conjugate matching)
Maximum Power: $P_{max} = \dfrac{|U_s|^2}{8R_s}$

Chapter 5

5.7 Circuit Frequency Response

5.7.1 Network Functions and Frequency Characteristics

Network Function: $H(j\omega) = \dfrac{\dot{Y}(j\omega)}{\dot{X}(j\omega)}$ (ratio of output phasor to input phasor)
Magnitude Response: The relationship between $|H(j\omega)|$ and frequency
Phase Response: The relationship between $\arg H(j\omega)$ and frequency

RC Low-Pass Filter Circuit

R in series, C in parallel for output

$H(j\omega) = \dfrac{1}{1+j\omega RC}$

Cutoff angular frequency: $\omega_c = 1/(RC)$

Passes low frequencies, attenuates high frequencies

RC High-Pass Filter Circuit

C in series, R in parallel for output

$H(j\omega) = \dfrac{j\omega RC}{1+j\omega RC}$

Cutoff angular frequency: $\omega_c = 1/(RC)$

Passes high frequencies, attenuates low frequencies

Frequency Response RC Low-Pass Filter Analysis

$\omega \ll \omega_c$: $|H| \approx 1$, signal passes through
$\omega = \omega_c$: $|H| = 1/\sqrt{2}$ (-3dB point)
$\omega \gg \omega_c$: $|H| \to 0$, signal is suppressed

Chapter 5

5.8 Resonance Circuits

Series Resonance 5.8.1 Series Resonance

Definition: Resonance in an R-L-C series circuit when inductive reactance equals capacitive reactance

$$\omega_0 L = \frac{1}{\omega_0 C} \quad \Rightarrow \quad \omega_0 = \frac{1}{\sqrt{LC}}$$

Minimum impedance: $Z = R$ (purely resistive)
Maximum current: $I = U/R$
Voltage and current are in phase (power factor = 1)
Inductor voltage and capacitor voltage are equal in magnitude and opposite in direction; they can be much larger than the source voltage
Quality factor: $Q = \dfrac{\omega_0 L}{R} = \dfrac{1}{\omega_0 CR}$
Higher Q means a sharper resonance curve and better selectivity

Parallel Resonance 5.8.2 Parallel Resonance

Definition: Resonance in an R-L-C parallel circuit

$$\omega_0 = \frac{1}{\sqrt{LC}}$$

Minimum admittance (maximum impedance)
Minimum total current
Branch currents may be much larger than total current
Circuit is purely resistive
Quality factor: $Q = R\sqrt{\dfrac{C}{L}} = \dfrac{R}{\omega_0 L} = \omega_0 RC$

Series Resonance vs. Parallel Resonance

Series Resonance

Series

Impedance: Minimum ($Z=R$)

Current: Maximum

$Q = \dfrac{\omega_0 L}{R}$

Q is inversely proportional to R

Smaller R, less loss, higher Q

Parallel Resonance

Parallel

Impedance: Maximum

Total current: Minimum

$Q = \omega_0 RC$

Q is directly proportional to R

Larger R, less shunting, higher Q

Chapter 6

6.1 Coupled Inductors

6.1.1 Mutual Inductance

Definition: When the current in one coil changes, it not only produces an induced voltage in that coil but also in an adjacent coil
Mutual Inductance M: $M_{12} = M_{21} = M$
Coupling Coefficient: $k = M/\sqrt{L_1 L_2}$, $0 \leq k \leq 1$
- $k = 0$: No coupling
- $k = 1$: Perfect coupling (ideal case)

6.1.2 V-I Relations of Coupled Inductors

V-I Relations Time-Domain and Phasor Forms

Voltage Equations:

$$u_1 = L_1\frac{di_1}{dt} \pm M\frac{di_2}{dt}$$ $$u_2 = L_2\frac{di_2}{dt} \pm M\frac{di_1}{dt}$$

The sign before M depends on the relationship between current direction and dot convention (positive for same-side, negative for opposite-side)

Phasor Form:

$$\dot{U}_1 = j\omega L_1\dot{I}_1 \pm j\omega M\dot{I}_2$$ $$\dot{U}_2 = j\omega L_2\dot{I}_2 \pm j\omega M\dot{I}_1$$

6.1.3 Dot Convention of Coupled Inductors

Definition: When currents in both coils simultaneously enter (or leave) the dotted terminals, the mutual flux and self-flux have the same direction
Determination Methods:
- Marking method: Mark the dotted terminals with "•" or "*"
- Right-hand screw rule: Determine based on winding direction
Purpose: To determine the sign of mutual inductance voltages

6.1.4 Equivalent Circuit Models of Coupled Inductors

Decoupling Equivalent: Replacing coupled inductors with an equivalent circuit without mutual inductance
T-Equivalent (when dotted terminals are connected): Equivalent inductances are $L_1-M$, $L_2-M$, and $M$
Application: Converts circuits with mutual inductance into circuits without mutual inductance for easier analysis

Chapter 6

6.2 Analysis of Sinusoidal Steady-State Circuits with Coupled Inductors

6.2.1 Direct Equation Method

Steps:

1. Assign reference directions for each branch current

2. Determine the sign of mutual inductance voltages based on dot convention

3. Write KVL equations (include mutual inductance voltage terms)

4. Solve simultaneously

6.2.2 Decoupling Equivalent Analysis

Steps:

1. Draw the decoupling equivalent circuit based on dot positions

2. The equivalent circuit has no mutual inductance; analyze using standard methods

3. After solving the equivalent circuit, return to the original circuit

Chapter 6

6.3 Air-Core Transformers

Air-Core Transformer Circuit Model and Reflected Impedance

Structure: Two coupled coils wound on the same non-ferromagnetic core

Primary: Voltage source $\dot{U}_s$, resistance $R_1$, inductance $L_1$
Secondary: Resistance $R_2$, inductance $L_2$, mutual inductance $M$
Reflected Impedance: The effect of the secondary on the primary can be represented by a reflected impedance $Z_r = (\omega M)^2/Z_{22}$

Chapter 6

6.4 Ideal Transformers

6.4.1 Main Characteristics of Ideal Transformers

Ideal Transformer Main Performance Parameters

Idealization Conditions: $k = 1$ (perfect coupling), $L_1, L_2 \to \infty$, lossless

$$n = \frac{N_1}{N_2} \quad \text{(turns ratio)}$$

Voltage Relationship: $u_1/u_2 = n$
Current Relationship: $i_1/i_2 = -1/n$ (negative sign indicates opposite current directions)
Impedance Transformation: $Z_{in} = n^2 Z_L$ (secondary impedance $Z_L$ reflected to primary is $n^2 Z_L$)
Power Relationship: Input power equals output power, $u_1 i_1 + u_2 i_2 = 0$

6.4.2 Circuit Analysis with Ideal Transformers

Determine turns ratio n
and dot convention

Write voltage and current
relationship equations

Impedance transformation:
reflect secondary to primary

Combine with KCL/KVL
and solve simultaneously

Analysis Methods Two Analysis Methods

Method 1: Use voltage-current relationships with KCL and KVL to write equations
Method 2: Impedance transformation method (reflect secondary impedance to primary)

Note that dot convention affects voltage and current direction relationships Observe the dot convention when reflecting impedances