Sinusoidal Steady-State and Phasor Analysis

Introduction

Alternating-current systems dominate electric-power generation, transmission, machines, communication circuits, instrumentation and analog electronics. After switching transients have decayed, a linear circuit driven by sinusoidal sources operates in sinusoidal steady state: every voltage and current is sinusoidal at the source frequency, although amplitudes and phase angles may differ.

Direct time-domain analysis requires repeated differentiation and integration. Phasor analysis replaces these operations with algebra involving complex numbers. Resistors, inductors and capacitors can then be represented by impedances or admittances, allowing DC analysis methods—including nodal analysis, mesh analysis, superposition, Thévenin and Norton equivalents—to be applied to AC circuits.

Unless otherwise stated, this chapter uses cosine-reference phasors and RMS magnitudes.

Learning Objectives

After studying this chapter, students should be able to:

1. Represent sinusoidal voltages and currents using complex numbers and phasors.
2. Convert between rectangular and polar complex-number forms.
3. Derive and calculate resistor, inductor and capacitor impedances and admittances.
4. calculate RMS values, complex power, power factor and reactive power.
5. Apply nodal and mesh analysis to sinusoidal steady-state circuits.
6. Apply superposition, source transformation, Thévenin and Norton theorems in AC networks.
7. Analyze practical AC filters and loads while recognizing the limitations of phasor methods.

1. Sinusoids and Complex Representation

A general sinusoidal voltage is

where:

• V_m is the peak amplitude in volts,
• ω is angular frequency in radians per second,
• t is time in seconds,
• φ is the phase angle in radians or degrees.

Angular frequency and ordinary frequency are related by

where f is measured in hertz. For example, a 50 Hz supply has ω = 2π(50) = 314.16 rad/s.

1.1 Phase relationships

For two sinusoids of the same frequency,

their relative phase is φ₁ − φ₂. If this quantity is positive, v₁ leads v₂. Sinusoids of different frequencies do not have a constant phase difference and cannot be combined using a single phasor system.

A sine can be converted to the cosine reference using

Thus, 10 sin(ωt + 20°) = 10 cos(ωt − 70°).

1.2 Euler's identity and phasors

Euler's identity is

where j = √(−1). Therefore,

The common time factor e^jωt can be suppressed, leaving the phasor. Using RMS magnitude,

For v(t)=V_m cos(ωt+φ), the corresponding RMS phasor is

A phasor is not itself a time-domain signal. Frequency must be known separately to reconstruct the waveform.

1.3 Rectangular and polar forms

A complex quantity can be written as

with

and

Rectangular form is convenient for addition and subtraction. Polar form is convenient for multiplication, division, powers and roots.

Figure: A phasor resolved into real and imaginary components.

2. From Differential Equations to Impedance

For steady-state sinusoids, differentiation corresponds to multiplication by jω:

Likewise, integration corresponds to division by jω, provided only the sinusoidal steady-state response is considered. Initial-condition transients are excluded.

2.1 Resistor

For a resistor, v(t)=Ri(t). Therefore,

Voltage and current are in phase. Resistance R and impedance Z_R are measured in ohms.

2.2 Inductor

For an ideal inductor,

Hence,

The inductive reactance is X_L=ωL. Current lags voltage by 90°. Inductance L is measured in henrys.

2.3 Capacitor

For an ideal capacitor,

Thus,

The magnitude of capacitive reactance is X_C=1/(ωC). Current leads voltage by 90°. Capacitance C is measured in farads.

These ideal models may become inaccurate at high frequency because physical components include parasitic resistance, inductance and capacitance.

3. Impedance, Admittance and Frequency Response

The general impedance is

where R is resistance and X is reactance. Positive reactance is inductive; negative reactance is capacitive.

Admittance is

where conductance G and susceptance B are measured in siemens. Positive susceptance is capacitive and negative susceptance is inductive.

Series impedances add:

Parallel admittances add:

For two parallel impedances, provided their sum is nonzero,

3.1 Series resonance

For an ideal series RLC circuit,

Resonance occurs when the net reactance is zero:

At resonance, Z=R, current is maximized for a fixed source voltage, and the input power factor is unity. These conclusions assume ideal L and C and a nonzero series resistance.

4. RMS Values and AC Power

For a periodic waveform x(t) with period T, its RMS value is

For a sinusoid,

RMS voltage or current equals the DC value that would produce the same average heating in a resistor.

4.1 Complex power

Using RMS phasors, complex power absorbed by a load is

where:

• S is complex power in volt-amperes,
• P is average or real power in watts,
• Q is reactive power in vars,
• I^* is the complex conjugate of current.

Under the passive sign convention, current enters the terminal marked positive for voltage. If θ=∠V−∠I, then

The power factor is

A positive Q indicates net inductive absorption and lagging power factor. A negative Q indicates net capacitive absorption and leading power factor.

For Z=R+jX,

For Y=G+jB,

Consequently,

The minus sign in the susceptance expression is essential.

Worked Example 1: Power of a series load

A 120 V RMS source supplies a load with Z=8+j9.42 Ω. Determine current, complex power and power factor.

The impedance in polar form is approximately

Therefore,

Complex power is

Thus,

The positive reactive power identifies an inductive, lagging load.

Worked Example 2: Parallel resistor–inductor load

A 100 V RMS, 50 Hz source supplies a 20 Ω resistor in parallel with a 7.65 H ideal inductor. Determine input admittance, current and power.

First,

The admittance is

Hence,

The power quantities are

The power factor is approximately 1.00 lagging. It is close to unity because the inductor's current is small relative to the resistor current.

5. Systematic AC Circuit Analysis

A reliable workflow is:

1. Confirm that the circuit is linear and operating in sinusoidal steady state.
2. Choose one cosine reference and one frequency.
3. Convert source magnitudes consistently to RMS phasors.
4. Replace R, L and C by their frequency-dependent impedances or admittances.
5. Apply Kirchhoff's laws or a network theorem using complex algebra.
6. Convert requested results to polar or rectangular form.
7. Check units, signs, phase relationships, KCL/KVL and power balance.

6. Nodal Analysis in the Phasor Domain

Nodal analysis applies KCL using admittances. For a branch of admittance Y_km between nodes k and m, the current from node k is

The matrix equation is

Each diagonal matrix entry is the sum of admittances incident on that node. Each off-diagonal entry is the negative of the admittance directly connecting the two nodes.

Worked Example 3: Two-node AC network

At a specified frequency, a 1∠0° A RMS source injects current into node 1. A 10 Ω resistor connects node 1 to ground. A capacitor with impedance −j10 Ω connects nodes 1 and 2. Node 2 is connected to ground through a 20 Ω resistor and an inductor with impedance j20 Ω. Find V₁ and V₂.

The relevant admittances are

The nodal equations are

In matrix form,

Solving gives

A branch-by-branch KCL check verifies the result. The fact that |V₂|>|V₁| does not violate energy conservation; reactive networks can produce voltage magnification.

Voltage sources between two nonreference nodes require a supernode. The supernode KCL equation must be supplemented by the source-voltage constraint.

7. Mesh Analysis in the Phasor Domain

Mesh analysis applies KVL to planar circuits using impedances. For a shared impedance Z_s, its voltage in mesh 1 is Z_s(I₁−I₂). In matrix form, diagonal entries are sums of impedances around each mesh, and off-diagonal entries are negative shared impedances.

Worked Example 4: Coupled mesh currents

A two-mesh circuit has clockwise mesh currents I₁ and I₂. The left mesh contains a 10∠0° V source, a 2 Ω exclusive resistor and a shared 2 Ω resistor. The right mesh contains the shared resistor and an exclusive inductor having impedance j2 Ω. Determine both currents.

KVL gives

From the second equation, I₁=(1+j)I₂. Substitution into the first gives

Therefore,

Substitution into both KVL equations confirms the solution. A current source shared by two meshes is handled using a supermesh plus the current-source constraint.

8. AC Network Theorems

The familiar linear network theorems remain valid after the circuit is transformed into the phasor domain.

8.1 Superposition

In a linear circuit, the response to multiple independent sources equals the phasor sum of the responses to each source. When suppressing other independent sources, ideal voltage sources are replaced by short circuits and ideal current sources by open circuits. Dependent sources remain active.

Sources at different frequencies must be analyzed separately. Their time-domain responses may then be added, but phasors at different frequencies must not be directly added.

8.2 Source transformation

A voltage source V_s in series with impedance Z_s is externally equivalent to a current source

in parallel with the same impedance, provided the source and impedance form a valid two-terminal source network.

8.3 Thévenin and Norton equivalents

Any linear two-terminal AC network at one frequency can be represented by:

• a Thévenin voltage V_Th in series with Z_Th, or
• a Norton current I_N in parallel with Z_Th.

Their relationship is

The open-circuit terminal voltage gives V_Th. The equivalent impedance may be found by suppressing independent sources and looking into the terminals. If dependent sources are present, retain them and use a test source or the ratio Z_Th=V_oc/I_sc when defined.

Worked Example 5: Load connected to an AC equivalent

A network has V_Th=20∠0° V and Z_Th=4+j3 Ω. A load Z_L=6−j2 Ω is connected. Then

The load voltage is

8.4 Maximum average power transfer

If Z_Th=R_Th+jX_Th and both load resistance and reactance may be selected, maximum average load power occurs for

For R_Th>0, the maximum power is

This uses RMS V_Th. Conjugate matching is not automatically the optimum engineering choice in power systems because it gives only 50% efficiency when source resistance represents real loss. Constraints on load magnitude or reactance require a different optimization.

9. Applied Case: Passive Sensor Filter

Illustrative case. An engineering team must attenuate high-frequency noise before a measurement device. A source drives a 1 kΩ series resistor. The output is taken across a parallel combination of a 10 kΩ input resistance and a 100 nF capacitor. Determine the no-additional-load transfer ratio at 1 kHz.

Background and problem

The instrument input resistance cannot be treated as infinite because it is part of the filter. The output impedance is therefore the parallel RC impedance rather than the capacitor impedance alone.

Analysis

At 1 kHz,

The parallel impedance is

Using the voltage divider,

An equivalent compact calculation is

Outcome

At 1 kHz, the output amplitude is approximately 78.9% of the source amplitude and lags by 29.7°. Ignoring the 10 kΩ input resistance would predict a different gain and phase.

Learning points

• Loading must be included in the equivalent impedance.
• Filter gain and phase are frequency dependent.
• Component tolerances and source impedance affect measured performance.
• A simulation or AI-generated result should be checked against limiting cases and an independent hand calculation.

10. Limitations, Professional Practice and Contemporary Tools

Phasor analysis is powerful but subject to clear limitations:

• It describes sinusoidal steady state, not startup or switching transients.
• A single phasor circuit applies to only one frequency.
• Nonlinear devices can generate harmonics, requiring separate frequency analysis or time-domain methods.
• Ideal component models may omit losses, saturation, skin effect and parasitics.
• RMS meters may report incorrectly for nonsinusoidal signals unless they are true-RMS instruments with sufficient bandwidth.

Professional calculations should state whether phasors use RMS or peak magnitude and whether angles use degrees or radians. Engineers should preserve passive sign conventions, include equipment voltage and current ratings, and assess power quality, heating and insulation requirements under applicable standards. Incorrect RMS interpretation can create safety hazards through underestimated conductor heating or component stress.

Modern circuit simulators solve large AC networks efficiently and can sweep frequency to obtain Bode plots. Measurement systems can estimate phasors through Fourier methods, while power-system phasor measurement units provide synchronized voltage and current data. AI-assisted design can identify candidate topologies or detect anomalies, but numerical outputs require validation through KCL, KVL, units, limiting behavior and power balance. Accountability for engineering decisions remains with qualified practitioners.

Chapter Summary

Sinusoidal steady-state analysis replaces time-domain differentiation and integration with complex algebra. RMS phasors encode sinusoidal magnitude and phase, while frequency-dependent impedances represent resistors, inductors and capacitors. Impedance is especially convenient for series networks; admittance is convenient for parallel networks and nodal analysis.

Complex power unifies real and reactive power through S=VI^*=P+jQ. Positive reactive power denotes an inductive load, while negative reactive power denotes a capacitive load. For Y=G+jB, the correct power relations are P=|V|²G and Q=−|V|²B.

Nodal and mesh methods retain their DC structure when conductances and resistances are replaced by complex admittances and impedances. Superposition, source transformation, Thévenin and Norton equivalents also remain valid for linear networks at a fixed frequency. Successful professional analysis requires consistent RMS conventions, correct frequency, accurate loading models and independent checks of calculated results.

Key Terms

Admittance: Reciprocal of impedance, measured in siemens.

Angular frequency: Sinusoidal frequency ω=2πf in radians per second.

Apparent power: Magnitude |S| of complex power, measured in volt-amperes.

Complex power: Quantity S=P+jQ describing average and reactive power.

Conductance: Real part of admittance.

Conjugate matching: Selection Z_L=Z_Th^* for maximum average power transfer under stated conditions.

Impedance: Complex ratio of voltage phasor to current phasor.

Mesh analysis: KVL-based method using independent mesh currents.

Nodal analysis: KCL-based method using node voltages.

Passive sign convention: Convention in which current enters the positive-voltage terminal of an absorbing element.

Phasor: Complex representation of a sinusoid's magnitude and phase at a known frequency.

Power factor: Ratio of average power to apparent power.

Reactance: Imaginary part of impedance.

Reactive power: Reversible energy-exchange rate represented by Q, measured in vars.

Resonance: Condition in which inductive and capacitive effects cancel at the input.

RMS value: Effective value producing the same resistor heating as an equivalent DC quantity.

Sinusoidal steady state: Long-term sinusoidal response after transients have decayed.

Susceptance: Imaginary part of admittance.

Thévenin equivalent: Voltage source and series impedance equivalent of a linear two-terminal network.

Review Questions

Short Answer Questions

1. What information is retained in a phasor, and what information is suppressed?
2. Why must sinusoids be expressed at the same frequency before their phasors are added?
3. State the impedances of an ideal resistor, inductor and capacitor.
4. Distinguish leading and lagging power factor.
5. Why does the formula for reactive power contain a minus sign when expressed using susceptance?

Long Answer Questions

1. Derive the impedance of an ideal inductor from its time-domain voltage–current relationship.
2. Explain the physical meanings of real, reactive, apparent and complex power.
3. Develop the nodal-admittance matrix rules for a general linear AC network.
4. Explain how Thévenin and Norton theorems are applied when dependent sources are present.
5. Discuss the assumptions and practical limitations of sinusoidal steady-state phasor analysis.

Analytical / Application Questions

1. A 230 V RMS, 50 Hz source supplies a series combination of a 30 Ω resistor and a 120 mH inductor. Calculate current, complex power and power factor.
2. Form and solve nodal equations for a two-node AC circuit containing at least one resistor, capacitor and inductor. Verify KCL numerically at each node.
3. Design a conjugately matched load for a network having V_Th=40∠15° V and Z_Th=8+j6 Ω. Calculate maximum average load power and comment on efficiency.

Multiple Choice Questions with Answers

Question 1

The RMS phasor corresponding to v(t)=10cos(100t+30°) V is:

A. 10∠30° V

B. 7.07∠30° V

C. 7.07∠−30° V

D. 14.14∠30° V

Correct Answer: B. 7.07∠30° V

Question 2

An ideal 20 mH inductor operates at ω=500 rad/s. Its impedance is:

A. j10 Ω

B. −j10 Ω

C. j0.04 Ω

D. 10 Ω

Correct Answer: A. j10 Ω

Question 3

The impedance corresponding to Y=0.02−j0.04 S is:

A. 20−j10 Ω

B. 10−j20 Ω

C. 10+j20 Ω

D. 20+j10 Ω

Correct Answer: C. 10+j20 Ω

Question 4

A load has V=120∠0° V and I=5∠−30° A using RMS phasors and passive sign convention. Its complex power is:

A. 519.6+j300 VA

B. 519.6−j300 VA

C. 600+j0 VA

D. 300+j519.6 VA

Correct Answer: A. 519.6+j300 VA

Question 5

Under the passive sign convention, an ideal capacitor has:

A. Positive real power and zero reactive power

B. Positive reactive power

C. Negative reactive power

D. Negative real power

Correct Answer: C. Negative reactive power

Question 6

In a nodal-admittance matrix, the off-diagonal entry for two nodes connected by admittance Y is:

A. Y

B. −Y

C. 1/Y

D. −1/Y

Correct Answer: B. −Y

Question 7

If Z_Th=5+j4 Ω and both load resistance and reactance are adjustable, the load for maximum average power is:

A. 5+j4 Ω

B. −5+j4 Ω

C. 4−j5 Ω

D. 5−j4 Ω

Correct Answer: D. 5−j4 Ω

Question 8

A 100 V RMS, 50 Hz source supplies a 20 Ω resistor in parallel with a 7.65 H ideal inductor. The total reactive power is approximately:

A. −4.16 var

B. 4.16 var

C. 500 var

D. 0.416 var

Correct Answer: B. 4.16 var

Question 9

For the illustrated passive-filter case with a 1 kΩ series resistor and a parallel 10 kΩ–100 nF output network at 1 kHz, V_o/V_s is approximately:

A. 0.789∠−29.7°

B. 0.789∠29.7°

C. 0.500∠−90°

D. 0.909∠0°

Correct Answer: A. 0.789∠−29.7°

Question 10

For a sinusoid with peak current 14.14 A, the RMS current is:

A. 5.00 A

B. 7.07 A

C. 10.00 A

D. 20.00 A

Correct Answer: C. 10.00 A