Sinusoidal Steady-State AC Circuits

Introduction

Alternating-current (AC) systems are fundamental to electric power networks, transformers, motors, filters, communication circuits, and electronic instrumentation. Unlike direct current, sinusoidal voltage and current continuously change magnitude and direction. In a linear circuit driven by a sinusoidal source, the transient response eventually decays, leaving a sinusoidal steady-state response at the same frequency. Phasors and complex numbers provide an efficient method for analyzing this condition.

This chapter develops the relationships among sinusoidal waveforms, RMS values, phase, impedance, admittance, resonance, frequency response, and complex power. The methods assume linear, time-invariant circuit elements and sinusoidal excitation unless stated otherwise.

Learning objectives

By the end of this chapter, you should be able to:

1. Convert sinusoidal quantities between time-domain, RMS-phase, and phasor forms.
2. Apply impedance and admittance to series and parallel AC circuits.
3. Calculate real, reactive, and apparent power and interpret power factor.
4. Analyze series and parallel resonance, bandwidth, and quality factor.
5. Determine frequency-response quantities such as cutoff frequencies and gain.
6. Select and evaluate power-factor correction in practical installations.

1. Sinusoidal quantities and RMS values

A sinusoidal voltage may be written as

where V_m is peak voltage in volts, ω = 2πf is angular frequency in rad/s, f is frequency in hertz, and φ_v is phase angle. The period is T = 1/f. Current is represented similarly.

The RMS value is the DC value that would produce the same average heating effect in a resistor. For a sinusoid,

Here V and I denote RMS magnitudes. Unless explicitly identified as peak values, AC power-system voltages and currents are normally RMS quantities.

The average value of a sinusoid over a complete cycle is zero, but its RMS value is not zero. For a resistor, average power is

where RMS values must be used in these equations.

Phase difference

If voltage and current are

then the phase difference of voltage relative to current is φ = φ_v − φ_i. A positive phase means voltage leads current; a negative phase means voltage lags current. In an inductor, voltage leads current by 90°. In a capacitor, current leads voltage by 90°.

2. Phasors

A phasor represents the magnitude and phase of a sinusoid at a specified frequency. Using RMS phasors,

The time dependence e^jωt is suppressed. Rectangular and polar forms are

with

The symbol j satisfies j² = −1. Rectangular form is convenient for addition; polar form is convenient for multiplication and division.

For example, 10∠30° equals 10(cos30° + j sin30°) = 8.66 + j5.00. Phasor methods apply only when all quantities in the circuit have the same angular frequency.

3. Impedance and admittance

Impedance is the phasor ratio

Its unit is the ohm (Ω). For ideal elements:

The reactance is the imaginary part of impedance. Inductive reactance is positive, X_L = ωL, while capacitive reactance is negative, X_C = −1/(ωC). Real components and parasitic effects can make practical inductors and capacitors nonideal.

Admittance is the reciprocal of impedance:

Its unit is the siemens (S). G is conductance and B is susceptance. Admittances add directly in parallel circuits, just as impedances add directly in series circuits.

Basic circuit relationships

For series elements,

For parallel elements,

For two impedances in parallel,

Kirchhoff’s voltage and current laws remain valid for phasors because the circuit is linear.

Worked Example 1: Series RL circuit

A 120 V RMS, 60 Hz source supplies a series circuit with R = 8 Ω and L = 20 mH. Find current and the phase angle.

First,

Thus,

The current is

Current lags the source voltage by 43.3°. The real power is P = I²R = (10.91)²(8) ≈ 952 W.

4. AC power and power factor

For RMS voltage and current phasors, complex power is

where the asterisk denotes complex conjugation. Its unit is volt-amperes (VA). The components are

Real power P is measured in watts and represents average energy transfer. Reactive power Q is measured in var and represents energy exchanged periodically with electric or magnetic fields. Apparent power |S| is measured in VA.

Power factor is

A lagging power factor indicates inductive current; a leading power factor indicates capacitive current. For a load impedance Z = R + jX, another useful expression is

Equivalently, S = I̅I̅*Z. The magnitude notation is important: writing I²Z is valid only when I explicitly denotes a real RMS magnitude, not a complex phasor.

For a balanced three-phase load, total power is

where V_L and I_L are line RMS voltage and current. This formula assumes a balanced sinusoidal system and the appropriate line quantities.

Worked Example 2: Complex power

A load draws 15 A RMS from a 230 V RMS supply at 0.80 lagging power factor. Find P, Q, and S.

Since cosφ = 0.80, sinφ = √(1−0.80²) = 0.60. For a lagging load, Q is positive:

Therefore S = 2760 + j2070 VA and |S| = 3.45 kVA.

5. Power-factor correction

Power-factor correction reduces reactive current supplied by the source. For an inductive load corrected from pf₁ to pf₂, with unchanged real power,

where φ_k = cos⁻¹(pf_k). A shunt capacitor supplies leading reactive power of magnitude Q_c. For a single-phase system,

provided V is the RMS voltage across the capacitor. In three-phase systems the capacitance formula depends on whether capacitors are connected in wye or delta; the corresponding phase voltage must be used.

Worked Example 3: Capacitor correction

A single-phase 75 kW load operates at 0.78 lagging power factor on a 400 V, 50 Hz supply. Correct it to 0.95 lagging. Find the capacitor reactive power and capacitance, assuming one shunt capacitor across the supply.

Thus,

Using ω = 2π(50) = 314.16 rad/s,

The corrected source apparent power is 75/0.95 = 78.95 kVA, compared with approximately 75/0.78 = 96.15 kVA before correction. The capacitor must be selected with appropriate voltage, harmonic, switching, discharge, and protection ratings.

6. Resonance

Resonance occurs when inductive and capacitive reactances cancel. In a series RLC circuit,

The series resonant angular frequency is

At resonance, Z = R, source current is maximum, and voltage and current are in phase. The quality factor for a series RLC circuit is

For a lightly damped series circuit, bandwidth between the half-power frequencies is

The exact half-power frequencies satisfy |Z| = √2R, or |ωL − 1/(ωC)| = R. The simple bandwidth expression is exact for the ideal series RLC model.

For a parallel RLC circuit, resonance is most naturally found using admittance. For an ideal parallel circuit,

Resonance occurs when the susceptance is zero, giving the same ω₀ = 1/√(LC). Source current is minimum and input impedance is maximum at resonance. Practical coil resistance changes the exact resonant frequency and quality factor.

Worked Example 4: Series resonance and bandwidth

A series circuit has R = 10 Ω, L = 50 mH, and C = 10 μF. Find resonant frequency, quality factor, and bandwidth.

Thus the half-power frequencies are approximately f₁ ≈ 209.2 Hz and f₂ ≈ 241.0 Hz using the exact quadratic solution. The response is selective but not extremely narrow.

7. Frequency response

Frequency response describes how a circuit’s magnitude and phase vary as the source frequency changes. For a transfer function,

The magnitude |H(jω)| gives gain or attenuation, and the argument gives phase shift. Frequency response is essential in filters, sensor interfaces, control systems, and communication circuits.

First-order RC low-pass circuit

For a resistor in series with a capacitor, with output across the capacitor,

Therefore,

The cutoff angular frequency is ω_c = 1/(RC), and f_c = 1/(2πRC). At cutoff, magnitude is 1/√2 of its low-frequency value, equivalent to −3.01 dB, and phase is −45°.

A first-order high-pass circuit has

Its magnitude is ωRC/√(1+(ωRC)²) and its phase is 90° − tan⁻¹(ωRC).

Worked Example 5: RC frequency response

An RC low-pass circuit has R = 10 kΩ and C = 10 nF. Find cutoff frequency and output for a 2 V RMS input at 1 kHz and 10 kHz.

At 1 kHz,

Thus V_o ≈ 2(0.847) = 1.69 V RMS, with phase approximately −32.1°.

At 10 kHz, ωRC = 6.28, so |H| ≈ 0.157. The output is approximately 0.314 V RMS, and phase is approximately −80.96°. This illustrates both attenuation and increasing phase lag above cutoff.

For a second-order resonant network, the response may exhibit a peak near f₀. The bandwidth is the difference between frequencies at which power is half its maximum, or magnitude is 1/√2 of the peak magnitude when the reference is the resonant peak. Always identify the reference level used when defining bandwidth.

8. Practical case: industrial motor installation

Background

An industrial plant operates a 75 kW induction-motor group at 0.78 lagging power factor from a 400 V, 50 Hz system.

Problem

The plant operator wants to improve the power factor to 0.95 lagging to reduce source current and avoid utility penalties, without changing mechanical output.

Analysis and solution

The required capacitor rating is found from

A practical design would use a switched capacitor bank rated near 35.5 kvar, often divided into steps so that changing motor load does not cause overcorrection. The bank controller measures power factor and switches steps while observing minimum switching times.

The initial apparent power is approximately 96.2 kVA; after correction it is approximately 78.9 kVA. At fixed line voltage, line current decreases in approximately the same ratio, reducing upstream copper losses and voltage drop. Capacitors must not be installed without checking resonance with transformer leakage reactance and harmonic-producing loads such as variable-speed drives. Detuned reactors, harmonic studies, fuses, discharge resistors, and appropriate standards compliance may be required.

Learning points

Power-factor correction does not reduce the motor’s real-power demand. It supplies part of the reactive current locally. Overcorrection can produce leading power factor, excessive voltage, or resonance. Design decisions must therefore consider load variability, harmonics, switching transients, thermal conditions, and electrical safety.

9. Measurement, limitations, and professional practice

Oscilloscopes display instantaneous waveforms and phase differences; true-RMS meters measure heating-equivalent magnitude, but meter bandwidth and waveform crest factor matter. A wattmeter measures real power, while power analyzers can determine P, Q, apparent power, distortion power, harmonics, and displacement power factor.

The simple relation pf = cosφ describes displacement power factor for sinusoidal voltage and current. With significant harmonic distortion, total power factor is generally not equal to the cosine of the fundamental phase angle. Harmonic currents can increase RMS current and losses even when displacement power factor is corrected. Engineers should distinguish total power factor from displacement power factor.

Phasor analysis cannot directly represent nonsinusoidal steady state as one phasor. Fourier analysis can decompose a periodic waveform into harmonics, after which each harmonic may be analyzed separately using its own frequency-dependent impedance. Semiconductor converters, LED drivers, and data-center supplies make this distinction increasingly important.

Safety practice includes de-energization where possible, lockout/tagout, verification of absence of voltage, correctly rated instruments, discharge of capacitors, and compliance with applicable electrical codes. Capacitors can retain dangerous energy after disconnection.

Modern power analyzers and simulation tools automate phasor, harmonic, and frequency-sweep calculations, but software output must be checked against circuit topology, reference directions, RMS conventions, and component models. Artificial-intelligence tools may assist with parameter identification or anomaly detection, but engineering responsibility remains with the qualified practitioner.

Chapter Summary

Sinusoidal steady-state analysis represents voltage and current by RMS phasors, allowing differential relationships to become algebraic relationships. Resistors have real impedance, inductors positive imaginary impedance, and capacitors negative imaginary impedance. Impedances add in series, while admittances add in parallel.

Complex power is S = V̅I̅* = P + jQ. Real power performs net energy transfer, reactive power represents periodic field-energy exchange, and apparent power is the product of RMS voltage and current. Power factor is the ratio of real to apparent power; capacitors can correct lagging inductive power factor but must be selected with regard to harmonics and resonance.

Resonance occurs when inductive and capacitive effects cancel. Series resonance produces minimum impedance and maximum current; parallel resonance generally produces maximum impedance and minimum source current. Frequency response quantifies gain and phase versus frequency. Cutoff frequencies, half-power bandwidth, and quality factor are central to filter and selective-circuit design.

Key Terms

Phasor: Complex representation of a sinusoidal magnitude and phase.

RMS value: Effective value of a periodic quantity based on equivalent resistive heating.

Angular frequency: ω = 2πf, measured in rad/s.

Phase angle: Relative angular displacement between sinusoidal quantities.

Impedance: Ratio of phasor voltage to phasor current, measured in ohms.

Reactance: Imaginary component of impedance.

Admittance: Reciprocal of impedance, measured in siemens.

Susceptance: Imaginary component of admittance.

Complex power: S = P + jQ, measured in VA.

Real power: Average power converted to work, heat, or other net energy transfer.

Reactive power: Power exchanged with inductive or capacitive fields.

Apparent power: Product of RMS voltage and current.

Power factor: Ratio of real power to apparent power.

Resonance: Condition in which opposing reactances or susceptances cancel.

Quality factor: Measure of resonant selectivity, commonly Q = f₀/Δf.

Bandwidth: Frequency interval between specified response limits.

Cutoff frequency: Frequency at which a first-order response reaches 1/√2 of its passband magnitude.

Displacement power factor: Cosine of the fundamental voltage-current phase angle.

Total power factor: Real power divided by total RMS apparent power, including distortion effects.

Review Questions

Short Answer Questions

1. Why are RMS values used in AC power calculations?
2. State the impedances of an ideal resistor, inductor, and capacitor.
3. What is the physical meaning of reactive power?
4. Distinguish series resonance from parallel resonance.
5. Define the −3 dB cutoff point for a first-order filter.

Long Answer Questions

1. Derive the impedance and phase angle of a series RL circuit and explain the physical meaning of the result.
2. Explain complex power using a power triangle and derive the relationships among P, Q, and |S|.
3. Derive the resonant frequency and bandwidth of an ideal series RLC circuit.
4. Compare impedance-based analysis of series circuits with admittance-based analysis of parallel circuits.
5. Explain why correcting displacement power factor may not eliminate problems caused by harmonic currents.

Analytical / Application Questions

1. A 230 V, 50 Hz supply feeds a series circuit with R = 12 Ω and L = 30 mH. Calculate impedance, current, power factor, and real power.
2. A 10 kW inductive load has 0.70 lagging power factor and must be corrected to 0.90 lagging. Calculate the capacitor reactive-power rating required.
3. Design an RC low-pass filter with a cutoff frequency of 2 kHz using a 7.5 kΩ resistor. Calculate the required capacitance and determine the magnitude of gain at 6 kHz.

Multiple Choice Questions with Answers

Question 1

A 120 V RMS sinusoidal source has what peak voltage?

A. 84.9 V

B. 120 V

C. 169.7 V

D. 240 V

Correct Answer: C. 169.7 V

Question 2

The impedance of an ideal capacitor is

A. R

B. jωL

C. jωC

D. 1/(jωC)

Correct Answer: D. 1/(jωC)

Question 3

A load draws 230 V RMS and 15 A RMS at 0.80 lagging power factor. What is its real power?

A. 1.84 kW

B. 2.76 kW

C. 3.45 kW

D. 4.31 kW

Correct Answer: B. 2.76 kW

Question 4

For a load impedance Z and RMS current magnitude |I|, complex power is correctly written as

A. S = I²Z for every complex phasor I

B. S = |I|²Z

C. S = Z/I

D. S = I/Z

Correct Answer: B. S = |I|²Z

Question 5

At series RLC resonance, the circuit impedance is

A. purely resistive and minimum for the ideal model

B. purely capacitive and maximum

C. purely inductive and minimum

D. zero for every nonzero resistance

Correct Answer: A. purely resistive and minimum for the ideal model

Question 6

For a first-order RC low-pass circuit, the cutoff angular frequency is

A. RC

B. R/C

C. 1/(RC)

D. 2πRC

Correct Answer: C. 1/(RC)

Question 7

A 75 kW load is corrected from 0.78 to 0.95 power factor. Approximately how much capacitor reactive power is required?

A. 7.1 kvar

B. 22.0 kvar

C. 35.5 kvar

D. 43.8 kvar

Correct Answer: C. 35.5 kvar

Question 8

In a balanced three-phase system, total real power is

A. V_L I_L sinφ

B. 3V_L I_L cosφ

C. √3 V_L I_L cosφ

D. V_L I_L/√3

Correct Answer: C. √3 V_L I_L cosφ

Question 9

For a series RLC circuit, increasing resistance while L and C remain fixed generally

A. increases quality factor and narrows bandwidth

B. decreases quality factor and widens bandwidth

C. changes the ideal resonant frequency to zero

D. has no effect on bandwidth

Correct Answer: B. decreases quality factor and widens bandwidth

Question 10

A lagging power factor most directly indicates that the load is predominantly

A. capacitive

B. inductive

C. purely resistive

D. open-circuit

Correct Answer: B. inductive