AC Circuits and Three-Phase Systems

Introduction

Alternating-current (AC) systems are the foundation of modern electric power generation, transmission, distribution, and the operation of most industrial machines. Unlike direct current, whose magnitude and direction are constant, AC voltage and current vary periodically with time. Sinusoidal AC is especially important because generators naturally produce approximately sinusoidal waveforms and because sinusoidal excitation permits the use of powerful frequency-domain methods.

This chapter develops the analysis of sinusoidal circuits using phasors, complex impedance, admittance, resonance, and AC power concepts. It then extends these ideas to balanced three-phase systems, including star (Y) and delta (Δ) connections. The analysis assumes steady-state sinusoidal operation unless stated otherwise.

Learning Objectives

After studying this chapter, the reader should be able to:

• describe sinusoidal voltage and current using amplitude, frequency, angular frequency, phase, and rms value;
• convert sinusoidal quantities into phasor form and apply Kirchhoff’s laws in the phasor domain;
• calculate impedance, admittance, current, voltage, and phase angle in series and parallel AC circuits;
• explain series and parallel resonance and determine resonant frequency and circuit response;
• calculate real, reactive, apparent, and complex power and determine power factor;
• design a basic power-factor correction arrangement;
• analyze balanced star- and delta-connected three-phase loads;
• identify practical limitations, safety requirements, and professional applications of AC analysis.

1. Sinusoidal Quantities

A sinusoidal voltage may be written as

where V_m is the peak voltage in volts, ω is angular frequency in radians per second, and φ_v is the phase angle. Angular frequency is related to frequency by

where f is frequency in hertz. The period is T = 1/f. For a sinusoidal waveform, the rms value is

and similarly I = I_m/√2 for current. RMS quantities are used in power calculations because they produce the equivalent heating effect of DC quantities in a resistor.

A current such as i(t) = I_m sin(ωt + φ_i) leads or lags the voltage by the phase difference φ = φ_v − φ_i. A positive phase difference means that current lags voltage when voltage is used as the reference; the precise verbal description should always be checked from the selected reference convention.

Phasors

A phasor is a complex number representing the magnitude and phase of a sinusoidal quantity at a specified frequency. Using rms values, a voltage with magnitude V and phase φ is written

where j = √(−1). Phasors are applicable only to sinusoidal steady-state quantities of the same frequency. They do not directly represent transients or nonsinusoidal waveforms.

For example, if v(t) = 141.4 sin(ωt + 30°) V, its rms phasor is V̲ = 100∠30° V. A common error is to use the peak value in one calculation and the rms value in another. The convention must remain consistent.

2. Impedance and Admittance

In the phasor domain, the relationship between voltage and current is

where Z is impedance in ohms. For a resistor, inductor, and capacitor:

Here R is resistance in ohms, L is inductance in henries, and C is capacitance in farads. The inductive reactance is X_L = ωL, and capacitive reactance is X_C = 1/(ωC). Thus a general series impedance may be written

where X = X_L − X_C. The magnitude and phase are

A positive X indicates inductive behavior; a negative X indicates capacitive behavior.

Admittance is the reciprocal of impedance:

where Y is in siemens, G is conductance, and B is susceptance. Admittances add directly in parallel circuits, making them particularly useful for parallel-network analysis.

Kirchhoff’s Laws in AC Circuits

Kirchhoff’s voltage and current laws remain valid for phasors. In a series circuit, the same phasor current flows through each element and impedances add:

In parallel, admittances add:

The current is then I̲ = V̲/Z_total for a series-equivalent impedance or I̲ = V̲Y_total when admittance is used.

Worked Example 1: Series R–L Circuit

A 230 V rms, 50 Hz supply feeds a series circuit containing R = 20 Ω and L = 0.10 H. Determine impedance, current, and power factor.

First,

Therefore,

The current is

The impedance angle is

Thus current lags supply voltage by 57.5°, and the power factor is

The result is physically consistent: an inductive circuit has lagging current and a positive reactive component.

3. Resonance

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

At resonance, the imaginary part is zero:

Consequently,

and

where f₀ is resonant frequency in hertz. The formula assumes ideal lumped components and a series RLC topology. Resistance does not change the ideal resonant frequency, although it determines current magnitude and selectivity.

At series resonance, Z = R, current is maximum, and voltage and current are in phase. The inductor and capacitor may each have large voltages even though their phasor sum cancels. This is called voltage magnification and can damage components.

For a parallel RLC circuit, resonance is most conveniently identified by zero net susceptance. In an ideal parallel LC circuit,

At parallel resonance, source current is ideally minimum while circulating branch currents may be substantial. Real losses shift the exact response and limit the quality factor.

The quality factor of a series RLC circuit is

for the ideal lumped model. Larger Q indicates a narrower bandwidth and greater selectivity.

4. AC Power

For rms voltage and current phasors, complex power is

where the asterisk denotes complex conjugate. The components are:

Here P is real or active power in watts, Q is reactive power in var, and |S̲| is apparent power in volt-amperes. For an inductive load, Q > 0; for a capacitive load, Q < 0. Ideal resistors consume real power but no reactive power. Ideal inductors and capacitors exchange reactive energy but consume no average real power.

Power factor is

A lagging power factor usually indicates inductive equipment such as motors, transformers, and reactors. A leading power factor usually indicates capacitive behavior.

Power Triangle

The power triangle relates active, reactive, and apparent power through S² = P² + Q². Improving power factor reduces current for a specified real power and supply voltage. It does not, by itself, reduce the real power required by a mechanical load.

Worked Example 2: Power and Correction

A single-phase motor consumes 10 kW at 0.75 lagging power factor from a 400 V supply. Determine current, reactive power, and the capacitor rating required to improve the power factor to 0.95 lagging.

The initial phase angle is φ₁ = cos⁻¹(0.75), and the final angle is φ₂ = cos⁻¹(0.95). Reactive powers are

Correcting the arithmetic, tan(cos⁻¹0.95) ≈ 0.3287, so

The capacitor must supply leading reactive power

The initial apparent power is S₁ = P/0.75 = 13.33 kVA, so the supply current is

After correction, S₂ = 10,000/0.95 = 10.53 kVA and I₂ = 10,526/400 = 26.3 A. The correction reduces line current and associated conductor losses, but the capacitor must be selected with suitable voltage, harmonic, switching, and discharge protection.

For a single-phase capacitor connected across a supply, its reactive power is

where V is rms voltage. The formula assumes an ideal capacitor and sinusoidal voltage.

5. Three-Phase Systems

A balanced three-phase source consists of three sinusoidal voltages of equal magnitude and frequency, displaced by 120° electrical. A positive phase sequence may be represented by

The phase sequence is important in rotating machines: interchanging any two lines reverses the direction of rotation.

A three-phase system is balanced when the three source voltages are equal in magnitude and symmetrically displaced, and the load impedances are equal in corresponding phases. Under balance, the neutral current of a star-connected load is zero, so the neutral conductor may be omitted in an ideal system.

Star Connection

In a balanced star load, each impedance is connected between a line and the neutral point. The relationships are

where V_L and I_L are line voltage and line current, while V_Ph and I_Ph are phase voltage and phase current. Line voltage leads the corresponding phase voltage by 30° in the usual positive-sequence convention; phase-angle reference must be maintained consistently.

Delta Connection

In a delta load, each impedance is connected between two lines. For a balanced delta load,

The line current is the phasor difference of two phase currents, which produces the factor √3 and a 30° phase displacement.

Three-Phase Power

For any balanced three-phase load,

These expressions use rms line-to-line voltage and line current. They apply to balanced sinusoidal systems. For unbalanced systems, individual phase powers should be calculated and added.

6. Worked Example 3: Balanced Star Load

A balanced star-connected load has per-phase impedance Z_Ph = 12 + j9 Ω and is supplied by a 415 V line-to-line, three-phase system. Determine phase voltage, line current, power factor, and total active power.

The phase voltage is

The impedance magnitude is

Therefore,

The impedance angle is φ = tan⁻¹(9/12) = 36.87°, so the power factor is 0.80 lagging. Total active power is

Alternatively, each phase consumes I_Ph²R = (15.97)²(12) ≈ 3,060 W, giving approximately 9.18 kW for three phases. The agreement verifies the result.

7. Worked Example 4: Balanced Delta Load

A balanced delta load with per-phase impedance Z_Ph = 30∠30° Ω is connected to a 400 V three-phase supply. Determine phase current, line current, and total active power.

For delta,

Thus,

The line current magnitude is

The impedance angle is 30°, so power factor is cos30° = 0.866 lagging. Hence,

The same result follows from three phase branches: P = 3V_PhI_Phcosφ = 3(400)(13.33)(0.866) ≈ 13.86 kW.

8. Practical Application: Industrial Motor Feeder

Background

Consider an illustrative factory feeder supplying several induction motors from a 415 V, three-phase, 50 Hz distribution board. The motors perform pumping and conveyor duties. Their combined real power is 120 kW at an average power factor of 0.78 lagging.

Problem

The feeder is approaching its thermal current limit, and the utility applies a power-factor penalty below 0.90. The engineering team must determine whether correction is feasible without creating excessive capacitive current or resonance with the supply network.

Analysis

The initial apparent power is

The line current is

For a target power factor of 0.95,

Using tan(cos⁻¹0.78) ≈ 0.802 and tan(cos⁻¹0.95) ≈ 0.329,

The corrected apparent power is S₂ = 120/0.95 = 126.3 kVA, and the new line current is approximately

Approach and Outcome

A switched three-phase capacitor bank rated near 57 kvar could reduce current substantially. In practice, the bank would be divided into steps so that changing motor load does not produce overcorrection. Detuned reactors may be required where variable-speed drives or other nonlinear loads generate harmonics. The final design must verify short-circuit level, capacitor withstand voltage, switching transients, harmonic resonance, protection coordination, and discharge time.

Learning Points

Power-factor correction is a system-design decision rather than simply a mathematical exercise. Capacitors should not be installed blindly at the terminals of motors that are frequently disconnected, nor should correction be used to conceal undersized conductors or overloaded equipment. Measurements should be made over representative operating cycles.

9. Limitations, Safety, and Professional Practice

The phasor method assumes linear components, sinusoidal steady state, fixed frequency, and a single frequency per calculation. Power electronic converters, magnetic saturation, arc devices, and switching supplies may produce significant harmonics. In such cases, each harmonic frequency can be analyzed separately, but total rms current and losses require appropriate harmonic summation rather than simple fundamental-frequency formulas.

Three-phase formulas also require attention to the distinction between line-to-line and line-to-neutral voltage. Misidentifying these quantities causes errors by a factor of √3. Unbalanced loads require phase-by-phase analysis, and a neutral conductor may carry current even when the fundamental components appear balanced because of triplen harmonics.

Electrical work must follow applicable standards, local wiring regulations, equipment ratings, and lockout/tagout procedures. Capacitors can retain dangerous charge after disconnection. Safe design includes overcurrent protection, earthing, isolation, touch protection, thermal assessment, fault-current analysis, and appropriate personal protective equipment. Engineers must not infer that a low current calculated under ideal assumptions guarantees safe operation.

Contemporary installations increasingly use smart meters, digital protective relays, variable-speed drives, and automated capacitor controllers. These devices improve monitoring and efficiency but introduce measurement, cybersecurity, harmonics, and interoperability considerations. Artificial intelligence can assist in load forecasting and anomaly detection, but protection decisions require validated models, dependable sensors, human oversight, and compliance with engineering standards.

Best-Practice Checklist

1. State whether voltage and current values are peak or rms.
2. Define the phase reference and use a consistent phase sequence.
3. Convert inductive and capacitive elements to impedance before applying series methods.
4. Use admittance for parallel networks when it simplifies calculation.
5. Check units, signs, and limiting cases.
6. Distinguish line quantities from phase quantities in three-phase circuits.
7. Verify calculated power using both complex-power and phase-by-phase methods where possible.
8. Assess harmonics, resonance, transients, and thermal effects in practical installations.
9. Select correction equipment for the actual load profile, not merely its nominal rating.
10. Apply electrical safety procedures and relevant standards before testing or modifying equipment.

Chapter Summary

Sinusoidal steady-state AC circuits can be analyzed efficiently with phasors. A sinusoid is characterized by magnitude, frequency, and phase, while rms values provide the basis for practical power calculations. Impedance combines resistance and reactance; admittance is especially convenient for parallel circuits. Inductors contribute positive reactance and capacitors negative reactance.

Series resonance occurs when inductive and capacitive reactances cancel, producing minimum impedance and maximum current in an ideal series circuit. AC power is represented by complex power: real power P measures net energy conversion, reactive power Q represents energy exchange, and apparent power S determines equipment current capability. Power-factor correction reduces current and system losses but must be designed with regard to harmonics and resonance.

Balanced three-phase systems use three equal sinusoidal quantities separated by 120°. For star loads, line voltage is √3 times phase voltage and line current equals phase current. For delta loads, line voltage equals phase voltage and line current is √3 times phase current. Balanced three-phase active power is P = √3V_LI_Lcosφ. These principles support the analysis and design of motors, transformers, feeders, industrial power systems, and modern energy-conversion equipment.

Key Terms

Phasor: A complex representation of a sinusoidal quantity containing magnitude and phase.

RMS value: The effective value of an AC quantity used in heating and power calculations.

Angular frequency: Frequency expressed in radians per second, ω = 2πf.

Impedance: Opposition to sinusoidal current, measured in ohms.

Reactance: The imaginary component of impedance caused by inductance or capacitance.

Admittance: Reciprocal of impedance, measured in siemens.

Conductance: Real part of admittance.

Susceptance: Imaginary part of admittance.

Resonance: Condition in which net inductive and capacitive effects cancel.

Quality factor: Measure of resonant selectivity or energy circulation relative to loss.

Real power: Average power converted into work, heat, or other energy forms.

Reactive power: Power exchanged periodically by inductors and capacitors.

Apparent power: Product of rms voltage and current, measured in VA.

Power factor: Ratio of real power to apparent power.

Phase sequence: Order in which three-phase voltages reach their corresponding peaks.

Star connection: Three-phase connection in which one end of each phase is joined at a neutral point.

Delta connection: Three-phase connection in which the three phase impedances form a closed loop.

Line voltage: Voltage measured between two line conductors.

Phase voltage: Voltage across one phase element.

Balanced load: A three-phase load with equal corresponding impedances.

Review Questions

Short Answer Questions

1. What information is represented by the magnitude and angle of a phasor?
2. State the impedance of an ideal inductor and an ideal capacitor.
3. Define real power, reactive power, and apparent power.
4. What condition produces resonance in a series RLC circuit?
5. State the line-to-phase voltage and current relationships for a balanced star connection.

Long Answer Questions

1. Explain the phasor method for analyzing a series R–L–C circuit, including the construction of impedance and determination of current phase angle.
2. Derive the resonant frequency of an ideal series RLC circuit and discuss the effects of resistance and quality factor.
3. Explain the power triangle and show how power-factor correction changes current without changing the real power of an ideal load.
4. Compare balanced star and delta connections, including voltage, current, and power relationships.
5. Discuss the assumptions and limitations of phasor analysis in systems containing harmonics, nonlinear loads, and transient events.

Analytical / Application Questions

1. A 230 V, 50 Hz supply feeds a series circuit with R = 15 Ω, L = 80 mH, and C = 100 μF. Determine the impedance, current, phase angle, and whether the circuit is net inductive or capacitive.
2. A balanced 400 V three-phase star load has per-phase impedance 8 + j6 Ω. Determine line current, power factor, and total active power.
3. An industrial installation consumes 75 kW at 0.72 lagging power factor. Determine the reactive-power rating of capacitors needed to achieve 0.92 lagging power factor, and explain two practical checks required before installation.

Multiple Choice Questions with Answers

Question 1

A 50 Hz sinusoidal voltage has an rms value of 100 V. What is its peak value?

A. 70.7 V

B. 100 V

C. 141.4 V

D. 200 V

Correct Answer: C. 141.4 V

Question 2

The impedance of an ideal capacitor is

A. R

B. jωL

C. 1/(jωC)

D. jωC

Correct Answer: C. 1/(jωC)

Question 3

A series circuit has impedance 10 + j10 Ω. Its power factor is approximately

A. 0.50 leading

B. 0.707 lagging

C. 0.707 leading

D. 1.00 unity

Correct Answer: B. 0.707 lagging

Question 4

At ideal series resonance, the circuit impedance is

A. Maximum and purely inductive

B. Minimum and equal to R

C. Zero for every value of R

D. Purely capacitive

Correct Answer: B. Minimum and equal to R

Question 5

A load consumes 8 kW at 0.8 power factor. Its apparent power is

A. 6.4 kVA

B. 8.0 kVA

C. 10.0 kVA

D. 12.8 kVA

Correct Answer: C. 10.0 kVA

Question 6

Reactive power supplied by a capacitor used to correct an inductive load is conventionally

A. Positive

B. Negative

C. Always zero

D. Equal to real power

Correct Answer: B. Negative

Question 7

In a balanced star-connected load, the line current is

A. √3 times the phase current

B. One-third of the phase current

C. Equal to the phase current

D. Always zero

Correct Answer: C. Equal to the phase current

Question 8

In a balanced delta-connected load, the line voltage is

A. √3 times the phase voltage

B. Equal to the phase voltage

C. One-third of the phase voltage

D. Independent of phase voltage

Correct Answer: B. Equal to the phase voltage

Question 9

A balanced three-phase load operates at 400 V line voltage, 20 A line current, and 0.8 power factor. Its active power is approximately

A. 6.40 kW

B. 8.00 kW

C. 11.09 kW

D. 16.00 kW

Correct Answer: C. 11.09 kW

Question 10

Which condition is most directly associated with a lagging power factor?

A. Predominantly inductive load

B. Ideal capacitive load

C. Purely resistive load

D. Open circuit

Correct Answer: A. Predominantly inductive load