AC Circuits, Resonance and Three-Phase Systems

Introduction and Context

Alternating-current (AC) circuits are fundamental to electrical power systems, instrumentation, communication equipment, motor drives, and electronic interfaces. Unlike direct current, an AC voltage or current changes magnitude and direction with time. In a sinusoidal steady state, circuit variables can be represented efficiently by phasors, allowing differential relationships involving inductors and capacitors to be treated algebraically.

This chapter develops laboratory methods for measuring and interpreting phasors, impedance, power, resonance, transients, frequency response, and three-phase quantities. The emphasis is on connecting theory with measurement: selecting instruments, identifying assumptions, estimating uncertainty, and comparing measured results with calculated predictions.

Learning Objectives

After completing this chapter, a student should be able to:

1. Represent sinusoidal voltages and currents using phasors.
2. Calculate and measure impedance, phase angle, and admittance.
3. Determine real power, reactive power, apparent power, and power factor.
4. Identify series and parallel resonance and measure frequency response.
5. Explain transient behavior in first- and second-order circuits.
6. Measure balanced and unbalanced three-phase voltages, currents, and power.
7. Evaluate capacitor-based power-factor correction.
8. Recognize instrument, wiring, loading, and safety limitations in AC experiments.

1. Sinusoidal Quantities and Phasors

A sinusoidal voltage may be written as

where V_m is peak voltage in volts, ω = 2πf is angular frequency in radians per second, f is frequency in hertz, and φ is phase angle relative to a chosen reference. The root-mean-square (rms) value is

for a sinusoid. The rms value is especially useful because it gives the equivalent DC value that produces the same average heating effect in a resistor.

A phasor represents the magnitude and phase of a sinusoidal quantity at one frequency. Using rms phasors, a voltage can be represented as \underline{V} = V∠φ. Phasor analysis applies only to sinusoidal steady-state circuits, or to each sinusoidal component of a linear circuit operating at a single frequency. It does not directly describe switching transients.

For a series circuit, Kirchhoff’s voltage law becomes

The resistor voltage is in phase with current, the inductor voltage leads current by 90°, and the capacitor voltage lags current by 90°.

A practical laboratory procedure is to select current as the reference phasor, measure the rms voltage across each element, and determine phase angles using an oscilloscope or a power-quality analyzer. Oscilloscope grounds must be connected only to circuit points permitted by the instrument design; a grounded probe connected incorrectly can short a live node.

2. Impedance and Admittance

Impedance is the opposition offered by an AC circuit to sinusoidal current:

Its unit is the ohm (Ω). For ideal components at angular frequency ω,

Here, R is resistance in ohms, L is inductance in henries, C is capacitance in farads, and j = √(-1). These relationships assume ideal components and sinusoidal steady state. Real inductors have winding resistance and core losses; real capacitors have equivalent series resistance and leakage.

For a series RLC circuit,

Thus,

and

where θ is the impedance angle. A positive angle indicates net inductive behavior; a negative angle indicates net capacitive behavior.

Admittance is the reciprocal of impedance:

where G is conductance in siemens and B is susceptance in siemens. Admittance is particularly convenient for parallel circuits because branch admittances add directly.

Worked Example 1: Series RLC Impedance

A series circuit has R = 20 Ω, L = 50 mH, C = 10 μF, and f = 100 Hz. Since ω = 2π(100) = 628.3 rad/s,

ωL = 628.3 × 0.05 = 31.42 Ω

and

1/(ωC) = 1/(628.3 × 10×10⁻⁶) = 159.15 Ω.

Therefore,

Z = 20 + j(31.42 − 159.15) = 20 − j127.73 Ω.

Its magnitude is approximately 129.3 Ω, and its angle is approximately −81.1°. The circuit is strongly capacitive at this frequency.

3. AC Power and Power Factor

For rms voltage and current phasors, complex power is

where the asterisk denotes complex conjugation. The quantities are:

These equations apply to sinusoidal voltage and current. For distorted waveforms, displacement power factor and true power factor are not necessarily identical. True power factor includes waveform distortion and is calculated from actual average power divided by rms voltage times rms current.

Positive Q conventionally denotes an inductive load, while negative Q denotes a capacitive load. Power-factor correction aims to reduce the reactive current supplied by the source without changing the useful real power demanded by the load.

Power Measurement

A wattmeter measures real power, while a voltmeter and ammeter provide the quantities needed for apparent power. Digital power analyzers may measure P, Q, S, power factor, harmonics, and phase sequence directly. Before measurement, verify voltage range, current range, frequency range, polarity, and whether the instrument uses a common or isolated input.

Worked Example 2: Power-Factor Correction

A single-phase load consumes P = 2 kW at PF₁ = 0.70 lagging from a 230 V, 50 Hz supply. The desired power factor is PF₂ = 0.95 lagging.

The initial and final reactive powers are

Using tan(cos⁻¹0.70) ≈ 1.020 and tan(cos⁻¹0.95) ≈ 0.329,

Q₁ ≈ 2.04 kvar and Q₂ ≈ 0.658 kvar.

The capacitor must supply

For a shunt capacitor,

so

The practical capacitor selected must have an appropriate AC voltage rating, permissible tolerance, discharge provision, and harmonic suitability. Correction should not produce a leading power factor or resonance with the supply impedance.

4. Resonance and Frequency Response

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

Therefore,

and

At ideal series resonance, impedance is minimum and source current is maximum. The source voltage and current are in phase, so the power factor is unity. Individual inductor and capacitor voltages can nevertheless be much greater than the source voltage.

For a series RLC circuit, the quality factor is

where R is the effective series resistance. The approximate half-power bandwidth is

These bandwidth relations assume a lightly damped, linear, series-resonant circuit.

In a parallel resonant circuit, the input impedance is ideally maximum and source current is minimum at resonance. Real losses and source resistance modify the exact resonant frequency and response. Consequently, laboratory resonance should be identified from measured voltage, current, phase, or power rather than from a calculated value alone.

Frequency-Response Experiment

A suitable experiment uses a function generator, RLC network, oscilloscope, and frequency counter. Keep generator amplitude constant while sweeping frequency. At each frequency record input voltage, output voltage, current if available, and phase difference. Plot the transfer magnitude and phase against frequency. A logarithmic frequency axis is useful when the range spans several decades.

The −3 dB points correspond to a power ratio of one-half. For a voltage magnitude measured across a fixed resistance, the corresponding amplitude is approximately 0.707 of the peak-response amplitude. The bandwidth is the difference between upper and lower half-power frequencies.

5. Transients in AC-Related Circuits

Phasor methods describe steady-state sinusoidal behavior, not the interval immediately after switching. Transient analysis is required when a circuit is energized, de-energized, or subjected to a sudden change.

For a series RL circuit supplied by a DC step, the current is

where I_∞ = V/R is final current and τ = L/R is the time constant in seconds. After approximately five time constants, the current is within about 0.7% of its final value.

For a series RC charging circuit,

where τ = RC. During discharge through resistance R,

A series RLC circuit may be underdamped, critically damped, or overdamped. The natural response is governed by the roots of the characteristic equation. For a series RLC circuit, the damping condition can be expressed using α = R/(2L) and ω₀ = 1/√(LC). It is underdamped when α < ω₀, critically damped when α = ω₀, and overdamped when α > ω₀.

In the laboratory, use a suitably rated switching source and oscilloscope. Measure the time required to reach 63.2% of a final first-order value; this is one time constant. Probe capacitances, coil resistance, generator output resistance, and oscilloscope bandwidth affect the measured result.

6. Three-Phase Systems

A balanced three-phase source consists of three sinusoidal voltages of equal magnitude and frequency separated by 120 electrical degrees. For positive phase sequence ABC,

where V_P is the rms phase-to-neutral voltage. In a balanced system, the phasor sum is zero:

For a balanced positive-sequence system,

where V_L is line-to-line rms voltage. The line voltage leads the corresponding phase voltage by 30° for the stated sequence.

In a balanced star-connected load, line current equals phase current. In a balanced delta-connected load,

where I_P is branch current. The total real power of a balanced three-phase load is

Reactive and apparent powers are

These equations assume balanced sinusoidal voltages and equal load conditions in all phases.

Unbalanced Loads

For an unbalanced load, phase currents are unequal and their sum may not be zero. A four-wire star load has a neutral current

If the neutral is disconnected, the load neutral point can shift, causing unequal phase voltages. Unbalanced systems produce negative- and zero-sequence components, additional losses, motor torque pulsations, and possible overheating.

The two-wattmeter method measures total three-phase real power in a three-wire system, balanced or unbalanced:

For a balanced load, reactive power may be obtained from

and power factor from

The latter relationships require a balanced load and sinusoidal conditions. One wattmeter may read negative when the power factor is below 0.5; its connections or digital sign convention must be handled correctly rather than ignored.

Practical Three-Phase Procedure

First identify the supply configuration, nominal voltage, phase sequence, and protective device rating. Measure line-to-line and line-to-neutral voltages before connecting the load. Connect meters according to the laboratory wiring diagram, energize through approved isolation and protection, and compare measured phase angles with theoretical values. For unbalanced experiments, record every phase current separately and check Kirchhoff’s current law using phasor, not merely scalar, addition.

7. Detailed Applied Case: Industrial Motor Feeder

Background

This illustrative case concerns a balanced three-phase induction motor supplied at 400 V line-to-line, 50 Hz. The motor input power is 15 kW and its measured power factor is 0.78 lagging. The plant engineer is considering capacitor correction.

Problem

Determine the approximate line current before correction, the reactive power to be compensated if the target power factor is 0.95 lagging, and the approximate three-phase capacitor-bank rating.

Analysis

The initial current is obtained from

Thus,

I_L = 15,000/(√3 × 400 × 0.78) ≈ 27.8 A.

Initial reactive power is

Q₁ = P tan(cos⁻¹0.78) ≈ 15 × 0.802 = 12.0 kvar.

At 0.95 power factor,

Q₂ = 15 tan(cos⁻¹0.95) ≈ 15 × 0.329 = 4.94 kvar.

The required capacitor rating is therefore approximately

Q_C = 12.0 − 4.94 = 7.1 kvar.

The corrected current is approximately

I_L,new = 15,000/(√3 × 400 × 0.95) ≈ 22.8 A.

Approach and Outcome

A suitably rated three-phase capacitor bank of approximately 7 kvar would reduce feeder current by about 18%, subject to actual motor loading and capacitor tolerances. The bank should be switched in steps if the motor load varies. Harmonic measurements must be made before installation because capacitors can interact with transformer and feeder inductance, producing resonance and amplified harmonic currents.

Learning Points

Power-factor correction reduces current and voltage drop but does not increase motor mechanical output. It must be coordinated with motor switching, protection, discharge resistors, harmonic filters, and local electrical regulations. A correction system that is technically correct at one operating point may become excessive at light load.

8. Measurement Uncertainty, Safety, and Professional Practice

AC laboratory work involves shock, arc-flash, stored-energy, rotating-machine, and unexpected-start hazards. Students should use isolated low-voltage trainers whenever possible, de-energize before rewiring, discharge capacitors, keep one hand away from live circuits when authorized procedures require measurement, and use appropriately rated leads and probes. Protective earth conductors must never be removed to eliminate measurement noise.

Common sources of error include resistor tolerance, inductor winding resistance, capacitor tolerance, generator output impedance, oscilloscope probe loading, non-sinusoidal waveforms, frequency drift, phase-meter resolution, and loose connections. Report measured values with sensible significant figures and distinguish instrument resolution from actual accuracy.

A reliable report should include the circuit diagram, component ratings, instrument model and ranges, measured data, calculated predictions, uncertainty or tolerance discussion, plots, and an explanation of discrepancies. Never alter data to fit theory. If a result appears impossible, check wiring, reference polarity, instrument grounding, units, and assumptions.

9. Contemporary Developments

Modern power analyzers use digital sampling to calculate rms values, harmonics, real and reactive power, and unbalance indices. Smart inverters and active power-factor-correction converters can control current waveform rather than merely supplying fixed shunt capacitance. In renewable-energy systems, bidirectional converters may exchange both real and reactive power with the grid.

Artificial-intelligence tools can assist with anomaly detection, waveform classification, and predictive maintenance, but their outputs require validated measurements and engineering judgment. A model trained on balanced laboratory data may fail on unbalanced, distorted, or fault conditions. Engineers remain responsible for safety decisions, traceability, and compliance with applicable standards.

Best-Practice Checklist

• Define the reference direction and phase sequence before calculating phasors.
• Use rms values consistently in power equations.
• State whether the load is balanced and whether waveforms are sinusoidal.
• Include component losses when comparing measured and ideal resonance.
• Confirm that wattmeter polarity and current-coil orientation are correct.
• Never connect an oscilloscope ground clip to an unknown live node.
• Check capacitor ratings and possible harmonic resonance before correction.
• Compare calculations, measurements, and uncertainty rather than reporting only one value.

Chapter Summary

AC circuit analysis represents sinusoidal steady-state voltages and currents by phasors. Impedance combines resistance and reactance, while admittance is convenient for parallel networks. Complex power separates real power, reactive power, and apparent power; power factor is the ratio of real to apparent power and may be improved by controlled capacitive compensation.

Series resonance occurs when inductive and capacitive reactances cancel, producing minimum impedance and maximum current. Frequency-response measurements identify resonant frequency, bandwidth, and damping. Transient behavior requires time-domain analysis using time constants and damping conditions rather than phasors alone.

Balanced three-phase systems use 120° phase displacement and provide compact expressions for total power. Unbalanced systems require individual phasor measurements and may produce neutral current, voltage displacement, and equipment heating. Sound laboratory practice combines correct theory with safe wiring, calibrated instruments, uncertainty evaluation, and honest interpretation of discrepancies.

Key Terms

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

RMS value: The effective value of an AC quantity based on equivalent resistive heating.

Impedance: The complex ratio of sinusoidal voltage to current.

Reactance: The imaginary component of impedance associated with inductors or capacitors.

Admittance: The reciprocal of impedance.

Phase angle: The angular displacement between two sinusoidal quantities.

Real power: Average power transferred to useful work, heat, or light.

Reactive power: Power exchanged with electric or magnetic fields.

Apparent power: Product of rms voltage and rms current.

Power factor: Ratio of real power to apparent power.

Resonance: Condition in which inductive and capacitive effects cancel.

Quality factor: A measure of resonance sharpness relative to energy loss.

Bandwidth: Frequency interval between specified response limits, commonly the half-power points.

Transient: Time-dependent response following a switching or sudden-change event.

Time constant: Characteristic time of a first-order circuit response.

Phase sequence: The order in which three-phase voltage waveforms reach corresponding peaks.

Balanced system: A three-phase system with equal magnitudes and 120° displacement.

Unbalanced system: A system in which phase quantities are unequal or not symmetrically displaced.

Two-wattmeter method: A method for measuring total real power in a three-wire three-phase system.

Power-factor correction: Reduction of source reactive current by supplying reactive power locally.

Review Questions

Short Answer Questions

1. Why are rms values used in AC power calculations?
2. State the impedance of an ideal inductor and an ideal capacitor.
3. What is the physical meaning of reactive power?
4. Define series resonance.
5. Why can a three-phase neutral conductor carry current in an unbalanced system?

Long Answer Questions

1. Explain phasor representation and derive the impedance magnitude and phase angle of a series RLC circuit.
2. Discuss real, reactive, and apparent power, including their relationship in the power triangle.
3. Describe a laboratory procedure for measuring the frequency response and bandwidth of a resonant circuit.
4. Explain the line and phase relationships for star- and delta-connected balanced loads.
5. Discuss safety, measurement uncertainty, and sources of discrepancy in AC laboratory experiments.

Analytical / Application Questions

1. A series circuit has R = 15 Ω, L = 40 mH, C = 20 μF, and is supplied at 100 Hz. Calculate its impedance magnitude and phase angle.
2. A 230 V single-phase load consumes 1.5 kW at 0.65 lagging power factor. Calculate its current and the reactive power that must be supplied to reach 0.90 lagging power factor.
3. A balanced three-phase load supplied at 415 V draws 12 A at 0.82 lagging power factor. Calculate total real power, reactive power, and apparent power.

Multiple Choice Questions with Answers

Question 1

Which impedance represents an ideal capacitor at angular frequency ω?

A. R

B. jωL

C. 1/(jωC)

D. ωC

Correct Answer: C. 1/(jωC)

Question 2

A load consumes 4 kW and 3 kvar inductive. What is its apparent power?

A. 1 kVA

B. 5 kVA

C. 7 kVA

D. 12 kVA

Correct Answer: B. 5 kVA

Question 3

At series resonance in an ideal RLC circuit, the source current is:

A. Zero

B. Minimum

C. Maximum

D. 90° behind the voltage

Correct Answer: C. Maximum

Question 4

The time constant of a series RL circuit is:

A. R/L

B. L/R

C. RC

D. 1/(RC)

Correct Answer: B. L/R

Question 5

For a balanced three-phase system, the line-to-line voltage magnitude is:

A. V_P/√3

B. V_P

C. √3 V_P

D. 3V_P

Correct Answer: C. √3 V_P

Question 6

Which instrument directly measures average real power?

A. A wattmeter

B. A frequency counter

C. An ohmmeter

D. A phase-sequence indicator

Correct Answer: A. A wattmeter

Question 7

A positive reactive power conventionally indicates a predominantly:

A. Capacitive load

B. Inductive load

C. Purely resistive load

D. Disconnected load

Correct Answer: B. Inductive load

Question 8

For a balanced three-phase load, total real power is:

A. V_L I_L cosφ

B. √3 V_L I_L cosφ

C. 3V_L I_L sinφ

D. V_P I_P sinφ

Correct Answer: B. √3 V_L I_L cosφ

Question 9

Power-factor correction using a shunt capacitor primarily reduces:

A. Useful real power

B. Source reactive current

C. Supply frequency

D. Load resistance

Correct Answer: B. Source reactive current

Question 10

The two-wattmeter method gives total real power in a three-wire system as:

A. W₁ − W₂

B. W₁W₂

C. W₁ + W₂

D. √3(W₁ + W₂)

Correct Answer: C. W₁ + W₂