Sinusoidal AC Circuits and Resonance

Introduction and Context

Alternating-current (AC) circuits are fundamental to electrical power systems, electric machines, measurement instruments, filters, sensors, and mechatronic actuators. Unlike direct current, a sinusoidal voltage or current continuously changes magnitude and reverses direction. This time variation introduces phase difference between voltage and current, frequency-dependent impedance, reactive power, and resonance.

In this chapter, sinusoidal signals are represented using RMS values, phase angles, and phasors. Resistance, inductance, and capacitance are combined using complex impedance and admittance. The chapter then develops AC power relationships, power-factor correction, series and parallel resonance, bandwidth, quality factor, and frequency response.

Learning Objectives

After studying this chapter, you should be able to:

1. Describe sinusoidal voltage and current using amplitude, angular frequency, phase, and RMS value.
2. Convert between time-domain sinusoidal expressions and phasors.
3. Calculate impedance and admittance for R, L, and C elements and their combinations.
4. Apply Kirchhoff’s laws and voltage-divider/current-divider principles to phasor circuits.
5. Calculate real, reactive, and apparent power and interpret power factor.
6. Determine resonance frequency, bandwidth, and quality factor for RLC circuits.
7. Interpret frequency-response plots and apply these ideas to filters and mechatronic systems.

1. Sinusoidal Signals and RMS Values

A sinusoidal voltage may be written as

where V_m is the peak value in volts, ω is angular frequency in radians per second, φ_v is phase angle, and t is time in seconds. The relationship between angular frequency and frequency is

where f is frequency in hertz. The period is T = 1/f.

For a sinusoid, the RMS value is

and similarly I = I_m/√2. RMS values are especially useful because they produce the same heating effect in a resistor as equal-valued DC quantities. For a resistor,

when V and I are RMS values and are in phase. RMS conversion above applies to pure sinusoids; non-sinusoidal waveforms require the general definition X_rms = √((1/T)∫₀^T x²(t)dt).

The phase difference between two sinusoids of the same frequency is their angular displacement. If current reaches its positive maximum later than voltage, current lags voltage. In an inductor, current lags voltage by 90°; in a capacitor, current leads voltage by 90°.

2. Phasor Representation

A phasor is a complex-number representation of a sinusoid at one fixed frequency. Using RMS phasors, the sinusoid

v(t) = V_m cos(ωt + φ)

is represented by

where V = V_m/√2 and j = √(-1). The bar distinguishes a phasor from an instantaneous quantity. Phasor analysis is valid for linear circuits in sinusoidal steady state, in which all independent sources have the same frequency and transient effects have decayed.

A phasor can be expressed in polar or rectangular form:

Polar form is convenient for multiplication and division; rectangular form is convenient for addition and subtraction. For example,

while rectangular components must be added separately.

3. Impedance and Admittance

Impedance is the ratio of voltage phasor to current phasor:

Its unit is the ohm (Ω). Admittance is the reciprocal of impedance:

and is measured in siemens (S). Write impedance as Z = R + jX, where R is resistance and X is reactance.

Here L is inductance in henries and C is capacitance in farads. The formulas assume ideal components. Real inductors have winding resistance and parasitic capacitance; real capacitors have equivalent series resistance and leakage.

For impedances in series,

For two impedances in parallel,

More generally, parallel admittances add:

The phasor form of Ohm’s law is V̅ = I̅Z. Kirchhoff’s voltage and current laws retain their usual forms, but all quantities are phasors.

Worked Example 1: Series RL Circuit

A 230 V RMS, 50 Hz source supplies a series circuit containing R = 20 Ω and L = 100 mH. Find current and the phase angle between source voltage and current.

First,

ω = 2π(50) = 314.16 rad/s.

The inductive reactance is

X_L = ωL = 314.16(0.100) = 31.42 Ω.

Thus,

Z = 20 + j31.42 Ω.

Its magnitude and angle are

|Z| = √(20² + 31.42²) = 37.24 Ω,

∠Z = tan^-1(31.42/20) = 57.5°.

Therefore,

I̅ = 230∠0°/37.24∠57.5° = 6.18∠−57.5° A.

The current lags the voltage by 57.5°. The negative phase angle is consistent with an inductive circuit.

4. AC Power

For RMS voltage and current with phase difference φ, the complex power is

where I̅* is the complex conjugate of current. The units and meanings are:

A lagging power factor indicates an inductive load; a leading power factor indicates a capacitive load. For an impedance Z = R + jX,

The sign convention is important: Q > 0 for inductive loads and Q < 0 for capacitive loads.

Power-factor correction commonly uses a shunt capacitor across an inductive load. If the initial and desired power factors are cosφ₁ and cosφ₂, respectively, the required capacitive reactive power is

where φ₁ = cos^-1(pf₁) and φ₂ = cos^-1(pf₂). Since a capacitor supplies negative reactive power, this expression gives a negative result when the correction is from lagging to less-lagging operation. The capacitor magnitude is selected from |Q_C| = ωCV² for a single-phase shunt capacitor.

Worked Example 2: Power-Factor Correction

A 5 kW motor operates at 0.70 lagging power factor from a 240 V, 50 Hz single-phase supply. Improve the power factor to 0.95 lagging. Find the capacitor value.

The two angles are

φ₁ = cos^-1(0.70) = 45.57°,

φ₂ = cos^-1(0.95) = 18.19°.

Thus,

Q_C = 5000(tan45.57° − tan18.19°) ≈ 5000(1.020 − 0.329) = 3455 var.

The capacitor must supply approximately 3.46 kvar. Therefore,

C = Q_C/(ωV²) = 3455/[2π(50)(240²)] = 0.000191 S?

Care is needed with units: using Q_C = ωCV²,

C = 3455/[2π(50)(240²)] = 0.000191 F = 191 μF.

The first displayed intermediate unit is not capacitance; the final expression gives the correct unit, farads. In practice, capacitor voltage rating, tolerance, harmonic current, switching transients, and resonance with supply inductance must be considered.

5. Series and Parallel Resonance

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

At resonance, the imaginary part is zero:

Consequently,

where ω₀ and f₀ are resonant angular frequency and frequency. At series resonance, impedance is minimum and equal to R, source current is maximum, and voltage and current are in phase. Although the source voltage may be moderate, the individual inductor and capacitor voltages can be large and opposite in phase.

For a parallel RLC circuit, resonance is most clearly identified through admittance. For an ideal parallel LC network,

Y = 1/(jωL) + jωC,

and the susceptances cancel when ω₀ = 1/√(LC). Parallel resonance generally produces maximum input impedance and minimum source current. Practical parallel circuits contain losses, so the exact impedance peak depends on the topology and component resistances.

Quality Factor and Bandwidth

For a series RLC circuit, the quality factor is

These equivalent forms apply to the idealized series model with resistance R. A high Q indicates sharp selectivity and large circulating reactive energy relative to energy dissipated per cycle.

The half-power frequencies f₁ and f₂ are frequencies at which power is half its resonant value. The bandwidth is

For a series RLC circuit,

The bandwidth formula assumes the series RLC model and defines the −3 dB points. At these points, voltage or current magnitude is 1/√2 of its resonant magnitude, while power is one-half.

Worked Example 3: Resonant Circuit

A series circuit has R = 10 Ω, L = 50 mH, and C = 20 μF. Find resonant frequency, quality factor, and bandwidth.

The resonant angular frequency is

ω₀ = 1/√[(0.050)(20×10^-6)] = 1000 rad/s.

Therefore,

f₀ = 1000/(2π) = 159.2 Hz.

The quality factor is

Q = ω₀L/R = (1000)(0.050)/10 = 5.

The bandwidth is

BW = R/(2πL) = 10/[2π(0.050)] = 31.83 Hz.

The approximate half-power frequencies are centered around 159.2 Hz, although exact values should be found from the complete impedance equation when precise limits are required.

6. Frequency Response and Filters

A frequency response describes how a circuit’s output magnitude and phase vary with input frequency. For a transfer function

|H(jω)| gives magnitude ratio and ∠H(jω) gives phase shift. A Bode plot uses logarithmic frequency axes and displays magnitude in decibels:

For power ratios, the factor is 10 rather than 20. The −3 dB frequency corresponds to an amplitude ratio of approximately 0.707.

Common passive filters include:

For a first-order RC low-pass circuit with output across the capacitor,

The cutoff angular frequency is ω_c = 1/(RC), and f_c = 1/(2πRC). At cutoff, the magnitude is −3 dB relative to the low-frequency value.

In mechatronics, frequency response is used to assess sensor bandwidth, motor-current control, vibration isolation, signal conditioning, and feedback stability. A filter that removes noise may also delay a measurement or attenuate useful transients. Thus, filter design must consider amplitude, phase, sampling, sensor dynamics, and control-loop stability together.

7. Applied Case: Vibration-Sensor Signal Conditioning

Background

Consider an illustrative industrial case in which an accelerometer mounted on a robotic gearbox produces a useful vibration component near 1.2 kHz but also receives high-frequency electrical noise from a switching motor drive.

Problem

The data-acquisition system accepts signals up to 2 kHz. Unfiltered noise causes false alarm events and may saturate the input amplifier. However, excessive filtering could distort the phase of the vibration signal and reduce fault-detection sensitivity.

Analysis

An analog low-pass filter is selected before sampling. If a second-order resonant filter is used, its quality factor must be limited. A high Q provides sharp selectivity but creates peaking near resonance and a longer transient response. A lower-Q response gives a flatter passband and is often safer for measurement.

The engineer first measures the sensor’s source impedance, amplifier input impedance, expected signal amplitude, and noise spectrum. The cutoff frequency is placed above 1.2 kHz but below the dominant switching-noise band. The complete transfer function, not only the nominal cutoff frequency, is evaluated for magnitude and phase.

Approach and Solution

A practical design uses a buffered active low-pass filter with a cutoff near 1.5 kHz, followed by an anti-aliasing stage. The design is verified using a frequency sweep. The engineer checks that the desired vibration component is not excessively attenuated, that the amplifier remains within voltage and current limits, and that component tolerances do not move the cutoff into the diagnostic band.

Outcome

The filtered signal has improved signal-to-noise ratio and fewer false alarms. The measured phase delay is included in the diagnostic algorithm, and the sampling frequency is selected sufficiently above the highest retained frequency. Periodic calibration is specified because sensor mounting, temperature, and aging affect the response.

Learning Points

1. Resonance and bandwidth are measurement-design parameters, not merely theoretical properties.
2. Magnitude response alone is insufficient; phase response can affect timing and control.
3. Real components and interconnections alter the ideal transfer function.
4. Anti-aliasing, grounding, shielding, and calibration are part of a complete AC measurement system.

8. Practical Design Issues, Safety, and Limitations

Ideal phasor calculations do not describe startup transients, nonlinear magnetic saturation, switching waveforms, or circuits containing several unrelated frequencies. Superposition may be applied to linear circuits, but each frequency must be analyzed separately. RMS values of different frequencies cannot generally be added as ordinary signed quantities; total RMS is obtained from the mean-square sum when components are orthogonal over the measurement interval.

At resonance, circulating currents and component voltages can exceed source values. Capacitors can retain hazardous charge after disconnection, and inductors can generate high voltages when current is interrupted. Measurements should use correctly rated probes, isolation procedures, fuses, discharge resistors, and lockout practices. Power-factor correction capacitors must be rated for voltage, current, temperature, harmonic environment, and switching duty. Capacitor banks can interact with system inductance and amplify harmonics.

Professional practice requires stating whether values are RMS or peak, identifying reference polarity and phase convention, checking units, and validating calculations with simulation or measurement. Circuit simulators are useful for frequency sweeps, but a simulation is only as reliable as its component models and assumptions. AI-assisted design tools may help generate parameter sweeps or detect trends, but engineers must independently verify equations, ratings, safety margins, and compliance with applicable electrical standards.

Chapter Summary

Sinusoidal AC quantities are described by amplitude, frequency, and phase. RMS values permit direct comparison of sinusoidal heating and power effects with DC values. Phasors convert differential time-domain relationships into algebraic complex-number relationships for linear sinusoidal steady-state circuits.

Impedance combines resistance and reactance: inductors contribute positive imaginary reactance and capacitors negative imaginary reactance. Admittance is the reciprocal of impedance and is especially convenient for parallel circuits. Complex power separates real power, reactive power, and apparent power, while power factor indicates how effectively current transfers useful power.

Series resonance occurs when inductive and capacitive reactances cancel, producing minimum impedance and maximum current. Parallel resonance generally produces maximum impedance. Resonant frequency, quality factor, and bandwidth quantify frequency selectivity. Frequency-response methods, including transfer functions and Bode plots, connect circuit behavior to filters, sensors, actuators, and control systems. Real designs must account for losses, tolerances, harmonics, safety, phase delay, and measurement limitations.

Key Terms

Sinusoid: A waveform described by a sine or cosine function.

RMS value: The effective value of an AC quantity based on its heating equivalence.

Phase angle: Angular displacement between sinusoidal quantities of the same frequency.

Phasor: A complex representation of a sinusoid at a fixed frequency.

Impedance: The complex ratio of voltage phasor to current phasor.

Reactance: The imaginary part of impedance associated with inductance or capacitance.

Admittance: The reciprocal of impedance.

Susceptance: The imaginary part of admittance.

Inductive reactance: The quantity X_L = ωL.

Capacitive reactance: The quantity X_C = 1/(ωC).

Real power: Average power converted into work or heat.

Reactive power: Energy exchanged periodically 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: Measure of resonant sharpness or selectivity.

Bandwidth: Difference between upper and lower half-power frequencies.

Frequency response: Variation of output magnitude and phase with frequency.

Bode plot: Logarithmic plot of transfer-function magnitude and phase.

Anti-aliasing filter: A filter that limits input bandwidth before sampling.

Review Questions

Short Answer Questions

1. What is the relationship between peak and RMS values for a sinusoidal voltage?
2. Why is a phasor useful in sinusoidal steady-state circuit analysis?
3. State the impedance of an ideal resistor, inductor, and capacitor.
4. Distinguish real power, reactive power, and apparent power.
5. What happens to the impedance of a series RLC circuit at resonance?

Long Answer Questions

1. Explain how rectangular and polar phasor forms are used in AC circuit calculations. Include the advantages of each form.
2. Derive the resonant frequency of a series RLC circuit from the condition that its net reactance is zero.
3. Explain power-factor correction using a shunt capacitor and discuss practical limitations.
4. Describe the meaning of bandwidth and quality factor for a resonant circuit.
5. Explain how magnitude and phase frequency responses influence the design of a mechatronic sensor-conditioning circuit.

Analytical / Application Questions

1. A 120 V RMS, 60 Hz source supplies a series circuit with R = 15 Ω, L = 80 mH, and C = 47 μF. Calculate total impedance, current magnitude, and phase angle.
2. A series RLC circuit has L = 25 mH, C = 10 μF, and R = 4 Ω. Calculate resonant frequency, quality factor, and bandwidth. State the assumptions used.
3. An induction motor consumes 8 kW at 0.72 lagging power factor from a 400 V, 50 Hz single-phase equivalent supply. Determine the reactive power before correction and the capacitor size required for 0.92 lagging power factor. Discuss two practical issues that must be checked before installation.

Multiple Choice Questions with Answers

Question 1

A sinusoidal voltage has a peak value of 100 V. What is its RMS value?

A. 50 V

B. 70.7 V

C. 100 V

D. 141.4 V

Correct Answer: B. 70.7 V

Question 2

What is the impedance of an ideal capacitor at angular frequency ω?

A. R

B. jωL

C. 1/(jωC)

D. ωC

Correct Answer: C. 1/(jωC)

Question 3

In a purely inductive circuit, the current

A. leads voltage by 90°

B. lags voltage by 90°

C. is in phase with voltage

D. has zero frequency

Correct Answer: B. lags voltage by 90°

Question 4

Which expression gives complex power using RMS phasors?

A. S = V̅I̅*

B. S = V̅ + I̅

C. S = V̅/I̅*

D. S = VI²

Correct Answer: A. S = V̅I̅*

Question 5

A load consumes 4 kW at an apparent power of 5 kVA. Its power factor is

A. 0.20

B. 0.80

C. 1.25

D. 9.00

Correct Answer: B. 0.80

Question 6

The resonant angular frequency of an ideal LC circuit is

A. √(LC)

B. 1/(LC)

C. 1/√(LC)

D. L/C

Correct Answer: C. 1/√(LC)

Question 7

At series resonance in a series RLC circuit, the source sees

A. maximum impedance

B. zero resistance in every practical circuit

C. minimum impedance equal to R in the ideal model

D. purely capacitive impedance

Correct Answer: C. minimum impedance equal to R in the ideal model

Question 8

For a series RLC circuit, increasing R while L remains constant generally causes the bandwidth to

A. increase

B. decrease

C. become zero

D. equal the resonant frequency

Correct Answer: A. increase

Question 9

A −3 dB voltage magnitude corresponds approximately to an amplitude ratio of

A. 0.10

B. 0.50

C. 0.707

D. 2.00

Correct Answer: C. 0.707

Question 10

The primary purpose of an anti-aliasing filter before an ADC is to

A. increase the sampling frequency automatically

B. limit frequency content before sampling

C. convert AC into DC

D. eliminate all phase shift

Correct Answer: B. limit frequency content before sampling