Lessons in Electric Circuits Volume II AC Study Guide and Chapter Summaries – Tony R. Kuphaldt (6th Edition, 2021) BSEE/BSET

BSEE and BSET students starting AC circuit analysis can use this guide to study Tony R. Kuphaldt’s “Lessons in Electric Circuits, Volume II: AC”. It is the second book of his open textbook series, with 14 chapters and 3 appendices, and the current edition (last update November 6, 2021) is published under the Creative Commons Attribution licence (CC BY 4.0). It expects you to know the basics of DC circuits from Volume I.

The book begins with sine waves, RMS values and phase, then teaches complex numbers so that voltage, current and impedance can be handled as phasors. From there it covers inductive and capacitive reactance, resonance, filters, transformers, three-phase power, power factor, AC meters, motors and transmission lines. Most sections are backed by SPICE simulations and worked examples, which suits a first AC course and revision before an exam. There are no end-of-chapter problem sets, so plan to pair it with the exercises from your class.

Book Overview

Chapter List

Chapter 1: Basic AC Theory

Difficulty: Easy · Key topics: alternator, sine wave, period, frequency, peak, peak-to-peak, average, RMS, crest factor, form factor, phase shift, radio waves

The book opens by contrasting DC with AC and explaining why AC dominates power systems: alternators and AC motors need no brushes, and transformers can step voltage up for efficient long-distance transmission. It then describes the sine wave an alternator produces, with cycle, period and frequency, using musical pitch as an analogy. Most of the chapter is about how to give a changing quantity one number: peak, peak-to-peak, average and RMS, with the bandsaw and jigsaw analogy for RMS. It closes with phase shift, leading and lagging, and a short introduction to antennas and electromagnetic waves.

Key Points:

• An alternator produces a sine wave because the induced voltage follows the rate of change of magnetic flux, as Faraday’s Law describes.
• Frequency is 1/period: a 16 ms period gives 62.5 Hz, and power lines run at 60 Hz in the USA and 50 Hz in Europe.
• RMS is the DC-equivalent value that gives the same heating in a resistor; for a sine wave RMS is about 0.707 of peak and the practical average is about 0.637 of peak.
• Crest factor is peak divided by RMS, and form factor is RMS divided by average (about 1.11 for a sine wave).
• Analog meter movements respond to the average value and are scaled to show RMS, so they read correctly only on pure sine waves.
• Ohm’s Law, Kirchhoff’s Laws and the table method still work for purely resistive AC circuits, as long as every value uses the same measure (peak, RMS and so on).
• Phase shift is always relative between two waves of the same frequency; the source voltage is usually taken as the 0 degree reference.

Memory Tip: RMS spells out the method: square the points, take the Mean, then the square Root.

Common Mistake: Mixing measures in one calculation, such as combining a peak voltage with an RMS current. Pick one measure and keep it throughout; if none is stated, assume RMS.

Important Questions:

• What does “10 volts RMS” mean? It is the AC voltage that heats a resistor exactly as much as a 10 volt DC source would.
• Why can transformers not work with steady DC? A transformer relies on a changing magnetic field to induce a voltage in the second coil, and steady DC only makes a steady field.

Chapter 2: Complex Numbers

Difficulty: Medium · Key topics: scalar vs complex quantity, vectors and phasors, polar form, rectangular form, j operator, complex arithmetic, polarity and phase angle, series AC sources

The chapter argues that scalar numbers are not enough for AC, because an AC quantity has both amplitude and phase shift, like a trip that needs a distance and a direction. A vector represents this: its length is the amplitude and its angle is the phase relative to a reference waveform. The author shows how in-phase voltages add, opposing ones subtract, and a 6 V and an 8 V source 90 degrees apart give 10 V. Polar and rectangular notation, the conversions between them and the four basic operations follow, then a clear section on why polarity marks matter for phase angles, and worked series examples confirmed with SPICE.

Key Points:

• A complex number holds amplitude and phase in one quantity; the term phasor is used when the vector angle stands for phase.
• Polar form is magnitude and angle (5 at 36.87 degrees); rectangular form is real plus j times imaginary (4 + j3).
• Polar to rectangular uses cosine and sine; rectangular to polar uses the Pythagorean theorem and the arctangent.
• Add and subtract in rectangular form; multiply and divide in polar form (multiply magnitudes and add angles, or divide magnitudes and subtract angles).
• Reversing the polarity of an AC source is the same as adding 180 degrees to its phase angle.
• All DC laws still hold in AC, except the power calculation, if every quantity is complex and all are at the same frequency.
• A voltmeter or ammeter shows only the polar magnitude; the angle needs an oscilloscope.

Memory Tip: Rectangular is for adding and subtracting, polar is for multiplying and dividing. Convert to the form that makes the next operation easy.

Common Mistake: Expecting AC series voltages to add like batteries. A 6 V source and an 8 V source 90 degrees apart give 10 V, not 14 V, and a meter across the pair would read 10 V.

Important Questions:

• Why are complex numbers used for AC analysis? An AC voltage or current has both amplitude and phase shift, and a complex number (a vector) can carry both in a single quantity.
• How do you convert 4 + j3 to polar form? The magnitude is the square root of 4 squared plus 3 squared, which is 5, and the angle is the arctangent of 3/4, about 36.87 degrees, so 5 at 36.87 degrees.

Chapter 3: Reactance and Impedance – Inductive

Difficulty: Medium · Key topics: AC resistor circuit, inductive reactance, angular velocity, impedance, series R-L, parallel R-L, table method, effective resistance, Q factor, skin effect

This is where the complex numbers are put to work. A resistor’s voltage and current stay in phase and it always dissipates power. An inductor opposes changes in current, so its voltage leads its current by 90 degrees, it returns as much energy as it absorbs, and its opposition (reactance) grows with frequency. Resistance and reactance combine into impedance, shown first in a series R-L circuit and then a parallel one, both solved with the familiar table method and checked with SPICE. The chapter ends with the quirks of real inductors: wire resistance, eddy currents, hysteresis, the Q factor and the skin effect.

Key Points:

• In a resistor, voltage and current are in phase; in an ideal inductor, voltage leads current by 90 degrees.
• Inductive reactance is XL = 2πfL = ωL, so it rises with frequency; a 10 mH inductor has about 3.77 ohms at 60 Hz.
• Impedance Z is the complex sum of resistance and reactance; an inductor’s impedance angle is +90 degrees and a resistor’s is 0 degrees.
• Ohm’s Law for AC is E = IZ, I = E/Z and Z = E/I, with all quantities complex.
• In a series R-L circuit the impedance angle lies between 0 and +90 degrees and the current lags the source; in parallel R-L, branch currents add and the reciprocal formula gives Z.
• Real inductors have effective resistance from wire, radiation, eddy currents and hysteresis; Q is the ratio of reactance to this resistance.
• Skin effect pushes AC toward the surface of a conductor, so AC resistance exceeds DC resistance and rises with frequency.

Memory Tip: For an inductor, voltage leads current. A common way to hold it: “ELI” – E (voltage) comes before I (current) in L.

Common Mistake: Adding resistance and reactance as plain numbers. 5 ohms plus 3.77 ohms of reactance is 6.26 ohms of impedance (at an angle), not 8.77 ohms.

Important Questions:

• Why does an ideal inductor dissipate no net power? Its voltage and current are 90 degrees apart, so it absorbs energy in half of each cycle and returns exactly the same amount in the other half.
• How does inductive reactance change with frequency? It is directly proportional to frequency (XL = 2πfL), so doubling the frequency doubles the reactance.

Chapter 4: Reactance and Impedance – Capacitive

Difficulty: Medium · Key topics: capacitor current and voltage, capacitive reactance, negative impedance angle, series R-C, parallel R-C, lossy capacitors, dielectric heating, electrolytic capacitors

The chapter mirrors Chapter 3 for the capacitor. Because a capacitor opposes changes in voltage, its current leads its voltage by 90 degrees, and its reactance falls as frequency rises. That reactance has an impedance angle of -90 degrees, the opposite of an inductor. A 100 microfarad capacitor gives about 26.5 ohms at 60 Hz, and the same table method solves the series and parallel R-C circuits. The last section on real capacitors explains that they are usually purer than inductors, but dielectric loss, heating and electrolytic types need care.

Key Points:

• Capacitor current is proportional to the rate of change of voltage, so current leads voltage by 90 degrees.
• Capacitive reactance is XC = 1/(2πfC); it falls as frequency rises, the opposite of an inductor.
• A capacitor’s impedance angle is exactly -90 degrees, so series R-C impedance has an angle between 0 and -90 degrees.
• In series R-C, 5 ohms with 26.53 ohms of reactance gives Z of 5 – j26.53, about 26.99 ohms at -79.3 degrees.
• Series impedances add and parallel impedances use the reciprocal formula, whatever the component types are.
• A lossy capacitor wastes power as dielectric heating; this is stronger at high frequency and is used industrially to heat plastics.
• Unless made for AC, electrolytic capacitors should never see reverse voltage, so they need a DC bias.

Memory Tip: Pair it with the inductor rule: in a capacitor, current leads voltage, so the voltage lags (the “ICE” half of ELI the ICE man: I before E in C).

Common Mistake: Expecting capacitive reactance to rise with frequency like inductive reactance. It is the reverse: higher frequency means less opposition from a capacitor.

Important Questions:

• What is the phase relationship between voltage and current in a capacitor? Current leads voltage by 90 degrees (voltage lags current), so a capacitor’s impedance angle is -90 degrees.
• How does capacitive reactance depend on frequency? XC = 1/(2πfC), so it is inversely proportional to frequency and capacitance: higher frequency or a bigger capacitor means lower reactance.

Chapter 5: Reactance and Impedance – R, L, and C

Difficulty: Medium · Key topics: R, X and Z review, series R-L-C, parallel R-L-C, series-parallel R-L-C, susceptance, admittance, table method

This chapter puts resistors, inductors and capacitors in the same circuit. It begins by reviewing R, X and Z and the impedance angles of 0, +90 and -90 degrees, then works a series R-L-C circuit on a 120 V, 60 Hz supply: find each reactance, turn everything into impedance, add, and apply E = IZ. The same components are then rearranged in parallel and in a series-parallel network, always with the table method and a SPICE check. A short section introduces susceptance and admittance as reciprocals, and the summary states the main message: AC analysis equals DC analysis done with complex numbers, except for power.

Key Points:

• First step of any AC analysis: convert every R, L and C into an impedance at the source frequency.
• Series inductive and capacitive impedances partly cancel, so total impedance can be smaller than one of its parts.
• That cancelling can leave a component voltage above the supply: in the worked series example the capacitor drops about 137.5 V from a 120 V source.
• Parallel branches share the voltage; branch currents add as complex numbers, and Z total comes from E/I or the reciprocal formula.
• Series-parallel circuits are reduced step by step exactly as in DC, once every component is written as Z.
• Susceptance B is 1/X and admittance Y is 1/Z; both are measured in siemens and add in parallel.
• Power is the one DC rule that does not carry over; it needs its own chapter (Chapter 11).

Memory Tip: Reactance first, impedance second, analysis third. Convert, then treat the circuit like a DC circuit.

Common Mistake: Combining R, L and C values directly or skipping the conversion to impedance. Inductance in henrys and capacitance in farads cannot be added to resistance in ohms.

Important Questions:

• Can a component voltage be larger than the supply voltage in a series R-L-C circuit? Yes. The inductor and capacitor voltages are in opposite directions and largely cancel in the total, so each can be larger than the source.
• What is admittance? It is the reciprocal of impedance (Y = 1/Z), a complex quantity measured in siemens that shows how easily AC flows through a circuit.

Chapter 6: Resonance

Difficulty: Medium · Key topics: tank circuit, resonant frequency, parallel resonance, series resonance, applications, antiresonance, Q factor, bandwidth, half-power points

The chapter starts with an “electric pendulum”: a charged capacitor and an inductor trade energy back and forth, just as a pendulum swaps potential and kinetic energy, at a frequency set only by L and C. It then derives the resonant frequency where the two reactances are equal, using 10 microfarads and 100 millihenrys (159.155 Hz) as the running example. A parallel tank circuit has very high impedance there, and a series circuit has very low impedance, which can produce large component voltages. After applications such as oscillators and tuners, it shows how added resistance shifts resonance (antiresonance), and ends with Q and bandwidth measured between the half-power points.

Key Points:

• Resonance occurs when XL = XC, which gives f = 1/(2π√(LC)); the frequency does not depend on the amount of stored energy.
• An ideal parallel LC circuit has infinite impedance at resonance (it draws no line current); an ideal series LC circuit has zero impedance.
• In series resonance, L and C voltages can be Q times the supply (100 V across each from a 1 V source in the book’s example), though they cancel in the total.
• Q is stored power divided by dissipated power (X/R); higher Q means a narrower bandwidth.
• Bandwidth is measured between the half-power points (0.707 of the peak current), and BW = fc/Q.
• Below resonance a series circuit looks capacitive and above it looks inductive; a parallel circuit is the reverse.
• Resistance in series with L or C of a tank circuit shifts the impedance peak (antiresonance), but resistance in a pure series R-L-C circuit does not shift resonance.

Memory Tip: Think of a pendulum: the energy swings between two stores (electric field and magnetic field), and the swing rate depends only on L and C, as a pendulum’s rate depends only on its length.

Common Mistake: Assuming the simple formula always gives the peak or dip. With extra resistance in a parallel tank circuit, the minimum current can move well away from 159 Hz, because of antiresonance.

Important Questions:

• What happens to the impedance of a series LC circuit and of a parallel LC circuit at resonance? The series circuit’s impedance falls to a minimum (zero when ideal), while the parallel circuit’s impedance rises to a maximum (infinite when ideal).
• How are Q and bandwidth related? Bandwidth equals the resonant frequency divided by Q, so a high-Q circuit is more selective, with a narrower response.

Chapter 7: Mixed-Frequency AC Signals

Difficulty: Hard · Key topics: non-sinusoidal waveforms, fundamental and harmonics, square wave series, Fourier analysis, FFT, spectrum analyzer, nonlinear components, superposition, DC offset

Up to now every circuit had one sine-wave source. This chapter widens the picture to signals that mix several frequencies, including DC (frequency zero) and non-sinusoidal shapes. It defines the fundamental, harmonics and overtones, using musical tones as the example. A square wave is shown to equal a sine wave plus odd harmonics of falling amplitude, and any repeating waveform can be rebuilt the same way. The author then compares oscilloscope and spectrum-analyzer views of a sine wave, chords, a trumpet, triangle and sawtooth waves, explains which shapes lack even harmonics, and shows how to analyze circuits with superposition.

Key Points:

• Any repeating waveform equals a DC level plus sine and cosine waves of different frequencies, phases and amplitudes; finding them is Fourier analysis, often done with the FFT.
• A square wave is a fundamental sine plus odd harmonics (3rd, 5th, 7th and so on) with amplitudes falling as 1/3, 1/5, 1/7.
• Harmonics are whole-number multiples of the fundamental; overtones are the extra frequencies a particular device actually produces.
• A waveform whose positive and negative halves mirror each other (square, triangle, sine) has no even harmonics; a sawtooth does contain them.
• A DC offset or bias does not change the harmonic content of a wave.
• Nonlinear parts such as SCRs and rectifiers distort the sine wave and are a common source of harmonics in power systems.
• Use superposition for mixed frequencies: analyze one source at a time, then add the results in the time domain, not as complex numbers.

Memory Tip: Time domain versus frequency domain: the oscilloscope shows what the wave looks like over time, while the spectrum analyzer shows which sine waves it is made of.

Common Mistake: Adding voltages of different frequencies as complex numbers. Complex numbers carry amplitude and phase only, so each frequency must be solved on its own and the answers combined at the end.

Important Questions:

• What sine waves make up a square wave? A sine wave at the fundamental frequency plus odd multiples of that frequency, with amplitudes that shrink in proportion to 1/3, 1/5, 1/7 and so on.
• How do you analyze a circuit that has several source frequencies? Replace the source with a series set of single-frequency sources, use the superposition theorem for each one, and then sum the results as time-domain waveforms.

Chapter 8: Filters

Difficulty: Medium · Key topics: low-pass, high-pass, band-pass, band-stop, cutoff frequency, Bode plot, Twin-T notch filter, resonant filters, selectivity, attenuation

The chapter builds on resonance and reactance to show how circuits pick some frequencies out of a mix. It starts with everyday uses such as equalizers, speaker crossover networks and conditioning non-sinusoidal signals. Low-pass and high-pass filters are built from a single inductor or capacitor with a resistor, and the cutoff frequency is defined at 70.7 percent of the input voltage. Band-pass filters join a low-pass and a high-pass stage in series, and band-stop filters, including the Twin-T notch, combine them in parallel. The chapter finishes with resonant LC filters, a warning about resonance peaks in combined L and C low-pass filters, and a summary that leans on SPICE and Bode plots.

Key Points:

• A low-pass filter passes low frequencies and blocks high ones; a high-pass filter does the opposite.
• The cutoff frequency is where the output falls to 70.7 percent of the input; for a simple RC filter it is 1/(2πRC), such as 45.47 Hz for 500 ohms and 7 microfarads, but the load changes the real value.
• Capacitor-based filters are usually preferred over inductor-based ones because capacitors are more nearly pure reactances.
• A band-pass filter is a low-pass and a high-pass in series; a band-stop (notch) filter puts them in parallel, as in the Twin-T design.
• Series LC passes its resonant frequency (band-pass) or shorts it out (band-stop); parallel LC does the reverse.
• Resonant filters need no series resistor, so they pass the wanted band with almost no attenuation, but stray resistance in inductors reduces selectivity.
• Bode plots and SPICE are the main tools for judging a filter’s frequency response.

Memory Tip: Think of what each part does to the signal path. A series capacitor blocks low frequencies (high-pass), and a capacitor to ground shorts out high frequencies (low-pass).

Common Mistake: Expecting the textbook cutoff formula to hold with any load. The load resistance changes the real cutoff, and an L-and-C “improved” low-pass can resonate and add a peak.

Important Questions:

• How is the cutoff frequency of a filter defined? It is the frequency where the load voltage is 70.7 percent of the source voltage (the half-power point). A low-pass filter passes more than that below the cutoff, and a high-pass filter passes more than that above it.
• How can a band-pass filter be built from simpler filters? Put a low-pass filter and a high-pass filter in series so that only frequencies that are neither too high nor too low get through.

Chapter 9: Transformers

Difficulty: Hard · Key topics: mutual inductance, magnetizing current, turns ratio, isolation, dot convention, multiple windings, autotransformer, voltage regulation, impedance matching, instrument transformers, losses and saturation

This is the longest chapter on a single device. It starts with an iron-core coil on AC and the flux that lags the source voltage by 90 degrees, then adds a second winding to explain mutual inductance and how load current is reflected back to the primary. Next come step-up and step-down ratios, electrical isolation, phasing and the dot convention, and transformers with several or tapped windings, including the autotransformer and Variac. Voltage regulation follows, then special types: impedance matching, potential and current transformers, air-core and Tesla coils, saturable reactors, the Scott-T and the LVDT. The last section covers practical limits such as losses, leakage inductance, saturation, inrush current and noise.

Key Points:

• The primary draws magnetizing current to set up core flux, which lags the supply voltage by 90 degrees; exciting current also covers core losses and is never perfectly sinusoidal.
• The voltage ratio follows the turns ratio, and the current ratio is its inverse, so a step-down of 10:1 raises current ten times.
• Isolation transformers couple power with no direct wire connection, so a DC voltage on one side does not appear on the other.
• The dot convention marks which winding ends share the same instantaneous polarity, and ideally there is no phase shift between primary and secondary.
• An autotransformer is a single tapped coil that changes voltage without isolation; a Variac is a variable autotransformer.
• Voltage regulation measures how little the secondary voltage sags from no load to full load; a good power transformer is under about 3 percent.
• A transformer can match a load’s impedance to a source; potential and current transformers scale voltage and current down for meters.
• Losses and limits include winding resistance, eddy currents, hysteresis, leakage inductance, stray capacitance, core saturation and inrush current.

Memory Tip: Power in equals power out (ideally), so if voltage is stepped down by a factor, current must be stepped up by the same factor.

Common Mistake: Trying to run a transformer from DC. A steady current makes a steady flux, so no voltage is induced; DC through a winding can even saturate the core.

Important Questions:

• What is the relationship between turns ratio, voltage and current in a transformer? Secondary voltage equals primary voltage times secondary turns over primary turns, and current changes in the opposite proportion, so apparent power stays the same.
• What is voltage regulation? It is the measure of how well a power transformer holds a constant secondary voltage as the load current changes; a lower percentage means better regulation.

Chapter 10: Polyphase AC Circuits

Difficulty: Hard · Key topics: single-phase and split-phase, three-phase, 120/208 system, phase rotation, polyphase motors, Y and delta, line and phase quantities, three-phase transformers, open delta, harmonics, triplen harmonics, phase sequence

The chapter uses one running example, two or three 10 kW loads on 120 V, to compare power systems by the wire they need. Single-phase needs the largest conductors, split-phase halves the current with a neutral wire, and sources 120 degrees apart cut current and shock risk further, leading to the three-phase 120/208 system. It then covers phase rotation, why polyphase motors start in a fixed direction while single-phase motors need help, and the Y and delta connections with their line and phase relationships. The second half applies these to three-phase transformer banks, and then to harmonics from nonlinear loads, including triplen harmonics in the neutral and positive, negative and zero sequences.

Key Points:

• Split-phase uses two in-phase sources in series with a neutral, giving lower conductor current and a lower load voltage for safety.
• Three-phase sources are 120 degrees apart; with 120 V phases the line-to-line voltage is about 208 V, written as a 120/208 system.
• Phase sequence (1-2-3 or 3-2-1) sets motor direction; swapping any two lines reverses it.
• In a balanced Y circuit, line voltage = √3 times phase voltage and line current = phase current; in a delta circuit, line current = √3 times phase current and line voltage = phase voltage.
• Three-phase transformers can be Y-Y, Y-delta, delta-Y or delta-delta; delta gives better reliability, and an open delta (the “V” connection) works with two transformers that must be oversized.
• Nonlinear loads (rectifiers, SCRs, discharge lamps, saturated cores) draw non-sinusoidal current, which means harmonics.
• Triplen harmonics (3rd, 9th, 15th and so on) are in phase in all three lines, so they add in a 4-wire Y neutral and circulate inside a delta loop.
• Harmonics rotate as positive (1st, 7th, 13th), zero (3rd, 9th) or negative (5th, 11th) sequence, and the 5th harmonic is the main worry for motors.

Memory Tip: Y keeps the voltage factor √3 on the line voltage, delta keeps it on the line current. Remember which quantity is the same in each shape.

Common Mistake: Assuming the neutral of a balanced three-phase system carries no current when harmonics are present. Triplen harmonics add in the neutral instead of cancelling.

Important Questions:

• What is the relationship between line and phase voltage in a balanced Y connection? The line voltage equals the phase voltage times the square root of 3, while line current equals phase current.
• Why are triplen harmonics a problem in three-phase systems? The 3rd harmonic and its multiples are in phase in all three lines, so in a 4-wire Y system they add up in the neutral conductor, and in a delta they circulate in the loop and cause heating.

Chapter 11: Power Factor

Difficulty: Medium · Key topics: true power, reactive power, apparent power, power triangle, power factor, power factor correction, parallel capacitor, kVA

The chapter starts from a 120 V, 60 Hz source feeding a 60 ohm resistor. The load takes 2 A and 240 W, and the power waveform is always positive and swings at twice the line frequency. Reactive loads instead absorb and return energy, so a new vocabulary is needed: true power in watts, reactive power in VAR, and apparent power in volt-amps, tied together by the power triangle. Power factor is the cosine of the impedance angle, or true over apparent power, and it matters because a poor factor forces the wiring to carry extra current. The chapter then corrects a lagging load with a parallel capacitor, first in a SPICE-checked example and then in a practical 240 V, 1.5 kW case.

Key Points:

• True power P (watts) is dissipated; reactive power Q (VAR) is only absorbed and returned; apparent power S (VA) is voltage times current with no phase angle.
• P, Q and S form a right triangle, with P adjacent, Q opposite and S as the hypotenuse; the angle equals the impedance angle.
• Power factor = P/S = cos of the phase angle; it is 1 for a resistive load and 0 for a purely inductive or capacitive one, and it has no unit.
• With a poor power factor the wiring carries more current than the true power alone needs, so there are more heat losses.
• Most real loads are inductive, so a parallel capacitor of the right size is used to correct the power factor.
• Correcting leaves the true power unchanged but reduces the line current: in the practical example the current falls from 9.615 A to 6.25 A.
• Measured true power (wattmeter) and apparent power (volts times amps) give reactive power and the power factor when the load’s inductance is unknown.

Memory Tip: Picture a beer glass: the liquid is true power (useful), the foam is reactive power, and the glass you must carry is apparent power.

Common Mistake: Adding true and reactive power as plain numbers. They are 90 degrees apart, so apparent power is the square root of P squared plus Q squared.

Important Questions:

• What is power factor and why is a low value a problem? It is the ratio of true power to apparent power (cos of the phase angle); a low value means more current must flow for the same useful power, which wastes energy in the wires.
• How is the power factor of an inductive load corrected? A capacitor is connected in parallel with the load, supplying the reactive power that the inductance needs, so the line current and apparent power fall while true power stays the same.

Chapter 12: AC Metering Circuits

Difficulty: Medium · Key topics: rectifier meters, true-RMS, frequency meters, Lissajous figures, Hall effect power sensor, power quality, harmonics, AC bridges, Wien and Maxwell bridges, LVDT, synchro, resolver

The chapter covers how AC quantities are actually measured. It starts with voltmeters and ammeters: DC-style meter movements need diodes to rectify the signal, and electromechanical movements respond to the average value unless they are calibrated for a known waveshape, while true-RMS meters avoid that limit. Frequency measurement follows, from vibrating reeds and tuned LC circuits to Lissajous figures on an oscilloscope. Power measurement uses the electrodynamometer and a Hall-effect multiplier, and power quality meters act as low-frequency spectrum analyzers for harmonics. The second half explains AC bridges, including the Wien and Maxwell-Wien types, and AC transducers such as the LVDT, RVDT, synchro and resolver.

Key Points:

• Permanent-magnet moving-coil movements cannot read AC directly, so diodes rectify the signal first.
• Electromechanical AC meters show the average value and are scaled to read RMS only for a known waveshape; true-RMS meters give the actual RMS.
• Frequency can be measured with tuned reeds (mechanical resonance), a calibrated LC tank, or Lissajous figures against a reference frequency.
• A Lissajous oval becomes a circle at 90 degrees of phase shift, and a stable pattern means the two frequencies are in a whole-number ratio.
• An electrodynamometer multiplies instantaneous voltage and current; a Hall-effect sensor can produce a DC signal proportional to power.
• Nonlinear loads fill power systems with harmonics that overheat transformers and overload neutral conductors, so power quality meters display the harmonic spectrum.
• An AC bridge balances a ratio of impedances, usually needing more than one adjustment to match both magnitude and phase angle; the Maxwell-Wien bridge measures an inductor against a capacitor standard.
• An LVDT senses position without sliding contact wear, and synchros used in pairs make one shaft follow another.

Memory Tip: Think of each instrument by what it reads: meters give the size, Lissajous figures compare frequency and phase, spectrum meters show the harmonics, and bridges find unknown impedances by balancing to zero.

Common Mistake: Trusting a simple analog AC meter on a distorted waveform. It is calibrated for a pure sine wave and will not show the true RMS of anything else.

Important Questions:

• Why do average-responding AC meters give wrong RMS readings on non-sinusoidal waves? Their scale is skewed by the sine-wave form factor, so the reading is correct only when the waveform has that exact shape.
• What does an LVDT do and what is its advantage over a potentiometer? It gives an AC voltage that indicates the position of a movable core, and since the core does not touch the windings there is no wear or arcing.

Chapter 13: AC Motors

Difficulty: Medium · Key topics: synchronous motor, synchronous condenser, reluctance motor, stepper motor, brushless DC motor, Tesla induction motor, slip, wound rotor, single-phase induction, shaded pole, selsyn, AC commutator motor

This chapter is a tour of AC machines. It begins with the history: AC motors first copied DC designs and ran into trouble until Steinmetz studied hysteresis and Tesla introduced the rotating-field induction motor. It then covers synchronous motors and the synchronous condenser, which corrects lagging power factor, followed by reluctance, stepper and brushless DC motors. About a quarter of the chapter is the Tesla polyphase induction motor: construction, rotating field, synchronous speed, slip, power factor, starting and speed control, and the linear version. The remaining sections cover wound-rotor motors, single-phase types (split-phase, capacitor-start, shaded-pole), selsyns and series or repulsion commutator motors.

Key Points:

• A synchronous motor runs at a fixed speed set by line frequency and pole count, and it can draw a leading power factor.
• A synchronous condenser is an unloaded, over-excited synchronous motor used to supply reactive power and cancel lagging power factor.
• About 90 percent of industrial motors are induction motors, and large ones are three-phase with a squirrel-cage rotor.
• Synchronous speed is 3600 rpm for a two-pole machine at 60 Hz (3000 at 50 Hz), and it halves each time the number of poles is doubled.
• Slip is the difference between synchronous speed and rotor speed divided by synchronous speed, and torque comes from the rotor slipping behind the stator field.
• Squirrel-cage motors can draw 500 to over 1000 percent of full-load current when starting; autotransformer starters or rotor resistance in a wound-rotor motor reduce the surge.
• A single-phase winding makes only a pulsating field, so single-phase induction motors need a starting aid such as a capacitor, a split-phase winding or shading coils.
• Steppers move in discrete steps and hold position; brushless DC motors resemble synchronous motors but have trapezoidal back EMF and electronic drives.

Memory Tip: Rotor speed in a synchronous motor equals field speed; in an induction motor the rotor always runs a little slower, and that difference (slip) is what makes torque.

Common Mistake: Expecting an induction motor to reach synchronous speed under load. If it did, no flux would cut the rotor conductors and there would be no torque.

Important Questions:

• What is slip in an induction motor? It is the amount by which the rotor speed falls behind the speed of the rotating stator field, expressed as a fraction of synchronous speed; it grows with the mechanical load.
• Why does a single-phase induction motor need a starting mechanism? One winding creates a pulsating field (two equal counter-rotating fields), which gives no starting torque, so a capacitor, split-phase winding or shading coil adds a phase-shifted field.

Chapter 14: Transmission Lines

Difficulty: Hard · Key topics: characteristic impedance, surge impedance, velocity factor, reflections, electrically short and long lines, wavelength, standing waves, SWR, quarter-wave resonance, impedance transformation, waveguides

The final chapter starts with a puzzle: a coaxial cable labelled 50 ohms measures an open circuit with an ohmmeter. The answer is that signals travel at close to the speed of light, so a long enough line shows a resistance-like characteristic impedance, set by its distributed capacitance and inductance, until the first reflection returns. The chapter covers velocity factor, reflections from open, short or mismatched ends, and the rule that a line counts as long when it is at least a quarter wavelength. It then develops standing waves, nodes, antinodes and SWR, quarter-wave impedance transformation using Z0 = √(Zsource x Zload), and ends with waveguides and their TE and TM modes.

Key Points:

• Characteristic (surge) impedance is the resistance an infinitely long line would show; it depends only on distributed L and C, not on wire resistance or leakage.
• Velocity factor is the propagation speed relative to light and equals 1/√(relative permittivity) of the insulation; typical cables fall between 0.66 and 0.80.
• A line terminated with a resistor equal to Z0 behaves as if it were infinite, with no reflected wave; any mismatch makes a reflected wave.
• Wavelength = velocity / frequency; a line is electrically long when it is at least a quarter wavelength, which is about 775 miles at 60 Hz but only a few feet at 100 MHz.
• Standing waves occur only with a mismatched termination; SWR is the ratio of maximum to minimum amplitude, equal to 1 when matched and infinite for an open or short.
• The lowest resonance of an open or shorted line occurs at a quarter wavelength, and further resonant points occur at harmonics.
• A quarter-wave line with Z0 = √(Zsource x Zload) matches different impedances: the book’s example matches a 300 ohm load to a 75 ohm source using a 150 ohm line.
• Waveguides are hollow metal tubes for very high frequencies, carrying TE or TM modes; the TEM mode of two-conductor cables cannot exist in them.

Memory Tip: Match the load to Z0 and there is nothing to reflect. No reflection means no standing wave, and SWR is 1.

Common Mistake: Trying to measure characteristic impedance with an ohmmeter. The meter is far too slow to see the brief resistive behavior, so a 50 ohm cable simply reads open or short depending on its far end.

Important Questions:

• What is the characteristic impedance of a transmission line? It is the impedance an infinitely long line presents, determined by the line’s distributed inductance and capacitance; terminating a line in this value prevents reflections.
• When is a transmission line considered electrically long? When its length is at least about a quarter of the signal wavelength, so propagation delay matters and the line’s own impedance dominates the circuit.

Appendices A-1 to A-3: About This Book, Contributor List and Licence

Appendix · Difficulty: Easy · Key topics: purpose of the book, copyleft, SPICE method, acknowledgements, contributor credits, licence text

The three appendices explain where the book came from and how it may be reused. In “About This Book” the author says he began as an industrial electronics instructor frustrated with an error-ridden, expensive textbook, and decided to write an affordable one and make it free and “open” in the way free software is. He explains why SPICE circuit simulations appear throughout, so that readers can learn by observing results as well as by reading explanations, and he thanks the free software used to produce the book. The Contributor List explains how to contribute and how credit must be given, and then lists the people who corrected or extended the text. The last appendix reprints the licence that sets the terms for copying, distributing and modifying the book: the Design Science License in older builds, and the Creative Commons Attribution licence in the current edition.

Key Points:

• The book is copyrighted but “copylefted”: anyone may copy, distribute or modify it as long as credit is given and the same freedom is passed on.
• Changes to the text must be documented in the credits with the contributor’s name, and the book is not in the public domain.
• The author uses SPICE inside the book so that readers can learn inductively, by watching simulated circuits, as well as deductively.
• The author warns that electricity can injure or kill, and recommends experiments only with small battery-powered circuits.
• The licence history is GPL first, then the Design Science License (written by Michael Stutz), and finally Creative Commons Attribution in the current edition; both later licences provide the work “as is”, with no warranty.

Memory Tip: Remember the three-part deal: you may copy it, you may change it, and you must credit the authors and let others do the same.

Common Mistake: Treating a copylefted book as public domain. You can reuse it freely, but only on the licence terms, which include credit and documenting your changes.

Important Questions:

• What does “copyleft” mean for this book? The author keeps the copyright but gives everyone the right to copy, distribute and modify the text, provided that credit is given and the same terms are passed to whoever receives it.
• Why does the book use SPICE simulations? They let students see circuit behavior for themselves and check the hand calculations, which supports learning by observation as well as by explanation.

How to Study This Book

The book declares no parts, so your course outline decides which chapters you need. For a first AC Circuit Analysis course, the core is Chapters 1 to 6 (waveforms, complex numbers, reactance, impedance and resonance), plus Chapter 8 on filters and Chapter 11 on power factor. Power-systems and machines courses add Chapters 9, 10 and 13, and instrumentation or communications courses add Chapters 12 and 14. Chapter 7 on mixed-frequency signals sits between circuit analysis and signals courses, so check whether your syllabus treats Fourier series elsewhere.

The book assumes you have finished Volume I (DC): Ohm’s Law, Kirchhoff’s Laws, series-parallel analysis and the basic idea of inductors and capacitors. It even says that SPICE is introduced early in Volume I. If any of those feel shaky, revise them first, because Chapter 1 only touches on them briefly before moving on to phase.

Spend the most time on Chapter 2. Polar and rectangular form, and the habit of adding in rectangular and multiplying in polar, are used on every page after it. Practise the conversions on a scientific calculator until they are automatic. Then work through Chapters 3, 4 and 5 using the table method: write every component as an impedance, fill the table in the same order each time, and finish with E = IZ. The same steps solve series, parallel and series-parallel circuits, and they are the steps that examiners expect to see.

Read Chapters 6, 7 and 8 as one story. Resonance (Chapter 6) explains why a tank circuit picks one frequency, Chapter 7 shows that real signals are mixtures of frequencies, and Chapter 8 uses both ideas to build filters. For exams, learn the resonant frequency formula, the meaning of Q and the half-power bandwidth, and the cutoff frequency of simple RC filters. The Fourier and spectrum material in Chapter 7 is conceptual, and it is the best place to start before a signals course.

Chapter 9 is the longest chapter in the book. For a circuits course, concentrate on Sections 9.1 to 9.6 (mutual inductance, turns ratios, isolation, phasing, tapped windings and regulation). Treat Section 9.7 on special transformers and Section 9.8 on practical limits as a reference you can return to. Chapter 10 builds on it: the Y and delta relationships in Section 10.5 are the part to memorize, and the harmonics sections near its end matter most to students heading into power systems. Chapter 11 is short and very examinable; draw the power triangle for every problem.

Chapters 12, 13 and 14 read more like guided tours than step-by-step methods. In Chapter 13 learn the induction motor first (Section 13.7: rotating field, synchronous speed, slip and starting) and the single-phase starting methods that follow, then browse the stepper, brushless and commutator sections as needed. Chapter 14 introduces characteristic impedance, reflections, standing waves and quarter-wave matching. It is the bridge to an electromagnetics or RF course, and waveguides get only a short introduction at the end.

Know what the book does not offer. It has worked examples, SPICE listings and short REVIEW lists throughout the book, but no end-of-chapter problem sets, so write your own practice questions from the examples. It has no Laplace transforms, operational amplifiers or Smith chart, and no section on variable-frequency motor drives. Its instrument discussions in Chapter 12 are written around analog meters and CRT oscilloscopes, so take current instrument details from your lab manual. The AC theory itself has not aged.

Used In These Programs

This book is part of the AC Circuit Analysis course in: BS Electrical Engineering · BS Engineering Technology. Browse all Circuit Analysis books or all Engineering books.

Who Should Read This

This book is written for undergraduate students of BSEE and BSET who are taking AC Circuits or Circuit Analysis II, and for technicians and self-learners who want a plain-language route into phasors, impedance, transformers and three-phase power. It assumes DC circuit analysis from Volume I of the same series and some trigonometry; no other background is needed. Its steady, example-driven style suits self-study. It is not the right choice if you need problem sets with solutions, or Laplace-transform and op-amp analysis. If transformers, induction and electromagnetic waves are new to you, a physics text such as University Physics Volume 2 covers the underlying magnetism and induction in more depth than Chapters 1 and 9 do. For students who want to see where the oscillation in Chapter 6 comes from mathematically, Elementary Differential Equations by Trench treats second-order equations and their applications. Phasors, impedance, power factor and three-phase relationships are also standard material in entry tests for engineering and technology programs.

Applicable Universities

This book is useful for students at Pakistani universities offering BS Electrical Engineering and BS Engineering Technology including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

Official Copy of Lessons in Electric Circuits Volume II AC

The author gives the whole book away free on his own site, with a PDF of the full volume and chapter-by-chapter web pages. Click below to open the official page and download the PDF.

FAQs

Related Books

• University Physics Volume 2 – OpenStax (electricity, magnetism and induction at calculus level)
• College Physics 2e – OpenStax (algebra-based physics for a gentler start)
• University Physics Volume 1 – OpenStax (oscillations and waves, the idea behind resonance)
• Elementary Differential Equations – Trench (the mathematics behind RLC circuits)

Lessons in Electric Circuits, Volume II: AC gives BSEE and BSET students a free, plain-language route from the sine wave through complex numbers, resonance and filters to transformers, three-phase power, motors and transmission lines. Browse more Engineering books for the rest of your semester.

Lessons in Electric Circuits, Volume II: AC, by Tony R. Kuphaldt, copyright 2000-2023, Sixth Edition, last update November 6, 2021. Licensed under the Creative Commons Attribution 4.0 International licence (CC BY 4.0); copies from before the 2021 revision carry the Design Science License. Chapter 13 was originally written by Dennis Crunkilton. Official source: https://www.ibiblio.org/kuphaldt/electricCircuits/AC/index.html