1.6 — AC Analysis: Phasors, Impedance, Power Factor and Resonance

A sine wave goes into a circuit with a capacitor in it. What comes out? Chapter 1.5 says you must solve a differential equation. Do that for a circuit with four capacitors and two inductors and you have a fourth-order differential equation and a ruined afternoon.

There is a better way, and it is one of the great engineering shortcuts. Represent each sine wave by a single complex number, and every derivative turns into a multiplication. The differential equations become ordinary algebra, and you can solve an AC circuit with exactly the same techniques — series, parallel, nodal, Thevenin — that you learned for resistors.

Charles Proteus Steinmetz introduced the method to American engineering in 1893, and it changed power system design from an art into a calculation.

The letters this chapter uses, said out loud

AC analysis introduces more new symbols than any other chapter in this Part, so here they are first.

\omega — "omega", angular frequency in radians per second, \omega = 2\pi f. It appears instead of f because the sine function's argument must be an angle, and the 2\pi converts cycles into radians.

\phi — "phi", the phase angle: how far one sine wave is shifted relative to another, measured as an angle rather than as a time, so that the answer does not change when the frequency does. A quarter-cycle shift is \phi = 90° whether the signal is at 50 Hz or 50 MHz, and that frequency-independence is the entire reason phase is measured in degrees.

Z — impedance, the AC generalisation of resistance, in ohms. It is a complex number: its real part is genuine resistance that turns energy into heat, and its imaginary part is reactance that stores energy and gives it back. The letter is from the German Zweipol.

X — reactance, the imaginary part of Z, also in ohms. X_L = \omega L for an inductor and X_C = 1/\omega C for a capacitor, and section 3 derives both.

j — the square root of -1, written j rather than i because i already means current. In this chapter it does one job: multiplying by j rotates a phasor by a quarter turn, which is exactly the quarter-cycle delay that an inductor or capacitor imposes. That is why complex numbers appear in circuit theory at all — they are the shortest way to write "and it happens a quarter cycle later".

\theta — "theta", the angle of the impedance, and therefore the angle between voltage and current. Its cosine is the power factor.

Q — the quality factor, a pure number measuring how sharply a resonant circuit selects one frequency. High Q means a narrow peak and a long ring. It collides with Q for charge from Chapter 1.1, and context is the only thing separating them.

\omega_0 — the resonant frequency, where the inductive and capacitive reactances cancel exactly. Section 5 derives \omega_0 = 1/\sqrt{LC}.

1. What a sine wave actually is

A general sinusoidal voltage is written

v(t) = V_m\cos(\omega t + \phi)

with three pieces of information and no more:

• V_m is the amplitude or peak value, in volts.
• \omega (omega) is the angular frequency in radians per second, related to the ordinary frequency f in hertz by \omega = 2\pi f. The 2\pi is there because one full cycle is 2\pi radians.
• \phi (phi) is the phase, in radians or degrees, saying where in its cycle the wave is at t=0.

The period is T = 1/f seconds — the time for one complete cycle. Mains electricity in India and Europe is 50 Hz, so T = 20 ms and \omega = 2\pi(50) = 314 rad/s. In North America it is 60 Hz, T = 16.7 ms, \omega = 377 rad/s.

The whole reason sines matter more than any other shape is a fact proved in Volume II, Chapter 9.1 and used constantly from Part 4 onward: put a sine into a linear circuit and you get a sine out, at exactly the same frequency, with only the amplitude and phase changed. No other waveform survives a circuit unchanged. Feed in a square wave and you get out something that is not a square wave. Feed in a sine and the shape is untouched. That property is what makes the whole phasor idea possible.

RMS — the number your meter shows

Mains is called 230 V, but its peak is 325 V. Which is right? Both, and the difference matters.

The useful measure of an alternating voltage is the one that delivers the same heating power as an equivalent DC voltage. Power in a resistor is v^2/R, so you want the average of v^2, then the square root to get back to volts. That is the root mean square, in exactly that order backwards: square it, take the mean, take the root.

For a sine, \overline{v^2} = \overline{V_m^2\cos^2(\omega t)}, and the average of \cos^2 over a whole cycle is exactly \tfrac12 (because \cos^2\theta = \tfrac12(1+\cos2\theta) and the \cos 2\theta part averages to zero). So

V_{rms} = \frac{V_m}{\sqrt2} \approx 0.707\,V_m

So 230 V rms has a peak of 230\times\sqrt2 = 325 V, and a peak-to-peak swing of 650 V. Insulation must survive the peak; heating is set by the rms. Every AC voltage quoted in an equipment specification is rms unless it says otherwise, and every oscilloscope measures peak unless you tell it otherwise, which is a mismatch that has caused a lot of confusion and some destroyed equipment.

The \sqrt2 factor is only true for sines. A square wave has V_{rms} = V_m; a triangle wave has V_m/\sqrt3.

2. The phasor

Here is the trick. Euler's formula (Volume II, Chapter 2.5) says

e^{j\theta} = \cos\theta + j\sin\theta

where j is the square root of -1. Electrical engineering writes j rather than i, because i was already taken by current, and this convention is universal in the field.

So \cos\theta is the real part of e^{j\theta}, and our voltage can be written

v(t) = \operatorname{Re}\left\{V_me^{j(\omega t+\phi)}\right\} = \operatorname{Re}\left\{V_me^{j\phi}\cdot e^{j\omega t}\right\}

Now notice: if every signal in the circuit is at the same frequency \omega, then the factor e^{j\omega t} is the same everywhere and can be dropped. What remains is a single complex number carrying the amplitude and phase:

\mathbf{V} = V_me^{j\phi} = V_m\angle\phi

This is the phasor. The notation V_m\angle\phi, read "V-m at an angle of phi", is standard shorthand for the same complex number in polar form.

Picture it as an arrow of length V_m sitting at angle \phi, and the real sine wave as the shadow that arrow casts on the horizontal axis while it spins at \omega radians per second. Everything spins together, so you can freeze the picture and look only at the relative angles.

Left: two phasors frozen mid-rotation, the angle between them being the phase difference. Right: the same two signals plotted against time. The whole point of the left picture is that it holds the same information with no time axis at all.

Adding two sine waves of the same frequency, which is horrible trigonometry in the time domain, becomes adding two complex numbers in phasor form. That alone would justify the method.

3. Impedance — Ohm's law for AC

Now apply the phasor idea to the three elements.

Resistor. v = iR has no derivative, so nothing changes: \mathbf{V} = \mathbf{I}R. Voltage and current stay exactly in phase.

Capacitor. i = C\,dv/dt. In phasor form the voltage is \mathbf{V}e^{j\omega t}, and differentiating e^{j\omega t} just multiplies it by j\omega. So:

\mathbf{I} = j\omega C\,\mathbf{V} \qquad\Longrightarrow\qquad \mathbf{V} = \frac{1}{j\omega C}\mathbf{I}

Inductor. v = L\,di/dt becomes, by the same step:

\mathbf{V} = j\omega L\,\mathbf{I}

All three now have the form \mathbf{V} = \mathbf{Z}\mathbf{I}, and \mathbf{Z} is called the impedance, measured in ohms:

Element	Impedance \mathbf{Z}	Magnitude	What the phase means
Resistor	R	R	current in phase with voltage
Inductor	j\omega L	\omega L	current lags voltage by 90°
Capacitor	\dfrac{1}{j\omega C} = \dfrac{-j}{\omega C}	\dfrac{1}{\omega C}	current leads voltage by 90°

This is the payoff. Impedances combine in series and parallel by exactly the same rules as resistances, dividers work the same way, and nodal analysis, mesh analysis, Thevenin and Norton all carry across unchanged — you just do the arithmetic with complex numbers.

Making sense of the j

The j is not decoration. Multiplying by j rotates a phasor by +90^\circ; dividing by j rotates by -90^\circ. So the j in an inductor's impedance is a compact way of saying "the voltage happens a quarter cycle ahead of the current".

Why ahead? Because v = L\,di/dt means the voltage tracks the slope of the current, and the slope of a sine peaks a quarter-cycle before the sine itself does. The mathematics and the physics say the same thing.

The mnemonic taught everywhere is ELI the ICE man: in an inductor (L), voltage E comes before current I — "ELI". In a capacitor (C), current I comes before voltage E — "ICE". Silly, effective, and it has survived a century of teaching.

Frequency dependence, which is the whole point of filters

Look at the magnitudes again:

|Z_L| = \omega L \qquad |Z_C| = \frac{1}{\omega C}

An inductor's impedance rises with frequency; a capacitor's falls. At DC (\omega=0) the inductor is a short and the capacitor is an open, which is exactly the steady-state rule from Chapter 1.5, now recovered as a special case. At very high frequency they swap roles.

This is the entire mechanism of filtering. Put a resistor and capacitor in series across a signal and take the output across the capacitor: at low frequency the capacitor's impedance is large so it takes most of the voltage, and at high frequency it is small so it takes almost none. You have built a low-pass filter without adding anything but a divider. Its cut-off frequency, where the output power has fallen to half, is

f_c = \frac{1}{2\pi RC}

which is the same RC time constant from Chapter 1.5 seen from the frequency side. Swap the two components and take the output across the resistor and you have a high-pass filter with the same cut-off. Part 5 develops this into proper filter design.

Impedance, resistance and reactance

Write the impedance in rectangular form:

\mathbf{Z} = R + jX

R is the resistance (the real part, which dissipates energy as heat) and X is the reactance (the imaginary part, which stores and returns energy without dissipating it). Inductive reactance is positive, capacitive reactance is negative.

Magnitude and angle:

|\mathbf{Z}| = \sqrt{R^2+X^2}, \qquad \theta = \arctan\frac{X}{R}

The angle \theta is the phase by which the current lags the voltage. Positive means inductive and lagging; negative means capacitive and leading; zero means purely resistive.

The reciprocal is the admittance \mathbf{Y} = 1/\mathbf{Z} = G + jB, where G is conductance and B is susceptance. It exists for the same reason conductance did in Chapter 1.3: parallel combinations are easier when you add rather than take reciprocals.

Worked example — a series RC at two frequencies

R = 1\ \text{k}\Omega, $C = 0.1\ \mu$F, driven by 10 V rms.

At 1 kHz:

|Z_C| = \frac{1}{2\pi(1000)(10^{-7})} = \frac{1}{6.283\times10^{-4}} = 1592\ \Omega

\mathbf{Z} = 1000 - j1592, \qquad |\mathbf{Z}| = \sqrt{10^6 + 2.53\times10^6} = 1880\ \Omega

\theta = \arctan\frac{-1592}{1000} = -57.9^\circ

I = \frac{10}{1880} = 5.32\ \text{mA, leading the voltage by } 57.9^\circ

The voltage across the capacitor is 5.32\ \text{mA} \times 1592 = 8.47 V, and across the resistor 5.32\ \text{mA}\times1000 = 5.32 V.

Notice: 8.47 + 5.32 = 13.8, not 10. The voltages do not add arithmetically, because they are 90° apart. They add as vectors: \sqrt{8.47^2+5.32^2} = 10.0 ✓. This surprises people the first time they measure it on a real circuit and think their meter is broken.

At 10 kHz: the capacitor's reactance drops to 159 Ω, so |\mathbf{Z}| = \sqrt{10^6+25300} = 1013\ \Omega, the phase is only -9^\circ, and the capacitor now takes just 1.57 V of the 10. Ten times the frequency has almost eliminated the capacitor's effect. That is the low-pass action in numbers.

4. AC power, and why the electricity company cares

In DC, power was simply VI. In AC it splits into three quantities, and the split is worth real money.

Take v = V_m\cos\omega t and i = I_m\cos(\omega t - \theta), with the current lagging by \theta. Instantaneous power is the product, and using the identity \cos A\cos B = \tfrac12[\cos(A-B) + \cos(A+B)]:

p(t) = \frac{V_mI_m}{2}\left[\cos\theta + \cos(2\omega t - \theta)\right]

The second term oscillates at twice the supply frequency and averages to zero over a cycle. The first term is constant. So the average power is

\boxed{P = V_{rms}I_{rms}\cos\theta}

The three quantities:

• Real power P = V_{rms}I_{rms}\cos\theta, in watts. The energy actually converted to heat, light or motion. This is what you pay for.
• Reactive power Q = V_{rms}I_{rms}\sin\theta, in volt-amperes reactive (var). Energy that sloshes into the magnetic or electric field and comes back out again each cycle, doing no net work.
• Apparent power S = V_{rms}I_{rms}, in volt-amperes (VA). Simply the product of the readings, the thing cables and transformers must actually be sized for.

They form a right triangle, S^2 = P^2 + Q^2, called the power triangle, and the ratio

\text{power factor} = \cos\theta = \frac{P}{S}

is the fraction of the delivered current that does useful work.

Why a bad power factor costs money

Consider a factory drawing 100 kW of real power at 400 V with a power factor of 0.7, which is typical for a plant full of induction motors, since a motor is largely an inductor.

S = \frac{100\ \text{kW}}{0.7} = 143\ \text{kVA}, \qquad I = \frac{143{,}000}{400} = 357\ \text{A}

At a power factor of 1.0 the same 100 kW would need only 250 A. The extra 107 A does no work at all, but it still heats every cable, transformer and switch on the way there, and the loss goes as I^2R (Chapter 1.1). The supply company must build infrastructure for 357 A while selling energy for 250 A worth of work — which is why industrial tariffs include a power factor penalty.

The fix is power factor correction: connect capacitors across the load. The motor's inductive reactive power is positive, the capacitor's is negative, and they cancel. The required capacitance follows directly from the reactive power you need to remove:

Q_C = P(\tan\theta_1 - \tan\theta_2), \qquad C = \frac{Q_C}{\omega V_{rms}^2}

To take the factory above from 0.7 (\theta_1 = 45.6^\circ, \tan = 1.02) to 0.95 (\theta_2 = 18.2^\circ, \tan = 0.329):

Q_C = 100{,}000(1.02 - 0.329) = 69.1\ \text{kvar}

C = \frac{69{,}100}{314\times400^2} = 1.38\ \text{mF}

A bank of capacitors totalling about 1.4 millifarads, which is a physically large but entirely ordinary piece of switchgear, and it pays for itself in months.

5. Resonance

Now the phenomenon that this chapter has been building towards, and one of the most consequential ideas in all of engineering.

Put R, L and C in series. The total impedance is

\mathbf{Z} = R + j\omega L + \frac{1}{j\omega C} = R + j\left(\omega L - \frac{1}{\omega C}\right)

The two reactances have opposite signs, so at some frequency they cancel exactly. Set the bracket to zero:

\omega L = \frac{1}{\omega C} \quad\Rightarrow\quad \omega^2 = \frac{1}{LC} \quad\Rightarrow\quad \boxed{f_0 = \frac{1}{2\pi\sqrt{LC}}}

At that resonant frequency:

• The impedance is purely resistive and at its minimum, just R.
• The current is therefore at its maximum, V/R.
• Voltage and current are exactly in phase, so the power factor is 1.

The part that surprises people

At resonance the individual voltages across L and across C can be enormously larger than the supply voltage. They are equal and opposite, so they cancel in the total, but each one individually is real and measurable.

The multiplier is the quality factor:

Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

and the voltage across each reactive element at resonance is Q times the supply voltage.

Worked example. L = 100 mH, C = 100 nF, R = 10\ \Omega, driven by 1 V.

f_0 = \frac{1}{2\pi\sqrt{0.1\times10^{-7}}} = \frac{1}{2\pi\times10^{-4}} = 1592\ \text{Hz}

\omega_0 L = 2\pi(1592)(0.1) = 1000\ \Omega, \qquad Q = \frac{1000}{10} = 100

Current at resonance: 1/10 = 100 mA. Voltage across the inductor: 0.1 \times 1000 = 100 V.

One volt in, one hundred volts across the coil. This is not a violation of anything — the capacitor has -100 V and the two cancel — but it is entirely real, and it is why a resonant circuit driven carelessly can destroy components rated well above the supply voltage.

Q also sets the sharpness of the resonance. The bandwidth, the width of the frequency band over which the response stays within 1/\sqrt2 of its peak, is

\text{BW} = \frac{f_0}{Q}

Our example has a bandwidth of just 16 Hz around 1592 Hz. High Q means a sharp, selective peak; low Q means a broad, gentle one. That trade-off is the design decision in every radio tuner ever built: too sharp and you cut off the sidebands that carry the actual audio, too broad and you hear the adjacent station.

Parallel resonance

The dual circuit — L and C in parallel, called a tank circuit — behaves in mirror image. At f_0 its impedance is at a maximum rather than a minimum, and the current drawn from the source is at a minimum, while a large current circulates round and round the loop between the inductor and the capacitor.

That circulating current is the physical picture worth holding: energy sloshing from the magnetic field of the coil into the electric field of the capacitor and back, twice per cycle, forever if there were no resistance. It is exactly the pendulum trading height for speed. The resistance is what stops it, and Q measures how many cycles it takes to die away.

Where resonance shows up

• Radio tuning. A variable capacitor changes f_0; the station at that frequency produces a big response and the rest are ignored. That is literally what tuning is, and it is why old radios had a big rotating dial connected to a capacitor with interleaved plates.
• MRI. The R stands for resonance. Hydrogen nuclei absorb energy strongly at one precise frequency in a given magnetic field, and the machine listens for what they emit on the way back down.
• Wireless charging. Two coils tuned to the same frequency transfer power far better than two untuned ones. Chapter 9.4.
• The Tacoma Narrows bridge, and the standard warning about soldiers breaking step on a bridge. Mechanical resonance is the same mathematics with mass and spring in place of L and C.
• A quartz crystal is a mechanical resonator with a Q in the tens of thousands, which is why it keeps time far better than any LC circuit and why it sits inside every clock and microcontroller.

6. AC analysis in practice — the method in five steps

1. Convert everything to phasors. Sources become V_m\angle\phi; resistors stay R; inductors become j\omega L; capacitors become 1/(j\omega C).
2. Solve the circuit exactly as if it were resistive. Series, parallel, dividers, nodal, mesh, Thevenin — all valid, all with complex arithmetic.
3. Read off magnitude and angle of whatever you wanted.
4. Convert back to the time domain if you need to: \mathbf{V} = V_m\angle\phi means v(t) = V_m\cos(\omega t + \phi).
5. Check by confirming that the real powers balance and that a purely resistive branch has zero phase.

Two limits to keep in mind. This method only works for a single frequency at a time, because the whole derivation dropped a common e^{j\omega t}. For a signal containing many frequencies you solve at each one separately and add — which is legitimate by superposition, and is precisely what a Fourier analysis in Part 4 is for. And it only works in steady state, after transients have died out; the once-off switching behaviour still belongs to Chapter 1.5.

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Every element so far has been passive and linear, and every circuit has just moved energy around. Part 2 introduces the components that made electronics a subject in its own right — devices built from doped silicon, which do not obey Ohm's law at all, and which can turn a small signal into a large one.

Every formula above, built from scratch

None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.

AC quantities

v(t) = V_m\cos(\omega t+\phi), \qquad \omega = 2\pi f, \qquad T = \frac1f

V_{rms} = \sqrt{\overline{v^2}}

For a sine: \overline{\cos^2} = \tfrac12 because \cos^2\theta = \tfrac12(1+\cos2\theta) and the second term averages to zero, so

V_{rms} = \frac{V_m}{\sqrt2} = 0.707V_m, \qquad V_m = 1.414V_{rms}

Square wave: V_{rms}=V_m. Triangle: V_m/\sqrt3. Half-wave rectified sine: V_m/2. Full-wave rectified sine: V_m/\sqrt2.

Average of a full-wave rectified sine (needed in Chapter 2.2):

V_{avg} = \frac{1}{\pi}\int_0^\pi V_m\sin\theta\,d\theta = \frac{2V_m}{\pi} = 0.637V_m

Phasors and impedance

v(t)=V_m\cos(\omega t+\phi) \;\longleftrightarrow\; \mathbf{V}=V_m\angle\phi = V_me^{j\phi}

Why it works: by Euler's formula the cosine is the real part of V_me^{j\phi}e^{j\omega t}; if every signal shares the same \omega, the common factor e^{j\omega t} cancels out of every equation.

\frac{d}{dt} \;\longrightarrow\; \times\, j\omega \qquad\qquad \int dt \;\longrightarrow\; \div\, j\omega

Z_R = R, \qquad Z_L = j\omega L, \qquad Z_C = \frac{1}{j\omega C} = \frac{-j}{\omega C}

Derivation of Z_L: v = L\,di/dt becomes \mathbf V = L(j\omega)\mathbf I. Derivation of Z_C: i = C\,dv/dt becomes \mathbf I = C(j\omega)\mathbf V, then divide.

\mathbf Z = R+jX, \qquad |\mathbf Z| = \sqrt{R^2+X^2}, \qquad \theta = \arctan\frac XR

\mathbf Y = \frac{1}{\mathbf Z} = G+jB

Series and parallel combination rules are unchanged from resistances, with complex arithmetic.

AC power

p(t) = V_mI_m\cos\omega t\cos(\omega t-\theta) = \frac{V_mI_m}{2}\left[\cos\theta+\cos(2\omega t-\theta)\right]

using \cos A\cos B = \tfrac12[\cos(A-B)+\cos(A+B)]. The second term averages to zero over a cycle, leaving

P = V_{rms}I_{rms}\cos\theta \ \text{(W)}, \quad Q = V_{rms}I_{rms}\sin\theta \ \text{(var)}, \quad S = V_{rms}I_{rms} \ \text{(VA)}

S^2 = P^2+Q^2, \qquad \text{pf} = \cos\theta = \frac PS

\mathbf S = \mathbf V\mathbf I^* = P+jQ

where \mathbf I^* is the complex conjugate of the current phasor.

Power factor correction

Q_C = P(\tan\theta_1-\tan\theta_2), \qquad C = \frac{Q_C}{\omega V_{rms}^2}

Derivation: you need to cancel the difference between the present reactive power P\tan\theta_1 and the target P\tan\theta_2. A capacitor supplies Q_C = V^2/X_C = V^2\omega C, so rearrange for C.

Resonance

Series RLC impedance:

\mathbf Z = R+j\left(\omega L-\frac{1}{\omega C}\right)

Set the imaginary part to zero:

\omega_0L = \frac{1}{\omega_0C} \;\Rightarrow\; \omega_0 = \frac{1}{\sqrt{LC}} \;\Rightarrow\; f_0 = \frac{1}{2\pi\sqrt{LC}}

Q = \frac{\omega_0L}{R} = \frac{1}{\omega_0CR} = \frac1R\sqrt{\frac LC}

Voltage magnification: at resonance I = V/R, so V_L = I\omega_0L = V\cdot\frac{\omega_0L}{R} = QV. Each reactive element carries Q times the supply voltage, in opposite phase.

\text{BW} = \frac{f_0}{Q} = \frac{R}{2\pi L}, \qquad f_{1,2} = f_0\left(\sqrt{1+\frac{1}{4Q^2}} \mp \frac{1}{2Q}\right)

f_0 = \sqrt{f_1f_2}

The centre frequency is the geometric mean of the two half-power points, not the arithmetic one — a detail that matters when a filter's band is wide.

First-order filter cut-off

f_c = \frac{1}{2\pi RC} \qquad\text{(RC)}, \qquad f_c = \frac{R}{2\pi L} \qquad\text{(RL)}

Derivation: the cut-off is defined where the output power is half the input, so the magnitude ratio is 1/\sqrt2. For the RC low-pass, \left|\dfrac{1/j\omega C}{R+1/j\omega C}\right| = \dfrac{1}{\sqrt{1+(\omega RC)^2}}, which equals 1/\sqrt2 when \omega RC = 1, so \omega_c = 1/RC and f_c = \omega_c/2\pi.

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Every one of these is used again from Part 2 onward. The next page puts them to work on sixteen problems.

What the next chapter fixes

Part 1 has treated every component as an ideal object obeying a law. None of them explains why a resistor resists or where a diode's one-way behaviour comes from. Part 2 opens the components up: what silicon actually is, what doping does, and how a junction between two doped regions becomes the device that everything else in this volume is built from.