AC Circuits

13.1 Introduction

• DC (direct current): the polarity of the supply voltage is fixed — the positive and negative ends of the supply do not change.
• AC (alternating current): the polarity of the voltage keeps reversing periodically.
• Electricity is supplied mostly as AC because it is cheap to generate and convenient to transport.
• Appliances such as TV, computer and radio use a rectifier to convert AC to DC; fans, fridges, air conditioners and heaters run directly on AC.
• Remember this: an AC shock is attractive while a DC shock is repulsive, so 220 V AC is more dangerous than 220 V DC. Also 220 V AC has a peak value of $\pm\sqrt2\times220 \approx \pm311$ V, whereas DC has a fixed value of 220 V only.

13.2 AC Generator

• An AC generator produces a time-dependent emf $e = e_0\sin\omega t$, where $e_0$ is the peak emf and $\omega$ the angular frequency of rotation of the coil.
• The current in the connected circuit has the same time dependence: $i = i_0\sin(\omega t + \alpha)$, with $\alpha$ the phase difference between current and emf.
• Graph of $e$ against $\omega t$ is a sine curve: the direction of the emf reverses after every half revolution of the coil. Such an emf is called alternating emf and the corresponding current alternating current.
• Peak value: the maximum value of the alternating current (or emf) in either direction.
• The average value of alternating current or emf over a complete cycle is zero.

13.3 Average and RMS Values

• Average value of AC is defined over one half cycle, because over a full cycle the average of $\sin\omega t$ — and hence of the AC — is zero and carries no meaning.
• Average value of $\sin\theta$ over $0$ to $\pi$ is $2/\pi = 0.637$, so $i_{av} = 0.637\,i_0$ and $e_{av} = 0.637\,e_0$.
• A moving-coil ammeter or voltmeter responds to the average value, so it cannot measure AC. What is needed is a property independent of the direction of flow — the heating effect, which depends on the square of the current.
• RMS (effective, virtual) value: the value of that steady current which produces the same heat in the same resistance in the same time as the given AC does.
• $i_{rms} = i_0/\sqrt2 = 0.707\,i_0$ and $e_{rms} = e_0/\sqrt2 = 0.707\,e_0$.
• Heat produced by a sinusoidal AC over a complete cycle is $H = i_{rms}^2Rt$ — the same as that produced by a DC of magnitude $i_{rms}$ for the same time.
• All AC ammeters, voltmeters and quoted supply ratings (220 V, 110 V) are rms values.

13.4 Phasors

• A rotating vector that represents a sinusoidally varying quantity is called a phasor; the diagram representing it is a phasor diagram.
• The length of the arrow represents the peak value of the quantity ($i_0$ or $e_0$); the phasor is inclined at $\omega t$ (or $\omega t + \alpha$) to the horizontal axis and rotates anticlockwise.
• The projection of the phasor on a fixed axis gives the instantaneous value: on the Y-axis for the sine form, on the X-axis for the cosine form.
• Representing emf and current as rotating vectors lets us add them by the laws of vector addition, which is why phasors simplify AC circuit analysis.

13.5 Different Types of AC Circuits

Resistor alone

• $e = e_0\sin\omega t$ gives $i = (e_0/R)\sin\omega t = i_0\sin\omega t$, with $i_0 = e_0/R$.
• Current and emf are in phase — both reach zero, minimum and maximum at the same instants.
• $R$ behaves identically for AC and DC, and can reduce both equally effectively.
• Phasor diagram: the phasors for $i$ and $e$ point in the same direction, making an angle $\omega t$ with OX; phase angle between them is zero.

Inductor alone

• Applied emf opposes the self-induced emf $e' = -L\,di/dt$. With no resistance, Kirchhoff's voltage law gives $di = (e_0/L)\sin\omega t\,dt$; integrating (the constant of integration is zero, since $i$ must oscillate about zero) gives $i = i_0\sin(\omega t - \pi/2)$ with $i_0 = e_0/\omega L$.
• Current lags the emf by $\pi/2$ rad (90°); equivalently the voltage across $L$ leads the current by 90°.
• Inductive reactance: $X_L = \omega L = 2\pi fL$, the opposing nature of an inductor to the flow of AC. SI unit ohm; dimensions the same as those of resistance.
• $X_L \propto f$ and $X_L \propto L$. For DC, $f = 0$, so $X_L = 0$ — a pure inductor cannot reduce DC. It passes DC and blocks high-frequency AC.
• Phasor diagram: the $i_0$ phasor is turned 90° clockwise from the $e_0$ phasor.

Capacitor alone

• $q = Ce$, so $i = dq/dt = C\,d(e_0\sin\omega t)/dt = \omega Ce_0\cos\omega t = i_0\sin(\omega t + \pi/2)$ with $i_0 = e_0/(1/\omega C)$.
• Current leads the emf by $\pi/2$ rad (90°).
• Capacitive reactance: $X_C = 1/\omega C = 1/2\pi fC$. It is defined as the ratio of the rms voltage across the capacitor to the corresponding rms current. SI unit ohm.
• $X_C \propto 1/f$ and $X_C \propto 1/C$. For DC, $f = 0$, so $X_C = \infty$ — a capacitor blocks DC and acts as an open circuit, while it passes high-frequency AC.
• Phasor diagram: the $i_0$ phasor is turned 90° anticlockwise from the $e_0$ phasor.

Series LCR circuit

• The same current $i = i_0\sin\omega t$ flows through all three; the voltages are $e_R = iR$ (in phase with $i$), $e_L = iX_L$ (leads $i$ by 90°) and $e_C = iX_C$ (lags $i$ by 90°).
• $e_L$ and $e_C$ are 180° out of phase, so the net reactive voltage is $(e_L - e_C)$.
• Phasor addition gives $e = \sqrt{e_R^2 + (e_L-e_C)^2} = i\sqrt{R^2+(X_L-X_C)^2} = iZ$.
• Impedance $Z = \sqrt{R^2+(X_L-X_C)^2}$, in ohm; it is the ratio of rms voltage to rms current. Its reciprocal is admittance, unit ohm$^{-1}$ or siemens.
• $\tan\phi = (X_L-X_C)/R$, where $\phi$ is the angle by which the voltage leads the current.
• Impedance triangle: base $R$, perpendicular $(X_L-X_C)$, hypotenuse $Z$, angle at the base $\phi$.

13.6 Power in an AC Circuit

• Instantaneous power $P = ei$; because both change every instant, the useful quantity is the average over one full cycle.
• Pure resistor: $P = e_0i_0\sin^2\omega t$, and since $\int_0^T\sin^2\omega t\,dt = T/2$, $P_{av} = e_0i_0/2 = e_{rms}i_{rms}$. This product is also called the apparent power.
• Pure inductor: $i = i_0\sin(\omega t - \pi/2)$, so $P = -e_0i_0\sin\omega t\cos\omega t$, and $\int_0^T\sin\omega t\cos\omega t\,dt = 0$, giving $P_{av} = 0$.
• Ideal capacitor: $i = i_0\cos\omega t$, so $P = e_0i_0\sin\omega t\cos\omega t$, and again $P_{av} = 0$.
• Series LCR: $P_{av} = e_{rms}i_{rms}\cos\phi$ — the true power.
• Power factor $\cos\phi = P_{av}/(e_{rms}i_{rms}) =$ true power / apparent power $= R/Z$.
• $\cos\phi = 1$ ($\phi = 0$) for a non-inductive circuit; $\cos\phi = 0$ ($\phi = 90^\circ$) for a pure inductor or ideal capacitor.
• Wattless (idle) current: the current through a pure inductor or an ideal capacitor, which consumes no power for its maintenance in the circuit. Power is dissipated only in the resistance.
• A low power factor in a transmission line means a large current is needed to deliver the same power, and the $i^2R$ loss in the line therefore becomes large.

13.7 LC Oscillations

• A charged capacitor connected across a resistanceless inductor sets up electrical oscillations of constant amplitude and frequency, called LC oscillations.
• Energy surges back and forth between the electric field of the capacitor, $U_E = q^2/2C$, and the magnetic field of the inductor, $U_B = \dfrac{1}{2}LI^2$.
• Stage by stage: capacitor fully charged with $q_0$ (all energy electrostatic, $I = 0$); capacitor discharges and $I$ grows (energy shared); capacitor fully discharged and $I = I_0$ (all energy magnetic, $\dfrac{1}{2}LI_0^2$); flux decreases and the induced emf drives the current in the same direction, charging the capacitor with reversed polarity; the process then repeats in the opposite direction and the circuit returns to its initial state.
• Frequency: $f = \dfrac{1}{2\pi\sqrt{LC}}$. Maximum current from energy conservation: $\dfrac{1}{2}\dfrac{q_0^2}{C} = \dfrac{1}{2}LI_0^2$.
• If there is no loss of energy the amplitude stays constant and the oscillations are undamped.
• In practice LC oscillations are damped, for two reasons: (a) every inductor has some resistance, so energy is lost as heat and the amplitude dies out; (b) even with zero resistance, energy is radiated away as electromagnetic waves — the basis of radio and TV transmitters.

13.8 Electric Resonance

• Resonance occurs when a system with a natural frequency of oscillation is driven by a source whose frequency equals that natural frequency; the amplitude of oscillation then becomes large.
• Series resonance circuit: $L$, $C$ and $R$ in series, admitting maximum current for a given AC frequency.
• As $f$ rises, $X_L = \omega L$ increases and $X_C = 1/\omega C$ decreases; at $\omega_r$ they are equal, so $Z = \sqrt{R^2+0} = R$, the least possible value.
• $\omega_r L = 1/\omega_r C$ gives $\omega_r = 1/\sqrt{LC}$ and $f_r = \dfrac{1}{2\pi\sqrt{LC}}$.
• Series resonance curve: rms current plotted against frequency, peaking sharply at $f_r$. The peak height is $e_0/R$, so a smaller $R$ gives a taller, sharper peak.
• Parallel resonance circuit: coil $L$ and condenser $C$ in parallel across the source. The total current is $i = e_0\left(\omega C - \dfrac{1}{\omega L}\right)\cos\omega t$, minimum when $\omega C = 1/\omega L$, i.e. at the same $f_r = \dfrac{1}{2\pi\sqrt{LC}}$.
• Do you know: for resonance to occur, both $L$ and $C$ must be present, because only then do the mutually 180°-out-of-phase voltages across them cancel and the whole source voltage appear across $R$. There is no resonance in an LR or a CR circuit.

13.9 Sharpness of Resonance — Q Factor

• Current amplitude in a series LCR circuit is $i_0 = e_0\Big/\sqrt{R^2+\left(\omega L - \dfrac{1}{\omega C}\right)^2}$; it is maximum when $\omega_r L - \dfrac{1}{\omega_r C} = 0$.
• Half-power frequencies: the two values of $\omega$ at which the current amplitude falls to $1/\sqrt2$ of its maximum, so that the power dissipated is halved. They lie symmetrically about $\omega_r$: $\omega_1 = \omega_r + \Delta\omega$ and $\omega_2 = \omega_r - \Delta\omega$.
• Bandwidth $= \omega_1 - \omega_2 = 2\Delta\omega$.
• Q factor of a series resonant circuit is defined as the ratio of the resonant frequency to the difference between the two frequencies, taken one on either side of resonance, at which the current amplitude is $1/\sqrt2$ times its value at resonance.
• $Q = \dfrac{\omega_r}{\omega_1-\omega_2} = \dfrac{\omega_r}{2\Delta\omega}$; it is dimensionless.
• Larger $Q$ means smaller bandwidth, a sharper peak and a more selective circuit.
• On the resonance curve the low-frequency side is dominated by the capacitor's reactance and the high-frequency side by the inductor's reactance; resonance occurs in the middle.
• Tuning a radio or TV is an LCR resonance: the antenna picks up many frequencies, but maximum current flows for the one equal to $f_r = \dfrac{1}{2\pi\sqrt{LC}}$. A large $Q$ keeps other stations weak, and varying the adjustable capacitor $C$ tunes in the desired station.

13.10 Choke Coil

• Reducing AC with an ordinary resistance wastes energy as Joule heat ($i^2Rt$). A choke coil reduces the current without much loss of energy.
• Construction: thick insulated copper wire wound closely in a large number of turns over a soft-iron laminated core.
• It offers a large reactance $X_L = \omega L$ to AC, so the current is reduced; the laminated core reduces eddy-current loss.
• Power dissipated $P = e_{rms}i_{rms}\cos\phi$ with $\cos\phi = \dfrac{R}{\sqrt{R^2+\omega^2L^2}}$.
• $L$ is very large and $R$ very small, so $\cos\phi$ is nearly zero and the power loss is very small.
• The only real loss is hysteresis loss in the core, reduced by using soft iron.

Standard One-Mark and Two-Mark Answers

• Brightness of a lamp in series with a capacitor when $C$ is increased: $X_C = 1/2\pi fC$ decreases, so $Z$ decreases, current increases and the lamp glows brighter.
• Why the total impedance falls when a capacitor is added in series with $L$ and $R$: $X_C$ acts opposite to $X_L$, so the net reactance $(X_L - X_C)$ is smaller than $X_L$ alone and hence $Z$ is smaller.
• Why a capacitor behaves like a pure conductor at very high frequency: $X_C = 1/2\pi fC \to 0$ as $f$ becomes very large, so it offers almost no opposition.
• Wattless current: the current in a pure inductor or ideal capacitor, where $\phi = 90^\circ$ and $P_{av} = e_{rms}i_{rms}\cos 90^\circ = 0$, so it consumes no power.
• Natural frequency of an LC circuit is $f = \dfrac{1}{2\pi\sqrt{LC}}$, and at this frequency $X_L = X_C$, so the net reactance of the circuit is zero.
• In an LR circuit with $X_L = R$, power factor $P_1 = R/\sqrt{R^2+R^2} = 1/\sqrt2$. Adding a capacitor with $X_C = X_L$ makes the circuit purely resistive, so $P_2 = 1$ and $P_1/P_2 = 1/\sqrt2 = 0.707$.

Traps That Cost Marks

• Writing the average value of AC over a full cycle as $0.637\,i_0$. Over a full cycle it is zero; $0.637\,i_0$ is the half-cycle average.
• Adding the voltages across $R$, $L$ and $C$ arithmetically. They must be added as phasors: $e = \sqrt{e_R^2+(e_L-e_C)^2}$. The reading across $L$ or $C$ alone can even exceed the source voltage.
• Confusing $X_L$ and $X_C$ behaviour with frequency. $X_L$ rises with $f$ (inductor blocks high-frequency AC, passes DC); $X_C$ falls with $f$ (capacitor blocks DC, passes high-frequency AC).
• Forgetting the $2\pi$ when converting: $X_L = 2\pi fL$, not $fL$; $X_C = 1/2\pi fC$, not $1/fC$.
• Using the peak value where the question gives an rms value. A supply described as "220 V, 50 Hz" is rms; convert with $e_0 = \sqrt2\,e_{rms}$ before using $i_0 = e_0/Z$.
• Treating the impedance at resonance as zero. It is minimum and equals $R$, not zero.
• Interchanging the two resonance circuits: series is minimum $Z$, maximum current, acceptor; parallel is maximum $Z$, minimum current, rejector.
• Saying resonance can happen in an LR or CR circuit. Both $L$ and $C$ are essential.
• Forgetting to convert mH, $\mu$F and pF to henry and farad before substituting. 1 mH $= 10^{-3}$ H, 1 $\mu$F $= 10^{-6}$ F, 1 pF $= 10^{-12}$ F.
• Quoting the power in a pure inductor or capacitor as $e_{rms}i_{rms}$. It is zero — the power factor is zero.
• Dropping $\cos\phi$ from $P_{av} = e_{rms}i_{rms}\cos\phi$ in an LCR numerical, which turns true power into apparent power.
• Leaving $\phi$ in degrees when a time lag is asked. $\Delta t = \phi/\omega$ needs $\phi$ in radian.