Alternating Current

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 7.1 to 7.8 with no gaps.

LC oscillations have been removed, yet Exercise 7.6 still asks for them. Older editions had a full section deriving the free oscillation of charge between a capacitor and an inductor. A search of this chapter returns zero hits for "LC oscillation".

Exercise 7.6 nonetheless reads: "A charged $30\mu$F capacitor is connected to a $27$ mH inductor. What is the angular frequency of free oscillations of the circuit?" — an exercise that has outlived its own section.

It is still answerable. The resonance treatment in 7.6.2 gives ${\omega }_0=1/\sqrt{LC}$, and that is the same quantity. What you have lost is the derivation showing energy sloshing between the capacitor's electric field and the inductor's magnetic field.

The sharpness of resonance has been removed too. There is no section on it, and zero hits for "sharpness" or "bandwidth". But the end-of-chapter symbol table still lists Quality factor $Q$, with the formula $Q={\omega }_{0}L/R$ — a quantity defined in a table that no section of the chapter teaches.

Two terms are used here that other chapters no longer define. The list of transformer energy losses in section 7.8 names eddy currents and hysteresis. Eddy currents were removed from Chapter 6, and hysteresis from Chapter 5. Both are now used without ever being taught.

Two symbol traps if you work from an extracted PDF. The ohm sign $\Omega$ extracts as a capital "W", and the micro symbol as a plain "m", so "$R=20\Omega$" reads as "20 W" and "$35\mu$F" as "35 mF". Both appear in this chapter's exercises.

2. RMS Values, and the Resistor (Textbook 7.2)

An alternating voltage $v=v_m\sin \omega t$ averages to zero over a cycle, so the mean value is useless for describing it. What matters is the heating effect, which depends on $i^2$ and is therefore always positive.

Root mean square values are defined so that an AC current delivers the same average power as a DC current of that value:

$I_{\text{rms}}=\frac{i_m}{\sqrt{2}}=0.707i_m,V_{\text{rms}}=\frac{v_m}{\sqrt{2}}$

Every unlabelled AC value is an rms value. The 220 V mains has a peak of $220\sqrt{2}\approx 311$ V. Exercise 7.2 tests the conversion both ways.

A pure resistor puts current and voltage exactly in phase: both peak together and both cross zero together. Ohm's law applies to the rms values directly, and the average power is:

$P=V_{\text{rms}}{I}_{\text{rms}}=I_{\text{rms}}^{2}R$

which is Exercise 7.1 in full.

Phasors (7.3). A sinusoid is represented as a rotating vector whose length is the amplitude and whose angle is the phase. Adding voltages that are out of step then becomes vector addition rather than trigonometry, which is what makes the LCR circuit tractable.

3. Inductors and Capacitors in AC (Textbook 7.4 to 7.5)

Each element opposes current in its own way, and each shifts the phase by a quarter cycle — in opposite directions.

Inductive reactance rises with frequency, because a faster-changing current induces a larger back emf. Capacitive reactance falls with frequency, because the plates have less time to charge up and oppose the flow.

The mnemonic. In an inductor the current lags; in a capacitor it leads. Exercises 7.3 and 7.4 are direct substitutions into $X_L$ and $X_C$ followed by $I=V/X$.

Neither stores energy permanently, and neither dissipates any. Over a complete cycle, energy drawn during one quarter is returned during the next, so the average power is exactly zero in both cases. That is Exercise 7.5, and the reason is that the phase difference is $90^{\circ }$, making $\cos \varphi =0$.

4. The Series LCR Circuit and Resonance (Textbook 7.6)

With all three in series, the same current flows through each, but their voltages are out of step. The phasor diagram adds $V_R$ along the current, $V_L$ at $+90^{\circ }$ and $V_C$ at $-90^{\circ }$.

Since $V_L$ and $V_C$ are exactly opposite, they subtract, and Pythagoras gives:

$Z=\sqrt{R^2+(X_L-X_C{)}^2},\tan \varphi =\frac{X_L-X_C}{R}$

Reading the phase angle. If $X_L>X_C$ the circuit is inductive and the current lags; if $X_C>X_L$ it is capacitive and the current leads.

Resonance (7.6.2). When the two reactances are equal they cancel entirely:

$X_L=X_C⟹{\omega }_0=\frac{1}{\sqrt{LC}}$

At this frequency:

• $Z$ falls to its minimum, equal to $R$ alone.
• The current reaches its maximum, $V/R$.
• The circuit is purely resistive, $\varphi =0$ and the power factor is 1.
• Power is maximum, $P=V^2/R$ — which is Exercise 7.7, giving 2000 W.

The result that surprises everyone. At resonance the individual voltages across $L$ and $C$ can each far exceed the supply voltage, because they cancel each other rather than the source. In Exercise 7.8, a 230 V supply produces 1437.5 V across each of $L$ and $C$ — more than six times the source — while their sum is exactly zero.

Note that ${\omega }_0$ depends only on $L$ and $C$. Resistance does not shift the resonant frequency; it only controls how sharp the peak is, which is the topic the current edition no longer covers.

5. Power and the Power Factor (Textbook 7.7)

The instantaneous product $vi$ averaged over a cycle gives:

$P=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$

where $\cos \varphi$ is the power factor. The product $V_{\text{rms}}{I}_{\text{rms}}$ alone is the apparent power; only the fraction $\cos \varphi$ of it is actually consumed.

Reading the power factor:

• Pure resistor: $\varphi =0$, $\cos \varphi =1$, all the power is dissipated.
• Pure inductor or capacitor: $\varphi =90^{\circ }$, $\cos \varphi =0$, no power is consumed at all.
• At resonance: $\cos \varphi =1$, as the circuit is purely resistive.

Wattless current is the name for the component $I\sin \varphi$, at right angles to the voltage. It flows, it can be measured, and it transfers no net energy over a cycle.

Why industry cares. A low power factor means large currents for little useful power, and those currents still cause $I^{2}R$ losses in the supply cables. Capacitor banks are installed to correct it.

6. Transformers (Textbook 7.8)

A transformer changes an alternating voltage using mutual inductance between two coils on a shared soft iron core. It works only on AC, because a steady current produces no changing flux.

For an ideal transformer, with no flux leakage and no losses:

$\frac{V_s}{V_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}$

Voltage and current move in opposite directions. A step-up transformer with more secondary turns raises the voltage and lowers the current by the same factor, so that $V_p{I}_p=V_s{I}_s$. A transformer never creates power.

Four sources of energy loss, which section 7.8 lists:

• Flux leakage — not all the primary flux links the secondary. Reduced by winding one coil over the other.
• Winding resistance — $I^{2}R$ heating in the copper. Reduced by using thicker wire.
• Eddy currents — the alternating flux induces circulating currents in the core, which heats it. Reduced by laminating the core.
• Hysteresis — repeated reversal of the core's magnetisation dissipates energy each cycle. Reduced by choosing a soft magnetic material.

As flagged above, the last two are named here but no longer defined anywhere in the book.

Why transmission uses high voltage. Stepping up before transmission cuts the current for the same power, and since line loss goes as $I^{2}R$, halving the current quarters the loss. The voltage is stepped down again near the consumer.

Summary

• $I_{\text{rms}}=i_m/\sqrt{2}$ and $V_{\text{rms}}=v_m/\sqrt{2}$; an unlabelled AC value is always rms.
• The 220 V mains has a peak of about 311 V.
• A resistor keeps current and voltage in phase, with $P=V_{\text{rms}}{I}_{\text{rms}}$.
• Inductive reactance $X_L=\omega L$ rises with frequency; the current lags by $90^{\circ }$.
• Capacitive reactance $X_C=1/\omega C$ falls with frequency; the current leads by $90^{\circ }$.
• An inductor passes DC and blocks high frequencies; a capacitor blocks DC and passes high frequencies.
• A pure inductor or capacitor consumes zero average power, because $\cos \varphi =0$.
• Series LCR: $Z=\sqrt{R^2+(X_L-X_C{)}^2}$ and $\tan \varphi =(X_L-X_C)/R$.
• $X_L>X_C$ means inductive and lagging; $X_C>X_L$ means capacitive and leading.
• Resonance at ${\omega }_0=1/\sqrt{LC}$: $Z$ minimum and equal to $R$, current maximum, $\varphi =0$, power maximum at $V^2/R$.
• ${\omega }_0$ depends only on $L$ and $C$ — resistance does not shift it.
• At resonance $V_L$ and $V_C$ can each vastly exceed the supply, cancelling each other rather than the source.
• $P=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$, with $\cos \varphi$ the power factor; the component $I\sin \varphi$ is the wattless current.
• Transformer: $V_s/V_p=N_s/N_p=I_p/I_s$, working by mutual inductance and only on AC.
• Stepping voltage up steps current down; a transformer never creates power.
• Transformer losses: flux leakage, winding resistance, eddy currents, hysteresis.
• Transmission at high voltage cuts $I^{2}R$ line loss, since the loss goes as the square of the current.
• LC oscillations and the sharpness of resonance have been removed, though Exercise 7.6 and the symbol table's quality factor still refer to them.