Alternating Current notes: ISC Class 12 Physics

1. Sinusoidal AC; mean and rms values

$i = I_0\sin(\omega t + \phi_0)$ : peak $I_0$ , period $T$ , frequency $f = 1/T$ , angular frequency $\omega = 2\pi f$ , phase $(\omega t + \phi_0)$ . Indian mains: 230 V rms, 50 Hz, $T = 20$ ms.
A phasor is a vector of length equal to the peak, turning anticlockwise at $\omega$ ; its projection gives the instantaneous value. The quantity whose phasor is ahead leads.
Mean over a half cycle: $\bar I = \dfrac{2}{T}\int_0^{T/2} I_0\sin\omega t\,dt = \dfrac{2I_0}{\pi} \approx 0.637\,I_0$ . Over a full cycle the mean is zero.
rms value: the steady current giving the same heating in a resistor. $\overline{\sin^2\omega t} = \tfrac12$ , so $I_\text{rms} = I_0/\sqrt2 \approx 0.707\,I_0$ (sine waves only). AC meters and ratings use rms values.

i / I₀ = sin ωt
(i / I₀)² = sin² ωt

Mean of i² = I₀²/2, so I_rms = I₀/√2.

Formulas

Half-cycle mean
$\bar I = \dfrac{2I_0}{\pi}$
rms values
$I_\text{rms} = \dfrac{I_0}{\sqrt2},\qquad V_\text{rms} = \dfrac{V_0}{\sqrt2}$

Key terms

rms value: The steady (DC) value that would produce the same heat in a given resistor in the same time; I₀/√2 for a sinusoidal current.
Phasor: A rotating vector whose length is the peak value of a sinusoidal quantity and whose projection gives its instantaneous value.

2. AC through pure R, L and C

Resistor: $i$ in phase with $v$ ; $I_\text{rms} = V_\text{rms}/R$ .
Inductor: $v = L\,di/dt$ gives $i = I_0\sin(\omega t - \pi/2)$ : current lags by 90°. Inductive reactance $X_L = \omega L = 2\pi fL$ .
Capacitor: $i = dq/dt = C\,dv/dt$ gives $i = I_0\sin(\omega t + \pi/2)$ : current leads by 90°. Capacitive reactance $X_C = 1/(\omega C) = 1/(2\pi fC)$ .
Graphs against f: $X_L$ a straight line through the origin; $X_C$ a rectangular hyperbola. A coil passes DC and low frequencies; a capacitor blocks DC and passes high frequencies.

v / V₀ across L
i / I₀ in L

v / V₀ across C
i / I₀ in C

X_L = 2πfL, L = 0.35 H
X_C = 1/(2πfC), C = 15 μF

X_L ∝ f; X_C ∝ 1/f.

Formulas

Reactances
$X_L = \omega L = 2\pi fL,\qquad X_C = \dfrac{1}{\omega C} = \dfrac{1}{2\pi fC}$

Key terms

Reactance: The opposition of an inductor (ωL) or a capacitor (1/ωC) to alternating current, in ohms; it depends on frequency and causes no heating.

3. The series LCR circuit

Same current through all; phasors: $V_R$ along $I$ , $V_L$ 90° ahead, $V_C$ 90° behind. Add as phasors: $V^2 = V_R^2 + (V_L - V_C)^2$ .
Impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$ , $I = V/Z$ ; phase angle $\tan\phi = (X_L - X_C)/R$ . $X_L > X_C$ : inductive, $I$ lags; $X_C > X_L$ : capacitive, $I$ leads.
RL: $Z = \sqrt{R^2 + \omega^2L^2}$ , $\tan\phi = \omega L/R$ (lags). RC: $Z = \sqrt{R^2 + 1/\omega^2C^2}$ , $\tan\phi = 1/\omega CR$ (leads).
Against frequency: $Z$ falls to a minimum $R$ where $X_L = X_C$ and rises again; $I = V/Z$ peaks there. The voltage across L or C can exceed the supply voltage.

Z, with L = 25 mH, C = 4.0 μF, R = 10 Ω

I = V/Z is greatest where Z is least.

Formulas

Series LCR
$Z = \sqrt{R^2 + (X_L - X_C)^2},\qquad \tan\phi = \dfrac{X_L - X_C}{R}$

Key terms

Impedance: The total opposition of a circuit to AC, Z = V/I; for a series LCR circuit √(R² + (X_L − X_C)²).
Phase angle: The angle φ by which the supply voltage leads (or lags) the current; tan φ = (X_L − X_C)/R.

4. Power, power factor, wattless current and the choke

$p = vi$ ; average over a cycle: $P = V_\text{rms}I_\text{rms}\cos\phi = I_\text{rms}^2R$ . $V_\text{rms}I_\text{rms}$ is the apparent power (VA).
Power factor $\cos\phi = R/Z$ : 1 for a pure resistor (and at resonance), 0 for a pure L or C.
Wattless current: the component $I\sin\phi$ in quadrature with $V$ ; it does no work over a cycle but still heats the supply cables. The active component is $I\cos\phi$ .
Choke coil: large L, very small R, laminated iron core; power factor nearly zero, so it limits an AC with almost no loss (a resistor would waste $I^2R$ ).

v / V₀
i / I₀ (lags by 60°)
p / V₀I₀ = vi / V₀I₀

p = vi; the average is V_rms I_rms cos φ.

Table of the phase angle, power factor and average power for a pure resistor, a pure inductor or capacitor, a general series LCR circuit and an LCR circuit at resonance
Load	Phase angle φ	Power factor cos φ	Average power
Pure resistor	0	1	$V_\text{rms}I_\text{rms}$
Pure inductor or pure capacitor	90°	0	zero (wattless)
Series LCR	$\tan^{-1}\dfrac{X_L - X_C}{R}$	$R/Z$	$I_\text{rms}^2R$
Series LCR at resonance	0	1	$V_\text{rms}^2/R$

Formulas

Average power
$P = V_\text{rms}I_\text{rms}\cos\phi,\qquad \cos\phi = \dfrac{R}{Z}$

Key terms

Power factor: cos φ, the ratio of the true power to the apparent power V_rms I_rms; equal to R/Z for a series circuit.
Wattless current: The component I sin φ of an alternating current that is 90° out of phase with the voltage and so delivers no average power.
Choke coil: An inductor of large inductance and negligible resistance used to limit an alternating current without wasting energy.

5. Resonance, bandwidth, Q factor and LC oscillations

Resonance when $X_L = X_C$ : $\omega_0 = 1/\sqrt{LC}$ , $f_0 = \dfrac{1}{2\pi\sqrt{LC}}$ . Then $Z = R$ , $I_0 = V/R$ , $\phi = 0$ , power factor 1; $V_L = V_C$ , each possibly much larger than $V$ .
Half-power frequencies $f_1$ , $f_2$ : $I = I_0/\sqrt2$ . Bandwidth $\Delta\omega = R/L$ ( $\Delta f = R/2\pi L$ ). Q factor $Q = \omega_0/\Delta\omega = \omega_0L/R = \tfrac1R\sqrt{L/C}$ ; $V_L = QV$ at resonance. (Formulae only; derivations not required.)
Smaller R: taller, sharper peak, larger Q, same $f_0$ . Used in radio tuning.
LC oscillations (qualitative): a charged capacitor across a coil; energy passes back and forth between the electric field of C and the magnetic field of L at $f = \dfrac{1}{2\pi\sqrt{LC}}$ ; resistance makes the oscillations die away.

R = 10 Ω
R = 30 Ω

Bandwidth f₂ − f₁ = R/(2πL); smaller R gives a sharper peak.

energy in C (electric)
energy in L (magnetic)

Qualitative picture: energy shuttles between C and L.

Formulas

Resonant frequency
$f_0 = \dfrac{1}{2\pi\sqrt{LC}}$
Bandwidth and Q
$\Delta\omega = \dfrac{R}{L},\qquad Q = \dfrac{\omega_0L}{R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}$

Key terms

Resonant frequency: The frequency 1/(2π√(LC)) at which X_L = X_C in a series LCR circuit, so that the impedance is least and the current greatest.
Q factor: ω₀/Δω = ω₀L/R for a series LCR circuit: a measure of the sharpness of resonance.
Bandwidth: The gap between the two half-power frequencies, at which the current is I₀/√2.

6. The AC generator; AC and DC

Principle: electromagnetic induction in a coil turned in a magnetic field. Parts: armature coil, field magnet, slip rings, carbon brushes; driven by a turbine or engine.
Theory: with the normal along B at $t = 0$ , $\Phi = NBA\cos\omega t$ and $e = -d\Phi/dt = NBA\omega\sin\omega t$ ; peak $e_0 = NBA\omega$ when the plane is along B; $e = 0$ when the plane faces B. Frequency = rotations per second.
AC against DC: AC reverses periodically and is easily transformed for efficient long-distance transmission; DC flows one way and is needed for charging, electroplating and electronics. AC is measured by its rms value.

flux Φ / NBA = cos ωt
emf e / e₀ = sin ωt

e = −dΦ/dt: the emf is largest where the flux changes fastest.

Table comparing alternating and direct current by direction, frequency, ease of changing the voltage, sources, typical uses and the meters that measure them
	Alternating current (AC)	Direct current (DC)
Direction and size	reverses periodically; size varies	one direction; usually steady
Frequency	50 Hz in India	zero
Change of voltage	easily stepped up or down by transformers	needs electronic converters
Sources	AC generators, inverters	cells, batteries, solar cells, DC dynamos
Needed for	transmission over long distances, motors, heaters	charging batteries, electroplating, electronics
Measured by	hot-wire or moving-iron meters (rms)	moving-coil meters

Formulas

Generator emf
$e = NBA\omega\sin\omega t,\qquad e_0 = NBA\omega$

Key terms

Slip rings: Two rings on the axle of an AC generator, joined to the ends of the coil, which with the brushes connect the turning coil to the external circuit.
Armature: The coil of many turns that is rotated in the magnetic field of a generator.

Exam tips

AC values: mains and meters give rms; peak = √2 × rms; half-cycle mean = 2/π × peak.
Reactances: X_L = 2πfL, X_C = 1/(2πfC); in series circuits add voltages as phasors.
Power: P = V_rms I_rms cos φ = I_rms² R; a pure L or C takes no power.
Resonance: f₀ = 1/(2π√LC), Z = R; quote bandwidth R/L and Q = ω₀L/R without derivation.
Generator: e = NBAω sin ωt; say where the angle is measured from.

1. Sinusoidal AC; mean and rms values

Formulas

Key terms

2. AC through pure R, L and C

Formulas

Key terms

3. The series LCR circuit

Formulas

Key terms

4. Power, power factor, wattless current and the choke

Formulas

Key terms

5. Resonance, bandwidth, Q factor and LC oscillations

Formulas

Key terms

6. The AC generator; AC and DC

Formulas

Key terms

Exam tips

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