A.C. Through a Resistor, a Capacitor and an Inductor: MDCAT Physics notes
A.C. Through a Resistor, a Capacitor and an Inductor for MDCAT: phase relations, reactance X_L = 2πfL and X_C = 1/2πfC, rms values and resonance.
Alternating current basics
An alternating voltage varies sinusoidally: $V = V_0\sin\omega t$ with $\omega = 2\pi f$. The rms (effective) values are
$$V_{\text{rms}} = \frac{V_0}{\sqrt2} \approx 0.707V_0 \qquad I_{\text{rms}} = \frac{I_0}{\sqrt2}$$
Mains in Pakistan is 220–230 V rms at 50 Hz.
Pure resistor
- Current and voltage are in phase.
- Opposition is the resistance $R$, independent of frequency.
- Instantaneous power $P = v^2/R = i^2R$; average power $= V_{\text{rms}}I_{\text{rms}}$.
Worked example: $V_0 = 40$ V, $f = 25$ Hz, $R = 20\ \Omega$. At $t = 1/300$ s, $\omega t = 2\pi(25)/300 = \pi/6 = 30^\circ$, so $v = 40\sin30^\circ = 20$ V and instantaneous power $= 20^2/20 = 20$ W.
Pure capacitor
- The capacitor must charge before voltage builds up across it, so current leads voltage by $90^\circ$ (a quarter cycle).
- Opposition is the capacitive reactance:
$$X_C = \frac{1}{\omega C} = \frac{1}{2\pi fC}$$
$X_C \propto 1/f$: doubling the frequency halves $X_C$ and doubles the current. A capacitor blocks DC ($f = 0$, $X_C$ infinite) and passes high-frequency AC easily.
Pure inductor
- The back emf opposes changes in current, so current lags voltage by $90^\circ$.
- Opposition is the inductive reactance:
$$X_L = \omega L = 2\pi fL$$
$X_L \propto f$: doubling the frequency doubles $X_L$ and halves the current. An inductor passes DC freely ($X_L = 0$) but opposes high-frequency AC.
| Element | Opposition | Depends on $f$ | Phase | Average power |
|---|---|---|---|---|
| Resistor | $R$ | No | $I$ in phase with $V$ | $V_{\text{rms}}I_{\text{rms}}$ |
| Capacitor | $X_C = 1/2\pi fC$ | $\propto 1/f$ | $I$ leads $V$ by $90^\circ$ | Zero |
| Inductor | $X_L = 2\pi fL$ | $\propto f$ | $I$ lags $V$ by $90^\circ$ | Zero |
Memory aid "CIVIL": in a Capacitor I comes before V; in an inductor (L) V comes before I.
Resonance
In an RLC series circuit the reactances cancel when $X_L = X_C$, at the resonant frequency
$$f_r = \frac{1}{2\pi\sqrt{LC}}$$
At resonance the impedance is minimum ($= R$) and the current is maximum. Tuning a radio is electrical resonance: the capacitor is adjusted until the circuit's natural frequency matches the station frequency.
Key formulas
- $V_{\text{rms}} = V_0/\sqrt2$
- $X_C = 1/2\pi fC$, $X_L = 2\pi fL$
- $I = V/X$
- $f_r = 1/2\pi\sqrt{LC}$
Common MDCAT traps
- Capacitor: current leads. Inductor: current lags. Do not swap.
- Raising frequency increases $X_L$ but decreases $X_C$.
- Phase shift in pure C or L is $90^\circ$, not $45^\circ$.
- For instantaneous values, convert $\omega t$ to degrees before taking the sine.
Quick revision
- Resistor: $V$ and $I$ in phase.
- Pure L or C consumes no average power.
- Capacitor blocks DC; inductor passes DC.
- Radio tuning uses electrical resonance.