Why a 230 V Household Supply Actually Peaks Above 325 V

The mains supply in Indian homes keeps reversing direction, completing $50$ cycles every second. Such alternating current is easy to step up and down and to send over long distances — but describing it needs new ideas: RMS values, phasors and reactance.

This part covers peak and RMS values, AC in a resistor with phasors, AC in an inductor, and AC in a capacitor.

**An alternating voltage varies as $v=v_m\sin \omega t$, where $v_m$ is the peak value, and its RMS value — the steady voltage that would produce the same heating — is $V=\frac{v_m}{\sqrt{2}}\approx 0.707v_m$; the same relation holds for current.

Why RMS.** The average of a sine wave over a cycle is zero, but heating depends on $i^2$, whose average is $\frac{i_m^2}{2}$:

$$\overline{P}=\frac{i_m^{2}R}{2}=I^{2}R\Rightarrow I=\frac{i_m}{\sqrt{2}}$$

Worked example. The household supply is $230$ V RMS at $50$ Hz:

$$v_m=\sqrt{2}\times 230\approx 325\text{ V},\omega =2\pi \times 50=100\pi {\text{ rad s}}^{-1}$$

A current of $2.3$ A RMS has a peak value of $\sqrt{2}\times 2.3\approx 3.25$ A.

Measuring AC. AC voltmeters and ammeters read RMS values, and appliance ratings are RMS values too.

An everyday example. **The $230$ V rating on an electric iron** is an RMS value — the voltage actually swings between about $+325$ V and $-325$ V in every cycle.

The substance. Insulation must withstand the peak voltage, not just the RMS value, which is why cables are rated well above $230$ V.

**Across a pure resistor, $v=v_m\sin \omega t$ drives $i=\frac{v_m}{R}\sin \omega t$, so current and voltage are in phase, and each can be drawn as a phasor — a vector rotating anticlockwise at angular speed $\omega$ whose projection on the vertical axis gives the instantaneous value.

Resistor:**

$$i=i_m\sin \omega t,i_m=\frac{v_m}{R}$$

- Current and voltage reach their peaks at the same instant
- Average power is $\overline{P}=VI=I^{2}R$

Phasors:

- The length of the phasor is the peak value
- It rotates anticlockwise at angular frequency $\omega$
- Its projection on the vertical axis gives the instantaneous value
- For a resistor, the voltage and current phasors point the same way

Worked example. A $230$ V RMS supply drives a $46\Omega$ heater:

$$I=\frac{230}{46}=5\text{ A},i_m=5\sqrt{2}\approx 7.07\text{ A},\overline{P}=230\times 5=1150\text{ W}$$

An everyday example. The coil of an electric kettle or geyser behaves almost like a pure resistor, so its current rises and falls exactly in step with the supply voltage.

The substance. Phasors are not vectors in space — they are a drawing tool that turns phase differences into angles.

**In a pure inductor the current lags the voltage by $\frac{\pi }{2}$, and the opposition to current is the inductive reactance $X_L=\omega L$, so $i_m=\frac{v_m}{X_L}$.

Derivation.** The applied voltage balances the induced emf, $v_m\sin \omega t=L\frac{di}{dt}$. Integrating,

$$i=-\frac{v_m}{\omega L}\cos \omega t=i_m\sin \left(\omega t-\frac{\pi }{2}\right),i_m=\frac{v_m}{\omega L}$$

Inductive reactance:

$$X_L=\omega L=2\pi fL$$

It rises with frequency, and for steady DC, $X_L=0$.

Power. The average power over a cycle is zero — energy stored in the magnetic field is returned to the source.

Worked example. A $0.5$ H inductor on a $230$ V, $50$ Hz supply:

$$X_L=2\pi \times 50\times 0.5=50\pi \approx 157\Omega ,I=\frac{230}{157}\approx 1.46\text{ A}$$

An everyday example. The choke of an old fluorescent tube light is an inductor that limits the current with very little heating, because an ideal inductor uses no average power.

The substance. An inductor opposes high frequencies but lets steady DC pass freely — the reverse of a capacitor.

**In a pure capacitor the current leads the voltage by $\frac{\pi }{2}$, and the opposition to current is the capacitive reactance $X_C=\frac{1}{\omega C}$, so $i_m=\frac{v_m}{X_C}$.

Derivation.** The charge is $q=C{v}_m\sin \omega t$, so

$$i=\frac{dq}{dt}=\omega C{v}_m\cos \omega t=i_m\sin \left(\omega t+\frac{\pi }{2}\right),i_m=\omega C{v}_m$$

Capacitive reactance:

$$X_C=\frac{1}{\omega C}=\frac{1}{2\pi fC}$$

It falls as frequency rises; for steady DC it is infinite, so a capacitor blocks DC. The average power is again zero.

Worked example. A $10\mu$F capacitor on a $230$ V, $50$ Hz supply:

$$X_C=\frac{1}{2\pi \times 50\times 10\times 10^{-6}}\approx 318\Omega ,I=\frac{230}{318}\approx 0.72\text{ A}$$

Comparison:

- Resistor — in phase; $R$ does not depend on frequency
- Inductor — current lags by $90^{\circ }$; $X_L$ rises with frequency
- Capacitor — current leads by $90^{\circ }$; $X_C$ falls with frequency

An everyday example. Capacitor-type fan regulators control the AC current reaching a ceiling fan's motor while wasting very little energy as heat.

The substance. A capacitor passes AC even though no charge crosses its dielectric — charge simply surges back and forth, charging and discharging the plates.