Why a Current Can Flow Through a Coil Yet Consume No Power

In an AC circuit, current and voltage do not always rise and fall together, so multiplying their RMS values overstates the real power. The fix is the power factor, which also explains why some current does no useful work at all.

This lesson covers average power and wattless current, the AC generator, and how AC compares with DC.

**Average power in an AC circuit is $P=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$, where $\cos \varphi$ is the power factor, and the component of current $I_{\text{rms}}\sin \varphi$, at $90^{\circ }$ to the voltage, is wattless because it consumes no average power.

Power factor in each case:**

- Pure resistor: $\varphi =0$, $\cos \varphi =1$, full power
- Pure inductor or capacitor: $\varphi =90^{\circ }$, $\cos \varphi =0$, zero average power
- Series LCR: $\cos \varphi =R/Z$

Why zero power is possible. In a pure inductor, energy flows in during one quarter-cycle and returns to the source in the next, so the average is zero.

Worked example 1. An LCR circuit has $R=30\Omega$ and $Z=50\Omega$ on 220 V, so $I=4.4$ A and $\cos \varphi =0.60$:

$$P=220\times 4.4\times 0.60=581\text{ W}{I}^{2}R=4.4^2\times 30=581\text{ W}$$

Worked example 2 — wattless current. A current of 10 A lags the voltage by $60^{\circ }$:

$$I\cos 60^{\circ }=5.0\text{ A (active)}I\sin 60^{\circ }=8.66\text{ A (wattless)}$$

An everyday example. Factories add capacitor banks to cancel the lagging current drawn by their motors, raising the power factor and cutting losses in the wiring.

The substance. Wattless current still heats the supply wires, so a low power factor wastes energy even though the load itself uses none of that current.

**An AC generator rotates a coil in a magnetic field so its flux linkage keeps changing, inducing an emf $\epsilon =NBA\omega \sin \omega t$ with peak value ${\epsilon }_0=NBA\omega$.

Construction:

- Armature — a coil of N turns wound on a soft-iron core
- Field magnets — provide a strong, steady field B
- Slip rings — rotate with the coil, each joined to one end
- Brushes — carbon contacts pressing on the slip rings to carry current out

Working.** The flux is $\Phi =NBA\cos \omega t$, so $\epsilon =-\frac{d\Phi }{dt}=NBA\omega \sin \omega t$. The emf is greatest when the coil's plane is parallel to the field, where flux is changing fastest, and zero when the plane is perpendicular.

Worked example. A 100-turn coil of area 0.050 m${}^2$ rotates at 50 Hz in a 0.40 T field:

$${\epsilon }_0=100\times 0.40\times 0.050\times 2\pi \times 50=628\text{ V}{\epsilon }_{\text{rms}}=\frac{628}{\sqrt{2}}=444\text{ V}$$

An everyday example. Hydroelectric plants use falling water to spin turbines that turn generator coils, or in large plants the magnets, at a steady speed.

The substance. Zero flux does not mean zero emf — when the coil's plane lies along the field, flux is zero but changing fastest, so the emf is at its peak.

AC can be stepped up or down with transformers and transmitted over long distances with small losses, while DC is needed for charging batteries, electroplating and electronic circuits.

Advantages of AC:

- Transformers change its voltage easily and efficiently
- High-voltage transmission means low current and low $I^{2}R$ loss
- AC generators and motors are simple and robust

Advantages of DC:

- Needed for electrolysis, electroplating and battery charging, where the direction must stay fixed
- Electronic devices need steady DC, obtained from AC through rectifiers

Disadvantage of AC. A 220 V RMS supply peaks at 311 V, so insulation must withstand the peak, and a shock from it can be more dangerous.

Worked example — why transmit at high voltage. Deliver 1.0 MW through a line of $2.0\Omega$.

$$\text{At 11 kV: }I=\frac{1.0\times 10^6}{1.1\times 10^4}=90.9\text{ A},I^{2}R=90.9^2\times 2.0=16.5\text{ kW}$$

$$\text{At 220 kV: }I=4.55\text{ A},I^{2}R=4.55^2\times 2.0=41\text{ W}$$

Raising the voltage 20 times cuts the loss $20^2=400$ times.

An everyday example. Your phone charger converts the AC from the socket to low-voltage DC, because the battery can only be charged with current in one direction.

The substance. The choice is not AC or DC but where each fits — AC for generation and transmission, DC at the device.