AC Generator

13.2 AC Generator

The source of AC — the generator — produces a time-dependent emf. From the study of electromagnetic induction in the previous chapter, the emf induced in a coil rotating uniformly in a magnetic field is

$$e = e_0\sin\omega t$$

where $e_0$ is the peak value of the emf and $\omega$ is the angular frequency of rotation of the coil in the AC generator.

Since the time variation of the current follows that of the emf, the current in a circuit connected to this generator is of the form

$$i = i_0\sin(\omega t + \alpha)$$

where $i_0$ is the peak value of the current and $\alpha$ is the phase difference between the current and the emf. The value of $\alpha$ is fixed by what the circuit contains: zero for a resistor, $-\pi/2$ for a pure inductor, $+\pi/2$ for an ideal capacitor, and $\pm\phi$ with $\tan\phi = (X_L-X_C)/R$ for a series LCR combination.

The Graph of $e$ versus $\omega t$

Plot $e$ on the vertical axis against $\omega t$ on the horizontal axis. The curve is a sine curve: starting at zero, rising to $+e_0$ at $\omega t = \pi/2$, returning to zero at $\pi$, falling to $-e_0$ at $3\pi/2$ and back to zero at $2\pi$. One such complete cycle takes a time $T = 2\pi/\omega$.

From the graph it is seen that the direction of the emf is reversed after every half revolution of the coil. An emf of this kind is called an alternating emf, and the corresponding current is called an alternating current.

Standard Results

• $e = e_0\sin\omega t$, $i = i_0\sin(\omega t + \alpha)$.
• $\omega = 2\pi f = 2\pi/T$; for the Indian mains, $f = 50$ Hz, $T = 0.02$ s, $\omega = 100\pi \approx 314.2$ rad/s.
• Peak value: the maximum value of the alternating current (or emf) in either direction.
• The average value of alternating current over a complete cycle is zero. This is the board's own one-line answer to that question.