One of the examples of Faraday’s law cited in Chapter 30 was an electric generator. A turbine, which might be powered by expanding steam or falling water, causes a coil of wire to rotate in a magnetic field. As the coil spins, the emf and the induced current oscillate sinusoidally. The emf is alternately positive and negative, causing the charges to flow in one direction and then, a half cycle later, in the other. The oscillation frequency of the grid in North and South America is $f=60\text{Hz,}$ whereas most of the rest of the world uses a 50 Hz oscillation.

The generator’s peak emf—the peak voltage—is a fixed, unvarying quantity, so it might seem logical to call a generator an alternating-voltage source. Nonetheless, circuits powered by a sinusoidal emf are called AC circuits, where AC stands for alternating current. By contrast, the steady-current circuits you studied in Chapter 28 are called DC circuits, for direct current.

AC circuits are not limited to the use of 50 Hz or 60 Hz power-line voltages. Audio, radio, television, and telecommunication equipment all make extensive use of AC circuits, with frequencies ranging from approximately ${10}^2\text{Hz}$ in audio circuits to approximately ${10}^9\text{Hz}$ in cell phones. These devices use electrical oscillators rather than generators to produce a sinusoidal emf, but the basic principles of circuit analysis are the same.

You can think of an AC generator or oscillator as a battery whose output voltage undergoes sinusoidal oscillations. The instantaneous emf of an AC generator or oscillator, shown graphically in FIGURE 32.1a, can be written

where ${\epsilon }_0$ is the peak or maximum emf and $\omega =2\pi f$ is the angular frequency in radians per second. Recall that the units of emf are volts. As you can imagine, the mathematics of AC circuit analysis are going to be very similar to the mathematics of simple harmonic motion.

FIGURE 32.1 An oscillating emf can be represented as a graph or as a phasor diagram.

Full Alternative Text

An alternative way to represent the emf and other oscillatory quantities is with the phasor diagram of FIGURE 32.1b. A phasor is a vector that rotates counterclockwise (ccw) around the origin at angular frequency $\omega$. The length or magnitude of the phasor is the maximum value of the quantity. For example, the length of an emf phasor is ${\epsilon }_0$. The angle $\omega t$ is the phase angle, an idea you learned about in Chapter 15, where we made a connection between circular motion and simple harmonic motion.

The quantity’s instantaneous value, the value you would measure at time t, is the projection of the phasor onto the horizontal axis. This is also analogous to the connection between circular motion and simple harmonic motion. FIGURE 32.2 helps you visualize the phasor rotation by showing how the phasor corresponds to the more familiar graph at several specific points in the cycle.

FIGURE 32.2 The correspondence between a phasor and points on a graph.