In an AC generator, a loop of wire rotates within a magnetic field. As it spins, the voltage induced doesn't stay constant. It rises to a peak, falls to zero, reverses to a negative peak, and returns to zero again. This completes one full 360-degree rotation of the loop and traces out a perfect sine wave.

The highest point of this wave is the peak voltage, written as $V_{max}$. This value corresponds to the moment the wire loop is cutting through the magnetic field lines at the fastest rate. The entire repeating pattern is called a cycle.

This smooth, continuous oscillation is the defining characteristic of alternating current. Unlike DC, which has a constant voltage, AC voltage is always in motion, swinging rhythmically between its positive and negative peaks.

To describe this oscillation, we need to quantify its speed. The most direct way is frequency ($f$), which measures how many full cycles occur in one second. The unit for frequency is Hertz (Hz).

A closely related concept is the period ($T$), which is the time it takes to complete one full cycle. Frequency and period are simply reciprocals of each other. If a wave has a frequency of 50 Hz, it completes 50 cycles every second. Its period is therefore 1/50th of a second, or 20 milliseconds.

While frequency is intuitive, for mathematical calculations, it's often more convenient to use angular velocity, represented by the Greek letter omega ($ω$). Angular velocity measures the rate of rotation in radians per second. Since one full circle (or one full cycle of a wave) contains $2\pi$ radians, the relationship is straightforward.

With these tools, we can create a powerful formula to find the voltage at any specific moment. This is called the instantaneous voltage, denoted as $v(t)$. It describes the exact value of the voltage at time $t$. The shape of the AC waveform is sinusoidal because it follows the sine function.

Let's break down the argument of the sine function, $\omega t$. Angular velocity $\omega$ has units of radians/second, and time $t$ has units of seconds. When multiplied, the seconds cancel out, leaving an angle in radians. The sine of this angle, a value between -1 and 1, is then scaled by the peak voltage $V_{max}$ to give the instantaneous voltage.

For example, at one-quarter of a period ($t = T/4$), the angle is $\omega t = (2\pi/T) \times (T/4) = \pi/2$ radians (or 90°). Since $\sin(\pi/2) = 1$, the voltage at that instant is $v(T/4) = V_{max} \times 1 = V_{max}$. This makes sense; at a quarter of the way through its rotation, the generator loop is inducing the maximum possible voltage.

What if we need to compare two different AC signals that don't start at the same time? This is where the concept of phase angle comes in. A phase angle, represented by the Greek letter phi ($φ$), is an offset that shifts the entire wave forward or backward in time.

We modify our equation to include it:

Phase is crucial in AC circuits containing capacitors and inductors, because these components introduce phase shifts between voltage and current. The amount of this shift depends directly on the frequency ($f$) of the AC signal. A higher frequency can cause a greater phase shift in certain components, fundamentally changing how the circuit behaves.

Understanding this relationship between frequency, time, and phase is the key to analysing any AC system, from a simple power grid to a complex radio transmitter.