Instantaneous Rate of Change

A sine wave with a period of 1 second. It thus has a frequency of 1 Hz.

Above figure shows that there are times the voltage is increasing, and times it is decreasing. Increasing, in this context, means “getting more positive,” and decreasing means “getting more negative.” The most rapid increase in voltage occurs when t = 0.0 and t = 1.0. The most rapid decrease takes place when t = 0.5.

When t = 0.25, and also when t = 0.75, the instantaneous voltage neither increases nor decreases. But this condition exists only for a vanishingly small moment, a single point in time. Suppose n is some whole number. Then the situation at t = n.25 is the same as it is for t = 0.25; also, for t = n.75, things are the same as they are when t = 0.75. The single cycle shown in above figurerepresents every possible condition of the ac sine wave having a frequency of 1Hz and a peak value of 1 V. The whole wave recurs, over and over, for as long as the ac continues to flow in the circuit.

Now imagine that you want to observe the instantaneous rate of change in the voltage of the wave in above figure, as a function of time. A graph of this turns out to be a sine wave, too—but it is displaced to the left of the original wave by 1⁄ 4 of a cycle. If you plot the instantaneous rate of change of a sine wave against time (following figure), you get the derivative of the waveform. The derivative of a sine wave is a cosine wave. This wave has the same shape as the sine wave, but the phase is different by 1⁄ 4 of a cycle.

A sine wave representing the rate of change in the instantaneous voltage of the wave shown in above figure