3.1 Signals Described in Math Form

Consider the network of Figure 3.1 where the switch is closed at time t=0.

Figure 3.1: A switched network with open terminals

We wish to describe v _out in a math form for the time interval ???. To do this, it is convenient to divide the time interval into two parts, ???.

For the time interval ??out is zero. In other words,

For the time interval 0?, the switch is closed. Then, the input voltage v _S appears at the output, i.e.,

Combining (3.1) and (3.2) into a single relationship, we obtain

We can express (3.3) by the waveform shown in Figure 3.2.

Figure 3.2: Waveform for v _out as defined in relation (3.3)

The waveform of Figure 3.2 is an example of a discontinuous function. A function is said to be discontinuous if it exhibits points of discontinuity, that is, the function jumps from one value to another without taking on any intermediate values.

3.2 The Unit Step Function u ₀(t)

A well known discontinuous function is the unit step function u ₀