Step and impulse functions

When analyzing the transition process of a linear circuit, some singular functions are often used to describe the excitation or response in the circuit. Step function and impulse function are the two most commonly used and important functions.

1. Unit step function.

The unit step function is defined as:

(Formula 8-2-1)

Figure 8-2-1

Its waveform is shown in Figure 8-2-1. The unit step function has a jump at , which is a discontinuous point. Multiplying the unit step function by a constant gives a step function , also known as a switching function . Because it can be used to describe the switching action in the circuit, as shown in Figure 8-2-2. The circuit shown in Figure 8-2-2 switches the switch S from 1 to 2 at time , then the input voltage of the one-port network can be expressed by a step function: , as shown in Figure 8-2-2 .

Figure 8-2-2

The unit step function of the delay is defined as:

(Formula 8-2-2)

Its waveform is shown in Figure 8-2-3. Taking Figure 8-2-2 as an example, if the switch S is switched from 1 to 2 at time , then the input voltage of the one-port network can be expressed as a delayed step function: .

2. Unit Impulse Function

The unit impulse function is defined as:

(Formula 8-2-3)

Its waveform is shown in Figure 8-2-5. In order to better understand the unit impulse function, let's first look at the unit pulse function . The unit impulse function is defined as:

(Formula 8-2-4)

Figure 8-2-5

Its waveform is shown in Figure 8-2-5 . The width of the unit pulse function is , the height is , and the area is 1. When the pulse width decreases, its height will increase, while the area remains 1. When the pulse width tends to be infinitely small, its height will tend to be infinitely large, but the area is still 1. When the pulse width tends to zero, the pulse function becomes a unit impulse function.

Multiplying the unit impulse function by the constant K gives an impulse function with an impulse intensity of K, expressed as .

The delayed unit impulse function is defined as:

(Formula 8-2-5)

Its waveform is shown in Figure 8-2-6.

Figure 8-2-6

The impulse function is not a general function, but a generalized function. For its more rigorous definition, please refer to the discussion in relevant mathematics books.