Zero-input response of a first-order circuit

A circuit containing only one independent energy storage element is called a first-order circuit. When there is no excitation in the circuit, the response caused only by the initial energy storage of the energy storage element is called a zero-input response.

1. Zero input response of RC circuit

As shown in Figure 8-4-1, the switch S is originally in position 1, the circuit has reached a steady state, and the voltage source voltage is , then . At time , S switches from 1 to 2. The zero input response and are deduced below .

Figure 8-4-1

When S switches to 2, we get:

(Formula 8-4-1)

Substituting into the above formula, we get:

(Formula 8-4-2)

(Equation 8-4-2) is a first-order linear homogeneous differential equation with constant coefficients, and its characteristic equation is:

The characteristic roots are:

The general solution to the homogeneous equation is:

(Formula 8-4-3)

The integral constant A in (Formula 8-4-3) is determined by the initial conditions. According to the path replacement rule, we get: , and substitute it into (Formula 8-4-3), we get:

then:

(Formula 8-4-4), (Formula 8-4-5)

The negative sign in (Equation 8-4-5) indicates that the actual capacitor discharge current direction is opposite to the assumed reference direction. It can also be calculated as follows:

Figure 8-4-2 plots the curves of and , both of which decay exponentially.

Figure 8-4-2

In (Equation 8-4-4) and (Equation 8-4-5), contains , which has the dimension of time, and is therefore called the time constant of the circuit. When C is 1 farad and R is 1 ohm , it is 1 second . The size of the time constant reflects the speed of the transition process. The larger it is, the longer the transition process lasts and the slower the transition process proceeds; the smaller it is, the shorter the transition process lasts and the faster the transition process proceeds.

The following uses the attenuation curve of capacitor voltage as an example to introduce the graphical method for finding the time constant . In Figure 8-4-3, draw a tangent from any point P on the attenuation curve, and its intersection with the t-axis is , draw a perpendicular line from point P to the t-axis, and its intersection with the t-axis is P', then:

The decay curve of or i is obtained through experiments and then calculated by graphical method , which is a useful method in practical work.

Figure 8-4-3

The size of the time constant depends on the structure and parameters of the circuit and has nothing to do with the excitation. The time constant of the series circuit . The larger the R and C , the larger the time constant. When R is constant and C is larger, the initial energy stored on the capacitor C is larger and the discharge time is longer; when C is constant and R is larger, the discharge current is smaller and the discharge time is longer.

2. Zero input response of the circuit

As shown in Figure 8-4-4, the voltage source voltage is , the switch S is originally placed in position 1, and the circuit has reached a steady state. At this time, the inductor is equivalent to a short circuit, . When , the switch S switches from 1 to 2. Find the zero input response , .

Figure 8-4-4

After the switch is switched to 2, the reference direction is selected as shown in Figure 8-4-4, and the following is obtained:

(Formula 8-4-7)

(Equation 8-4-7) is a first-order linear homogeneous differential equation with constant coefficients, and its characteristic equation is:

The characteristic roots are:

(Formula 8-4-8)

The general solution to the homogeneous equation is:

(Formula 8-4-9)

The integral constant A in (Equation 8-4-9) is determined by the initial conditions.

According to the switching rule:

(Formula 8-4-10)

From (Formula 8-4-9):

then:

, (Formula 8-4-11)

It can be seen that after switching, the current decays from the initial value according to the exponential law and finally decays to zero, as shown in Figure 8-4-5.

Figure 8-4-5

Resistor voltage:

(Formula 8-4-12)

Inductor voltage:

(Formula 8-4-13)

, differ by a negative sign, and the change patterns of both are the same as that of the current. Their time-varying curves are also shown in Figure 8-4-5.

In (Equation 8-4-11) to (Equation 8-4-13), contains , has the dimension of time, when R is 1 ohm , L is 1 Henry , then is 1 second , just like the time constant in the circuit, it reflects the speed of the transition process.