Total Response of First-Order Circuits and the Three-Element Method-EEWORLD

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The response caused by external excitation and the initial energy storage of the energy storage element in a non-zero initial state is called the total response. The total response is the total solution of the differential equation, which is the sum of the particular solution of the equation and the general solution of its homogeneous equation.

Figure 8-6-1

As shown in the circuit in Figure 8-6-1, before the switch S is closed, there is an initial voltage across the capacitor. At time , the switch S is closed. Then, write the KVL equation of the circuit:

(Formula 8-6-1)

(Formula 8-6-1) is the same as (Formula 8-5-1) in the previous section. By the same token, we can get:

(Formula 8-6-2)

According to the route switching rules:

From (Formula 8-6-2) we get:

therefore:

Finally got the full response:

(Formula 8-6-3)

Now let's transform (Formula 8-6-3), namely:

(Formula 8-6-4)

Looking back at the steps of solving the transition process of the first-order circuit using the classical method, it is found that the full response of the first-order circuit is always equal to the full solution of the corresponding first-order linear constant coefficient differential equation, denoted as , there is always:

(Formula 8-6-5)

In the formula, represents the specific solution of the equation, represents the general solution of the homogeneous equation, and is always in exponential form , then:

(Formula 8-6-6)

Get the value of time:

，

So we get:

(Formula 8-6-7)

(Formula 8-6-7) is the famous three-element formula. It is a simple and effective tool for solving first-order dynamic circuits. (Formula 8-6-7) contains the three elements of a first-order dynamic circuit:

: is a special solution of the first-order linear differential equation with constant coefficients, and is the forced component of the first-order dynamic circuit under excitation. When the excitation is a DC or sinusoidal AC power supply, the forced component is the steady-state component. At this time, it can be obtained according to the solution method of the DC circuit and the sinusoidal AC steady-state circuit ;

: is the initial value of the response immediately after the switch, solved using the method described in §8-3:

: is the time constant. A first-order circuit has only one time constant. Or , is the equivalent resistance of the ports at both ends of the circuit energy storage element.

Example 8-6-1 For the circuit shown in Figure 8-6-2, , originally is open, and there is no charge on C. When is closed, find ; when is closed again, find .

Figure 8-6-2 Example 8-6-1 Attached Figure

Solution: From the question, we know:

According to the route switching rules:

Here the excitation is DC. When , it is closed, and the steady-state value of is, that is,

Time constant:

Using the three-element formula (Formula 8-6-11), we get:

（V）

When , closed, we have:

At the switching time of , the switching rule is still satisfied:

The steady-state value of is still :

Time constant:

Because the route switching is in progress, it is delayed , so according to the three-element formula, we get:

(V),

Example 8-6-2 In the circuit shown in Figure 8-6-3, , the circuit has reached a steady state. When , switch S is closed. What is the transition current in switch S ?

Figure 8-6-3 Example 8-6-2 Attached Figure

Solution: , the circuit has reached a steady state and can be calculated using phasor. From KVL we get:

time:

According to the route switching rules:

,and:

Draw the equivalent circuit at time (figure omitted), and you can get:

When , it is the steady-state switch current. At this time , the series branch is short-circuited by S , and the charge at both ends of the capacitor C has been discharged. Therefore: