Zero-state response of a first-order circuit-EEWORLD

Collect

Zero-state response of a first-order circuit

When all energy storage elements have no initial energy storage and the circuit is in a zero initial state, the response generated in the circuit by external excitation is called a zero-state response.

The following discusses the zero-state response of the circuit when the excitation is DC and sinusoidal AC respectively .

1. Zero-state response under DC excitation.

1. Series circuit

As shown in Figure 8-5-1, switch S is originally placed in position 2, and the circuit has reached a steady state, that is , there is no initial energy storage on the capacitor. At time , switch S is switched from 2 to 1, and the circuit is connected to a DC voltage source. Find the zero-state response , , and after the switch .

Figure 8-5-1

When the switch S is switched to 1, we get:

(Formula 8-5-1)

This is a first-order linear non-homogeneous differential equation with constant coefficients. Based on the knowledge of solving differential equations, we can get the special solution:

General solution to a homogeneous equation:

The full solution is:

(Formula 8-5-2)

According to the route switching rules:

From (Formula 8-5-2):

therefore:

Finally, we get:

(Formula 8-5-3)

(Formula 8-5-4)

(Formula 8-5-5)

According to (Equation 8-5-3) - (Equation 8-5-5), draw the zero-state response and the curve of changing with time, as shown in Figure 8-5-2.

Figure 8-5-2

In the circuit shown in Figure 8-5-1, when , the voltage source charges the capacitor . The capacitor gradually increases from the initial voltage of zero and finally charges to the steady-state voltage , while the current gradually decreases from the initial value and finally decays to the steady-state value of zero.

2. Series circuit.

As shown in Figure 8-5-3, switch S is placed in position 2, and the circuit has reached a steady state, that is , there is no initial energy storage on the inductor L. At time , switch S is switched from 2 to 1, and the circuit is connected to a DC voltage source . Find the zero-state response , and after the switching .

Figure 8-5-3

After , switch S is switched to 1, and we get:

(Formula 8-5-6)

(Equation 8-5-6) is a first-order linear non-homogeneous differential equation with constant coefficients. The full solution of this equation is the sum of the particular solution and the general solution of the homogeneous equation, that is:

(Formula 8-5-7)

represents the complete solution, represents the special solution, and represents the general solution. The steady-state current that the circuit reaches a new stable state after switching is the special solution, that is:

(Formula 8-5-8)

The general solution is:

(Formula 8-5-9)

So, the full solution is:

(Formula 8-5-10)

The integral constant A in (Equation 8-5-10) is determined by the initial conditions. At time , according to the switching rule:

From (Formula 8-5-10):

therefore:

Finally, we get:

(Formula 8-5-11)

(Formula 8-5-12)

(Formula 8-5-13)

Obviously, , satisfies . Figure 8-5-4 plots the zero-state response , and curves.

Figure 8-5-4

2. Zero-state response under sinusoidal AC excitation

1. Series circuit

Still taking the circuit shown in Figure 8-5-1 as an example, the DC voltage source is changed to a sinusoidal AC voltage source . After that, the differential equation of the circuit is obtained by:

(Formula 8-5-14)

The total solution of is equal to the sum of the particular solution and the general solution , that is:

Since the excitation is sinusoidal AC excitation, it is a steady-state component, which is a transient component. The steady-state component can be calculated using phasor:

Where:

The transient component is still , so the full solution is:

(Formula 8-5-15)

At the moment, according to the switching rule , determine the integral constant:

From (Formula 8-5-15):

Finally, we get:

(Formula 8-5-16)

(Formula 8-5-17)

(Formula 8-5-18)

(Equation 8-5-16) to (Equation 8-5-18) show that the initial phase angle of the power supply has an impact on the size of the transient component, usually called the switch-on angle. When or , the transient component of the capacitor voltage is the largest. It is not difficult to see from (Equation 8-5-16) that the maximum value of the capacitor transition voltage will never exceed twice the amplitude of the steady-state voltage. However, it can be seen from (Equation 8-5-17) that in some cases, the maximum value of the transition current will greatly exceed the amplitude of the steady-state current .

2. RL series circuit

Still taking the circuit shown in Figure 8-5-3 as an example, the DC voltage source is changed to a sinusoidal AC voltage source . After that, the differential equation of the circuit obtained by KVL is:

(Formula 8-5-19)

The initial condition is still . As mentioned before, the total solution of a non-homogeneous differential equation is the sum of the particular solution and the general solution , that is:

The right side of (Equation 8-5-19) is a sinusoidal function, and the special solution is also a sinusoidal function. The special solution is the steady-state current under sinusoidal AC excitation, which can be solved using phasor: