0 input response of first order circuit

In the dynamic circuit, there is no external excitation power supply, only the response generated by the initial energy storage of the dynamic element is called the 0 input response of the dynamic circuit.

0 input response of RC circuit

Before the switch S is closed, the capacitor C is charged and its voltage is uc=uo. The switch action time is now taken as the starting point of the timer (t=0). When the switch is closed, i.e. t≥0+, the KVL can be

Ur-uc=0

The Ur=ri (to r, current, voltage reference direction is the associated reference Direction ), i=-C (DUC/DT) (to the capacitance C, current, voltage reference direction is a non-associative reference direction ) into the above equation, there are

RC (DUC/DT) +uc=0

(Homogeneous linear equation dy/dx+p (x) y=0

General solution for Y=CE-∫P (x) dx (C=±EC1)

For RC (DUC/DT) +uc=0,uc=ae-∫ (1/RC) dt=ae-(1/RC) t

(rcp+1) aept=0)

This is the first order homogeneous differential equation, the initial conditions of UC (0+) =uc (0-) =u0, so that the general solution of this equation uc=aept, substituting the upper formula has

(rcp+1) Aept=0

The corresponding characteristic equation is

p=-(1/RC)

according to UC (0+) =uc (0-) =u0 to take this as a substitute for UC =aept , the integral constant can be obtained A=UC (0+) =u0.

The current in the circuit is

I= (U0/R) e (1/RC) t

Voltage on the resistor

ur=uc=u0e-(1/R) t

because of the P=-1/RC, this is the characteristic root of the circuit characteristic equation, only depends on the circuit structure and the component parameter. When the unit of resistance is Ω, the unit of the capacitor is f, the product of the RC unit is S, which is called the RC circuit time constant, denoted by Tau. After the introduction of tau, the capacitance voltage UC and current I can be expressed separately as

Uc=u0e-t/τ

I= (U0/R) e-t/τ

0 input response of the RL circuit

　　

The process is similar to the RC circuit, withheld

Τ=l/r

I=i0e-t/τ

Ur=ri0e-t/τ

Ul=-ri0e-t/τ

0 input response of first order circuit