Kevin D. Donohue, University of Kentucky

1

Transient Analysis - First Order Circuits

Switches, Transient Response, Steady-State Response, and

Differential Equations

Kevin D. Donohue, University of Kentucky

2

Transient Response

DC analysis of a circuit only provides a description of voltages and currents in steady-state behavior.

When the applied voltage or current changes at some time, say t0, a transient response is produced that dies out over a period of time leaving a new steady-state behavior.

The circuit’s differential equation must be used to determine complete voltage and current responses.

Kevin D. Donohue, University of Kentucky

3

Examples

Describe v0 for all t. Identify transient and steady-state responses.

+ v0 -

+ v0 -

VDC

t=0

t=0

R

C

Show:

0for volts0

0for voltsexp1)(0

t

tRC

tV

tv DC

For steady-state response, let t , for transient response subtract out steady-state response.

Kevin D. Donohue, University of Kentucky

4

Instantaneous Voltage and Current Changes in Capacitors and Inductors:

What would be the required current, ic , in this circuit for the voltage on the capacitor to change instantaneously?

What would be the required voltage, vL , in this circuit for the current in the inductor to change instantaneously?

+ vC -

VDC

ic

t=0

C

+ vL -

IDC

iL

t=0

L

Conclusion: If the source cannot produce infinite instantaneous power, then neither the capacitor voltage, nor the inductor current can change instantaneously.

Kevin D. Donohue, University of Kentucky

5

Switch Notation and Initial Conditions:

In order to denote the time right before t=0 (limit from the left as t0), and the time right after t=0 (limit from the right as t0), the following notation will be used:

Let t=0+ be the moment after the switch is closed and t=0- be the moment before the switch is closed.

For circuits with practical sources, the voltage across a capacitor cannot change instantaneously,

and the current in an inductor cannot change instantaneously

)0()0( cc vv

)0()0( LL ii

t=0

Kevin D. Donohue, University of Kentucky

6

Complete Solution by the Differential Equation Approach

5 major steps in finding the complete solution: Determine initial conditions on capacitor voltages

and/or inductor currents. Find the differential equation for either capacitor

voltage or inductor current (mesh/loop/nodal …. analysis).

Determine the natural solution (complementary solution).

Determine the forced solution (particular solution). Apply initial conditions to the complete solution to

determine the unknown coefficients in the natural solution.

Kevin D. Donohue, University of Kentucky

7

Example

Find the complete solution for iL for V10sv

vs

t=0

0.25 H

25

Show for t 0: ))100exp(1(4.0 tiL

+

vL

-

Kevin D. Donohue, University of Kentucky

8

Example

Find the complete solution for vc when mA 1si

t=0

t=0

R

100 1 mF

is + vc -

)10exp(1.0 tvc Show for t 0:

Kevin D. Donohue, University of Kentucky

9

The solution of circuits containing energy storage elements can be divided into a steady-state and transient component. In addition, when only one energy storage element is present, the Thévenin resistance can be obtained with respect to the terminal of the energy storage element and used to compute the time constant for the transient component.

Assume solution is of the form

Assume steady-state before the switch is thrown and let either or , and find initial condition for quantity of interest

Let = steady-state solution after switch is thrown,

and , or

t

KKtx exp)( 21

)0()0( cc vv

)0()0( LL ii

1K

)0( x

12 )0( KxK

thCR thRL

Step-by-Step Method