1. First Order Circuit
Capacitors and inductors
RC and RL circuits

2. RC and RL circuits (first order circuits)
Circuits containing no independent sources
‘source-free’ circuits
Excitation from stored energy
Natural response
Circuits containing independent sources
DC source (voltage or current source)
Sources are modeled by step functions
Step response
Forced response
Complete response = Natural response + forced response

3. RC circuit – natural response+ vc  ic iR R C
Assume that capacitor is initially charged at t = 0
 vc(0) = Vo
Objective of analysis: to find expression for vc(t) for t >0
i.e. to get the voltage response of the circuit
Taking KCL,    OR

4. RC circuit – natural response
Can be written as
,  = RC  time constant
This response is known as the natural response Vo
t= t
vC(t)
0.3768Vo
 Voltage decays to zero exponentially
 At t=, vc(t) decays to 37.68% of its initial value
 The smaller the time constant the faster the decay

5. RC circuit – natural response
The capacitor current is given by: 
And the current through the resistor is given by
The power absorbed by the resistor can be calculated as:
The energy loss (as heat) in the resistor from 0 to t:

6. RC circuit – natural response
As t  , ER 
As t  , energy initially stored in capacitor will be dissipated in the resistor in the form of heat

7. RC circuit – natural response PSpice simulation1
RC circuit
c e-6 IC=100 r
.tran 7e-6 7e-3 0 7e-6 UIC
.probe
.end + vc  ic iR R C

8. RC circuit – natural response PSpice simulation
.param c=1
c1 1 0 {c} IC=100 r
.step param c list 0.5e-6 1e-6 3e-6
.tran 7e-6 7e-3 0 7e-6
.probe
.end 1 + vc  ic iR R C
c1 = 1e-6
c1 = 3e-6
c1 = 0.5e-6

9. RL circuit – natural responseiL  vL + R L vR
Assume initial magnetic energy stored in L at t = 0
 iL(0) = Io
Objective of analysis: to find expression for iL(t) for t >0
i.e. to get the current response of the circuit
Taking KVL,    OR

10. RL circuit – natural response
Can be written as
,  = L/R  time constant
This response is known as the natural response Io
t= t
iL(t)
0.3768Io
 Current exponentially decays to zero
 At t=, iL(t) decays to 37.68% of its initial value
 The smaller the time constant the faster the decay

11. RL circuit – natural response
The inductor voltage is given by: 
And the voltage across the resistor is given by
The power absorbed by the resistor can be calculated as:
The energy loss (as heat) in the resistor from 0 to t:

12. RL circuit – natural response
As t  , ER 
As t  , energy initially stored in inductor will be dissipated in the resistor in the form of heat

13. RL circuit – natural response PSpice simulation1
RL circuit
L IC=10 r
.tran 7e-6 7e-3 0 7e-6 UIC
.probe
.end  vL + R L vR

14. RL circuit – natural response PSpice simulation
.param L=1H
L1 0 1 {L} IC=10 r
.step param L list
.tran 7e-6 7e-3 0 7e-6 UIC
.probe
.end 1  vL + + vR  L R
L1 = 1H
L1 = 3H
L1 = 0.5H