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9.1
Time Domain Response of First-Order Circuit:
A phasor analysis of a circuit only provides a description of voltage and current
steady-state behavior. When the source waveform changes at some time t0, a
transient response is produced, which dies out over a period of time leaving the new
steady state behavior. The circuit’s differential equation must be used to determine
complete voltage and current responses.
Example: Describe v0 for all t. Identify transient and steady-state responses.
VDC
t=0
R
C
t=0
Show:
+
v0
-
t  
 
 VDC 1  exp
for t  0
  volts
v0 (t )   
 RC  
0 volts
for t  0
For steady-state response, let t  , for transient response subtract
{complete response - steady-state response} in each t-interval.
9.2
Instantaneous Voltage and Current Change in Capacitors and Inductors:
What would be the required current, ic , in this circuit for the voltage on the
capacitor to change instantaneously?
ic
VDC
t=0
C
+
vC
-
What would be the required voltage, vL , in this circuit for the current in the
inductor to change instantaneously?
iL
IDC
t=0
L
+
vL
-
Conclusion: If the source cannot produce infinite instantaneous power, then
neither the capacitor voltage, nor the inductor current can change instantaneously.
9.3
Switch Notation and Initial Conditions:
In order to denote the time right before t=0 (limit from the left as t0), and the time
right after t=0 (limit from the right as t0), the following notation will be used:
t=0
where t=0+ is the moment after the switch is thrown
and t=0- is the moment before the switch is thrown.
In order to determine initial conditions for solving the differential equations, the
following statement can be used:
For circuits with practical sources:
 The voltage across a capacitor cannot change instantaneously:
vc (0  )  vc (0  )
 The current in an inductor cannot change instantaneously:
iL (0  )  iL (0  )
9.4
Finding the Complete Solution:
There are 5 major steps in finding the complete solution:
1. Find the differential equation for either capacitor voltage or inductor current.
2. Determine the natural solution (homogenous solution).
3. Determine the forced solution (steady-state solution).
4. Determine initial conditions.
5. Apply initial conditions to the complete solution to determine the unknown
coefficients in the natural solution.
Example: Find the complete solution for the iL for the following cases where:
a) vs  10 b) vs  3490 cos(360t ) c) vs  exp 100t 
iL
t=0
vs
25 
0.25 H
+
vL
-
Show for t  0: a) iL  0.4(1  exp( 100t )) ,
b) i  10 cos(360t )  36 sin(360t )  10 exp( 100t )
L
c) i  4t exp( 100t )
L
9.5
Example: Find the complete solution for the vc for the following cases where
a) i  1 mA b) i  cos(10t ) mA
c) i  exp  10t 
s
s
s
t=0
is
t=0
100 
R
Show for t  0: a) v  01
. (1  exp( 10t )) ,
01
.

b) v 
cos10t    0.05 exp( 10t )
2 
4
1 mF
+
vc
-
c
c
c) v  100t exp( 10t )
c
When solving circuits with one energy storage element, the Thevenin or Norton
equivalent circuit can be found with respect to the inductor or capacitor
terminals so the equivalent circuit has the same form as the preceding examples.
This technique can simplify some circuit problems. But the most general way to
solve these problems is to derive the differential equation directly from the
circuit.
9.6
SPICE - Transient Analysis
Initial Conditions can be optionally set for energy storage elements:
Inductors:
LXXX 2
5
6
iL(0+)
5
3
IC=9
2
iL(0+)
Component Value
in Henries
Terminal
Nodes
Capacitors:
CXXX 2
IC=0.8
5
2
vC(0+)
VC(0+)
Terminal
Nodes
Component Value
in Farads
6H
3F
5
9.7
SPICE will perform transient analysis when the following statement is included:
.TRAN .01 2.5 0
Time Step for
computing
waveform points
UIC
Optional analysis
start time in
seconds- Default
start time is zero
Analysis end time in
seconds
Use Specified ICs, otherwise
SPICE will compute from circuit
9.8
Sources can be constructed to have piece-wise linear waveform (i.e. can model
switches or changes in the source behavior that generate transients)
SPICE Source Description (Voltage Source Example):
VXXX 2
3
PWL(0 0 0.1 7 0.2 0)
applied voltage waveform
8
7
6
t1 v(t1) t2 v(t2) t3 v(t3)
volts
5
4
3
2
1
0
-0.1
0
0.05
0.1
seconds
0.15
0.2
0.25
0.3
applied current waveform
SPICE Source Description (Current Source Example):
IXXX 0
1
PWL(0 10 2 10)
12
10
8
amps
t1 i(t1) t2 i(t2)
-0.05
6
4
2
0
-1
-0.5
0
0.5
1
seconds
1.5
2
2.5
3
9.9
SPICE Example:
Write a SPICE code to plot the complete response for vo(t) in the circuit below for
0<t5:
1
10 v
t=0
2
1 k
1 mF
t=0
0
* This SPICE code will compute the waveform of
the
* complete response for the circuit between
* 0 and 5 seconds.
V1 1
0
PWL(0 1 5 1)
R1 1
2
1k
C1 2
0
1m IC=0
.TRAN
.05 5.0 0
UIC
.PRINT
TRAN V(2)
.PROBE
.END
+
v0
-