EEE1012
Introduction to Electrical &
Electronics Engineering
Chapter 4: Capacitors and Inductors
Transient Analysis
by Muhazam Mustapha, August 2010
Learning Outcome
By the end of this chapter students are
expected to be able to sketch transient
graph of single capacitor and single inductor
circuit time response.
Chapter Content
• Thevenin’s equivalence
• Rules for sketching capacitor & inductor
transient response
• Sketching Example
Thevenin’s Equivalence
in L & C Circuit
Thevenin’s Equivalence for C & L
• If a circuit is not a simple series or parallel type,
we can convert the rest of the circuit into the
Thevenin’s equivalent and solve for the C
(capacitor) or L (inductor) transient problem.
Replace with
Thevenin’s
equivalent
Transient Curve
Sketching Rules
Transient Curve Sketching Rules
• For the horizontal axis, draw the curve for 5τ,
after which it should then be flat.
• For the vertical axis, the curve should be drawn
– roughly – to change exponentially from the
initial value with percentage of reaching the
steady state as follows:
–
–
–
–
–
1τ: 65%
2τ: 85%
3τ: 95%
4τ: 98%
5τ: 100%
Transient Curve Sketching Rules
• The exponential sketch should be drawn with
highest slope at the initial stage, but,
– for inductor, current graph cannot have vertical step
(infinity slope)
– for capacitor, voltage graph cannot have vertical step
(infinity slope)
• The sketch should then slowly flatten to the
steady state after 5τ.
Transient Curve Sketching Rules
Steady state
98%
95%
85%
65%
Initial state
τ
Highest
slope at
beginning
2τ
3τ
4τ
5τ
t
zero slope
after 5τ
Transient Curve Drawing Rules
• For capacitor,
– current graph converges to horizontal axis
– any transition from initial to steady state in voltage
will trigger a transition in current in opposite slope
sign
– all current transition will be followed by a reverse
decay graph
• indicated by a wage-like shaped graph
Transient Curve Drawing Rules
• For inductor,
– voltage graph converges to horizontal axis
– any transition from initial to steady state in current
will trigger a transition in voltage in opposite slope
sign
– all voltage transition will be followed by a reverse
decay graph
• indicated by a wage-like shaped graph
Example of Wrong Sketch
Initial state 1
Vertical slope should not
be at initial stage unless
followed by a wage
Steady state 1 =
initial state 2
Small slope should
not be at initial
stage
Steady state 2
Highest slope should not
be at steady state
Example of Correct Sketch
Voltage transitions –
no sudden changes
(For capacitor)
Voltage
Time
Current
Current wages
Sketching Example
Sketching Example
Consider the following circuit of Boylestad’s Introductory Circuit
Analysis 10th Edition, Example 10.10 page 405 (modified)
60kΩ
21V
10kΩ
1
30kΩ
2
10kΩ
0.2μF
Sketch the time response when the
switch was thrown to position 1 at t = 0 s
then to position 2 at t = 40ms
Sketching Example
Convert the resulting circuit with Thevenin’s equivalent by
excluding the capacitor:
30kΩ
7V
0.2μF
Time constant, τ = RC = (30kΩ)(0.2μF) = 6 ms
Sketching Example
• Voltage: initial 0V, final 7V (Thevenin)
• Current:
– Final: 0A  open circuit
– Initial: capacitor reduces to short circuit
I=V/R
= 7 / 30k
= 0.233 mA
Sketching Example
• Sketch up to 30ms (5τ)
V
7V
0V
30ms
I
0.233mA
0A
30ms
Sketching Example
Resulting circuit when switch thrown to position 2:
0.2μF
10kΩ
Time constant, τ = RC = (10kΩ)(0.2μF) = 2 ms
Sketching Example
• Voltage: initial 7V, final 0V
• Current:
– Final: 0A  open circuit
– Initial: capacitor reduces to short circuit
I = −V / R
= −7 / 10k
= −0.7 mA
Sketching Example
• Adding sketch from 30ms to 50ms
V
7V
0V
30ms
50ms
40ms
I
0.233mA
0A
0.7mA
40ms
50ms
30ms