1. EEE1012 Introduction to Electrical & Electronics Engineering Chapter 4: Capacitors and Inductors Transient Analysis by Muhazam Mustapha, August 2010

2. 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.

3. Chapter Content Thevenin’s equivalence
Rules for sketching capacitor & inductor transient response
Sketching Example

4. Thevenin’s Equivalence in L & C Circuit

5. 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

6. Transient Curve Sketching Rules

7. 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%

8. 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.

9. Transient Curve Sketching Rules
Steady state
98%
95%
85%
65%
Initial state τ 2τ 3τ 4τ 5τ t
zero slope after 5τ
Highest slope at beginning

10. 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

11. 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

12. Example of Wrong Sketch
Vertical slope should not be at initial stage unless followed by a wage
Initial state 1
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

13. Example of Correct Sketch
(For capacitor)
Voltage transitions – no sudden changes
Voltage
Time
Current
Current wages

14. Sketching Example

15. Sketching Example
Consider the following circuit of Boylestad’s Introductory Circuit Analysis 10th Edition, Example page 405 (modified)
60kΩ
10kΩ 1 2
21V
30kΩ
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

16. Sketching Example
Convert the resulting circuit with Thevenin’s equivalent by excluding the capacitor:
30kΩ
0.2μF 7V
Time constant, τ = RC = (30kΩ)(0.2μF) = 6 ms

17. Sketching Example Voltage: initial 0V, final 7V (Thevenin) Current:
Final: 0A  open circuit
Initial: capacitor reduces to short circuit I = V / R = 7 / 30k = mA

18. Sketching Example Sketch up to 30ms (5τ) V 7V 0V 30ms I 0.233mA 0A

19. Sketching Example Resulting circuit when switch thrown to position 2:
0.2μF
10kΩ
Time constant, τ = RC = (10kΩ)(0.2μF) = 2 ms

20. 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

21. Sketching Example Adding sketch from 30ms to 50ms V 7V 0V 30ms 50ms
0.233mA
40ms
50ms 0A
30ms
0.7mA