1. ELECTRICAL CIRCUIT THEORY (BEL10103) – LECTURE #06
Oleh:
Nabiah Bte Zinal
Electronic Engineering Department
Faculty of Electrical and Electronic Engineering
Universiti Tun Hussein Onn Malaysia

2. Topic: Step Response of RC and RL circuit

3. Lecture Contents Introduction The step response of an RC circuit
The step response of an RL circuit
The step response of an RC circuit

4. “Revision”

5. We use the step function to represent an abrupt change in voltage or current occurs in electric circuit as a result of switching operation
u(t) = (6.23)
The unit step function t
u(t) 1

6. Strength of impulse function=10
We use the unit impulse function to represent the impulsive current/voltage occurs in electric circuit as a result of switching operation
(6.28)
Strength of impulse function=10 t -2 -1 1 2 3
Three impulse function

7. Unit ramp function r(t) is a function that changes at a constant rate.
Integrating the unit step function u(t) results in the
unit ramp function r(t), we write
r(t) 1
(6.29) 1 t
The unit ramp function
Eg.
( t-4 ) u(t-4)
r(t-4)

8. We use the step function to represent an abrupt change in voltage or current,
eg. changes occurs in the circuits of control systems and digital computers.
v(t) = (6.26)
v(t) = V0u(t –t0)
If to=0  v(t) = V0u(t)

9. Step Response of an RC circuit

10. Step Response
When the dc source of an RC circuit is suddenly applied, the voltage or current source can be modeled as a step function, and the response is known as a step response
The step response is the response of the circuit due to a sudden application of a dc voltage or current source

11. -(3) - becomes Eq(2) 0, t For ) capacitor across voltage v ( -(2) or
have we ,
KCL
Applying
(1)
ously,
instantane
change
cannot a of
the
Since RC V dt dv
u(t) R C v( s = +
> + v - R C Vs
t=0
(a) R C + -
Vsu(t) v
(b)
Figure 1: An RC circuit with
voltage step input

14. Assuming that Vs >V0, a plot of v(t) is shown in Figure 2.
This is known as the complete response (or total response) of the RC circuit to a sudden application of a dc voltage source, assuming the capacitor is initially charged.
Assuming that Vs >V0, a plot of v(t) is shown in Figure 2.
v(t) Vs V0 t
Figure 2. Response of an RC circuit with initially charged capacitor

15. If we assume that capacitor is uncharged initially, we set V0=0 in Eq(7) so that
This is the complete step response of the RC circuit
when the capacitor is initially uncharged.

16. v(t)
i(t)
(b)
(a) Vs
Vs/R t t
Figure 3.Step response of an RC circuit with initially uncharged capacitor
(a) voltage response (b) current response

17. Short-cut method for finding the step response of an RC or RL circuit.
Let us reexamine Eq(6)
 it has two components
Two ways of decomposing
(i) Complete Response = natural response + forced response
stored energy independent source
(ii) Complete Response = transient response + steady-state response

18. (i) Complete Response = natural response + forced response stored energy independent source
vf : Forced response because it is produced by the circuit when an external “force” (a voltage source in this case) is applied.

19. One temporary and the other permanent, i.e.
The natural response eventually dies out along with the transient component of the forced response,
leaving only the steady- state component of the forced response.
Another way of looking at the complete response is to break into components –
One temporary and the other permanent, i.e.
Complete Response
transient response + steady-state response
temporary part permanent part

20. vt : Transient response is temporary:
(ii) Complete Response = transient response + steady-state response temporary part permanent part
vt : Transient response is temporary:
it is the portion of the complete response that decays to zero as
time approaches infinity.
vss: Steady-state response :
it is the portion of the complete response that remains after the
transient response has died out.

21. The transient response is the circuit’s temporary response that will die out with time.
The steady-state response is the behavior of the
circuit a long time after an external excitation is applied.

24. Note that if the switch changes position at time t=t0 instead of a t=0,
there is a time delay in the response so that Eq.(12) becomes,
where, v(to) is the initial value at t=t0+.

25. Example 1
Find the capacitor voltage for t >0 and t <0 for the circuit below. + v - 2Ω
t =0
4 V
3 F 3Ω
12 V

26. Solution + v - 2Ω
t =0
4 V
3 F 3Ω
12 V

27. Example 2
The switch in the figure below has been in position a for a long time. At t =0, the switch moves to b.
Determine i(t) for all t >0. 6Ω a i
t =0 b
30 V 3Ω
12 V
2 F

28. Solution
30 V
t =0
12 V 3Ω a
2 F 6Ω b i

29. The step response of an RL circuit

30. Step Response of RL Circuit
The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source.

31. Consider the RL circuit in Figure 4 as the circuit response
 our goal is to find the inductor current, i
as the circuit response
t=0 R R i i +
v(t) - +
v(t) - L + - Vs L + -
Vsu(t)
(a)
(b)
Figure 4: An RL circuit with step input voltage

32. Let the response to be the sum of the natural current and the forced current,
i = in + if (16)
We know that the natural response is always a decaying exponential, that is
where A is a constant to be determined.

33. The forced response is the value of the current a long time after the switch in Figure 4(a) is closed.
We know that the natural response essentially dies out after five time constants.
At that time, the inductor becomes a short circuit , and the voltage across it is zero.
The entire source voltage Vs appears across R.
Thus, the forced response is,

35. v(t) t
Figure 5: Total response of the RL circuit with initial inductor current I0

36. The response in Eq.(21) may be written as
where i(0) and i() are the initial and final values of i.
Thus, to find the step response of an RL circuit requires three things:
1. The initial inductor current i(0) at t=0+
(obtained for the given circuit for t<0)
2. The final inductor i( ).
(obtained for the given circuit for t>0)
3. The time constant .

37. This is the step response of the RL circuit
Again, if the switching takes place at time t=t0 instead of t=0,Eq (12) becomes
If I0 =0, then
This is the step response of the RL circuit
with no initial inductor current.

38. The voltage across the inductor is obtained from Eq.24 using v=L di/dt.
We get

39. Figure 6 shows the step response in Eq.24 and 25
i(t)
v(t)
(b)
(a)
Vs/R Vs t t
Figure 3. Step response of an RL circuit with no initial inductor
current
(a) current response (b) voltage response

40. Example 3 For the circuit below find i(t) for t >0. 10Ω i t =0 20 V
40Ω

41. Solution
5 H
20 V i
t =0
40Ω
10Ω

42. Example 4
Obtain the inductor current for both t <0 and t>0 for the circuit below. i 4Ω
12Ω
2 A 4Ω
t =0
3.5 H

43. Solution

44. Summary - behavior of the circuit after an independent source
The steady- state response
- behavior of the circuit after an independent source
has been applied for a long time
The transient response
- component of the complete response that dies with
time
Total response
steady-state response + transient response

45. Summary…
The step response is the response of the circuit to a sudden application of a dc current or voltage.
The step response may be written as
Or more generally
Or we may write it as
instantaneous value