1. CIRCUIT ANALYSIS USING LAPLACE TRANSFORM

2. METHODOLOGY If the circuit is a linear circuit
Laplace transform of the sources
of excitation: s(t)  S(s)
Laplace transform of the all the
elements in the circuit
Find the output O(s) in the
Laplace freq. domain
Obtain the time response O(t) by
taking the inverse Laplace
transform
Stop or approximate
the circuit into a linear
circuit and continue NO
YES
Examples of nonlinear circuits:
logic circuits, digital circuits,
or any circuits where the
output is not linearly
proportional to the input.
Examples of linear circuits:
amplifiers, lots of OPM
circuits, circuits made of
passive components (RLCs).

3. THE s-DOMAIN CIRCUITS
Equation of circuit analysis: integrodifferential equations.
Convert to phasor circuits for AC steady state.
Convert to s-domain using Laplace transform.
KVL, KCL, Thevenin,etc.

4. KIRCHHOFF’S VOLTAGE LAW
Consider the KVL in time domain:
Apply the Laplace transform:

5. KIRCHHOFF’S CURRENT LAW
Consider the KCL in time domain:
Apply the Laplace transform:

6. OHM’S LAW Consider the Ohm’s Law in time domain 
Apply the Laplace transform 

7. INDUCTOR
Inductor’s voltage
In the time domain:
In the s-domain:

8. INDUCTOR
Inductor’s current
Rearrange VL(s) equation:

9. CAPACITOR
Capacitor’s current
In the time domain:
In the s-domain:

10. CAPACITOR
Capacitor’s voltage
Rearranged IC(s) equation:

11. RLC VOLTAGE
The voltage across the RLC elements in the s-domain is the sum of a term proportional to its current I(s) and a term that depends on its initial condition.

12. CIRCUIT ANALYSIS FOR ZERO INITIAL CONDITIONS (ICs = 0)

13. IMPEDANCE
If we set all initial conditions to zero, the impedance is defined as:
[all initial conditions=0]

14. IMPEDANCE & ADMITANCE The impedances in the s-domain are
The admittance is defined as:

15. Ex.
Find vc(t), t>0

16. Obtain s-Domain Circuit
All ICs are zero since there is no source for t<0

17. Convert to voltage sourced s-Domain Circuit

18. Find I(s)

19. Find Capacitor’s Voltage
The capacitor’s voltage:
Rewritten:

20. Using PFE
Expanding Vc(s) using PFE:
Solved for K1, K2, and K3:

21. Find v(t)
Using look up table:

22. Ex.
Find the Thevenin and Norton equivalent circuit at the terminal of the inductor.

23. Obtain s-domain circuit

24. Find ZTH

25. Find VTH or Voc

26. Draw The Thevenin Circuit
Using ZTH and VTH:

27. Obtain The Norton Circuit
The norton current is:

28. Ex.
Find v0(t) for t>0.

29. s-Domain Circuit Elements
Laplace transform all circuit’s elements

30. s-Domain Circuit

31. Apply Mesh-Current Analysis
Loop 1
Loop 2

32. Substitute I1 into eqn loop 1

33. Find V0(s)

34. Obtain v0(t)

35. Ex.
The input, is(t) for the circuit below is shown as in Fig.(b). Find i0(t) 1 2
t(s)
is(t)
(a)
(b)

36. s-Domain Circuit

37. Using current divider:

38. Derive Input signal, Is t
is1(t) 1 2
t(sec)
is(t) 2 t
is2(t)

39. Obtain Is(t) and Is(s) Expression for is(t):
Laplace transform of is(t):

40. Substitute eqn. (2) into (1):

41. Inverse Laplace transform

42. CIRCUIT ANALYSIS FOR NON-ZERO INITIAL CONDITION (ICs ≠ 0)

43. TIME DOMAIN TO s-DOMAIN CIRCUITS
s replaced t in the unknown currents and voltages.
Independent source functions are replaced by their s-domain transform pair.
The initial condition serves as a second element, the initial condition generator.

44. THE ELEMENTS LAW OF s-DOMAIN

45. THE ELEMENTS LAW OF s-DOMAIN

46. TRANSFORM OF CIRCUITS- RESISTOR
In the time domain:
In the s-domain:

47. TRANSFORM OF CIRCUITS- INDUCTOR
In the time domain:

48. TRANSFORM OF CIRCUITS- INDUCTOR
Inductor’s voltage:
Inductor’s current:

49. TRANSFORM OF CIRCUITS- CAPACITOR
In the time domain:

50. TRANSFORM OF CIRCUITS- INDUCTOR
Capacitor’s voltage:
Capacitor’s current:

51. Ex.
Find v0(t) if the initial voltage is given as v0(0-)=5 V

52. s-Domain Circuit

53. Apply nodal analysis method

54. Cont’d

55. Using PFE
Rewrite V0(s) using PFE:
Solved for K1 and K2:

56. Obtain V0(s) and v0(t) Calculate V0(s):
Obtain V0(t) using look up table: