1. DEPT.:-ELECTRONICS AND COMMUNICATION SUB: - CIRCUIT & NETWORK
Prepared By:-
Trivedi Arpita ( )
III-EC

2. Applications of Laplace Transforms

3. Contents Introduction Circuit Element Models Circuit Analysis
Transfer Functions
State Variables
Network Stability
Summary

4. Introduction
To learn how easy it is to work with circuits in the s domain
To learn the concept of modeling circuits in the s domain
To learn the concept of transfer function in the s domain
To learn how to apply the state variable method for analyzing linear systems with multiple inputs and multiple outputs
To learn how the Laplace transform can be used in stability analysis

5. Circuit Element Models
Steps in applying the Laplace transform:
Transform the circuit from the time domain to the s domain (a new step to be discussed later)
Solve the circuit using circuit analysis technique (nodal/mesh analysis, source transformation, etc.)
Take the inverse Laplace transform of the solution and thus obtain the solution in the time domain

6. s-Domain Models for R and L
Time domain
s domain
s domain

7. s-Domain Model for C
Time domain
s domain
s domain

8. Summary
For inductor:
For capacitor:

9. Summary Impedance in the s domain Admittance in the s domain
Z(s)=V(s)/I(s)
Admittance in the s domain
Y(s)=1/Z(s)=V(s)/I(s)
Element
Z(s)
Resistor R
Inductor sL
Capacitor
1/sC
*Assuming zero initial conditions

10. Example 1

11. Example 2

12. Example 2 (Cont’d)

13. Example 3

14. Circuit Analysis
Operators (derivatives and integrals) into simple multipliers of s and 1/s
Use algebra to solve the circuit equations
All of the circuit theorems and relationships developed for dc circuits are perfectly valid in the s domain

15. Example 1

16. Example 2

17. Example 3 Assume that no initial energy is stored.
Find Vo(s) using Thevenin’s theorem.
Find vo(0+) and vo() by apply the
initial- and final-value theorems.
(c) Determine vo(t).
=10u(t)

18. Example 3: (a)

19. Example 3: (b), (c)

20. Transfer Functions
The transfer function H(s) is the ratio of the output response Y(s) to the input excitation X(s), assuming all initial conditions are zero.

21. Transfer Functions (Cont’d)
Two ways to find H(s)
Assume an input and find the output
Assume an output and find the input (the ladder method: Ohm’s law + KCL)
Four kinds of transfer functions

22. Example 1

23. Example 2
Find
H(s)=V0(s)/I0(s).

24. Example 2 (The Ladder Method)

25. Example 3 Find H(s) = Vo/Vi, the impulse response,
the response when vi(t) = u(t) V,
the response when vi(t) = 8cos2t V.

26. Example 3: (a), (b)

27. Example 3: (c), (d)

28. State Variables
The state variables are those variables which, if known, allow all other system parameters to be determined by using only algebraic equations.
In an electric circuit, the state variables are the inductor current and the capacitor voltage since they collectively describe the energy state of the system.

29. State Variable Method

30. State Variable Method (Cont’d)

31. How to Apply State Variable Method
Steps to apply the state variable method to circuit analysis:
Select the inductor current i and capacitor voltage v as the state variables (define vector x, z)
Apply KCL and KVL to obtain a set of first-order differential equations (find matrix A, B)
Obtain the output equation and put the final result in state-space representaion (find matrix C)
H(s)=C(sI-A)-1B

32. Network Stability
A circuit is stable if its impulse response h(t) is bounded as t approaches ; it is unstable if h(t) grows without bound as t approaches .
Two requirements for stability
Degree of N(s) < Degree of D(s)
All the poles must lie in the left half of the s plane

33. Network Stability (Cont’d)
A circuit is stable when all the poles of its transfer function H(s) lie in the left half of the s plane.
Circuits composed of passive elements (R, L, and C) and independent sources either are stable or have poles with zero real parts.
Active circuits or passive circuits with controlled sources can supply energy, and can be unstable.

34. Example 1

35. Example 2
Find k for a stable circuit.

36. Summary The methodology of circuit analysis using Laplace transform
Convert each element to its s-domain model
Obtain the s-domin solution
Apply the inverse Laplace transform to obtain the t-domain solution

37. Summary
The transfer function H(s) of a network is the Laplace transform of the impulse response h(t)
A circuit is stable when all the poles of its transfer function H(s) lie in the left half of the s plane.

38. THANK YOU