1. Fundamentals of Electric Circuits Chapter 15
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2. Overview This chapter introduces the Laplace Transform.
The properties of the transform will be discussed.
The inverse transform and convolution integral will also be covered.
Finally, the general approach to using the Laplace transform in circuit analysis will be introduced.

3. Laplace Transform
We have seen that a powerful technique for circuit analysis is to perform a time domain to frequency domain transformation.
Now we will look at a more generalized approach to this, called the Laplace transformation.
It has the advantage of making solutions simpler.
It also is capable of providing the total response of the circuit with both the natural and forced responses.

4. Laplace Transform II Given a function f(t) its Laplace transform is:
Where s is a complex variable given by:
This parameter has the dimensions of frequency.
The lower limit is denoted as 0- to include any discontinuities present at t=0

5. Example 1
Determine the Laplace transform of each of the following functions shown below:

6. Solution:
The Laplace Transform of unit step, u(t) is given by

7. Solution:
The Laplace Transform of exponential function, e-atu(t),a>0 is given by

8. Solution:
The Laplace Transform of impulse function,
δ(t) is given by

9. Laplace Transform III
The Laplace transform is considered a transformation from t-domain to s-domain.
s can be seen as the complex frequency domain.
In circuit analysis, the differential equations that describe the circuit behavior are converted to algebraic equations in the s-domain
The complementary function is the inverse Laplace transform

10. Linearity:
If F1(s) and F2(s) are, respectively, the Laplace Transforms of f1(t) and f2(t)
Example:

11. Scaling:
If F (s) is the Laplace Transforms of f (t), then
Example:

12. Time Shift:
If F (s) is the Laplace Transforms of f (t), then
Example:

13. Frequency Shift:
If F (s) is the Laplace Transforms of f (t), then
Example:

14. Time Differentiation:
If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its derivative is
Example:

15. Time Integration:
If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its integral is
Example:

16. Frequency Differentiation:
If F(s) is the Laplace Transforms of f (t), then the derivative with respect to s, is
Example:

17. Properties of the Transform II
Frequency shift:
Time differentiation: Or
Or for the n’th derivative:

18. Properties of the Transform III
Time integration:
Frequency differentiation:
Time periodicity: The function can be considered to be composed of a sum of time shifted functions

19. Initial and Final Values
It is possible to get the initial value f(0) and final value f() of a function directly from the Laplace transform:

20. Summary of Properties

21. Summary of Properties II

22. Transform Pairs

23. Inverse Laplace Transform
Although it is always possible to use the formula for the inverse Laplace transform, it is preferable to use the table of transform pairs.
Suppose that F(s) has the form:
Where N(s) and D(s) are both polynomials.

24. Inverse Transform The steps to finding the inverse transform are:
Decompose F(s) into simple terms using partial fraction expansion.
Find the inverse from the transform pairs.
Next we will consider three possible forms F(s) may take and apply these two steps to them.

25. Simple Poles
If F(s) has only simple poles, then it can be expressed as:
Where s=-p1, -p2,…,-pn are the simple poles.
This can be expanded as:

26. Simple Poles II
With
Yielding:

27. Example 2
Find the inverse Laplace transform of
Solution:

28. Repeated Poles If F(s) has n repeated poles at s=-p, then:
Where F1(s) is the remaining part of F(s) that does not have a pole at s=-p.
The transform will be:

29. The Convolution Integral
Convolution means folding.
It is a valuable tool as it provides a means of viewing and characterizing physical systems.
For example it is used in finding the response y(t) of a system to an excitation x(t), knowing the impulse response h(t).
This is done using the convolution integral:

30. The Convolution Integral II
In the integral,  is a dummy variable.
The asterisk denotes convolution.
The convolution process is commutative, meaning the order in which the two functions are convolved does not matter.
The convolution of two signals consists of time-reversing one of the signals, shifting it, and multiplying it point by point with the second signal and integrating the product.

31. Simplified Integral
If x(t)=0 for t<0, and if the system’s impulse response is causal (h(t)=0 for t<0), then:
Here are some properties:

32. Integral Properties
Here are some more properties:

33. Relationship to Laplace
The convolution integral and the Laplace transformation can be related to each other:
Given two functions f1(t) and f2(t) with Laplace transforms F1(s) and F2(s).
Taking the Laplace transform gives:

35. t-domain vs. s-domain
From this we can see that many cases of convolution are simply multiplication in the s-domain.
But sometimes the product is too complicated to do the inverse transform of.
Or there may only be numerical data for the input and no Laplace transform exists. In these cases, the convolution is better to be done in the time domain.

36. Steps to Evaluate
Folding: Take the mirror image of h() about the ordinate axis to obtain h(-).
Displacement: Shift or delay h(-) by t to obtain h(t-).
Multiplication: Find the product of h(t-) and x().
Integration: For a given time t, calculate the area under the product h(t-)x() for 0< < t to get y(t) at t.

39. Application to Integrodifferential Equations
The Laplace transform is useful in solving linear integrodifferential equations.
Using the differential and integral properties of Laplace transforms, each term in the equation is transformed.
Initial conditions are automatically taken into account.
The resulting algebraic equation is solved in the s-domain.

40. Application to Integrodifferential Equations II
The solution is then converted back to time domain by using the inverse transform.

41. Example 3:
Use the Laplace transform to solve the differential equation
Given: v(0) = 1; v’(0) = -2

42. Solution:
Taking the Laplace transform of each term in the given differential equation and obtain