Week 6

1. Week 6

2. Key Concepts Circuit transformation Basic circuit analysis in the s domain Voltage division principle Current division principle Proportionality theorem Superposition theorem Node-Voltage analysis in the s domain Mesh-Current analysis in the s domain

3. Keywords Element constraints Impedance Admittance Equivalent impedance Equivalent admittance Network functions Zero-input response Zero-state response Circuit determinant Natural poles Forced poles Stable Marginally stable

4. Reading The Analysis & Design of Linear Circuits Sections 10-1 to 10-6 Pages 490-534 Roland E. Thomas, Albert J. Rosa Gregory J. Toussaint

5. Circuit Analysis in the S Domain Circuit Transformation to the S Domain Prior work has been in the time domain We worked with The basic elements – R,L, and C KVL and KCL i(t) and v(t) relationships Now move to the s-domain – that is…the frequency domain We will work with The basic elements – R,L, and C KVL and KCL – they are still valid I(S) and V(S) relationships

6. Circuit Analysis in the S-Domain Circuit Transformation to the S-Domain Voltage relations in the time and s-domains are given as:

7. Circuit Analysis in the S Domain Circuit Transformation to the S Domain Start with a basic RC circuit Set the input vS(t) = VAe–αtu(t) V Assume that the capacitor has an initial voltage V0 Find VC(t)

8. Circuit Analysis in the S Domain Circuit Transformation to the S Domain Using superposition, we first find the zero-state response Then we find the zero input response

9. Circuit Analysis in the S Domain Circuit Transformation to the S Domain Doing a partial fraction expansion of the zero-state response Finally converting back to the time domain

10. Circuit Analysis in the S Domain Finding the Thévenin Equivalent Circuit Find the Thévenin voltage VT and impedance ZT

11. Circuit Analysis in the S Domain Working with a Pole-Zero Plot

12. Circuit Analysis in the S Domain Working with a Pole-Zero Plot

13. Circuit Analysis in the S Domain Working with a Pole-Zero Plot

14. Key Concepts Network function Network functions of one- and two-port circuits Network functions and impulse response Network functions and step response Network functions and the sinusoidal steady-state response Impulse response and convolution Network function design

15. Keywords Network function Driving-point impedance Input impedance Equivalent impedance Transfer function Cascade connection Chain rule Impulse response Step response Step response DC steady-state response Asymptotically stable Blocking capacitor Rise time Delay time Overshoot Convolution Convolution integral Prototype

16. Reading The Analysis & Design of Linear Circuits Sections 11-1, 11-2. and 11-7 Pages 545-561, 581-596

17. Network Functions Network functions assume that all initial conditions are set to zero Since X(s) and Y(s) can be either voltage or current transforms there are four different types of transfer functions The most common is the ratio of a voltage transform output to a voltage transform input called TV(s)

18. Network Functions In analysis,we are given the input X(s) and the transfer functionT(s), and are asked to find the output Y(s). In analysis, there is only one correct solution  In design we are given the input X(s) and the output Y(s) and asked to create a transfer function T(s). In this case there may be one, many or no solutions In the case of many working solutions, one will usually be best for the constraints of the task. This optimum solution is found using judgment and evaluation

19. Network Functions Find the transfer function for the circuit Using nodal analysis

20. Network Functions Find the driving point impedance seen by the voltage source in the circuit The input impedance is the parallel combination of the capacitor and resistor R1

21. Network Functions Cascade Connections and the Chain Rule The transfer function of the first stage is The transfer function of the second stage is

22. Network Functions Cascade Connections and the Chain Rule If the chain rule were to hold then when we cascade the two stages together we would get

23. Network Functions Cascade Connections and the Chain Rule If we were to find the transfer function by actually calculating it we would find that we would get which is NOT the same as if we could cascade the transfer function of the two stages that make up the circuit The reason is that I2(s) is not zero

24. Network Functions Network Functions – Impulse and Step Response Introduce impulse and step responses and their corresponding transforms h(t), the impulse response that is used in convolution g(t), the step response H(s), the impulse response transform G(s), the step response transform T(s), the transfer function

25. Network Functions Network Functions – Impulse and Step Response

26. Network Functions Network Functions – Impulse and Step Response Start with the transfer function Find H(s), G(s), g(t), h(t)

27. Network Functions Network Functions – Impulse and Step Response The impulse response of a linear circuit is Find the circuit’s Step response g(t) Impulse response transform h(s) Step response transform g(s) Circuit’s transfer function T(s)

28. Network Functions Network Functions – Impulse and Step Response The results are

29. SUMMARY: In this unit, we first practiced working and analyzing circuits in the S domain. Of particular importance were the concepts of initial and final values for the states of the energy storage components and then of certain currents and voltages. The second portion of the unit, we analyzed the network transfer function in greater depth and utilized circuit poles and zeros as part of that analysis.

30. REMINDERS: Assignments from Unit 6 are due at start of next unit. Prepare for the next unit by reading: Thomas, Rosa, and Toussaint, Sections 12-1 to 12-5 and 13-1 to 13-5, prior to coming to class.

32. Laplace Transforms • A review of Laplace transform basics.

33. The Laplace transform is what we call an "operational" method for solving differential equations by algebraic means. This is a tremendous help, especially for higher order (two and above) circuits. The Laplace transform is called an "operational" method because it makes use of the operator, s, which refers to the first derivative with respect to time

34. Differential Equation Steps involved in using the Laplace transform. Transform differential equation to algebraic equation Solve equation by algebra Determine inverse transform Solution

35. Time domain circuit Steps involved in s-domain circuit analysis process using Laplace transformations. Convert circuit to s-domain form Solve for desired response in s-domain Determine inverse transform Time domain Solution

36. Laplace Transformation Inverse Transform

37. The Nature of the s-Domain Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations. dt Where a time domain signal, x (t), is related to an s-domain signal, X(s).

38. The Nature of the s-Domain dt • The term , is called a complex exponential. • Complex exponentials are a compact way of representing both sinusoids and exponentials in a single expression • The complex variable, s, where • j= notation indicating an imaginary number. t = time in seconds

39. 1.5 1.0 Amplitude 0.5 0.0 - 0.5 - 4 - 3 - 2 - 1 0 1 2 3 4 Time Rea l Part 2.5 2.0 1.5 1.0 Amplitude 0.5 0.0 - 0.5 - 1.0 - 16 - 12 - 8 - 4 0 4 8 12 16 Frequency Imaginary Part 2.5 2.0 1.5 Amplitude 1.0 0.5 0.0 - 0.5 - 1.0 - 16 - 12 - 8 - 4 0 4 8 12 16 Frequency Time, frequency, and s-domains Time Domain s-Domain Fourier Transform Frequency Domain

40. Continuity • Since the general form of the Laplace transform is: it makes sense that f(t) must be at least piecewise continuous for t ≥ 0. • If f(t) were very nasty, the integral would not be computable.

41. Piecewise Defined In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function which is defined by multiple sub functions, each sub function applying to a certain interval of the main function's domain (a sub-domain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial function: a function that is a polynomial on each of its sub-domains, but possibly a different one on each. The word piecewise is also used to describe any property of a piecewise-defined function that holds for each piece but may not hold for the whole domain of the function. A function is piecewise differentiable or piecewise continuously differentiable if each piece is differentiable throughout its subdomain, even though the whole function may not be differentiable at the points between the pieces. In convex analysis, the notion of a derivative may be replaced by that of the subderivative for piecewise functions. Although the "pieces" in a piecewise definition need not be intervals, a function isn't called "piecewise linear" or "piecewise continuous" or "piecewise differentiable" unless the pieces are intervals. From Wikipedia

42. Boundedness • This criterion also follows directly from the general definition:

43. Evaluating F(s) = L{f(t)} The Hard Way – do the integral let let let Integrate by parts

44. Evaluating F(s)=L{f(t)}- Hard Way remember let Substituting, we get: let It only gets worse…

45. Evaluating F(s) = L{f(t)} This is the easy way ... • Recognize a few different transforms • See Common Transform Pairs slide • Or see handout .... • Learn a few different properties • Do a little math

46. Laplace Transforms of Common Functions Name f(t) F(s) Impulse 1 Step Ramp Exponential Sine

47. Common Transform Pairs

49. Derive the Laplace transform of the exponential function

50. Example #1 A voltage in volts (V) starting at t = 0 is given below. Determine the Laplace transform. Time Domain Laplace (s) Domain Expand to a quadratic