1. EGR 2201 Unit 10 Second-Order Circuits  Read Alexander & Sadiku, Chapter 8.  Homework #10 and Lab #10 due next week.  Quiz next week.

2.  The circuits we studied last week are called first-order circuits because they are described mathematically by first-order differential equations.  We studied four kinds of first-order circuits: Source-free RC circuits Source-free RL circuits RC circuits with sources RL circuits with sources Review: Four Kinds of First-Order Circuits

3. Review: A General Approach for First-Order Circuits (1 of 3)

4. Review: A General Approach for First-Order Circuits (2 of 3)

5. Review: A General Approach for First-Order Circuits (3 of 3) Transient response Steady-state response

6.  The textbook’s Sections 7.8 and 8.9 discuss using PSpice simulation software to perform transient analysis of first-order and second- order circuits.  We can also do this with Multisim, as shown here. Transient Analysis with Multisim

7. Our Goal: A General Approach for Second-Order Circuits  Next we will develop a general approach for analyzing more complicated circuits called second-order circuits.  Unfortunately the general approach for second-order circuits is quite a bit more complicated than the one for first-order circuits.

8.  The circuits we’ll study are called second-order circuits because they are described mathematically by second-order differential equations.  Whereas first-order circuits contain a single energy-storing element (capacitor or inductor), second-order circuits contain two energy-storing elements. These two elements could both be capacitors or both be inductors, but we’ll focus on circuits containing one capacitor and one inductor. Second-Order Circuits

9.  We’ll study four kinds of second-order circuits: Source-free series RLC circuits Source-free parallel RLC circuits Series RLC circuits with sources Parallel RLC circuits with sources Four Kinds of Second-Order Circuits

10.  Recall that the term natural response refers to the behavior of source-free circuits.  And the term step response refers to the behavior of circuits in which a source is applied at some time.  So our goal in this unit is to understand the natural response of source-free RLC circuits, and to understand the step response of RLC circuits with sources. Natural Response and Step Response

11.  Our procedure will usually require us to find values of voltages or currents at the following three times: At t = 0 , just before a switch is opened or closed. At t = 0 +, just after a switch is opened or closed. As t  , a long time after a switch is opened or closed.  Usually the circuit looks different at these three times, so you’ll want to redraw the circuit for each of these times. Redraw, Redraw, Redraw!

12.  To completely solve a first-order differential equation, you need one initial condition, usually either: An initial inductor current i(0 + ), or An initial capacitor voltage v(0 + ).  To completely solve a second-order differential equation, you need two initial conditions, usually either: An initial inductor current i(0 + ) and its derivative di(0 + )/dt, or An initial capacitor voltage v(0 + ) and its derivative dv(0 + )/dt. Finding Initial Values

13.  To find initial derivative values such as dv(0 + )/dt, we’ll rely on the basic relationships for capacitors and inductors:  For example, if we know a capacitor’s initial current i(0 + ), then we can use the left-hand equation above to find the initial derivative of that capacitor’s voltage, dv(0 + )/dt. Finding Initial Derivative Values

14.  We’ll also rely on the fact that a capacitor’s voltage and an inductor’s current cannot change abruptly. Example: In this circuit, i(0 + ) must be equal to i(0  ), and v(0 + ) must be equal to v(0  ). Since these values must be equal, we don’t really need to distinguish between their values at time t = 0  and at time t = 0 +. So we could just write i(0) instead of i(0 + ) and i(0  ). Quantities that Cannot Change Abruptly

15.  Don’t assume that every quantity has the same value at times t = 0  and t = 0 +.  Example: In the same circuit, i C (t) changes abruptly from 0 A to 2 A at time t = 0.  So we must distinguish between i C (0  ) and i C (0 + ): i C (0  ) = 0 A i C (0 + ) = 2 A i C (0) is undefined. Caution: Some Quantities Can Change Abruptly

16.  Our procedure will sometimes also require us to find final or “steady-state” values, such as: A final inductor current i() A final capacitor voltage v().  Usually these final values are easier to find than initial values, because: 1. We don’t have to worry about abrupt changes as t  , so we never need to distinguish between t    and t   +. 2. We don’t have to find derivatives of currents or voltages as t  . Finding Final Values

17.  Consider the circuit shown. Assume that at time t=0, the inductor has initial current I 0, and the capacitor has initial voltage V 0.  As time passes, the initial energy in the capacitor and inductor will dissipate as current flows through the resistor.  This results in changing current i(t), which we wish to calculate. Natural Response of Source-Free Series RLC Circuit (1 of 2)

18. Natural Response of Source-Free Series RLC Circuit (2 of 2)

19. A Closer Look at Our Differential Equation

20. Solving Our Differential Equation (1 of 4)

21. Solving Our Differential Equation (2 of 4)

22. Solving Our Differential Equation (3 of 4)

23. Solving Our Differential Equation (4 of 4)

24. The Overdamped Case (  >  0 ) Real number

25. The Critically Damped Case (  =  0 ) Zero

26. The Underdamped Case (  <  0 ) Imaginary number

27. Graphs of the Three Cases  Details will differ based on initial conditions and element values, but the shapes shown here are typical.  Note the oscillation in the underdamped case.

28. Natural Response of Source-Free Parallel RLC Circuit (1 of 2)

29. Natural Response of Source-Free Parallel RLC Circuit (2 of 2)

30. Solving Our Differential Equation

31. The Overdamped Case (  >  0 )

32. The Critically Damped Case (  =  0 )

33. The Underdamped Case (  <  0 )

34. Graphs of the Three Cases  Details will differ based on initial conditions and element values, but the shapes shown here are typical.  Note the oscillation in the underdamped case.

35. A General Approach for Source-Free Series or Parallel RLC Circuits (1 of 3)

36. A General Approach for Source-Free Series or Parallel RLC Circuits (2 of 3)

37. A General Approach for Source-Free Series or Parallel RLC Circuits (3 of 3)

38. Making Graphs in Word 2013 (1 of 4) 1. Select Insert > Chart on Word’s menu bar. 2. Select X Y (Scatter). 3. Select Scatter with Smooth Lines. 4. Click OK.

39. Making Graphs in Word 2013 (2 of 4) 5. Type your data values in this window. 6. You can create a new plot on the same chart by typing a new column of data.

40. Making Graphs in Word 2013 (3 of 4) 7. Close the data-editing window by clicking X. 8. If you need to re-open that window to edit your data, select Edit Data on Word’s menu bar.

41. Making Graphs in Word 2013 (4 of 4) 9. Add axis titles and a chart title by clicking the + and checking the boxes. 10. Edit your axis titles and chart title by clicking them and typing.

42. Typing Equations in Word 2013 1. Select Insert > Equation on Word’s menu bar. 2. Use the toolbar’s Structures section to create fractions, exponents, square roots, and more. 3. Use the toolbar’s Symbols section to insert basic math symbols, Greek letters, special operators, and more.