Laplace Transforms Circuit Analysis Passive element equivalents Review of ECE 221 methods in s domain Many examples

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1 Laplace Transforms Circuit Analysis Passive element equivalents Review of ECE 221 methods in s domain Many examples J. McNames Portland State University ECE 222 Laplace Circuits Ver

2 Example 1: Circuit Analysis We can use the Laplace transform for circuit analysis if we can define the circuit behavior in terms of a linear ODE. For example, solve for v(t). Check your answer using the initial and final value theorems and the methods discussed in Chapter 7. i(0-) = -2 ma 5kΩ + 10 u(t) 5 mh v(t) - J. McNames Portland State University ECE 222 Laplace Circuits Ver

3 Example 1:Workspace Hint: [r,p,k] = residue([-2e-3 2e3],[1 1e6 0]) r = , , p = , 0 k=[] J. McNames Portland State University ECE 222 Laplace Circuits Ver

4 Example 1:Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

5 Laplace Transform Circuit Analysis Overview LPT is useful for circuit analysis because it transforms differential equations into an algebra problem Our approach will be similar to the phasor transform 1. Solve for the initial conditions Current flowing through each inductor Voltage across each capacitor 2. Transform all of the circuit elements to the s domain 3. Solve for the s domain voltages and currents of interest 4. Apply the inverse Laplace transform to find time domain expressions How do we know this will work? J. McNames Portland State University ECE 222 Laplace Circuits Ver

6 N v k (t) =0 k=1 M i k (t) =0 k=1 Kirchhoff s Laws N V k (s) =0 k=1 M I k (s) =0 k=1 Kirchhoff s laws are the foundation of circuit analysis KVL: The sum of voltages around a closed path is zero KCL: The sum of currents entering a node is equal to the sum of currents leaving a node If Kirchhoff s laws apply in the s domain, we can use the same techniques that you learned last term (ECE 221) Apply the LPT to both sides of the time domain expression for these laws Thelawsholdinthesdomain J. McNames Portland State University ECE 222 Laplace Circuits Ver

7 Defining s Domain Equations: Resistors i(t) R I(s) R + v(t) - + V(s) - v(t) =Ri(t) V (s) =RI(s) Generalization of Ohm s Law As with KCL & KVL, the relationship is the same in the s domain as in the time domain Note that we used the linearity property of the LPT for both Ohm s law and Kirchhoff s laws J. McNames Portland State University ECE 222 Laplace Circuits Ver

8 Defining s Domain Equations: Inductors i(t) + L v(t) - I 0 s I(s) Ls L I 0 I(s) Ls + V(s) - + V(s) - v(t) =L di(t) dt i(t) = 1 L t 0 - v(τ)dτ + I 0 V (s) =L [si(s) I 0 ] I(s) = 1 sl V (s)+1 s I 0 V (s) =sli(s) LI 0 Where I 0 i(0 - ) J. McNames Portland State University ECE 222 Laplace Circuits Ver

9 Defining s Domain Equations: Capacitors i(t) C CV 0 + v(t) - I(s) 1 sc V 0 s 1 I(s) sc + V(s) - + V(s) - i(t) =C dv(t) dt I(s) =C [sv (s) V 0 ] v(t) = 1 C V (s) = 1 C t i(τ)dτ + V 0 0 [ - ] 1 s I(s) + 1 s V 0 I(s) =scv (s) CV 0 V (s) = 1 sc I(s)+V 0 s Where V 0 v(0 - ) J. McNames Portland State University ECE 222 Laplace Circuits Ver

10 s Domain Impedance and Admittance Impedance: Admittance: Z(s) = V (s) I(s) Y (s) = I(s) V (s) The s domain impedance of a circuit element is defined for zero initial conditions This is also true for the s domain admittance We will see that circuit s domain circuit analysis is easier when we can assume zero initial conditions J. McNames Portland State University ECE 222 Laplace Circuits Ver

11 s Domain Circuit Element Summary Resistor V (s) = RI(s) V = RI R Inductor V (s) = sli(s) V = sli Ls Capacitor V (s) = 1 sc I(s) V = 1 sc I sc 1 AlloftheseareintheformV (s) =ZI(s) Note similarity to phasor transform Identical if s = jω Will discuss further later Equations only hold for zero initial conditions J. McNames Portland State University ECE 222 Laplace Circuits Ver

12 Example 2: Circuit Analysis i(0-) = -2 ma 5kΩ + 10 u(t) 5 mh v(t) - Solve for v(t) using s-domain circuit analysis. J. McNames Portland State University ECE 222 Laplace Circuits Ver

13 Example 2: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

14 Example 3: Circuit Analysis t = 0 1kΩ + sin(1000t) 1 μf v o - Given v o (0) = 0, solve for v o (t) for t 0. J. McNames Portland State University ECE 222 Laplace Circuits Ver

15 Hint: Example 3: Workspace [r,p,k] = residue([1e6],conv([1 0 1e6],[1 1e3])) r = [ , i, i] p = 1.0e+003 *[ , i, i] k=[] [abs(r) angle(r)*180/pi] ans = [ , , ] J. McNames Portland State University ECE 222 Laplace Circuits Ver

16 Example 3: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

17 Example 3: Plot of Results 1 Total Transient 0.8 Steady State 0.6 vo(t) (V) Time (ms) J. McNames Portland State University ECE 222 Laplace Circuits Ver

18 Example 4: Circuit Analysis 50 Ω 50 Ω 100 Ω t = 0 10 mf 175 Ω 40 V + v Ω 10 mh Solve for v(t). J. McNames Portland State University ECE 222 Laplace Circuits Ver

19 Hint: Example 4: Workspace [r,p,k] = residue([1e ],[ e3 10e3]) r = [ , ] p = [-21250, ] k = [0.0010] J. McNames Portland State University ECE 222 Laplace Circuits Ver

20 Example 4: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

21 Example 5: Parallel RLC Circuits + i(t) C L R v(t) - Find an expression for V (s). Assume zero initial conditions. J. McNames Portland State University ECE 222 Laplace Circuits Ver

22 Example 6: Circuit Analysis μf i L + 8 H v 20 kω - Given v(0) = 0 V and the current through the inductor is i L (0 - )= ma, solve for v(t). J. McNames Portland State University ECE 222 Laplace Circuits Ver

23 Hint: Example 6: Workspace [r,p,k] = residue([98e3],[ e6]) r = [ i, i] p = 1.0e+002 * [ i, i] k=[] [abs(r) angle(r)*180/pi] ans =[ , ] J. McNames Portland State University ECE 222 Laplace Circuits Ver

24 Example 6: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

25 Example 6: Plot of v(t) (volts) Time (ms) J. McNames Portland State University ECE 222 Laplace Circuits Ver

26 t = 0:0.01e-3:40e-3; v = 50*exp(-200*t).*sin(979.8*t); t = t*1000; h = plot(t,v, b ); set(h, LineWidth,1.2); xlim([0 max(t)]); ylim([-23 40]); box off; xlabel( Time (ms) ); ylabel( (volts) ); title( ); Example 6: MATLAB Code J. McNames Portland State University ECE 222 Laplace Circuits Ver

27 Example 7: Series RLC Circuits + v R (t) - + v L (t) - R L + v(t) C v C (t) - Find an expression for V R (s), V L (s), and V C (s). Assume zero initial conditions. J. McNames Portland State University ECE 222 Laplace Circuits Ver

28 Example 7: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

29 Example 8: Circuit Analysis 20 kω t = 0 + v 1 (t) 50 nf 80 V + v 2 (t) nf 2.5 nf There is no energy stored in the circuit at t =0. Solve for v 2 (t). J. McNames Portland State University ECE 222 Laplace Circuits Ver

30 Example 8: Workspace Hint: [r,p,k] = residue([320e3],[1 5e3 0]) r = [-64, 64] p = [-5000,0] k=[] J. McNames Portland State University ECE 222 Laplace Circuits Ver

31 Example 8: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

32 Example 9: Circuit Analysis 0.25 v 1 10 Ω v 1 20 H v u(t) 0.1 F 140 Ω Solve for V 2 (s). Assume zero initial conditions. J. McNames Portland State University ECE 222 Laplace Circuits Ver

33 Example 9: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

34 Example 9: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

35 Example 10: Circuit Analysis 10 mf u(t) i(t) 100 Ω 9 i(t) a b Find the Thevenin equivalent of the circuit above. Assume that the capacitor is initially uncharged. J. McNames Portland State University ECE 222 Laplace Circuits Ver

36 Example 10: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

37 Example 10: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

38 Example 11: Circuit Analysis v(t) R + v o (t) - i L L Find an expression for v o (t) given that v(t) =e αt u(t) and i L (0) = I 0 ma. J. McNames Portland State University ECE 222 Laplace Circuits Ver

39 Example 11: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

40 Example 11: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

41 Example 12: Circuit Analysis 200 Ω 100 mh 1 μf 1kΩ v s (t) + v o (t) - Find an expression for V o (s) in terms of V s (s). Assume there is no energy stored in the circuit initially. What is v o (t) if v s (t) =u(t)? J. McNames Portland State University ECE 222 Laplace Circuits Ver

42 Hint: Example 12: Workspace [r,p,k] = residue([-0.2 0],conv([1 2e3],[1 1e3])) r = [ , ] p = [-2000, -1000] k=[] J. McNames Portland State University ECE 222 Laplace Circuits Ver

43 Example 12: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

44 Example 13: Circuit Analysis C B C A R B R A v s (t) R L + v o (t) - Find an expression for V o (s) in terms of V s (s). Assume there is no energy stored in the circuit initially. J. McNames Portland State University ECE 222 Laplace Circuits Ver

45 Example 13: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

46 Example 13: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

47 Example 14: Circuit Analysis C R v s (t) R L + v o (t) - Find an expression for V o (s) in terms of V s (s). Assume there is no energy stored in the circuit initially. J. McNames Portland State University ECE 222 Laplace Circuits Ver

48 Example 14: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver

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