Laplace Transforms Circuit Analysis Example 1: Circuit Analysis We - PowerPoint PPT Presentation

Laplace Transforms Circuit Analysis Example 1: Circuit Analysis We can use the Laplace transform for circuit analysis if we can define • Passive element equivalents the circuit behavior in terms of a linear ODE. • Review of ECE 221 methods in s domain For example, solve for v ( t ) . Check your answer using the initial and • Many examples final value theorems and the methods discussed in Chapter 7. i (0-) = -2 mA 5 k Ω + 10 u(t) 5 mH v ( t ) - J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 1 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 2 Example 1:Workspace Example 1:Workspace Hint: ≫ [r,p,k] = residue([-2e-3 2e3],[1 1e6 0]) r = -0.0040, 0.0020, p = -1000000, 0 k = [] J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 3 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 4

Laplace Transform Circuit Analysis Overview Kirchhoff’s Laws • LPT is useful for circuit analysis because it transforms differential N N � � v k ( t ) = 0 V k ( s ) = 0 equations into an algebra problem k =1 k =1 • Our approach will be similar to the phasor transform M M 1. Solve for the initial conditions � � i k ( t ) = 0 I k ( s ) = 0 – Current flowing through each inductor k =1 k =1 – Voltage across each capacitor • Kirchhoff’s laws are the foundation of circuit analysis 2. Transform all of the circuit elements to the s domain – KVL: The sum of voltages around a closed path is zero 3. Solve for the s domain voltages and currents of interest – KCL: The sum of currents entering a node is equal to the sum 4. Apply the inverse Laplace transform to find time domain of currents leaving a node expressions • If Kirchhoff’s laws apply in the s domain, we can use the same • How do we know this will work? techniques that you learned last term (ECE 221) • Apply the LPT to both sides of the time domain expression for these laws • The laws hold in the s domain J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 5 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 6 Defining s Domain Equations: Resistors Defining s Domain Equations: Inductors L R R i ( t ) I ( s ) i ( t ) I 0 s v ( t ) V ( s ) v ( t ) + - + - + - L I 0 Ls Ls I ( s ) I ( s ) v ( t ) = R i ( t ) V ( s ) = R I ( s ) V ( s ) V ( s ) + - + - • Generalization of Ohm’s Law • As with KCL & KVL, the relationship is the same in the s domain � t v ( t ) = L d i ( t ) i ( t ) = 1 as in the time domain 0 - v ( τ ) d τ + I 0 d t L • Note that we used the linearity property of the LPT for both I ( s ) = 1 sLV ( s ) + 1 Ohm’s law and Kirchhoff’s laws V ( s ) = L [ sI ( s ) − I 0 ] sI 0 V ( s ) = sLI ( s ) − LI 0 Where I 0 � i (0 - ) J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 7 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 8

Defining s Domain Equations: Capacitors s Domain Impedance and Admittance C Z ( s ) = V ( s ) i ( t ) CV 0 Impedance: I ( s ) v ( t ) + - Y ( s ) = I ( s ) V 0 1 Admittance: 1 s V ( s ) sC sC I ( s ) I ( s ) • The s domain impedance of a circuit element is defined for zero V ( s ) V ( s ) + - + - initial conditions � t i ( t ) = C d v ( t ) v ( t ) = 1 • This is also true for the s domain admittance 0 - i ( τ ) d τ + V 0 d t C • We will see that circuit s domain circuit analysis is easier when we � 1 � V ( s ) = 1 + 1 can assume zero initial conditions I ( s ) = C [ sV ( s ) − V 0 ] sI ( s ) sV 0 C V ( s ) = 1 sC I ( s ) + V 0 I ( s ) = sCV ( s ) − CV 0 s Where V 0 � v (0 - ) J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 9 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 10 s Domain Circuit Element Summary Example 2: Circuit Analysis i (0-) = -2 mA R Resistor V ( s ) = RI ( s ) V = RI 5 k Ω + Inductor V ( s ) = sLI ( s ) V = sLI Ls 10 u(t) 5 mH v ( t ) 1 - 1 1 Capacitor V ( s ) = sC I ( s ) V = sC I sC Solve for v ( t ) using s -domain circuit analysis. • All of these are in the form V ( 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. 1.63 11 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 12

Example 2: Workspace Example 3: Circuit Analysis 1 k Ω t = 0 + sin(1000 t ) 1 µ F 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. 1.63 13 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 14 Example 3: Workspace Example 3: Workspace Hint: ≫ [r,p,k] = residue([1e6],conv([1 0 1e6],[1 1e3])) r = [ 0.5000, -0.2500 - 0.2500i, -0.2500 + 0.2500i] p = 1.0e+003 *[ -1.0000, 0.0000 + 1.0000i, 0.0000 - 1.0000i] k = [] ≫ [abs(r) angle(r)*180/pi] ans = [ 0.5000 0, 0.3536 -135.0000, 0.3536 135.0000] J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 15 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 16

Example 3: Plot of Results Example 4: Circuit Analysis 50 Ω Total 1 Transient 50 Ω 100 Ω Steady State 0.8 t = 0 0.6 0.4 175 Ω v o ( t ) (V) 0.2 175 Ω 0 + 10 mF −0.2 v 10 mH 40 V −0.4 - −0.6 Solve for v ( t ) . −0.8 0 5 10 15 20 25 Time (ms) J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 17 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 18 Example 4: Workspace Example 4: Workspace Hint: ≫ [r,p,k] = residue([1e-3 20 0],[1 21.25e3 10e3]) r = [-1.2496, -0.0004] p = [-21250,-0.4706] k = [0.0010] J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 19 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 20

Example 5: Parallel RLC Circuits Example 6: Circuit Analysis + + i ( t ) C L R v ( t ) 0.125 µ F 8 H v 20 k Ω i L - - Find an expression for V ( s ) . Assume zero initial conditions. Given v (0) = 0 V and the current through the inductor is i L (0 - ) = − 12 . 25 mA, solve for v ( t ) . J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 21 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 22 Example 6: Workspace Example 6: Workspace Hint: ≫ [r,p,k] = residue([98e3],[1 400 1e6]) r = [ 0 -50.0104i, 0 +50.0104i] p = 1.0e+002 * [ -2.0000 + 9.7980i, -2.0000 - 9.7980i] k = [] ≫ [abs(r) angle(r)*180/pi] ans =[ 50.0104 -90.0000, 50.0104 90.0000] J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 23 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 24

Example 6: MATLAB Code Example 6: Plot of v ( t ) t = 0:0.01e-3:40e-3; 80 v = 50*exp(-200*t).*sin(979.8*t); t = t*1000; h = plot(t,v,’b’); 60 set(h,’LineWidth’,1.2); xlim([0 max(t)]); ylim([-23 40]); box off; 40 xlabel(’Time (ms)’); ylabel(’(volts)’); (volts) title(’’); 20 0 −20 −40 0 5 10 15 20 25 30 35 40 Time (ms) J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 25 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 26 Example 7: Series RLC Circuits Example 7: Workspace 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. 1.63 27 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 28

Example 8: Circuit Analysis Example 8: Workspace Hint: ≫ [r,p,k] = residue([320e3],[1 5e3 0]) 20 k Ω r = [-64, 64] t = 0 + p = [-5000,0] v 1 ( t ) 50 nF k = [] - 80 V + v 2 ( t ) 10 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. 1.63 29 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 30 Example 8: Workspace Example 9: Circuit Analysis 0.25 v 1 v 1 20 H 10 Ω v 2 600 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. 1.63 31 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 32

Example 9: Workspace Example 9: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 33 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 34 Example 10: Circuit Analysis Example 10: Workspace 10 mF i ( t ) a 9 i ( t ) u ( t ) 100 Ω 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. 1.63 35 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 36

Example 10: Workspace 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. 1.63 37 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 38 Example 11: Workspace Example 11: Workspace J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 39 J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 40