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Phasor Analysis Phasors, Impedance, SPICE, and Circuit Analysis Kevin D. Donohue, University of Kentucky 1 Impedance The conversion of resistive, inductive, and capacitive elements to impedance for a sinusoidal excitation at frequency is given by: X L L (Reactance) 1 (Reactance) C R R (Resistance) XC In general impedance is a complex quantity with a resistive component (real) and a reactive component (imaginary): X Zˆ R jX R 2 X 2 tan 1 R Kevin D. Donohue, University of Kentucky 2 Phasors Sources can be converted to phasor notation as follows: A cos(t ) A A sin(t ) A 90 This can be applied to all sources of the same frequency, where is used in the impedance conversion of the circuit. If sources of different frequencies exist, superposition must applied to solve for a given voltage or current: 1. Select sources with a common and deactive all other sources. 2. Convert circuit elements to impedances. 3. Solve for desired voltage or current for selected . 4. Repeat steps 1 through 3 for new until all sources have been applied. 5. Add together all time-domain solutions solutions obtain in Step 3. Kevin D. Donohue, University of Kentucky 3 Loop Analysis Example Determine the steady-state response for vc(t) when vs(t) = 5cos(800t) V + vc(t) 3 kW vs(t) 114.86 nF 6 kW Show: Vˆc 2.5000 - j1.4434 2.8868 30 vc (t ) 2.8868 cos 800 t V 6 Kevin D. Donohue, University of Kentucky 4 Nodal Analysis Example Find the steady-state value of vo(t) in the circuit below, if vs(t) = 20cos(4t): 10 W vs ix 0.1 F 1H 2 ix + vo 0.5 H - Show: v0(t) = 13.91cos(4t + 198.3º) Kevin D. Donohue, University of Kentucky 5 Multiple Source Example Find io if is = 3cos(10t) and vs = 6cos(20t + 60º) io 5W vs 0.01 F 0.5 H is 3 2 1 Amps Show io = 0.54cos(20t+123.4º)+2.7cos(10t-153.4º) 0 -1 -2 Kevin D. Donohue, University of Kentucky 0 0.2 0.4 0.6 Seconds 0.8 1 1.2 6 Equivalent Circuit Example Find io steady-state using Norton’s Theorem, if vs(t) = 2sin(10t): 10 W vs io 0.01 F 5W .4 H Show is(t)= .2sin(10t); Zth = 3-j = 3.2-18.4º; io = 0.15cos(10t-153.4 º) Kevin D. Donohue, University of Kentucky 7 Equivalent Circuit Example Find vo steady-state using Thévenin’s Theorem, if vs(t) = 20cos(4t): 10 W vs ix 1H 0.1 F Iˆsc 53.333.7 Vˆoc 5.3 83.9 2 ix + vo 0.5 H - Zˆ th 0.1117 .6 vo (t ) 5.5 cos(20t 85.3 ) Kevin D. Donohue, University of Kentucky 8