EE201 Lecture 31 P 1 Phasors Phasors Complex number representations of sinusoidal signals at fixed frequency Complex number Aej t A ej t ej But if is constant Aej A Im For A 1 sin ej cos j sin cos Re Unit circle e j e j e j cos 2 sin 2 1 EE201 Lecture 31 P 2 A completely specifies the quantity A cos t for known Phasor voltage V Vm Denoted by bold type in text Phasor current I Im v t 15 sin t 60 v t 15 cos t 30 i t 25 cos t 45 V 15 30 I 25 45 Ohm s Law Relationships for SSS Exists for resistors inductors capacitors General form V z j I z j impedance units ohms EE201 Lecture 31 P 3 Impedance concept for resistors iR t IRe j t R IR V R z R j I R V R R I R vR t RIRe j t vR t VR z R j R Resistor impedance is independent of frequency Impedance concept for inductors iL t ILe j t IL vL t VLe j t VL EE201 Lecture 31 P 4 From I V relationship for inductors VLe j t L d ILe j t dt VLe j t j L ILe j t VL j L IL zL j j L When 0 inductor behaves as a short When inductor behaves as an open Consequence of multiplying IL by j IL IL cos jsin VL j L IL L IL sin j cos Taking dot period sin cos sin cos 0 Therefore VL and IL are orthogonal EE201 Lecture 31 VL P 5 Im IL Re VL leads IL by 90 IL lags VL by 90 Impedance concept for capacitors iC t ICe j t IC vc t Vce j t Vc EE201 IC e j t Lecture 31 C d P 6 VCe j t dt ICe j t j C Vce j t ICe j j C Vce j IC j C Vc VC 1 Ic zc jw IC j C zc j 1 j C When 0 capacitor behaves as an open When capacitor behaves as a short Consequences of multiplying Ic by 1 j C Note 1 j C j C EE201 Lecture 31 P 7 IC IC e j IC cos j sin j I C IC sin j cos C C Therefore VC and IC are perpendicular Im IC sin IC cos IC Re VC VC lags IC by 90 EE201 Lecture 31 P 8 KCL and KVL for Phasors i2 t i1 t i3 t i1 t 2 cos 25t 30 A i2 t 4 cos 25t 60 A Find i3 t I1 2 30 2 cos 30 jsin 30 1 73 j1 0 I2 4 60 4 cos 60 jsin 60 2 j3 46 I3 I1 I2 0 27 j 4 46 EE201 Lecture 31 P 9 0 27 2 4 46 2 4 47 tan 1 4 46 0 27 93 4 I3 4 47 93 4 Converting to real sinusoids i e cos function i3 t 4 47cos 25t 93 4 KVL Find v3 t by phasor methods v1 t v2 t v3 t vs t 10sin t EE201 Lecture 31 P 10 v1 t 5 cos t 45 v2 t 6 cos t 45 Convert real sinusoids to phasors v1 t 5 cos t 45 V1 5 45 3 53 j3 53 v2 t 6 cos t 45 V2 6 45 4 24 j4 24 vs t 10 sin t 10 cos t 90 Vs j10 10 90o Using KVL V3 Vs V1 V2 V3 j10 3 53 j3 53 4 24 j 4 24 V3 7 77 j 9 29 12 11 130 v3 t 12 11 cos t 130 in degrees
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