Any cosinusoidally time varying function xLH Acos wt 0 can be represented as
It Re Aeiht1013 The Aei0eJwt
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Time independent part i e X Aei or A201 is called the phaser counterpart
of It Then
a X IH 10 cos 1300T Be 10ed
X IOeJ0 I1
b Xslt 5cos lot 3140 5cos lot 3140 3601 5 cos lot 1460
usually the rageof the phaseangle is chosen between
180 to 1800 or IT to X
Re Sei lot 1469 Be Ses t e t X 5L
c Xslt 16sin150T 30 16cos Sot 30 900 16cos 150T 1200
Be lbe5 50T 120 3 Re If e
51200ejSot
Xg 16e Jk0 16L izo1
ECE 2020 AU22
Hw7 Solutions
For agiver angular frequency w time domain counterpart of a phaser Ae is
lH Re AeJ0eJwt A cos wt 0
where w 2Af Hence Ae A cos wt 10 For this problem
w 2Xf 271 60 1201T rad Is
a f the 32 eJ28 eJwt 32 cos wt 284 32 cos 1120kt 280
b 5120 Slew 51 hence
LH Be 51eJwt 51 cos wt 51 cos 1120kt
c 102 65 10 e Jbs hence
zlH The 10eJb e’wt locos wt 650 locos 120kt 650
d x It Re 1 e eJwt L cos Iwt t 132 cos 1120kt t 1320
a Za I 1 j 4 a
j w C j 50 10
b Z jwL j 5 103 2 10
c Za R IS r
Transfer to phaser domain
vsLt 12 cos 2500T 300 Vs 12 300 V and w 2500 radlsec
Z jWL j 2500 300 10
Zc 1 1 j 200 r
jw C j 2500 2 10
2 j7SOr Total impedance of the circuit
Zo 5200 r ZT ZL c Zc Zr
j 750 j 200 4001 til
400 j 550 r4001
ffooEssoTLtan 13 1
680.07 53.970
12_ 2300 53.970 17.66 L 23.970mA
680.07 253.970 680.07
I 200 900 17.66 10 3 L 23970 3.53 L 113970 V
veLt Be 3.53 e J
7 ed t 3 3.53 cos 2500T 113.970 V
Switch to phasor domain
vsLt 120 cos 12000T V 3 Vs 120 LO V and w 2000 rad Isee
2 L jw L j 2000 200 10
2 c j 100 R
u noon if soon
2 T 3001 Kj 3001
15052 4450 150 j 150 r x 212 13 245
1 120 00 2527 450 450 Ay 0.57
15052 450 5
a Complex power associated with the source is
S Vs Io I 120 e.JO 2 ed 120 Lo EL 45
24527 e 5450 VA
24527 cos 145 j sin 145
24 j 24 VA
the real power generated by the source is P 24W
the reactivepower of the source is Q 24 VAR
Pf 24 a 0.71 lagging
Since source current Io lags the source voltage Vs by 450 we have a
lagging pf
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