Phasors. A phasor is a complex number that represents the magnitude and phase of a sinusoid:. Complex Exponentials. A complex exponential is the mathematical tool needed to obtain phasor of a sinusoidal function. A complex exponential is e j w t = cos w t + j sin w t
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ejwt= cos wt + j sin wt
A = z q = z ejq = z cos q + j z sin q
What do you get when you multiple A withejwt
for the real part?
Aejwt = z ejqejwt = z ej(wt+q)
z ej(wt+q) = z cos (wt+q) + j z sin (wt+q)
Re[Aejwt] = z cos (wt+q)
z cos (wt+q)= Re[Aejwt]
Aejwt = z ej(wt+q), A= z ejq,
V = z q
+
i(t)
v(t)
C

Suppose that v(t) is a sinusoid:
v(t) = VM ej(wt+q)
Find i(t).
V = VMq
I = jwCV
V = jwLI
+
i(t)
v(t)
L

V = IZ
w = 377
Find VC
20kW
+
+
VC
10V 0
1mF


ZR = 20kW= 20kW 0
ZC = 1/j (377 1mF) = 2.65kW 90
20kW 0
+
+
VC
2.65kW 90
10V 0


W
Ð
°
æ
ö
2
.
65
k

90
=
Ð
°
ç
÷
V
10V
0
C
W
Ð
°
+
W
Ð
°
2
.
65
k

90
20
k
0
è
ø
Now use the voltage divider to find VC:
+
+
VC
1mF
–
V
+
1kW
VR
–
–
2mA 40
Imaginary Axis
Real Axis
V
VC
VR