The output voltage of an AC generator is given by ? v = 119 V sin (27 ? t ). The
ID: 1286710 • Letter: T
Question
The output voltage of an AC generator is given by ?v = 119 V sin (27?t). The generator is connected across a 0.007 H inductor. Find the following.
(a) frequency of the generator
Hz
(b) rms voltage across the inductor
V
(c) inductive reactance
?
(d) rms current in the inductor
A
(e) maximum current in the inductor
A
(f) average power delivered to the inductor
W
(g) Find an expression for the instantaneous current. (Use the following as necessary: t.)
(h) At what time after t = 0 does the instantaneous current first reach 1.00 A? (Use the inverse sine function.)
s
Explanation / Answer
Given data
The output voltage of an AC generator is
V = (119 V)sin (27t)
The inductance of the inductor, L = 0.007 H
a)
The equation for the output voltage of an AC generator is given by
V = V0sin (t)
Here, V0 is the maximum voltage of an AC generator and = 2f is the angular frequency.
Comparing the equations V = (119 V)sin (27t) andV = V0sin (t),
The angular frequency is given by,
= 27
Therefore, the frequency of the generator is
f = /2
= 27/2
= 13.5 Hz
b)
The maximum voltage, V0 = 119 V
The rms voltage across the inductor is given by,
Vrms = V0/2
= (119 V)/2
= 84.14 V
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c)
The inductive reactance of the inductance is given by,
XL = L
= (2f)L
= (2)(13.5 Hz)(0.007 H)
= 0.5937
= 0.594
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d)
The rms current in the inductor is given by,
Irms = Vrms/XL
= ( 84.14 V)/(0.5937 )
= 141.7 A
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e)
The maximum current in the inductor is given by,
Imax = Irms(2)
= (141.7 A)2
= 200.39 A
= 200A
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f)
The average power delivered to the inductor is
Pavg = IrmsVrms
= (141.7 A)(84.14 V)
= 11922.6 W
_____________________________________________________________
g)
The instantaneous current as function of t is given by,
i = imaxsin(t-/2)
= imaxsin(2ft-/2)
= (200.39 A)sin(27t-/2)
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h)
The instantaneous current is i = 1.00 A
1.00 A = (200.39 A)sin(27t-/2)
sin(27t-/2) = 1/(200.39 A)
(27t-/2) = sin-1[1/(200.39 A)]
Solve for t,
t = 0.0219 s
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