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ParoXsitiC
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Homework Statement
The AC power supply (maxium voltage 20.0V) in RCL circuit is run at resonance for the following parameters:
R = 20.0 Ω
L = 88.0 mH
C = 79.9 μF
When an iron core is added to the inductor, the self-inductance is doubled
A) Find the frequency of the generator
B) After the core had been added, find the phase angle and impedance in the circuit.
C) Draw phasor diagram at time 4.1267 ms
D) Find the voltages across each circuit element at this time and show that they follow Kirchoff's voltage rule.
Homework Equations
σ = tan-1 ([itex]\frac{X_{L}-X_{C}}{R}[/itex])
XL = ωL
XC = [itex]\frac{1}{ωC}[/itex]
@ Resonance XL = XC
frequency (f) = [itex]\frac{ω}{2\pi}[/itex]
impedance (z) = [itex]\sqrt{(R)^2+(X_{L}-X_{C})^2}[/itex]
The Attempt at a Solution
A)
ω= [itex]\frac{1}{\sqrt{LC}}[/itex] = 377 rad/s
frequency (f) = [itex]\frac{ω}{2\pi}[/itex] = [itex]\frac{377 rad/s}{2\pi}[/itex] = 592 Hz
B)
Assuming ω stays the same, L is now 2*88.0 mH = 176 mH
XL = ωL = (377 rad/s)(176 mH) = 66.3
XC = [itex]\frac{1}{ωC}[/itex] = [itex]\frac{1}{(377 rad/s)(79.9 μF)}[/itex] = 33.2
σ = tan-1 ([itex]\frac{X_{L}-X_{C}}{R}[/itex]) = tan-1 ([itex]\frac{66.3-33.2}{20 Ω}[/itex]) = 58.9 degrees
impedance (z) = [itex]\sqrt{(R)^2+(X_{L}-X_{C})^2}[/itex] = [itex]\sqrt{(20 Ω)^2+(66.3-33.2)^2}[/itex] = 38.7 Ω
C)
E = Emax sin (ωt) = 20.0 v sin(377 rad/s 4.167 ms) ≈ 20 v
Thus Emax should be vertical, and the phase angle of 58.9 degrees is between Vr (also Imax) and Emax
D)
This is where I am stuck...
I know the vertical components should all equal 0.
Emax = 20 v
VL = XL Sin(180-59.8) = 66.3 Sin(120.2) = 57.3 v
VC = XC Sin(0-59.8) = 33.2 Sin(-59.8) = -28.7 v
VR = R sin(90-59.8) = 20 Ω sin(30.2) = 10.1 v
-20 + 57.3 - 28.7 + 10.1 ≠ 0
OK, so this isn't the right way... maybe using Imax..
Imax = [itex]\frac{E_{max}}{z}[/itex] = [itex]\frac{20.0 v}{38.7 Ω}[/itex] = 0.517 A
Emax = 20 v
VL = XL Imax = 66.3 * 0.517 = 34.3 v
VC = XC Imax = 33.2 * 0.517 = 17.2 v
VR = R Imax = 20 Ω * 0.517 = 10.3 v
-20 + 34.3 - 17.2 + 10.3 ≠ 0
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