Various RLC Circuit Problems (Resonance Frequency, Phase, Current)

Spaghetti

1. The problem statement, all variables and given/known data
An RLC circuit has L = 20mH, R = 20kΩ, C = 250μF.

1) Calculate the resonance frequency for this circuit.

2) For a frequency ω=400 rad/s, determine the phase angle and the circuit impedance.

3) Using ε = 30[V]sin(ωt), calculate the current in the circuit. This will be a function of time.

4) Draw the phasor diagram for t=0 seconds.


2. Relevant equations
ω[itex]_{o} = \frac{1}{\sqrt{LC}}[/itex]

ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right][/itex]

[itex]X_{L}[/itex] = ωL

[itex]X_{C} = \frac{1}{ωC}[/itex]

Z = [itex]\sqrt{R^{2} + (X_{L} - X_{C})^{2}}[/itex]

I(t) = [itex]\frac{ε}{Z}[/itex]

3. The attempt at a solution
So I wasn't sure if I was doing any of this correctly; with all of the formulas/equations, it looked like the questions were mostly some simple plug-in questions, but I feel like my numbers just weren't coming out quite right.

1) Calculate the resonance frequency for this circuit.
ω[itex]_{o} = \frac{1}{\sqrt{LC}}[/itex]
L = 20mH = 0.02H; C = 250μF
ω[itex]_{o} = \frac{1}{\sqrt{(0.02H)(250μF)}} = \frac{1}{2.24 sec}[/itex] = 0.447 Hz

2) For a frequency ω=400 rad/s, determine the phase angle and the circuit impedance.
Phase Angle ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right][/itex]

[itex]X_{L} = ωL = (400)(0.02) =[/itex] 8
[itex]X_{C} = \frac{1}{(400)(250)} = \frac{1}{100,000} =[/itex] 0.00001
R = 20kΩ

ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right] = tan^{-1} \left[\frac{8-0.00001}{20,000}\right] = tan^{-1} \left[\frac{7.99999}{20,000}\right] = tan^{-1} \left[3.99x10^{-4}\right] =[/itex] 0.0229°

Impedance Z = [itex]\sqrt{R^{2} + (X_{L} - X_{C})^{2}} = \sqrt{20,000^{2} + (8 - 0.00001)^{2}} = \sqrt{4.0x10^{8} + 63.99} =[/itex] 20000.0016Ω

3) Using ε = 30[V]sin(ωt), calculate the current in the circuit. This will be a function of time.

I(t) = [itex]\frac{ε}{Z} = \frac{30[V]sin(400[rad/s]t)}{20000.0016[Ω]} = \frac{30[V]sin(400[rad/s]t)}{20000.0016[Ω]} = 0.00149sin(400t)[/itex]

So I(t) = 0.00149sin(400t)

4) Draw the phasor diagram for t=0 seconds.

I wasn't sure at all how to go about doing this one. As far as I can tell, this one ends up being a graph with voltage and current functions (current found in #3) drawn at t=0, showing how out of phase they are at that time, although I'm not sure how to find the voltage/what to use to find it. Of course, I don't want anyone to draw the graph for me, but some advice on phasor diagrams in general/a push in the right direction would be very nice.

Any help would be greatly appreciated. Thank you in advance!

ehild

The capacitance is given in microfarads: C=250 *10-6 F. Repeat the calculations.

ehild

Spaghetti

Alrighty. I guess my mind totally drew a blank around that capacitance.

I feel a lot better about the frequency now; I'm hoping the others are okay, too.

1) Calculate the resonance frequency for this circuit.
ω[itex]_{o} = \frac{1}{\sqrt{LC}}[/itex]
L = 20mH = 0.02H; C = 250x10^{-6}F
ω[itex]_{o} = \frac{1}{\sqrt{(0.02H)(250x10^{-6}F)}} = \frac{1}{.00223 sec}[/itex] = 447.21 Hz

2) For a frequency ω=400 rad/s, determine the phase angle and the circuit impedance.
Phase Angle ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right][/itex]

[itex]X_{L} = ωL = (400)(0.02) =[/itex] 8
[itex]X_{C} = \frac{1}{(400)(250x10^{-6})} = \frac{1}{.1} =[/itex] 10
R = 20kΩ

ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right] = tan^{-1} \left[\frac{8-10}{20,000}\right] = tan^{-1} \left[\frac{-2}{20,000}\right] = tan^{-1} \left[-1.0x10^{-4}\right] =[/itex] -0.0057°

Impedance Z = [itex]\sqrt{R^{2} + (X_{L} - X_{C})^{2}} = \sqrt{20,000^{2} + (8 - 10)^{2}} = \sqrt{4.0x10^{8} + 4} =[/itex] 20000.0001Ω

3) Using ε = 30[V]sin(ωt), calculate the current in the circuit. This will be a function of time.

I(t) = [itex]\frac{ε}{Z} = \frac{30[V]sin(400[rad/s]t)}{20000.0001[Ω]} = \frac{30[V]sin(400[rad/s]t)}{20000.0001[Ω]} = 0.0015sin(400t)[/itex]

So I(t) = 0.0015sin(400t)

4) Draw the phasor diagram for t=0 seconds.
I'm still a little confused about this phasor diagram, but I think I can get by. Again, if anyone could offer a little advice on this, it would be great.

And thank you for catching my mistake, ehild.