RLC circuit - total impedance,phasor analysis etc

[pat666]

Homework Statement

A resistor, R = 100 Ω, an inductor, L = 0.20 H and a capacitor, C = 3 µF are connected in series across an ac source of VS = 100 V and frequency, f = 200 Hz. Calculate the following for this series circuit:
a) the total circuit impedance (Zt)
b) the total circuit current (IS)
c) the voltage drop across each component (VR, VL and VC)
d) the phase angle between VS and It.
e) Draw the phasor diagram of voltages and current for the circuit.
f) What is the frequency of the ac source for maximum current to flow?
g) What is the quality factor, Q, of the circuit at the frequency calculated in part f)?

Homework Equations

in attachment

The Attempt at a Solution

attachment
a) I think is right.
b) Not sure about since i don't have vmax do I??
c) really doubt is right but I can't think of another way to do it
d) tan(phi)=X_L-X_C/R which I got -7.9 degrees?
e) I'm trying to understand the phasor analysis paragraph in the text and all I know is that I will need V_R,max,$$\epsilon$$_max and V_C,max all of which I should be able to calculate when I know if my I from b) is right.. even when I have that I am still not sure how to draw it?
f) and g) I am really stumped on...
Thanks for any and all help.

the d) I have in the pic is for another question-sorry I can't see how to remove that attachment.

 

[Andrew Mason]

The Attempt at a Solution

attachment
a) I think is right.
b) Not sure about since i don't have vmax do I??

Assume the AC voltage is 100 V RMS. Vmax is V RMS/.707. But I am not sure why you need Vmax here. Use: I = V/Z

c) really doubt is right but I can't think of another way to do it

The phasor sum of all the voltage drops adds (vectorally) to the applied voltage. Remember that voltage across the inductor is 180 degrees ahead of the voltage across the capacitor and the voltage across the resistor is perpendicular to that across the inductor and capacitor. So draw V_R pointing along the +x axis, V_L along the + y-axis and V_C along the -y axis. V_R = IR. |V_C| = IX_C and |V_L|= IX_L

d) tan(phi)=X_L-X_C/R which I got -7.9 degrees?

Looks right.

e) I'm trying to understand the phasor analysis paragraph in the text and all I know is that I will need V_R,max,$$\epsilon$$_max and V_C,max all of which I should be able to calculate when I know if my I from b) is right.. even when I have that I am still not sure how to draw it?
f) and g) I am really stumped on...

You don't actually need to use Vmax. Just use the given 100 VAC (rms) and add them using a phasor diagram. AM

 

[pat666]

Thanks Andrew,
for b I get the same answer, the reason I had Vmax is because the formula in my textbook says I = Vmax/Z but that's probably for Imax?
Still confused about c) do I need to draw a diagram? I used VR = IR. |VC| = IXC and |VL|= IXL so my numerical answers should be right?
e) I am still stumped on, how do i find the angle for V r max,E max,Vcmax or the magnitude?
f) not sure--thinking that I should find I_max using Irms/swrt(2) but that will give me the max current flow at 200Hz won't it?? If I am looking for max current flow I'm after the resonant frequency i think but I don't know any formula for that??
g)Q=X_L/R=wL/R but since I don't know how to find the frequency I can't do this yet??

 

[Andrew Mason]

Thanks Andrew,
for b I get the same answer, the reason I had Vmax is because the formula in my textbook says I = Vmax/Z but that's probably for Imax?

That would be Imax, correct.

Still confused about c) do I need to draw a diagram? I used VR = IR. |VC| = IXC and |VL|= IXL so my numerical answers should be right?

Right. But according to Kirchoff's laws, the voltages must add up to the applied voltage. They do add up at any given moment but to do that you have to take into account the phase differences between the voltages. To do that, you use a phasor diagram.

e) I am still stumped on, how do i find the angle for V r max,E max,Vcmax or the magnitude?

Draw a phasor diagram of the voltages. You don't need to use Vmax. You can use Vrms. Draw the voltage vectors as I suggested in my last post using the values you found in c). Then draw a phasor showing the current vectors. The lengths will be proportional to voltages but the phase angles of the inductor and capacitor currents will have different directions.

f) not sure--thinking that I should find I_max using Irms/swrt(2) but that will give me the max current flow at 200Hz won't it?? If I am looking for max current flow I'm after the resonant frequency i think but I don't know any formula for that??

Use I = V/Z. In order to maximize I, you need to minimize Z. How do you do that? (hint: express X_L and X_C in terms of f, L and C.)

g)Q=X_L/R=wL/R but since I don't know how to find the frequency I can't do this yet??

Find f and you will have no problem.

AM

 

[pat666]

Hey Andrew,
been a while but I finally got around to finishing this problem.
I'm really not sure about my phasor diagram, I ended up using Vmax just because I had it worked out from a previous question. my Vcmax is way bigger than both my V_r,max and emf max - not sure if this means its wrong or not?
for f) I got 205.5Hz and for g I got 2.58, sound right? also what are the units for quality factor?

Thanks