Determine the value of capacitor for an A.C Series Circuit?

[alex282]

Determine the value of capacitor to make sure that the current and voltage are in phase.

The diagram is a simple series circuit with;

Voltage Source = 30cos(377t+30°)v
then a resistor R = 10 Ω
then an inductor = 5mH
then capacitor C = ?

I'm not sure how to start this as the EMF has confused me, I think it needs to be changed to polar or rectangular form. If anyone could help it would be much appreciated

 

[alex282]

V = IZ
w=377

The formula for Z (impedance) of inductor is ZL = jwL

ZL = j1.89
ZR = 10
ZC = 1/jwC

so 30cos(377t+30°)v = I(10 + j1.89 + 1/j377C)

Does anyone understand the EMF on the LHS or know how I would find the current I?

 

[gneill]

The EMF is that of a cosine with amplitude 30V, an angular frequency of 377 radians per second, and a phase shift of 30°. The important bit is the angular frequency, as this will set the impedances of the capacitor and inductor (according to their values).

Suppose you were given all three component values, R, L, and C. How would you determine the angle of the current given some EMF E at frequency ω?

 

[alex282]

The EMF is that of a cosine with amplitude 30V, an angular frequency of 377 radians per second, and a phase shift of 30°. The important bit is the angular frequency, as this will set the impedances of the capacitor and inductor (according to their values).

Suppose you were given all three component values, R, L, and C. How would you determine the angle of the current given some EMF E at frequency ω?

I've got this far (using j instead of i to avoid confusion with current)

V=IZ
V=I(10 + j1.89 + 1/j377C)

Where j1.89 is the inductors impedance

I don't know where to go from there because I can't solve for the current unless I know the value of capacitance but the question asks to determine the value of capacitance for where the current and voltage are in phase

 

[gneill]

I've got this far (using j instead of i to avoid confusion with current)

V=IZ
V=I(10 + j1.89 + 1/j377C)

Where j1.89 is the inductors impedance

I don't know where to go from there because I can't solve for the current unless I know the value of capacitance but the question asks to determine the value of capacitance for where the current and voltage are in phase

Good. You can solve for the current symbolically. It will have real and imaginary parts. What condition must hold if the relative angle of the current is to be zero?

 

[alex282]

So I should try to substitute a value in for the capacitance that will give me a phase angle of 0? Would I be right in converting 30cos(377t+30) into polar as (30, 30) or should I be adding something like 90 degrees to the phase angle to convert cos to sin before I put it in polar?

 

[gneill]

So I should try to substitute a value in for the capacitance that will give me a phase angle of 0? Would I be right in converting 30cos(377t+30) into polar as (30, 30) or should I be adding something like 90 degrees to the phase angle to convert cos to sin before I put it in polar?

If you think about it, the magnitude of the voltage supply is independent of the phase; no matter what the voltage is it will not effect the phase shift that the circuit causes. Furthermore, the 'reference' phase of the voltage supply (the 30°) will also not effect the phase introduced by the circuit. After all, a sinewave is the same a cosinewave simply shifted in time. The same goes for a sinewave shifted by 30°; it's just another sinewave, and any phase shift introduced between the current and voltage by the circuit is independent of the initial phase of the voltage supply. So you can safely ignore both the voltage magnitude AND the initial phase of the supply, since you're only interested in the phase shift between the current and the voltage as introduced by the network it's connected to. Only the frequency of the supply matters here. You can simply represent the voltage symbolically as V.

Take your expression V=I(10 + j1.89 + 1/j377C). In order for there to be no phase difference between the voltage and current, what can you say about the components of the impedance? In other words, what would make the angle of the impedance zero?