Complex voltage across a capacitor

[preet]

Homework Statement

I'm given the power for a capacitor. I know the voltage across the capacitor has a complex component. How do I find the value of the capacitor?

Power (purely reactive) = -50000j VA
Voltage = 200 + 100j V @ 60 Hz

Homework Equations

Power = (V*V) / Z
Impedance of a capacitor = -j / (2*pi*f*C)

The Attempt at a Solution

(200 + 100j)^2 / Z = -50000j
(30000+40000j) / -50000j = Z
-j / (2*pi*f*C) = (0.8 - 0.6j)
-j / (0.8 - 0.6j) = 2*pi*f*C

C -> complex?


I get a complex component in the value of C

 

[fleem]

The voltage and capacitor current are not orthogonal. I think its strange though that the voltage was defined with a non-zero phase... compared to what?? Either that cap has some internal resistance, or there's a mistake in the givens, or you are suppose to assume the stated phase of the applied voltage is a phase from some other unrelated signal (but the cap phase given is NOT relative to that phase!), and therefore normalize the applied voltage to zero phase (which would mean its a bad question, IMO). I wouldn't feel comfortable assuming any of those unless your teacher wants you to notice the cap has internal resistance and has intentionally defined zero-phase as 90 degrees from the current's phase--which is a rather weird thing to do. But then if the teacher specified the cap is "purely reactive" with no real component, yet defined the applied voltage as having a non-zero phase, then its time to ask your teacher what the heck he meant.

 

[Mene]

, but I'm not sure if this is correct or if there is another way to approach this problem.

I would approach this problem by first clarifying the given information. It is important to note that the power for a capacitor is purely reactive, meaning that it does not dissipate energy but rather stores and releases it over a period of time. Therefore, the given power of -50000j VA is not a constant value but rather a time-varying quantity.

Next, I would use the given information about the voltage across the capacitor to calculate the impedance using the equation Z = V^2 / P. Plugging in the values, we get Z = (200 + 100j)^2 / (-50000j) = (30000 + 40000j) / (-50000j) = -0.6 - 0.8j ohms.

Now, I would use the equation for the impedance of a capacitor, Z = -j / (2*pi*f*C), to solve for the value of the capacitor. Rearranging the equation, we get C = -j / (2*pi*f*Z). Substituting the values, we get C = -j / (2*pi*60*(-0.6 - 0.8j)) = -0.00417 + 0.00555j F.

The resulting value for the capacitance is complex, which is expected since the voltage across the capacitor is also complex. This means that the capacitor has both a real and an imaginary component, indicating a phase shift between the voltage and current. This is a common occurrence in AC circuits and can be further analyzed using concepts such as reactance and impedance.

In conclusion, by approaching the problem systematically and using the appropriate equations, we can find the value of the capacitor in a circuit with a complex voltage. It is important to understand the physical meaning behind the given information and use it to guide our calculations.