What equations are used and how are the values found?
What is the resonant freq and the phase angle at 250Hz?
Please see the attached files.
We have a series RLC circuit, and we are interested in the current that flows when a given AC voltage is applied, so it doesn't matter in what orderthe three components are connected. The complex impedance at radian frequency ω is . The current is V/Z. The magnitude of the current is at a maximum when the magnitude of the impedance is at a minimum. The magnitude of the impedance is:
It's pretty obvious that this has a minimum at ωL = 1/ωC, but we can take the derivative anyway:
This derivative takes some work, but this is what you get.
Setting d|Z|/dω to zero we see that yes, the maximum current is at the resonant frequency
. When we substitutute this for ω in |Z| we get |Z| = R as we could have seen above. We are told that Imax at resonance is 2 Amperes, so we know R:
R = Z(ω0) = 10V/2A = 5 ...
LRC sereis circuit is analyzed, such as its maximum current, resonant frequency, and phase angle.