etotheipi
- Homework Statement
- A loop containing a ##20 \Omega## resistor in series with a ##3 \text{mH}## inductor is wired in parallel with another loop containing a capacitor of unknown capacitance ##C##. Calculate ##C## if the reactance of the combination is zero at ##50\text{Hz}##.
- Relevant Equations
- ##Z = R + iX##
If just found out about reactances and impedances today and came across this little problem. I have worked it through with a sort of brute force approach (that I'm not totally sure is correct!) but wondered if it could be done slightly more quickly?
I denoted the impedance of the top branch ##Z_{1} = 20 + \frac{3\pi}{10}i## and that of the bottom branch ##Z_{2} = -\frac{i}{100\pi C}##. The effective impedance follows from ##Z_{T} = \frac{Z_{1}Z_{2}}{Z_{1} + Z_{2}}##, and the effective reactance is the imaginary part of this which may then be set to zero after rationalising the denominator.
This all seems okay however does the fact that the voltage and current end up having zero phase difference suggest a slightly nicer method, or is this wishful thinking? Thanks!
I denoted the impedance of the top branch ##Z_{1} = 20 + \frac{3\pi}{10}i## and that of the bottom branch ##Z_{2} = -\frac{i}{100\pi C}##. The effective impedance follows from ##Z_{T} = \frac{Z_{1}Z_{2}}{Z_{1} + Z_{2}}##, and the effective reactance is the imaginary part of this which may then be set to zero after rationalising the denominator.
This all seems okay however does the fact that the voltage and current end up having zero phase difference suggest a slightly nicer method, or is this wishful thinking? Thanks!