MEL TECH
CIRCTRONICS
Nodal Analysis with Phasors
1. Convert the circuit to phasor domain:
Note that impedance of a resistor is the same as the time-domain resistance. Impedance of capacitors and inductors depend on the frequency, and sources are written in polar format as a magnitude and phase (the form sin/cos and the frequency are left out).
2. Solve for the desired value in the phasor domain.
Let's find the impedance on the right side of the voltage source first:
Then we'll use the total impedance on the right to find the current flowing through the entire right side, and then use a current divider equation to find the fraction of the total current to the right that ends up flowing through the capacitor. Thus the first part of the equation is just Ohm's law (I=V/Z) and the second part, in square brackets, is the faction o the current that flows through the capacitor. Note that the current divider fraction puts the total impedance of the two branches in the denominator, and the impedance of the other branch in the numerator.
3. Convert answer to time-domain. In this case, we only need magnitude:
Note that the magnitude of the voltage source is just Vm. We found the magnitude of the entire fraction by doing the magnitude of the numerator divided by the magnitude of the denominator. Magnitude of the denominator was found by taking the square root of the sum of the squares of the real and imaginary parts. Remember - the imaginary part does not include the j.