AC Network Analysis

Suppose we have two equivalent LR networks, one series and one parallel, such that they have the exact same total impedance (Z_total):

We may write an equation for the impedance of each network in rectangular form, like this:

Since we are told these two networks are equivalent to one another, with equal impedances, these two expressions in rectangular form must also be equal to each other:

Algebraically reduce this equation to its simplest form, showing how R_s, R_p, X_s, and X_p relate.

Challenge question: combine the result of that simplification with the equations solving for scalar impedance of series and parallel networks (\(Z^2_s= R^2_s+X^2_s\) for series and \(Z^2_p= \frac{1}{\frac{1}{R^2_p}+\frac{1}{X^2_p}}\) for parallel) to prove the following transformative equations, highly useful for “translating” a series network into a parallel network and visa-versa:

Determine the phase shift of the output voltage (V_out) with reference to the source voltage (0^o) for each of the two switch positions, assuming the source frequency is such that X_C = R:

Note: you should be able to do all the necessary math mentally, without the aid of a calculating device!

This interesting bridge circuit is a variable phase-shifter. It works best when the excitation frequency is such that X_C = R in each arm of the bridge:

Supposing that X_C does equal R in each arm of the bridge, and that the potentiometer resistance is sufficiently high to limit current through it to a negligible level (in other words, R_pot >> R). Calculate the phase shift of V_out with respect to the excitation source voltage when:

The potentiometer wiper is fully left:

The potentiometer wiper is fully right:

The potentiometer wiper is perfectly centered: