How Do You Solve a DC+AC Transformer Circuit Problem?

[ZetaOfThree]

Homework Statement

Find ##V_{out}## for the following circuit:

We are given that the transformers are ideal and ##\frac{n_1}{n_2}=a>1## where ##n_1## is the number of windings on the coil on the left and ##n_2## is the number of windings on the left.

Homework Equations

KVL
Transformer equation: ##\frac{n_1}{n_2}=\frac{V_1}{V_2}## where the V's are the voltages in the transformers.

The Attempt at a Solution

Just use superposition of the voltage sources. I've got the AC part down, but I'm not so sure how to effectively handle the DC part. We can use KVL to write down a system of differential equations that relate the currents that flow through the three loops in the circuit. I was able to solve it using Maple, but the solution is pretty messy and Maple was unable to evaluate the solutions in the limit as ##t \rightarrow \infty##. Is there any way to easily see a steady state solution for the DC part of this problem, without using software as a crutch?

Thanks in advance!

 

[gneill]

Transformers do not couple DC across windings; at steady state there's no change in flux from DC current in an inductor. No change in flux means no changing flux through the other windings to induce a current there.

 

[ZetaOfThree]

Thank you for the response!
The only problem I have with that is what's going on in the rightmost loop in the circuit on the left (the one with the capacitor and the inductor). KVL gives ##V_C + V_L = 0##, but if there is not change in current, then ##V_L=0##, so ##V_C=0##. How can it be that there is no charge buildup on the capacitor?

 

[gneill]

At DC steady state an ideal inductor looks like a short circuit, so zero volts potential across it. The capacitor is effectively shorted out and "disappears" from the circuit.

 

[rezeew]

I would suggest approaching this problem by breaking it down into smaller, more manageable parts. First, analyze the AC part of the circuit using superposition and the transformer equation. Once you have the AC voltage output, you can then use this as a voltage source in the DC part of the circuit. This will simplify the problem and make it easier to find the steady state solution without relying on software. Additionally, you can use KVL and Ohm's law to analyze the DC part of the circuit and find the voltage output. Remember to also consider the effects of the ideal transformers on the DC circuit. With this approach, you should be able to find a steady state solution for the DC+AC transformer problem without relying on software.