# How Would One Analyze a Circuit with Two Voltage Sources?

I'm studying Circuit Analysis and a sample problem is as follows:

I'm not really sure how to approach the problem. My intuition leads me to believe that the two voltage sources are acting against one another versus stacking their effect. When I search this problem online, I come across examples of superposition, which breaks the circuit apart, analyzes the individual parts, then brings the parts together in an additive or subtractive manner. Thus far in my studies I haven't learned superposition, thus I feel that I shouldn't need it to find the required values.

I'm not looking for a walkthrough for this problem, just somewhere to start. I've thus far learned KVL, KCL, and Ohm's Law. Apparently KVL should be sufficient for solving this problem, I'm just not seeing it.

What method would I use to find the current? I believe that if I know the current I can solve the remainder of the problem.

In this circuit KVL would be enough to solve. Applying KVL would sum all voltages and equal them to zero. Initially taking the given current direction as correct, we have: 5-i-9i-10-10i = 0, then i = -0.25, which tells us the current is in opposite direction regarding the one given in the image.

Found the current, Vab is the voltage across the 1ohm resitor. As current flows from B to A(we know that because we discovered its value in the topic a) ), the voltage Vab is negative, so Vab = -1*0.25 = -0.25V.

Finally,the voltage in node B is GREATER than the voltage in node A. We know that because if you measure the voltage from B to GND you will find Vb = 5 + 0.25 = 5.25V, while Va is 5V because is the source voltage.

All the components are in series so you can re-arrange in any manner providing you keep the voltage polarities the same. This means you can have the 10 ohm resistor that is in the ground limb moved up in the top limb. This should make it a lot easier to see the wood from the trees.

Apparently KVL should be sufficient for solving this problem, I'm just not seeing it.

It's as simple as adding up all the voltage drops around the loop, like you'd do in any KVL problem:

$$5\ {\rm V} - i (1\ {\rm \Omega}) - i (9\ {\rm \Omega}) - 10\ {\rm V} - i (10\ {\rm \Omega}) = 0$$

This is one equation with one unknown, and it's linear, so it will have a single unique solution.