S1E5: KVL-0

Joe was debugging part of an experimental apparatus, probing around with his voltmeter. Part of the apparatus had two obvious resistors in series with an unknown element, as shown in the diagram below:

The unknown element is hard to reach, so Joe put the negative (black) probe of his voltmeter at the interconnection of the two obvious resistors and then put the positive (red) probe at the other end of each resistor, measuring V and V.

What is the voltage (in Volts) measured across the unknown element?

This is KVL, so we construct an equation where the sum of the voltages is 0.

There are two important points to consider here:

1. Joe took the v1 and v2 measurements in opposite directions. This will change the sign of one of these measurements in our final equation.

2. The direction of going around the loop doesn't matter to the voltage (which will be 0) but it will matter for constructing the correct equation!

So, what we'll do is establish a convention:

• Moving from a - to + will be a "positive" term in our equation.
• Moving from a + to - will be a "negative" term in our equation
• We will start and return to the lower left-hand corner of the circuit.

Let's go around the loop clockwise:

(+v3) + (-v1) + (+v2) = 0

Now let's go around the loop counter-clockwise:

(-v2) + (+v1) + (-v3) = 0

Now let's solve for v3 in the clockwise case:

v3 + -(1.4) + (0.9) = 0

v3 = 0.5

Now let's solve for v3 in the counter-clockwise case:

(-0.9) + (1.4) + (-v3) = 0

(-v3) = -0.5

v3 = 0.5

We got a consistent answer for both: 0.5 V

As you can see the diagram is a little deceptive; it looks like v1 and v3 have the same orientation, but the equations tell a different story. Note that our - to + convention could have been the opposite; then our clockwise and counter-clockwise equations would have been opposite as well.

So it doesn't matter which - to + convention you use, or which counter/clockwise direction you take, but for each equation you make, you need to pick one and stick with it throughout the equation. If you don't, then your signs get messed up and then you don't get the right answer.