Sep 7, 2011

Series-Parallel Quiz Solution #2

Once again, one reply was all it took to get the right answer. I posted the circuit in Figure 1 just over a week ago after my second EE Fundamentals entry on the differences between series and parallel connections. The question asked you to find the total power dissipation in the circuit. You needed to apply concepts from all the previous EE Fundamentals segments to solve the problem.  

The correct answer: 3.6 watts.
 Figure 1. Quiz circuit from the last EE Fundamentals Post

Similar to my first quiz, the easiest way to solve this problem is to reduce the circuit to its simplest equivalent model. Step 1 is to look into the circuit from the perspective of the source, which is a voltage source in this case. Looking into the circuit from the left, we note that the source sees two 10 ohm resistors in series with a 20 ohm resistor on the far side. Adding up these values according to the rules of series connections we find that these three resistors can be reduced to a 40 ohm resistor. We now have a circuit that looks like Figure 2 below.
Figure 2. The circuit from Figure 1 after reducing the far right side

We want to keep reducing the circuit from right to left. After combing the first three resistors, we now have two 40 ohm resistors in parallel with each other. When two resistors of the same value are in parallel with each other, their equivalent resistance is half their nominal value. In this example, that means we can model them with a 20 ohm resistor like in Figure 3. If these resistances were different values, you would need to use the parallel resistance rules from my first series-parallel post.
Figure 3. Equivalent circuit after series and parallel resistance reductions

At this point, it looks just like what we started with in Figure 1 before reducing the circuit. I would hope that by now you could finish this up since you just do the same thing over and over until there is only one resistance left. When you are finished, you should end up with the circuit in Figure 4.
 Figure 4. Final equivalent model of the circuit in Figure 1

Cool. So we found the equivalent circuit….but that wasn’t the question I asked. To find the power dissipation you will have to make use of the hints I gave you in the quiz question. The power loss in a circuit is found by summing the power losses of all its elements. Power dissipation in a circuit element is equal to the voltage across the element multiplied by the current through the element. If we look at our circuit, we only have one element to worry about assuming the wires connecting the resistor to the source are ideal conductors.

So here is what we know:
  • Supply Voltage: 12 volts
  • Equivalent circuit resistance: 40 ohms
  • Power Loss = V * I in the resistor
To get the current, use ohms law (I = V/R) and divide the voltage across the resistor by its resistance. You should find that the current is equal to 300 milliamps. Therefore,

Power = V*I = 12 volts * 0.3 amps = 3.6 watts.

It is possible to go branch by branch and sum the various voltages, currents, and power losses, but that should not change the answer. Equivalent circuit models like the one we used in Figure 4 are great for looking at this type of circuit because they break it down to its simplest form while keeping the analysis completely valid. Next time, we solve RC circuits.


Noah Ryan said...

Next time, we eat RC circuits.

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