Solar Power Class: Formulae Roundup

Okay, so back to the classroom, while it’s fresh on your mind. Yesterday we talked about Kirchoff’s Laws, as well as a few other formulas that you’ll use when trying to figure out your electrical generation setup and capacity. To review, here’s a listing of formulas:

Voltage = “Electromotive Force” = E
Current = “Amperes” = I
Resistance = “Ohms” Ω = R
Power = “Watts” = P

Ohm’s Law (for DC power): voltage = current * resistance ( E = I * R ),
power = current * voltage ( P = I * E )

Total Circuit Resistance: SERIES: Rt = R1 + R2 + … + Rn
PARALLEL: 1/Rt = 1/R1 + 1/R2 + … + 1/Rn

Voltage Divider Formula (Series): Erx = Et ( Rx / Rt ) (solve for resistor “x”)
Current Divider Formula (Parallel): Irx = It ( Rt / Rx ) (solve for resistor “x”)

Kirchoff’s Voltage Law (Series): Et – E1 – E2 – … – En = 0 (circuit w/ “n” resistors)
Kirchoff’s Current Law (Parallel): It – I1 – I2 – … – In = 0 (circuit w/ “n” resistors)

Whoa, boy, that’s a lot of formulas! Using these, you can pretty much figure out whatever you want about a simple series or parallel circuit. Of course, the circuits designed for power generation are rarely simple like the above, but we’ll get to that in a second!

So now that you’ve beat these formulas in your head, how do you use them? Here’s a sample problem:

“A 220V series circuit has three resistors. The first has a resistance of 25Ω, the second has a resistance of 50Ω, and the third has a resistance of 35Ω. What is the current and resistance of the circuit, and what is the voltage drop across each resistor? How much power will it produce?”

First, you diagram everything to see what you’re working with. You see that it’s a series circuit, which means that the current is common and the total resistance is the sum of all the partial resistances.

Rt = 25Ω + 50Ω + 35Ω = 110Ω

and since E = I * R, 220v = I * 110Ω, or I = 220/110 = 2 amps.

So now you know the voltage (220 V), the resistance (110 Ω), and the current (2 amps). To figure out the power produced, you simply use Ohm’s Law, (P = I * E), or P = 220 V * 2 amps = 440 watts. We’ll add that to the list: power (440 W).

The final part of the work is to figure out the voltage drops across the circuit at each resistor. This will require the voltage divider formula, since it’s a series circuit. ( Ex = Et ( Rx / Rt ) )

Here are the values plugged in for resistor 1: Er1 = 220V ( 25Ω / 110Ω ) = 50V.
Here are the values plugged in for resistor 2: Er2 = 220V ( 50Ω / 110Ω ) = 100V.
Here are the values plugged in for resistor 2: Er3 = 220V ( 35Ω / 110Ω ) = 70V.

You can check your work on this by adding all the voltage drops together: they should equal the voltage given in the problem: 50V + 100V + 70V = 220V. Good! Now for extra credit you can figure out the power produced at each resistor by using your power formula and the voltage and resistance of each resistor in the circuit. If you do, post your answer in the comments section!

This is basically the same process as for parallel circuits. Let’s use the same problem as above, but now the circuit is hooked up in parallel:

In this case, to get total resistance, you use the formula 1/Rt = 1/R1 + 1/R2 + 1/R3. This looks scarier than it is. to get a reciprocal (one divided by a number) on a calculator, you simply plug in the number, hit the divided by sign, then the equals sign. The calculator will do all the dirty work for you, but you should try doing them in your head sometimes, ’cause you never know when the calculator might stop working when you need to figure something out! So:

1/Rt = 1/25Ω + 1/50Ω + 1/35Ω = .04 + .02 + 0.0286 = 0.0886

This is the reciprocal of the total resistance (this is also called the “conductance”), so we take the reciprocal of this and get a total resistance: 11.287Ω. A total resistance of less than each of the parts? Yes, this example illustrates two things about parallel circuits: first, the numbers are rarely as “pretty” as in series circuits. Secondly, the total resistance of a parallel circuit is always less than the lowest partial resistance. If it’s not, then you calculated something incorrectly.

So now, as before, we know the voltage (220V) and the resistance (11.287Ω). To figure out the total current, we use Ohm’s Law ( E = I * R ): 220V = I * 11.287Ω so I = 19.49 amps. You can now easily figure out the power produced: P = 19.49A * 220V = 4287.8 watts of power (or 4.3 kW).

Using the same resistors and voltage, you’re generating more than ten times the power by hooking things up in parallel!

Rounding out the lesson, you’re now trying to figure out the current drop across the resistors (remember, in a parallel circuit, current is divided, not voltage). Use the Current Divider Formula: Irx = It ( Rt / Rx )

Here are the values plugged in for the first resistor: Ir1 = 19.49A ( 11.28Ω / 25Ω ) = 8.794 A.
Here are the values plugged in for the second resistor: Ir2 = 19.49A ( 11.28Ω / 50Ω ) = 4.397 A.
Here are the values plugged in for the third resistor: Ir3 = 19.49A ( 11.28Ω / 35Ω ) = 6.281 A.

To check your numbers, use Kirchoff’s Current Law, and add up the current drops:
8.794 A + 4.397 A + 6.281 A = 19.472 A, which is the same as above, save my rounding. This is why it’s important to keep numbers true to the highest number of decimal places you can… your work will be that much more accurate.

Again, if you want to figure out the power available at any point, use the same process as for the series circuit.

Congratulations! You’ve now mastered simple series and parallel circuits. That’s a lot to know. Next time, we’ll take a look at how to deal with more complex circuits, like the ones you might encounter in your alternative energy generating system, and also talk about why all this math is important to you when all you want to do is drop off the grid and tune out. I found that this stuff didn’t really cement in my head until I tried figuring out practical problems with them, so I’d recommend you do a little practice session, perhaps using the problems found at Open DNS: Ohm’s Law Practice 1, and Ohm’s Law Practice 2. Or try this site if you like to “play and learn”: Ohm’s Law Battleship Game, where your shots won’t land unless you can solve some problems along the way. See you next time, captain!

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1 Response so far »

  1. 1

    Alex said,

    Your blog is interesting!

    Keep up the good work!


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