Parallel and Series-Parallel Resistance
Date: 05/24/2000 at 21:36:00 From: John Denning Subject: Physic/Electrical Question I am having problems calculating resistance in series circuits, parallel circuits, and series parallel circuits. I know the equation for parallel circuits is R_total = 1/r1 + 1/r2 + 1/r3 + ... I know that they are inverted but I am having trouble. For example: What is the total resistance in a circuit with 3 Ohm, 25 Ohm, 45 Ohm, and 67 Ohm resistors in parallel? Thank you, John Denning
Date: 05/25/2000 at 13:12:14 From: Doctor TWE Subject: Re: Physic/Electrical Question Hi John - thanks for writing to Dr. Math. Using the 'reciprocal rule' for parallel resistance, you would take the reciprocal (1/x) of each resistor, add them up, and then take the reciprocal of the total. For example, suppose we have a parallel circuit with resistors of 30 Ohms, 60 Ohms, 20 Ohms and 10 Ohms. Then the total resistance is: 1 1 1 1 1 -- = -- + -- + -- + -- Rt 30 60 20 10 2 1 3 6 = -- + -- + -- + -- 60 60 60 60 12 1 = -- = - 60 5 Thus 1 Rt = --- = 5 Ohms 1/5 Of course, if you're using a calculator you don't have to work with the fractions themselves. On most scientific calculators, you could enter it like this: (Note that [1/x] is the reciprocal key and may be labeled [x^-1] on some models.) 30 [1/x] [+] 60 [1/x] [+] 20 [1/x] [+] 10 [1/x] [=] [1/x] On some calculators, you'll need to press the [=] key again at the end. Try my example on your calculator and see if you get 5. If not, look in your owner's manual to see how to use the reciprocal function. An alternate way to solve for total resistance in parallel is the 'product-over-sum rule'. This rule is often easier to use if you don't have a calculator with the reciprocal function handy. For two resistors, the rule states: R1 * R2 <- their product Rt = ------- <- over R1 + R2 <- their sum If, for example, we had a 30-Ohm resistor in parallel with a 60-Ohm resistor, the total resistance would be: 30 * 60 Rt = ------- 30 + 60 1800 = ---- = 20 Ohms 90 This agrees with the reciprocal rule's answer. (Check it!) You have to be careful, however, when extending the rule to more than 2 resistors. For 3 resistors, the rule becomes: R1 * R2 * R3 Rt = --------------------- R1*R2 + R1*R3 + R2*R3 The 'product' part is simple, but the 'sum' part becomes the sum of the products of every 2-resistor combination. It gets even worse with more resistors. Of course, you can use the 'product-over-sum' rule to find the parallel resistance of R1 and R2, then use it again to find the parallel resistance of that with R3, etc. In the last part of your question, you asked about series-parallel circuits. Sometimes, a series-parallel circuit can be broken into series parts and parallel parts. In this case, each time you simplify, use the appropriate rule. For example, in the following circuit: let R1 = 30 Ohms, R2 = 10 Ohms, R3 = 40 Ohms and R4 = 20 Ohms. First, we can combine R1 and R2, which are in series. Call that Rx: Rx = R1 + R2 = 30 + 10 = 40 Ohms. (Series rule - add R's up) Now we can combine Rx and R3, which are in parallel. Call that Ry: 1/Ry = 1/Rx + 1/R3 (Parallel - use Reciprocal rule) = 1/40 + 1/40 = 2/40 = 1/20 so Ry = 1/(1/20) = 20 Ohms. Now we have a series circuit with Ry and R4. So our total resistance is the sum of these values: Rt = Ry + R4 = 20 + 20 = 40 Ohms If you can't break the circuit up into series and parallel parts, you have to use Kirchoff's laws and Thevenin's theorems to produce a set of simultaneous equations for the voltages and currents, then use algebra to solve the system of equations. This can get very messy. I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
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