Why Doesn't The Product Over Sum Equation (for 2 Resistors: 1/R=1/r1 + 1/r2=r1r2/(r1+r2)) Work For More Than 2 Resistors? (Ex: 1/3 + 1/3 + 1/6 Does Not Equal 4.5)

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2 Answers

Oddman Profile
Oddman answered
Your first equation can be extended to as many resistors as you like.
1/r = 1/r1 + 1/r2 + 1/r3 + ... + 1/rn

When you express that sum over a common denominator (r1*r2*r3*...*rn), you find the numerator becomes (for 3 resistors)
r2*r3 + r1*r3 + r1*r2

Inverting that makes your "product over sum" expression into
r = r1*r2*r3/(r2*r3 + r1*r3 + r1*r2)

Each element of the sum is the product with one term removed, just as it is in the 2-resistor case.

1/r = 1/3 + 1/3 + 1/6 = 5/6
r = 6/5

r = (3*3*6)/(3*6 + 3*6 + 3*3) = 54/(18+18+9) = 54/45 = 6/5
Anonymous Profile
Anonymous answered
If you first solve for a resistor pair, and then use the product over sum method for each additional resistor, it will still work, but it's just not the most straightforward way to go about it.  

You can first solve for the first two; R1*R2/(R1+R2)  = R12

And then add in each additional resistor;  R12R3/(R12+R3)...

But it's easier to add up the conductances, i.e. 1/Rtotal = 1/R1 + 1/R2 + 1/R3...

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Anonymous