Parallel resistors

[gazdkw82]

I'm getting conflicting information regarding parallel resistors. I originally thought the below was how to calculate total parallel resistance

1. Rt= 1/R1+1/R2+1/R3

2. 1/Rt

However in my pre exam workbook it says to

1. Calculate Rt as 1. above
2. Find the lowest common denominator
3. Calculate how many times each resistor fits within demonitor
4. Add up and devide by demonitor
5.invert both sides and this will give you Rt

This seems really long winded however I have found sometimes my original formula Iv always used can sometimes be inaccurate.....


Thoughts?

 

[Karin]

i think that's essentially the same, no? the written method just explains how you solve the first equation. you need the lowest common denominator to add fractions.

if you've got a calculator a really quick method is to use the x-¹ key

R1x-¹ + R2x-¹ + R3x-¹ = Rtx-¹


EDIT: ah the difference between what i've just written and your original formula is that in step 1. you are calculating 1/Rt, not Rt

 

[Richard Burns]

For resistors in parallel

1/Rt = 1/R1 + 1/R2 + 1/R3 ...

If you are calculating this with a series of values then the mathematical method of calculation is to convert each fraction into fractions with the same denominator so that you can then just add the numerators.

Then because the result you want is Rt and the sum of the fractions is 1/Rt then inverting the fraction gives you Rt.

So if you had three resistors in parallel 2 ohm, 5 ohm and 10 ohm
then 1/Rt = (1/2) + (1/5) + (1/10)
lowest common denominator is 10 so you convert the (1/2) and (1/5) to (5/10) and (2/10), the (1/10) stays as it is .
So 1/Rt = (5/10) + (2/10) + (1/10)
5 + 2 + 1 = 8 so 1/Rt = 8/10
turning over gives Rt = 10/8 = 1.25 ohms

They are the same means of calculation but the second method is simpler to work out without a calculator.

 

[gazdkw82]

Excellent explanation!!

So both ways are correct and will achieve the same result?

 

[Richard Burns]

Thank you.
Yes the same calculation just different approaches to it.
Same result every time, though if you do the calculation with a calculator then there will be a small amount of rounding going on but not a significant figure.

 

[Lucien Nunes]

This is one of those things that makes most sense if you have a clear visualisation of what the calculation is doing.

Resistance R = V/I. i.e volts per amp.
Conductance G = I/V. i.e. amps per volt.
Conductance and resistance are simply reciprocals of each other: G=1/R and R=1/G. One normally quotes things as resistances but you can always convert one into the other.

Resistances in series add because the same current goes through them all, while the voltages across each of them add to make the total voltage. Rt = R1 + R2 + R3 in series.
Conductances in parallel add because the same voltage appears across them all, while currents through each of them add to make the total current. Gt = G1 + G2 + G3 in parallel.
Substituting 1/R for G, this can be expressed as 1/Rt = 1/R1 + 1/R2 + 1/R3, which gives us a formula for calculating resistances in parallel.

When using this formula, you are converting resistances R1, R2, R3 into their equivalent conductances, adding them because conductances add in parallel, and then converting the total back to a resistance Rt. You will have no difficulty working out how to calculate conductances in series, although no-one is ever likely to ask you to do this.