Electronics Components: Parallel Resistors
So how do you calculate the total resistance for resistors in parallel on your electronic circuit? Put on your thinking cap and follow along. Here are the rules:
First, the simplest case: Resistors of equal value in parallel. In this case, you can calculate the total resistance by dividing the value of one of the individual resistors by the number of resistors in parallel. For example, the total resistance of two, 1 kΩ resistors in parallel is 500 Ω and the total resistance of four, 1 kΩ resistors is 250 Ω.
Unfortunately, this is the only case that’s simple. The math when resistors in parallel have unequal values is more complicated.
If only two resistors of different values are involved, the calculation isn’t too bad:
In this formula, R1 and R2 are the values of the two resistors.
Here’s an example, based on a 2 kΩ and a 3 kΩ resistor in parallel:
For three or more resistors in parallel, the calculation begins to look like rocket science:
The dots at the end of the expression indicate that you keep adding up the reciprocals of the resistances for as many resistors as you have.
In case you’re crazy enough to actually want to do this kind of math, here’s an example for three resistors whose values are 2 kΩ, 4 kΩ, and 8 kΩ:
As you can see, the final result is 1,142.857 Ω. That’s more precision than you could possibly want, so you can probably safely round it off to 1,142 Ω, or maybe even 1,150 Ω.
The parallel resistance formula makes more sense if you think about it in terms of the opposite of resistance, which is called conductance. Resistance is the ability of a conductor to block current; conductance is the ability of a conductor to pass current. Conductance has an inverse relationship with resistance: When you increase resistance, you decrease conductance, and vice versa.
Because the pioneers of electrical theory had a nerdy sense of humor, they named the unit of measure for conductance the mho, which is ohm spelled backward. The mho is the reciprocal (also known as inverse) of the ohm.
To calculate the conductance of any circuit or component (including a single resistor), you just divide the resistance of the circuit or component (in ohms) into 1. Thus, a 100 Ω resistor has 1/100 mho of conductance.
When circuits are connected in parallel, current has multiple pathways it can travel through. It turns out that the total conductance of a parallel network of resistors is simple to calculate: You just add up the conductances of each individual resistor.
For example, suppose you have three resistors in parallel whose conductances are 0.1 mho, 0.02 mho, and 0.005 mho. (These are the conductances of 10 Ω, 50 Ω, and 200 Ω resistors, respectively.) The total conductance of this circuit is 0.125 mho (0.1 + 0.02 + 0.005 = 0.125).
One of the basic rules of doing math with reciprocals is that if one number is the reciprocal of a second number, the second number is also the reciprocal of the first number. Thus, since mhos are the reciprocal of ohms, ohms are the reciprocal of mhos.
To convert conductance to resistance, you just divide the conductance into 1. Thus, the resistance equivalent to 0.125 mho is 8 Ω (1 ÷ 0.125 = 8).
It may help you remember how the parallel resistance formula works when you realize that what you’re really doing is converting each individual resistance to conductance, adding them up, and then converting the result back to resistance. In other words, convert the ohms to mhos, add them up, and then convert them back to ohms. That’s how — and why — the resistance formula actually works.