Using Kirchhoff’s Laws to Derive Resistor Combinations
Where do the formulas for the equivalent resistance of resistors in series and parallel come from?
Why combine resistors?
Sometimes, a resistor with the right amount of resistance we need might not be available. This is because most electronics manufacturers only produce a standard set of commonly used resistors. So, if we need to use an out-of-the-ordinary resistance, we will have to connect standard resistors in some pattern of series and parallel combinations.
Kirchhoff’s Laws will help us determine the equivalent resistance of our combination.
Resistors in series
Applying Kirchhoff’s voltage law in the direction of the loop,
Because they are connected in series, the same current I passes through each resistor.
Appplying Ohm’s Law, V = IR,
where Req is the equivalent resistance.
Substituting and simplifying, we get:
Resistors in Parallel
The battery forms a closed loop with each of the two resistors. According to Kirchhoff’s voltage law, that means the p.d. across each resistor is the same, specifically equal to the supply voltage, V.
At the junction where the current I splits into I₁ and I₂, we can apply Kirchhoff’s current law.
Using Ohm’s Law again,
Substituting,
Great job! Now we can apply these formulas to find the equivalent resistance of any combination of resistors.
If you would like to read more about the importance of Kirchhoff’s Laws in circuit analysis,
