Resistors in Series and Parallel Formula Derivation

Updated on February 21, 2020
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Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

Formulas for Resistors in Series and Parallel

Resistors are ubiquitous components in electronic circuitry both in industrial and domestic consumer products. Often in circuit analysis, we need to work out the values when two or more resistors are combined. In this tutorial, we'll work out the formulas for resistors connected in series and parallel.

A selection of resistors
A selection of resistors | Source

Some Revision: A Circuit With One Resistor

In an earlier tutorial, you learned that when a single resistor was connected in a circuit with a voltage source V, the current I through the circuit was given by Ohm's Law:

I = V / R ........... Ohm's Law

Ohms Law

I = V / R

Schematic of a simple circuit. A voltage source V drives a current I through the resistance R
Schematic of a simple circuit. A voltage source V drives a current I through the resistance R | Source

Two Resistors in Series

Now let's add a second resistor in series. Series means that the resistors are like links in a chain, one after another. We call the resistors R1 and R2.
The same voltage source V causes a current I to flow.

We notice that the same current I flows through both resistors.


Two resistors connected in series. The same current I flows through both resistors.
Two resistors connected in series. The same current I flows through both resistors. | Source

Let the voltage drop (also called the potential difference) measured across R1 be V1 and let the voltage measured across R2 be V2 as shown in the diagram below.

From Ohm's Law, we know that for a circuit with a resistance R and voltage V:

I = V / R

Therefore

V = IR

So for resistor R1

V1 = IR1

and for resistor R2

V2 = IR2

Voltage drop across resistors connected in series.
Voltage drop across resistors connected in series. | Source

From Kirchoff's Voltage Law, we know that the voltages around a loop in a circuit add up to zero. So in our example:

V = V1 + V2

Substitute for V1 + V2

V = IR1 + IR2 = I(R1 + R2)

Divide both sides by I

V / I = R1 + R2

But from Ohm's Law, we know V / I = total resistance of the circuit. Let's call it Rtotal

Therefore

Rtotal = R1 + R2

In general if we have n resistors:

Rtotal = R1 + R2 + ...... Rn

So to get the total resistance of resistors connected in series, we just add all the values.

Formula for resistors connected in series.
Formula for resistors connected in series. | Source

Two Resistors in Parallel

Next we'll derive the expression for resistors in parallel. Parallel means all the ends of the resistors are connected together at one point and all the other ends of the resistors are connected at another point.

When resistors are connected in parallel, the current from the source is split between all the resistors instead of being the same as was the case with series connected resistors. However, the same voltage is now common to all resistors.

Two resistors connected in parallel.
Two resistors connected in parallel. | Source

Let the current through resistor R1 be I1 and the current through R2 be I2

The voltage drop across both R1 and R2 is equal to the supply voltage V

Therefore from Ohm's Law

I1 = V / R1

and

I2 = V / R2

But from Kirchoff's Current Law, we know the current entering a node (connection point) is equal to the current leaving the node

Therefore

I = I1 + I2

Substituting the values derived for I1 and I2 gives us

I = V / R1 + V / R2

= V(1 / R1 + 1 / R2)

The lowest common denominator (LCD) of 1 / R1 and 1 / R2 is R1R2 so we can replace the expression (1 / R1 + 1 / R2) by

R2 / R1R2 + R1 / R1R2

Switching around the two fractions

= R1 / R1R2 + R2 / R1R2

and since the denominator of both fractions is the same

= (R1 + R2) / R1R2

Therefore

I = V(1 / R1 + 1 / R2) = V(R1 + R2) / R1R2

Rearranging gives us

V / I = R1R2 / (R1 + R2)

But from Ohm's Law, we know V / I = total resistance of the circuit. Let's call it Rtotal

Therefore

Rtotal = R1R2 / (R1 + R2)

So for two resistors in parallel, the combined resistance is the product of the individual resistances divided by the sum of the resistances.


Formula for two resistors connected in parallel.
Formula for two resistors connected in parallel. | Source

Multiple Resistors in Parallel

If we have more than two resistors connected in parallel, the current I equals the sum of all the currents flowing through the resistors.

So for n resistors

I = I1 + I2 + I3. ........... + In

= V / R1 + V / R2 + V / R3 + ............. V / Rn

= V ( 1 / R1 + 1 / R2 + V / R3 ........... 1 / Rn)

Rearranging

I / V = ( 1 / R1 + 1 / R2 + V / R3 ........... 1 / Rn)

If V / I = Rtotal then

I / V = 1 / Rtotal = ( 1 / R1 + 1 / R2 + V / R3 ........... 1 / Rn)

So our final formula is

1 / Rtotal = ( 1 / R1 + 1 / R2 + V / R3 ........... 1 / Rn)

We could invert the right side of the formula to give an expression for Rtotal , however it's easier to remember the equation for the reciprocal of resistance.
So to calculate the total resistance, we calculate the reciprocals of all the resistances first, sum them together giving us the reciprocal of the total resistance. The we take the reciprocal of this result giving us Rtotal

So

Multiple resistors in parallel.
Multiple resistors in parallel. | Source
Formula for multiple resistors in parallel.
Formula for multiple resistors in parallel. | Source

Recommended Books

Introductory Circuit Analysis by Robert L Boylestad covers the basics of electricity and circuit theory and also more advanced topics such as AC theory, magnetic circuits and electrostatics. It's well illustrated and suitable for high school students and also first and second year electric or electronic engineering students. This hardcover 10th edition is available from Amazon with a "good - used" rating. Later editions are also available.

Source

References

Boylestad, Robert L, Introductory Circuit Analysis (1968) published by Pearson
ISBN-13: 9780133923605

© 2020 Eugene Brennan

Comments

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    • eugbug profile imageAUTHOR

      Eugene Brennan 

      3 months ago from Ireland

      Thanks Umesh!

    • bhattuc profile image

      Umesh Chandra Bhatt 

      3 months ago from Kharghar, Navi Mumbai, India

      Interesting. You have revised my memories of high school science classes. Well presented. Keep it up.

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