Resistors in Series and Parallel: Formula Derivation and Examples

© Eugene Brennan

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.

Some Revision: A Circuit With One Resistor

In an earlier tutorial, 'How to Understand Electricity: Volts, Amps and Watts Explained on Appliances', you learned that when a single resistor with resistance R ohms was connected in a circuit with a voltage source V, the current I through the circuit was given by Ohm's Law:

Ohms Law

I = V/R

Example: A 240 V mains supply is connected to a heater with a resistance of 60 ohms. What current will flow through the heater?

Current = V/R = 240/60 = 4 amps

Schematic of a simple circuit. A voltage source V drives a current I through the resistance R. © Eugene Brennan

Derivation of Formula for Resistance When Resistors Are 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.
Because the resistors are linked together, the voltage source V causes the same current I to flow through both of them.

Two resistors connected in series. The same current I flows through both resistors. © Eugene Brennan

There will be a voltage drop or potential difference across both resistors.

Let the voltage drop measured across R₁ be V₁ and let the voltage measured across R₂ be V_2, as shown in the diagram below.

Voltage drop across resistors connected in series. © Eugene Brennan

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

I = V/R

Therefore, rearranging the equation by multiplying both sides by R

IR = V

or switching around

V = IR

So for resistor R₁

V₁ = IR₁

and for resistor R₂

V₂ = IR₂

Kirchoff's Voltage Law

From Kirchoff's Voltage Law, we know that the sum of voltages around a closed loop in a circuit adds up to zero. We decide on a convention, so voltage sources with arrows pointing clockwise from negative to positive are considered positive and voltage drops across resistors are negative. So, in our example:

V - V₁ - V₂ = 0

Rearranging

V = V₁ + V_{2 ................} (i.e., the voltage V equals the sum of the drops across the resistors)

Substitute for V₁ and V₂ calculated earlier

V = IR₁ + IR₂ = I(R₁ + R₂)

Divide both sides by I

V/I = R₁ + R₂

But from Ohm's Law, we know I = V/R, so rearranging:

V/I = R = total resistance of the circuit. Let's call it R_total

Therefore

V/I = R_total = R₁ + R₂

In general, if we have n resistors:

R_total = R₁ + R₂ + ...... R_n

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

Formula for resistors connected in series

Example 1

Five 10k resistors and two 100k resistors are connected in series. What is the combined resistance?

Answer

Resistor values are often specified in kiloohm (abbreviated to "k") or megaohms (abbreviated to "M")

1 kiloohm or 1k = 1000 ohms

1 megaohm or 1M = 1000,000 ohms

So total resistance = sum of the resistances

= 5 x (10k) + 2 x (100k)

= 50k + 200k

= 250k or 250,000 ohms

Example 2

Three 47 ohm, five 1.2k, four 100k and two 3.3M resistors are connected in series. What is the total resistance?

Answer

We often replace the decimal point in resistor values with the multiplier to avoid misreading if, e.g., the "dot" gets erased from the value printed on a component or in documents. So 1.2k becomes 1k2.

So total resistance = sum of the resistances

= 3 x 47 + 5 x 1k2 + 4 x 100k + 2 x 3M3

= 3 x 47 + 5 x 1200 + 4 x 100,000 + 2 x 3,300,000

= 141 + 6000 + 400,000 + 6,600,000

= 7,006,141 ohms

Derivation of Formula for Resistance of 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. © Eugene Brennan

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 R_total.

Therefore,

R_total = 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.

Example

A 100 ohm resistor and a 220 ohm resistor are connected in parallel. What is the combined resistance?

Answer

For two resistors in parallel, we just divide the product of the resistances by their sum.

So total resistance = 100 x 220 / (100 + 220) = 22000/320 = 8.75 ohms

Derivation of Formula for Resistance of 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.

Multiple resistors in parallel. © Eugene Brennan

So for n resistors

I = I₁+ I₂+ I₃. ........... + I_n

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

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

Rearranging

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

If V/I = R_total then

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

So, our final formula is

1/R_total = (1/R₁ + 1/R₂ + V/R₃ ........... 1/R_n)

We could invert the right side of the formula to give an expression for R_total; 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 and sum them together, giving us the reciprocal of the total resistance. Then, we take the reciprocal of this result, giving us R_total.

Formula for multiple resistors in parallel

Example

Calculate the combined resistance of three 100-ohm and four 200-ohm resistors in parallel.

Answer

Let's call the combined resistance R.

So

1/R = 1/100 + 1/100 + 1/100 + 1/200 + 1/200 + 1/200 + 1/200

We can use a calculator to work out the result for 1/R by summing all the fractions and then inverting to find R, but let's try and work it out "by hand".

So

1/R = 1/100 + 1/100 + 1/100 + 1/200 + 1/200 + 1/200 + 1/200

= 3/100 + 4/200

To simplify a sum or difference of fractions, we can use a lowest common denominator (LCD). The LCD of 100 and 200 in our example is 200

Therefore, multiply the top and bottom of the first fraction by 2 giving:

1/R = 3/100 + 4/200 = (2 x 3) / (2 x 100) + 4/200

= 6 / 200 + 4/200

= (6 + 4)/200 = 10/200

and inverting gives R = 200 / 10 = 20 ohms. No calculator needed!

Recommended Books

Introductory Circuit Analysis by Robert L Boylestad and available from Amazon 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. New and used versions of the hardcover 10th edition are available on Amazon. Later editions are also available.

References

Boylestad, Robert L. (1968) Introductory Circuit Analysis (6th ed. 1990) Merrill Publishing Company, London, England.

Disclaimer 

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2020 Eugene Brennan

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