Resistors in Series & Parallel Formula Derivation

A derivarion for the formulas for calculating total resistance of resistors in series and parellel.

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The formulas for calculating the total resistance for a number of resistors wired in series, and also for resistors in parallel are well known.

What may be less well known are the reasoning and derivation for the formulas.

Understanding how to derive the formulas for a set of resistors in series or parallel may be required on some occasions, and it also helps in understanding general circuit theory.

The basis for the derivation of the equations for both series and parallel resistor formulas, revolves around the use of Kirchoff's laws. Using these the derivations of the equations are relatively simple.

Derivation of total resistance of resistors in series

The equation for the total resistance of a series of resistors in parallel is the sum of all the resistors are given below.

R_{total} = R_{1} + R_{2} + R_{3} + . . . .

The first stage in proving the formula is to look at the case of two resistors in series, to see how the circuit behaves.

Resistors in series - the electronic circuit design showing the voltages present along with the current, etc. — Resistors in series - individual resistors, currents and voltages
*Two resistors are show here but the same derivation can easily be expanded to any number of resistors.*

There are two facts that need to be considered when starting the derivation of the equation for the total resistance of a set of resistors in series. The first is that the same current flows around the circuit. The same current flows through the voltage source as well as the resistors.

Secondly, Kirchoff's laws state that the sum of voltages around a circuit is zero. Thus the sum of the voltage drops across the resistors is equal to the voltage supplied by the source in the circuit shown.

From Ohm's law:

V_{1} = I R_{1} & V_{2} = I R_{2}

Then from Kirchoff's law:

V - V_{1} - V_{2} = 0 or V = V_{1} + V_{2}

Then substituting for V₁ and V₂

V = I R_{1} + I R_{2} = I (R_{1} + R_{2})

This simplifies to:

\frac{V}{I} = R 1 + R 2

But V/I = R_total, therefore

R_{total} = R_{1} + R_{2}

Using the same logic it is possible to expand this out to the general case of multiple resistors:

R_{total} = R_{1} + R_{2} + R_{3} + . . . .

Derivation of total resistance of resistors in parallel

It is often the case that multiple resistors are placed in parallel. There are many circumstances when this occurs when undertaking electronic circuit design, etc.

Resistors in parallel - these scenarios can be seen when undertaking electrical or electronic cirucuit design

The standard formula for calculating the total resistance for a number of resistors or restances in parallel is given below.

\frac{1}{Rtotal} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} + . . . . . .

The derivation for the overall equation for a set of multiple resistors in parallel is quite easy to work through. taking the basic aspects of the circuit, it os possible to easily derive the overall equation for a set of resistors in parallel.

Resistors in parallel - the electronic circuit design showing the voltages present along with the electric current definitions, etc. — Resistors in parallel - individual resistors, currents and voltages

When looking at deriving the formula for the total resistance of a set of resistors in parallel, it is necessary to consider the current flowing through each resistor in turn, and understand that each resistor has the same potential difference or voltage across it.

The first point to understand is that the sum of the currents flowing through the individual resistors equals the overall current provided by the voltage source as shown in the diagram:

I = I_{1} + I_{2} + I_{3} + . . . I_{n}

Knowing that I=V/R from Ohm's law, it is possible to relate the current levels flowing in terms of the voltage (which is the same for all as they are in parallel), and the resistance.

I = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}} + . . . \frac{V}{R_{n}}

Then by dividing both sides by V, we can see:

\frac{I}{V} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . \frac{1}{R_{n}}

But as I/V is 1/R_{total, this can be replaced in the equation to give:}

\frac{1}{R_{total}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . \frac{1}{R_{n}}

It can be seen that the derivation for the total resistance of a series of resistors in parallel is remarkably easy to derive.

Derivation of the formula for two resistors in parallel

It is often the case in various electrical and electronic design or installation that it is necessary to work out the total resistance for two resistors in parallel.

For this case the equation can be considerably simplified, making it much easier to calculate the total resistance.

R_{total} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}

Deriving this equation is relatively straightforward, requiring some simple manipulation of the general equation for parallel resistors, but simplifying it down to include only two electronic components.

\frac{1}{R_{total}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}

Multiplying through by R_total gives:

1 = \frac{R_{total}}{R_{1}} + \frac{R_{t}}{R_{2}}

Then multiple by R₁ and R₂

R_{1} R_{2} = R_{total} R_{2} + R_{total} R_{1}

Isolate R_total $R_{1} R_{2} = R_{total} (R_{2} + R_{1})$

Then divide by (R₁ + R₂)

R_{total} = \frac{R_{1} R_{2}}{R_{2} + R_{1}}

Using this formula, it is very easy to calculate the overall resistance of two resistors in parallel

The equations for determining the total resistance for sets of resistors in series and parallel are widely used n many areas from electrical work to electronic circuit design, and a host of other areas. Although it is not necessary to derive the equations from first principles allt he time, it is useful to understand how this can be done as it gives a much better understanding of what is happening.

Written by Ian Poole .
Experienced electronics engineer and author.

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Resistors in Series & Parallel Formula Derivation

A derivarion for the formulas for calculating total resistance of resistors in series and parellel.

Derivation of total resistance of resistors in series

Derivation of total resistance of resistors in parallel

Derivation of the formula for two resistors in parallel

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