# Resistors in Series & Parallel Formula Derivation

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

**Resistance Tutorial Includes:**

What is resistance
Ohms Law
Ohmic & Non-Ohmic conductors
Resistance of filament lamp
Resistivity
Resistivity table for common materials
Resistance temperature coefficient
Voltage coefficient of resistance, VCR
Electrical conductivity
Series & parallel resistors
Parallel resistors table

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.

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

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:

$V-{V}_{1}-{V}_{2}=0\phantom{\rule{1ex}{0ex}}\text{or}\phantom{\rule{1ex}{0ex}}V={V}_{1}+{V}_{2}$Then substituting for V_{1} and V_{2}

This simplifies to:

$\frac{V}{I}=R1+R2$But V/I = R_{total}, therefore

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

${R}_{\mathrm{total}}={R}_{1}+{R}_{2}+{R}_{3}+....$## 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.

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

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.

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.

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:}

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.

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.

Multiplying through by R_{total} gives:

Then multiple by R_{1} and R_{2}

Isolate R_{total}
${R}_{1}{R}_{2}={R}_{\mathrm{total}}({R}_{2}+{R}_{1})$

Then divide by (R_{1} + R_{2})

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.

**More Basic Electronics Concepts & Tutorials:**

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Current
Power
Resistance
Capacitance
Inductance
Transformers
Decibel, dB
Kirchoff's Laws
Q, quality factor
RF noise
Waveforms
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