Electric Resistance. Combinations of Resistors

In these revision notes for Electric Resistance. Combinations of Resistors, we cover the following key points:

• The definition of resistance
• The unit of resistance
• The factors affecting the resistance of a conductor
• Methods of combining resistors in a circuit (series & parallel)
• How to calculate the equivalent resistance is a series setup of resistors
• How to calculate the equivalent resistance is a parallel setup of resistors
• How to calculate the equivalent resistance is a complex setup of resistors

Electric Resistance. Combinations of Resistors Revision Notes

Electrical resistance R is a quantity related to the opposition that electrons encounter during their flow through a conductor.

Resistance is measured in Ohms [Ω], in honour to the German scientist Georg Simon Ohm.

All electrical appliances - some more and some less - make resistance to the electrons flow. Conductors in general provide low resistance while insulators a very high resistance to the flow of charges.

There are four factors affecting the resistance of a material. They are:

1. Length of material, L. Longer the material, higher the resistance it makes to the flow of electrons.
2. Thickness of material or cross-sectional area, A. Thicker the material, lower the resistance it makes to the flow of electrons.
3. Type of material. This feature is represented by a quantity known as resistivity, ρ, and shows how many ohms of resistance does a 1m³ cube of the given material make to the electrons flowing through it.
4. Temperature of material, T. Higher the temperature of material, higher the resistance it makes to the flow of electrons.

The equation used for calculating the resistance of a conductor at constant temperature (at 20°C) is

We can combine two or more resistors in series or in parallel to change the resistance of a circuit.

1. If two or more resistors are connected in series, they are permeated by the same current. As a result, electrons encounter resistance in two successive positions. This causes an accumulative effect in the total (otherwise known as "equivalent") resistance, R_eq of the circuit, i.e.
   R_eq = R₁ + R₂ + ⋯
   Therefore, we can replace the two above resistors connected in series by a single resistor whose resistance is the mathematical sum of each single resistor.
   A parallel setup is when there are at least two branches containing one or more resistors each
2. In a parallel setup the current comes as a whole from the source, then it divides into two smaller currents I₁ and I₂ flowing through the resistors R₁ and R₂ respectively. As a result, the equivalent resistance of a parallel setup is smaller than the resistance of each single resistor, as electric charges flow easier through two resistors than if there it was a single resistor. The concept of equivalent resistance is used to ease the study of such situations as in this case we assume it was only a single resistor in a single branch
   The equation for the equivalent resistance of a parallel setup is
   1/R_eq =1/R₁ + 1/R₂ + ⋯
3. When resistors are combined both in series and in parallel in the same circuit. In these cases, the equivalent resistance is calculated in steps (stages) and we start calculating it from the innermost part of the setup. Such combination is called "complex combination of resistors".
   If temperature of conductor is different from 20°C (which is known as reference temperature, T_ref), we calculate its resistance through the formula
   
   R = R_ref ∙ [1 + α ∙ (T - T_ref )]
   where α is the temperature coefficient of resistance for a given conducting material. It is an intrinsic property of the material itself and is given in tables.

The unit of temperature coefficient of resistance is [K-1] or [1/K].