The article discusses the four main factors that affect electrical resistance: length, cross-sectional area, the type of material, and temperature. It also explores the concept of superconductivity and its practical applications.
Ohmic materials are those that have a fixed resistance regardless of the applied voltage. The voltage-versus-current graph will be a straight line showing a constant resistance value. (As most common resistors are ohmic resistors, the word ‘ohmic’ is generally not used.)
In electronics, non-ohmic resistors change their resistance according to some other parameter, such as applied voltage, light, or temperature. (Again, the term ‘non-ohmic’ is not generally used.) Examples of non-ohmic resistors are tungsten lamp elements, photo-sensitive resistors, voltage-dependent resistors, and temperature-sensitive resistors.
All parts of an electric circuit (the supply source, the conductors, and the load) resist current flow. Even the chemicals inside a battery have a resistance that changes with temperature, state of charge, and the gases dissolved in the electrolyte. To understand resistance, and therefore, electric circuits, it is necessary to understand that resistance is determined by four factors:
- Length
- Cross-Sectional Area
- Type of Material
- Temperature
Length
Electrical resistance is associated with the collisions between moving electrons (the electric current) and the atoms of the conducting material. As with driving down a road, the risk of having a collision increases the further one travels. In fact, traveling twice the distance doubles the risk. So, the resistance of a conductor is proportional to its length:
$$R\ \propto\ l$$
This can be proved quite simply by measuring the resistance of the active conductor in a 100 m roll of 1mm^{2} cable (it should be about 1.7 Ω). Then, measure the resistance of the neutral conductor (again, about 1.7 Ω). Finally, measure the resistance of both the active and neutral conductors joined together at one end of the roll (now 200 m of wire). The resistance should be twice the resistance of either wire, as the total length is twice as long (approximately 3.4 Ω, or twice what was measured for one wire).
Cross-Sectional Area (CSA)
The roll of cable used in the previous section should be labeled ‘CSA=1mm^{2}’. That is the area of the end of the wire if it is cut across at 90°, which is known as the ‘cross-section’ of the wire. The area of the cross-section is called the ‘cross-sectional area’ or ‘CSA’ (and is not to be confused with the diameter).
If the CSA is doubled, it is easy to imagine that more electrons can pass easily down the wire, and therefore it will have less resistance. The resistance is said to be proportional to the inverse of the CSA. In other words, as the CSA increases, the resistance decreases, so resistance is inversely proportional to area:
$$R\ \propto\frac{l}{A}$$
Using the roll of cable again and knowing the resistance of the conductors, connect both ends (which is the same as increasing the CSA to 2 mm^{2}, or double what it was). The resistance of 100 m of 2 mm^{2} wire should be half the resistance of one wire (approximately 0.85 Ω).
Type of Material (Resistivity)
The copper used to make electrical wire (annealed copper) has a known value of resistance. A cube of copper 1 m on each face has a resistance of approximately 1.72×10^{−8} Ω.
The standard of one square meter of material one meter long is based on the SI system, but no lab would attempt to measure the resistance of such a large lump of material. Instead, a sample such as 100 m of 1 mm^{2} is used and the resistivity is calculated mathematically. (Resistivity is given the Greek letter ‘rho’, which is written as ρ and pronounced ‘roe’.)
Resistance is directly proportional to resistivity:
$$R\ \propto\rho$$
The resistivity of a material is defined as the resistance between the opposite faces of a 1 m cube at a specified temperature (e.g., 20°C) and is measured in ohm-meters (Ωm).
So
$$\rho=\frac{RA}{l}$$
Or transposing to define R:
$$R=\frac{\rho l}{A}$$
where ρ = resistivity.
Knowing the resistivity of any material, the resistance of any conductor can be calculated (with allowances being made for temperature differences where necessary). In Table 1, some electrical materials are listed, together with their resistivity values.
Conductor | Resistivity (ρ) @ 20°C | Use |
Aluminum | 2.83×10^{−8} Ωm | Pure metals used for conductors |
Copper | 1.72×10^{−8} Ωm | |
Gold | 2.44×10^{−8} Ωm | |
Lead | 2.04×10^{−8} Ωm | |
Platinum | 10.09×10^{−8} Ωm | |
Silver | 1.63×10^{−8} Ωm | |
German silver | 33×10^{−}8 Ωm | Alloys used as resistance wire |
Advance | 49×10^{−8} Ωm | |
Manganin | 48×10^{−8} Ωm | |
Nichrome | 112×10^{−8} Ωm |
Table 1. Resistivity of selected materials
The values in the table are given in ohm-meters because the formula $R(\Omega)\times\frac{A(m^2)}{l(m)}$, when simplified, becomes:
$$\frac{{\Omega m}^2}{m}=\Omega m$$
Resistivity also changes depending on whether the material is mechanically hard or soft. Annealed copper (copper, which has been heated to make it more flexible) has a higher resistivity than hard copper.
The resistivity of a conductor also depends on the purity of the material and the nature of any gaseous inclusions within it. Hi-fi speaker installers pay higher prices for ‘oxygen-free’ speaker leads.
Table 1 shows that silver has the least resistance, closely followed by copper; but copper is less expensive than silver, so it is used extensively as an electrical conductor.
The four materials listed at the end of the table are alloys that are generally used for making resistors—that is, they restrict the flow of electricity far more than those above them.
When calculating the resistance of a solid material, there are 1000×1000 square millimeters in a square meter
$$1m^2=1\times{10}^{-3}\times{10}^{-3}\times{10}^{-3}=1\times{10}^{-6}m^2$$
Example 1
Find the resistance of a copper cable 500 m in length if it has a cross-sectional area of 2.5 mm^{2}. Take the resistivity of copper to be 1.72×10^{−8} Ω/m.
Note that 2.5 mm^{2} is 2.5×10^{−6} m^{2}.
Solution
$$R=\frac{\rho l}{A}$$
$$R=\frac{1.72\times{10}^{-8}\times500}{2.5\times{10}^{-6}}=3.44\Omega$$
Example 2
To manufacture a 15 Ω resistor from 0.2 mm^{2} diameter manganin wire, what length of wire is required?
Solution
$$A=\frac{\pi d^2}{4}=\frac{\pi\times\left(0.2\times{10}^{-3}\right)^2}{4}=0.0314{mm}^2$$
$$R=\frac{\rho l}{A}$$
$$l=\frac{RA}{\rho}=\frac{15\times0.0314\times{10}^{-6}}{48\times{10}^{-8}}=0.981\ m=981\ mm$$
Temperature
In all these calculations, the resistance value is accurate only at 20°C. As the temperature increases or decreases, allowances may have to be made for a change in resistance.
For some materials, an increase in temperature causes an increase in resistance. These materials are said to have a ‘positive temperature coefficient’ (PTC). When a material has a lower resistance at higher temperatures, it is said to have a ‘negative temperature coefficient’ (NTC). Some resistors are made from specific materials to take advantage of these characteristics. The temperature coefficient of resistance is defined as the change in resistance per ohm per degree Celsius (or Kelvin).
Resistivity values are specified at a particular temperature because resistance can change with temperature. The resistance of most metallic conductors increases with temperature (PTC) and over a limited range. The increase is a linear function of temperature or very close to it within that range. This leads to what is called the ‘inferred zero’ method of calculating the resistance of conductors at another temperature. The inferred zero value varies for different materials, but the method is illustrated in Figure 1.
Figure 1. Effect of temperature on resistance
Copper has an inferred zero resistance at −234.5 °C, and the increase of resistance plotted against temperature is basically linear. The resistance of a length of copper wire can be given as a resistance of R0 at 0°C, and the increase in resistance per degree C will continue to be linear through R_{1} and R_{2}. Therefore, at any temperature significantly above −234.5 °C, the resistance can be calculated from any other known resistance at a known temperature. The calculation is a simple ratio:
$$\frac{R_2}{R_1}\propto\frac{t_2}{t_1}$$
Temperatures are usually taken as relative to 0°C, so the formula needs to recognize the inferred zero resistance at −234.5 °C plus the resistor temperature.
$$R_2=R_1\frac{{234.5+t}_2}{{234.5+t}_1}$$
where:
R_{1} = resistance at temperature t_{1}
R_{2} = resistance at temperature t_{2}
An electric motor may be tested to see how hot the windings become in full-load use. To measure the temperature directly would require the motor to be disassembled for a temperature probe to be inserted into the windings, but another method is often used.
Example 3
The resistance of a coil of copper wire is 34 Ω at 15 °C. What would be its resistance at 70 °C?
Solution
$$R=R_1\frac{{234.5+t}_2}{{234.5+t}_1}$$
$$R=34\times\frac{234.5+70}{234.5+15}=41.49\Omega$$
The motor winding resistance is measured when the motor is cold and then re-measured immediately after it is shut down. The temperature can be calculated from the cold temperature and the resistance change of the windings using the formula shown before.
Example 4
A motor at 20 °C has a winding resistance of 16 Ω. After running up to temperature at full load, the resistance is measured as 24.8 Ω. What is the temperature of the windings?
Solution
$$R_2=R_1\frac{{234.5+t}_2}{{234.5+t}_1}$$
By transposition:
$$t_2=\frac{R_2}{R_1}\times\left(234.5+t_1\right)-234.5=\frac{24.8}{16}\times\left(234.5+20\right)-234.5=160\ °C $$
Resistance values can also be calculated from the temperature coefficient of resistance, which is defined as the change in resistance per ohm per degree change in temperature (symbol α—alpha).
$$\alpha=\frac{Change\ in\ Resistance\ per\ Degree}{Resistance\ at\ t_1}$$
Table 2 lists a selection of conductors and the temperature coefficients of resistance for those conductors at 0°C and 20°C.
Temperature coefficient of resistance (/°C) | ||
Conductor | α_{0} | α_{20} |
Aluminum | 0.00423 | 0.0039 |
Copper | 0.00427 | 0.00393 |
Gold | 0.00368 | 0.00343 |
Lead | 0.00411 | 0.0039 |
Platinum | 0.00367 | 0.0039 |
Silver | 0.004 | 0.004 |
Zinc | 0.00402 | 0.004 |
German silver | 0.0004 | 0.0004 |
Advance | 0.00002 | 0.00002 |
Manganin | 0.00001 | 0.00001 |
Nichrome | 0.0002 | 0.0002 |
Table 2. Temperature coefficients of resistance
For most metals, the change in resistance per ohm per °C is relatively constant, but the coefficient changes as temperature changes. The temperature at which the value of the coefficient is effective is usually given by a subscript to the symbol (e.g., α_{0} and α_{20}, indicating the temperature coefficients at 0°C and 20°C respectively).
Mathematically, the new resistance can be calculated by adding the change in resistance to the original resistance, which in turn is the change in temperature multiplied by the temperature coefficient.
$$R_2=R_1\left[1+\alpha\left(t_2-t_1\right)\right]$$
where:
R_{1} = resistance at temperature t_{1}
R_{2} = resistance at temperature t_{2}
α = temperature coefficient of resistance.
Example 5
A 2.5 mm^{2} copper conductor has a resistance of 0.241 Ω at an ambient temperature of 20°C. Find its resistance at 75°C. Do the sums in brackets first.
Solution
$$R_2=R_1\left[1+\alpha\left(t_2-t_1\right)\right]$$
$$R_2=0.241\times\left[1+0.00393\times\left(75-20\right)\right]=0.293\Omega$$
Example 6
Copper conductors of 2.5 mm^{2} cross-sectional area are supplying a current of 15 A to an air-conditioner. If the conductors have a total resistance of 0.43 Ω, calculate the power lost in the conductors.
Solution
$$P=\ I^2R={15}^2\times0.43=96.75W$$
Temperature Effects on Superconductors
Many materials produce an effect known as ‘superconductivity’ when they are cooled below a certain temperature. Even lead is a superconductor at around −256.8°C. At the critical temperature, electrons can pass through the material with seemingly zero resistance. Other materials, mostly pure metals and some special alloys, have different critical temperatures but exhibit the same total lack of resistance below that temperature.
Since a superconductor has no resistance, once a current flow is initiated, the current will continue to flow at the same value without an applied potential. If a conductor has no resistance, then current flow through it generates no heat. If no heat is generated in a conductor, the amount of current passed through it can be increased far beyond normal values.
The current flowing through a superconductor creates a magnetic field that becomes equal and opposite to any applied magnetic field, with the result that extremely powerful electromagnets can be constructed.
Research has been carried out for many years to try to produce a superconductor effect at higher temperatures. Although breakthroughs could occur at any time, at the moment, superconductors still need to be chilled to temperatures well below freezing point, below the temperatures where most gases become liquids.
Although materials may be made superconductive, it has been discovered that their current capacity is not unlimited and current over a certain level destroys the superconductivity.
Factors Affecting Resistance Review Questions
What are the four factors that affect resistance?
The four factors that affect resistance are its length, cross-sectional area, resistivity based on the material used and its temperature.
What is a superconductor, and what are they used for?
Extremely powerful electromagnets can be constructed from superconductors and are used to levitate electric trains, are used to generate images of the human body (MRI), and in particle accelerators.
Key Takeaways of Factors Affecting Resistance
Understanding the factors affecting resistance—length, cross-sectional area, material type, and temperature—is crucial for designing and optimizing electrical systems. These principles are fundamental to ensuring efficient energy transmission, minimizing power loss, and tailoring materials for specific applications.