# Temperature Coefficient of Resistance : Formula and Measuring Method

In electrical or electronic engineering, when the flow of current supplies through a wire then it gets heat because of the wire’s resistance. In perfect condition, resistance must be ‘0’ however that doesn’t take place. When the wire gets heat up, then the wire resistance changes according to the temperature. Even though it is preferred that resistance must stay stable & it must be independent for the temperature. So, the resistance change for every degree change within temperature is termed as the temperature coefficient of resistance (TCR). Generally, it is denoted with a symbol alpha (α). The TCR of the pure metal is positive because when the temperature increases then resistance will be increased. Therefore, to make highly accurate resistances wherever resistance does not modify alloys is necessary.

## What is the Temperature Coefficient of Resistance (TCR)?

We know that there are many materials and they have some resistance. The resistance of material changes based on the variation of temperature. The main relation between the modify in temperature & modification in resistance can be given by the parameter called TCR (temperature coefficient of resistance). It is signified with the symbol α (alpha).

Based on the material obtainable, TCR is separated into two types such as a positive temperature coefficient of resistance (PTCR) and a negative temperature coefficient of resistance (NTCR).

In PTCR, when the temperature is increased, then the material resistance will be increased. For instance, in conductors when the temperature increases then the resistance also increases. For the alloys like constantan & manganin, the resistance is pretty low over a particular temperature range. For semiconductors such as insulators (rubber, wood), silicon & germanium & electrolytes. the resistance reduces then the temperature will be increased thus they have negative TCR.

In metallic conductors, when the temperature increases then the resistance will be increased due to the following factors which include the following.

• Straightly on the early resistance
• Rise of temperature.
• Based on the life of the material.

### The Formula for Temperature Coefficient of Resistance

The conductor resistance can be calculated at any specified temperature from the temperature data, it’s TCR, its resistance at the typical temperature & the operation of temperature. In general term, the temperature coefficient of the resistance formula can be expressed as

R = Rref (1+α (T−Tref))

Where

‘R’ is the resistance at ‘T’ temperature

‘Rref’ is the resistance at ‘Tref ‘temperature

‘α’ is the TCR of the material

‘T’ is the temperature of the material in ° Celsius

‘Tref’ is the reference temperature used for which the coefficient of temperature is stated.

The SI unit of the temperature coefficient of resistivity is per degree celsius or ( /°C)

The unit of the temperature coefficient of resistance is ° Celsius

Normally, the TCR (temperature coefficient of resistance) is consistent with a 20°C temperature. So normally this temperature is taken as normal room temperature. Thus the temperature coefficient of resistance derivation normally takes this into the description:

R = R20 (1+α20 (T−20) )

Where

‘R20’ is the resistance at 20°C

‘α20’ is the TCR at 20°C

The TCR of resistors is positive, negative otherwise constant over a fixed range of temperature. Selecting the correct resistor could stop the need for temperature compensation. A large TCR is required to measure temperature in some applications. Resistors intended for these applications are known as thermistors, which have a PTC (positive temperature coefficient of resistance) or NTC (negative temperature coefficient of resistance).

#### Positive Temperature Coefficient of Resistance

A PTC refers to some materials which experience once their temperature raised then electrical resistance also increased. The materials which have a higher coefficient then show a quick rise with temperature. A PTC material is designed to achieve the utmost temperature used for a given i/p voltage because at a particular point when the temperature increases then electrical resistance will be increased. The positive temperature coefficient of resistance materials is self-limiting naturally not like NTC materials or linear resistance heating. Some of the materials like PTC rubber also have exponentially rising temperature coefficient

#### Negative Temperature Coefficient of Resistance

An NTC refers to some materials which experience once their temperature raised then electrical resistance will be decreased. The materials which have a lower coefficient then they show a quick decrease with temperature. NTC materials are mainly used for making current limiters, thermistors and temperature sensors.

#### Measuring Method of TCR

The TCR of a resistor can be decided through calculating the resistance values over a suitable range of temperatures. The TCR can be measured when the normal slope of the resistance value is above this interval. For linear relations, this is precise as the temperature coefficient of the resistance is stable at each temperature. But, there are several materials that have a coefficient like non-linear. For example, a Nichrome is a popular alloy used for resistors, and the main relation among the TCR and temperature is not linear.

As the TCR is measured like normal slope thus, it is very significant to identify the interval of TCR & the temperature. The TCR can be calculated using a standardized method like MIL-STD-202 technique for the range of temperature from -55°C to 25°C and 25°C to 125°C. Because the maximum calculated value is identified as TCR. This technique frequently effects above indicating a resistor intended for low demanding applications.

### Temperature Coefficient of Resistance for Some Materials

The TCR of some materials at 20°C temperature is listed below.

• For Silver (Ag) material, the TCR is 0.0038°C
• For Copper (Cu) material, the TCR is 0.00386°C
• For Gold (Au) material, the TCR is 0.0034°C
• For Aluminum (Al) material, the TCR is 0.00429°C
• For Tungsten (W) material, the TCR is 0.0045°C
• For Iron (Fe) material, the TCR is 0.00651°C
• For Platinum (Pt) material, the TCR is 0.003927°C
• For Manganin (Cu = 84% + Mn = 12% + Ni = 4%) material, the TCR is 0.000002°C
• For Mercury (Hg) material, the TCR is 0.0009°C
• For Nichrome (Ni = 60% + Cr = 15% + Fe = 25%) material, the TCR is 0.0004°C
• For Constantan (Cu = 55% + Ni = 45%) material, the TCR is 0.00003°C
• For Carbon (C) material, the TCR is – 0.0005°C
• For Germanium (Ge) material, the TCR is – 0.05°C
• For Silicon (Si) material, the TCR is – 0.07°C
• For Brass (Cu = 50 – 65% + Zn = 50 – 35%) material, the TCR is 0.0015°C
• For Nickel (Ni) material, the TCR is 0.00641°C
• For Tin (Sn) material, the TCR is 0.0042°C
• For Zinc (Zn) material, the TCR is 0.0037°C
• For Manganese (Mn) material, the TCR is 0.00001°C
• For Tantalum (Ta) material, the TCR is 0.0033°C

### TCR Experiment

The temperature coefficient of the resistance experiment is explained below.

Objective

The main objective of this experiment is to discover the TCR of a given coil.

Apparatus

The apparatus of this experiment mainly include connecting wires, Carey foster bridge, resistance box, lead accumulator, one-way key, unknown low resistor, jockey, galvanometer, etc.

Description

A Carey foster bridge is mainly similar to a meter bridge because this bridge can be designed with 4 resistances like P, Q, R & X and these are connected to each other.

In the above Whetstone’s bridge, the galvanometer (G), a lead accumulator (E) & the keys of the galvanometer and accumulator are K1&K respectively.

If the resistance values are changed then there is no flow current through the ‘G’ and the unknown resistance can be determined by any of three known resistances like P, Q, R & X. The following relationship is used to determine the unknown resistance.

P/Q =R/X

The Carey foster bridge can be used to calculate the disparity among two almost equal resistances & knowing the one value, the other value can be calculated. In this kind of bridge, the last resistances are removed in computation. It is a benefit and thus it can easily use to calculate a known resistance.

The equal resistances like P & Q are connected in the internal gaps 2 & 3, the typical resistance ‘R’ can be connected within gap1 & the ‘X’ (unknown resistance) is connected within the gap4. The ED is the balancing length which can be calculated from the ‘E’ end. According to the Whetstone Bridge principle

P/Q = R + a + l1ρ/X + b + (100- l1)ρ

In the above equation, a & b are the end modifications at the E & F end & is the resistance for the length of every unit in bridge wire. If this testing is continual by changing X & R, the balancing length ‘l2’ is calculated from the end E.

P/Q = X + a + 12 ρ/ R + b +(100-12) ρ

From the above two equations,

X = R + ρ (11 -12)

Let l1 & l2 are the balancing lengths once the above testing is done through a typical resistance ‘r’ instead of ‘R’ & instead of X, a broad copper strip of ‘0’resistance.

0 = r + ρ (11’ -12’) or ρ = r/11’ -12’

If the coil resistances are X1 & X2 at the temperatures like t1oc & t2oc, then the TCR is

Α = X2 – X1/(X1t2 – X2t1)

And also if the coil resistances are X0 & X100 at the temperatures like 0oc & 100oc, then the TCR is

Α = X100 – X0/(X0 x 100)

Thus, this is all about the temperature coefficient of resistance. From the above information finally, we can conclude that this is the calculation of modify in any substance of electrical resistance for every level of temperature change. Here is a question for you, what is the unit of the temperature coefficient of resistance?