Measurement of Resistance

Resistance is a fundamental concept in electrical and electronics engineering, varying from the very low (like in transformer winding) to very high (such as the insulation resistance of those same windings). Although multimeters provide a rough estimate of resistance, specific methods are necessary for precise measurements, especially at extreme values. This article will explore various methods of measuring resistance, categorized into three classes.

Measurement of Low Resistance (<1Ω)

A major challenge in measuring low resistance values is the contact or lead resistance of the measuring instruments. Although small, this resistance is comparable to the resistance being measured, leading to significant error.
To eliminate this issue, resistors with small values are constructed with four terminals—two for current and two for potential.
Figure below shows the construction of low resistance.

The current is flown through current terminals C₁ and C₂ while the potential drop is measured across potential terminals V₁ and V₂. Hence we can find out the value of resistance under experiment in terms of V and I as indicated in the above figure. This method helps us to exclude the contact resistance due to current terminals and though contact resistance of potential terminals still comes into picture, it is very small fraction of high resistance potential circuit and hence induces negligible error.

The methods employed for measurement of low resistances are:-

• Kelvin’s Double Bridge Method
• Potentiometer Method
• Ducter Ohmmeter.

Kelvin’s Double Bridge

Kelvin’s double bridge is a modification of simple Wheatstone bridge. Figure below shows the circuit diagram of Kelvin’s double bridge.

As we can see in the above figure there are two sets of arms, one with resistances P and Q and other with resistances p and q. R is the unknown low resistance and S is a standard resistance. Here r represents the contact resistance between the unknown resistance and the standard resistance, whose effect we need to eliminate. For measurement we make the ratio P/Q equal to p/q and hence a balanced Wheatstone bridge is formed leading to null deflection in the galvanometer. Hence for a balanced bridge we can write

By substituting Equation 2 into Equation 1 and using the ratio P/Q = p/q, we derive the following result:

Hence we see that by using balanced double arms we can eliminate the contact resistance completely and hence error due to it. To eliminate another error caused due to thermo-electric emf, we take another reading with battery connection reversed and finally take average of the two readings. This bridge is useful for resistances in range of 0.1µΩ to 1.0 Ω.

Ducter Ohmmeter

The Ducter Ohmmeter, an electromechanical instrument, measures low resistances. It includes a permanent magnet, similar to a PMMC instrument, and two coils positioned within the magnetic field and at right angles to each other, rotating freely about a common axis. The diagram below illustrates a Ducter Ohmmeter and the necessary connections to measure an unknown resistance R.

One of the coil called current coil, is connected to current terminals C₁ and C₂, while the other coil called, voltage coil is connected to potential terminals V₁ and V₂. Voltage coil carries current proportional of the voltage drop across R and so is its torque produced. Current coil carries current proportional to the current flowing through R and so is its torque too. Both the torque acts in opposite direction and the indicator come to halt when the two are equal. This instrument is useful for resistance in range 100µΩ to 5Ω.

Measurement of Medium Resistance (1Ω – 100kΩ)

Following are the methods employed for measuring a resistance whose value is in the range 1Ω – 100kΩ –

• Ammeter-Voltmeter Method
• Wheatstone Bridge Method
• Substitution Method
• Carey- Foster Bridge Method
• Ohmmeter Method

Ammeter Voltmeter Method

This is the most crude and simplest method of measuring resistance. It uses one ammeter to measure current, I and one voltmeter to measure voltage, V and we get the value of resistance as

Now we can have two possible connections of ammeter and voltmeter, shown in the figure below.

Now in figure 1, the voltmeter measures voltage drop across ammeter and the unknown resistance, hence

Hence, the relative error will be,

For connection in figure 2, the ammeter measures the sum of current through voltmeter and resistance, hence

The relative error will be,

It can be observed that the relative error is zero for R_a = 0 in first case and R_v = ∞ in second case. Now the questions stand that which connection to be used in which case. To find out this we equate both the errors

Hence for resistances greater than that given by above equation we use the first method and for less than that we use second method.

Wheatstone Bridge Method

This is the simplest and the most basic bridge circuit used in measurement studies. It mainly consists of four arms of resistance P, Q; R and S. R is the unknown resistance under experiment, while S is a standard resistance. P and Q are known as the ratio arms. An EMF source is connected between points a and b while a galvanometer is connected between points c and d.

A bridge circuit always works on the principle of null detection, i.e. we vary a parameter until the detector shows zero and then use a mathematical relation to determine the unknown in terms of varying parameter and other constants. Here also the standard resistance, S is varied in order to obtain null deflection in the galvanometer. This null deflection implies no current from point c to d, which implies that potential of point c and d is same. Hence

Combining the above two equations we get the famous equation –

Substitution Method

The figure below shows the circuit diagram for resistance measurement of an unknown resistance R. S is a standard variable resistance and r is a regulating resistance.

First the switch is place at position 1 and the ammeter is made to read a certain amount of current by varying r. The value of ammeter reading is noted. Now the switch is moved to position 2 and S is varied in order to achieve the same ammeter reading as it read in the initial case. The value of S for which ammeter reads same as in position 1, is the value of unknown resistance R, provided the EMF source has constant value throughout the experiment.

Measurement of High Resistance (>100kΩ)

Following are few methods used for measurement of high resistance values-

• Loss of Charge Method
• Megger
• Megohm bridge Method
• Direct Deflection Method

We normally utilize very small amount of current for such measurement, but still owing to high resistance chances of production of high voltages is not surprising. Due to this we encounter several other problems such as-

1. Electrostatic charges can get accumulated on measuring instruments
2. Leakage current becomes comparable to measuring current and can cause error
3. Insulation resistance is one of the most common in this category; however a dielectric is always modeled as a resistor and capacitor in parallel. Hence while measuring the insulation resistance (I.R.) the current includes both the component and hence true value of resistance is not obtained. The capacitive component though falls exponentially but still takes very long time to decay. Hence different values of I.R. are obtained at different times.
4. Protection of delicate instruments from high fields.

Hence to solve the problem of leakage currents or capacitive currents we use a guard circuit. The concept of guard circuit is to bypass the leakage current from the ammeter so as to measure the true resistive current. Figure below shows two connections on voltmeter and micro ammeter to measure R, one without guard circuit and one with guard circuit.

In the first circuit the micro ammeter measures both capacitive and the resistive current leading to error in value of R, while in the other circuit the micro ammeter reads only the resistive current.

Loss of Charge Method

In this method we utilize the equation of voltage across a discharging capacitor to find the value of unknown resistance R. Figure below shows the circuit diagram and the equations involved are-


However the above case assumes no leakage resistance of the capacitor. Hence to account for it we use the circuit shown in the figure below. R₁ is the leakage resistance of C and R is the unknown resistance.
We follow the same procedure but first with switch S₁ closed and next with switch S₁ open. For the first case we get

For second case with switch open we get

Using R₁ from above equation in equation for R’ we can find R.

Megohm Bridge Method

In this method we use the famous Wheatstone bridge philosophy but in a slightly modified way. A high resistance is represented as in the figure below.

G is the guard terminal. Now we can also represent the resistor as shown in the adjoining figure, where R_AG and R_BG are the leakage resistances. The circuit for measurement is shown in the figure below.

It can be observed that we actually obtain the resistance which is parallel combination of R and R_AG. Although this causes very insignificant error.

Megger

Megger is one of the most important measuring device used by electrical engineers and is essentially used for measuring insulation resistance only. It consists of a generator which can be hand driven or nowadays we have electronic megger. Details of megger have been discussed in a separate article.