Measurement of Resistance ( FULL UNIT ) LOW &MEDIUM RESISTANCE MEASUREMENT

Measurement of Resistance

Resistance is one of the most basic elements encountered in electrical and electronics engineering. The value of resistance in engineering varies from very small value like, resistance of a transformer winding, to very high values like, insulation resistance of that same transformer winding. Although a multimeter works quite well if we need a rough value of resistance, but for accurate values and that too at very low and very high values we need specific methods. In this article we will discuss various methods of resistance measurement. For this purpose we categories the resistance into three classesMeasurement of Low Resistance (<1Ω)

The major problem in measurement of low resistance values is the contact resistance or lead resistance of the measuring instruments, though being small in value is comparable to the resistance being measured and hence causes serious error. Thus to eliminate this issue small valued resistance are constructed with four terminals. Two terminals are current terminals and other two are potential terminals. 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 Putting eqn 2 in 1 and solving and using P/Q = p/q, we get- 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

It is an electromechanical instrument used for measurement of low resistances. It comprises of a permanent magnet similar to that of a PMMC instrument and two coils in between the magnetic field created by the poles of the magnet. The two coils are at right angles to each other and are free to rotate about the common axis. Figure below shows a Ducter Ohmmeter and the connections required 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.

Induction Type Meters

The principle of working and construction of induction type meter is very simple and easy to understand that's why these are widely used in measuring energy in domestic as well as industrial world. In all induction meters we have two fluxes which are produced by two different alternating currents on a metallic disc. Due to alternating fluxes there is an induced emf, the emf produced at one point (as shown in the figure given below) interacts with the alternating current of the other side resulting in the production of torque.

Similarly, the emf produced at the point two interacts with the alternating current at point one, resulting in the production of torque again but in opposite direction. Hence due to these two torques which are in different directions, the metallic disc moves. This is basic principle of working of an induction type meters. Now let us derive the mathematical expression for deflecting torque. Let us take flux produced at point one be equal to F₁ and the flux and at point two be equal to F₂. Now the instantaneous values of these two flux can written as:

Where, F_m1 and F_m2 are respectively the maximum values of fluxes F₁ and F₂, B is phase difference between two fluxes. We can also write the expression for induced emf's at point one be

at point two. Thus we have the expression for eddy currents at point one is Where, K is some constant and f is frequency. Let us draw phasor diagram clearly showing F₁, F₂, E₁, E₂, I₁ and I₂. From phasor diagram, it clear that I₁ and I₂ are respectively lagging behind E₁ and E₂ by angle A. The angle between F₁ and F₂ is B. From the phasor diagram the angle between F₂ and I₁ is (90-B+A) and the angle between F₁ and I₂ is (90 + B + A). Thus we write the expression for deflecting torque as Similarly the expression for T_d2 is, The total torque is T_d1 - T_d2, on substituting the the value of T_d1 and T_d2 and simplying the expression we get Which is known as the general expression for the deflecting torque in the induction type meters. Now there are two types of induction meters and they are written as follows:

• Single phase type
• Three phase type induction meters.

Here we are going to discuss about the single phase induction type in detail. Given below is the picture of single phase induction type meter. Single phase induction type energy meter consists of four important systems which are written as follows: Driving System: Driving system consists of two electromagnets on which pressure coil and current coils are wounded, as shown above in the diagram. The coil which consisted of load current is called current coil while coil which is in parallel with the supply voltage (i.e. voltage across the coil is same as the supply voltage) is called pressure coil. Shading bands are wounded on as shown above in the diagram so as to make angle between the flux and and applied voltage equal to 90 degrees. Moving System: In order to reduce friction to greater extent floating shaft energy meter is used, the friction is reduced to greater extinct because the rotating disc which is made up of very light material like aluminium is not in contact with any of the surface. It floats in the air. One question must be arise in our mind is that how the aluminium disc float in the air? To answer this question we need to see the constructional details of this special disc, actually it consists of small magnets on both upper and lower surfaces. The upper magnet is attracted to an electromagnet in upper bearing while the lower surface magnet also attracts towards the lower bearing magnet, hence due to these opposite forces the light rotating aluminium disc floats. Braking System: A permanent magnet is used to produce breaking torque in single phase induction energy meters which are positioned near the corner of the aluminium disc. Counting System: Numbers marked on the meter are proportion to the revolutions made by the aluminium disc, the main function of this system is to record the number of revolutions made by the aluminium disc. Now let us look at the working operation of the single phase induction meter. In order to understand the working of this meter let us consider the diagram given below: Here we have assumed that the pressure coil is highly inductive in nature and consists of very large number of turns. The current flowin in the pressure coil is I_p which lags behind voltage by an angle of 90 degrees. This current produces flux F. F is divided into two parts F_g and F_p.

1. F_g which moves on the small reluctance part across the side gaps.
2. F_p: It is responsible for the production of driving torque in the aluminium disc. It moves from high reluctance path and is in phase with the current in the pressure coil. F_p is alternating in nature and thus emf E_p and current I_p. The load current which is shown in the above diagram is flowing through the current coil produces flux in the aluminium disc, and due this alternating flux there on the metallic disc, an eddy current is produced which interacts with the flux F_p which results in production of torque. As we have two poles, thus two torques are produced which are opposite to each other. Hence from the theory of induction meter that we have discussed already above the net torque is the difference of the two torques.

Advantages of Induction Type Meters

Following are the advantages of induction type meters:

1. They are inexpensive as compared to moving iron type instruments.
2. They have high torque is to weight ratio as compared to other instruments.
3. They retain their accuracy over wide range of temperature as well as loads.

Rectifier Type Instrument | Construction Principle of Operation

Rectifier type instrument measures the alternating voltage and current with the help of rectifying elements and permanent magnet moving coil type of instruments. However the primary function of rectifier type of instruments work as voltmeter. Now one question must arises in our mind why we use rectifier type of instruments widely in the industrial world though we have various other AC voltmeter like electrodynamometer type instruments, thermocouple type instruments etc? The answer to this question is very simple and is written as follows.

1. Cost of electrodynamometer type of instruments is quite high than rectifier type of instruments. However rectifier type of instruments as much accurate as electrodynamometer type of instruments. So rectifier type of instruments are preferred over electrodynamometer type instruments
2. The thermocouple instruments are more delicate than the rectifier type of instruments. However thermocouple type of instruments is more widely used at very high frequencies.

Before we look at the construction principle and working of rectifier type instruments, there is need to discuss in detail about the voltage current characteristics of ideal and practical rectifier element called diode. Let us first discuss the ideal characteristics of rectifying element. Now what is an ideal rectifying element? A rectifying element is one which offers zero resistance if it is forward biased and offers infinite resistance if it is reversed biased.

This property is used to rectify the voltages (rectification means to convert an alternating quantity into direct quantity i.e. AC to DC). Consider the circuit diagram given below.

In the given circuit diagram the ideal diode is connected in series with the voltage source and load resistance. Now when we make the diode forward biased it conducts perfectly offering zero electrical resistance path. Thus behaves as short circuited. We can make the diode forward biased by connecting the positive terminal of the battery with anode and negative terminal with cathode. The forward characteristic of rectifying element or diode is shown in the voltage current characteristic.

Now when we apply negative voltage i.e. connecting the negative terminal of the battery with the anode terminal of the diode and positive terminal of the battery to the cathode terminal of the diode. Due to reverse biased it offers infinite electrical resistance and thus it behaves as open circuit. The complete voltage current characteristics are shown below. Let us again consider the same circuit but the difference is here we are using the practical rectifying element instead of ideal one. Practical rectifying element is having some finite forward blocking voltage and high reverse blocking voltage. We will apply the same procedure in order to obtain the voltage current characteristics of practical rectifying element. Now when we make the practical rectifying element forward biased it does not conduct till the applied voltage is not greater the forward breakdown voltage or we can say knee voltage. When the applied voltage becomes greater than the knee voltage then diode or rectifying element will come under conduction mode. Thus behaves as short circuited but due to some electrical resistance there is voltage drop across this practical diode. We can make the rectifying element forward biased by connecting the positive terminal of the battery with anode and negative terminal with cathode. The forward characteristic of practical rectifying element or diode is shown in the voltage current characteristic. Now when we apply negative voltage i.e. connecting the negative terminal of the battery with the anode terminal of the diode and positive terminal of the battery to the cathode terminal of the rectifying element. Due to reverse biased it offers finite resistance and the negative voltage till the applied voltage becomes equal to reverse break down voltage and thus it behaves as open circuit. The complete characteristics are shown below Now rectifier type of instruments uses two types of rectifier circuits:

Half Wave Rectifier Circuits of Rectifier Type Instruments

Let us consider a circuit given below in which the rectifying element is connected in series with sinusoidal voltage source, permanent magnet moving coil instrument and the multiplier resistor. The function of this multiplier electrical resistance is to limit the current drawn by the permanent magnet moving coil type of instrument. It is very essential to limit the current drawn by the permanent magnet moving coil instrument because if the current exceeds the current rating of PMMC then it destructs the instrument. Now here we divide our operation in two parts. In first part we apply constant DC voltage to the above circuit. In the circuit diagram we are assuming the rectifying element as ideal one. Let us mark the resistance of multiplier be R, and that of permanent magnet moving coil instrument be R₁. The DC voltage produces a full scale deflection of magnitude I=V/(R+R1) where V is root mean square value of voltage. Now let us consider second case, in this case we will apply AC sinusoidal AC voltage to the circuit v =Vm × sin(wt) and we will get the output waveform as shown. In the positive half cycle the rectifying element will conduct and in the negative half cycle it does not conduct. So we will get a pulse of voltage at moving coil instrument which produces pulsating current thus pulsating current will produce pulsating torque. The deflection produced will corresponds to the average value of voltage. So let us calculate the average value of electric current, in order to calculate the average value of voltage we have integrate the instantaneous expression of the voltage from 0 to 2 pi. So the calculated average value of voltage comes out to be 0.45V. Again we have V is root mean square value of current. Thus we conclude that the sensitivity of the ac input is 0.45 times the sensitivity of DC input in case of half wave rectifier.

Full Wave Rectifier Circuits of Rectifier Type Instruments

Let us consider a circuit given below. We have used here a bridge rectifier circuit as shown. Again we divide our operation into two parts. In the first we analyze the output by applying the DC voltage and in another we will apply AC voltage to the circuit. A series multiplier resistance is connected in series with the voltage source which has the same function as described above. Let us consider first case here we applying DC voltage source to the circuit. Now the value of full scale deflection current in this case is again V/(R+R₁), where V is the root mean square value of the applied voltage, R is the resistance of the resistance multiplier and R₁ which is the electrical resistance of the instrument. The R and R₁ are marked in the circuit diagram. Now let us consider second case, in this case we will apply AC sinusoidal voltage to the circuit which is given v = Vmsin(wt) where Vm is the peak value of the applied voltage again if we calculate the value of full scale deflection current in this case by applying the similar procedure then we will get an expression of full scale current as .9V/(R+R1). Remember in order to obtain the average value of voltage we should integrate the instantaneous expression of voltage from zero to pi. Thus comparing it DC output we conclude that the sensitivity with AC input voltage source is 0.9 times the as in the case of DC input voltage source. The output wave is shown below. Now we are going to discuss the factors which affect the performance of Rectifier type instruments:

1. Rectifier type of instruments is calibrated in terms of root mean square values of sinusoidal wave of voltages and current. The problem is that the input waveform may or may not have same form factor on which the scale of these meter is calibrated.
2. There may be some error due to the rectifier circuit as we not included the resistance of the rectifier bridge circuits in both the case. The non linear characteristics of bridge may distort the current and voltage waveform.
3. There may variation in the temperature due to which the electrical resistance of the bridge changes hence in order to compensate this kind of errors we should apply multiplier resistor with high temperature coefficient.
4. Effect of capacitance of the bridge rectifier: Bridge rectifier has imperfect capacitance thus due to this it byp asses the high frequency currents. Hence there is decrement in the reading.
5. The sensitivity of Rectifier type instruments is low in case of AC input voltage.

Advantages of Rectifier Type Instruments

Following are the advantages of the rectifier type of instruments:

• The accuracy of rectifier type instrument is about 5 percent under normal operating condition.
• The frequency range of operation can be extended to high value.
• They have uniform scale on the meter.
• They have low operating value of current and voltages.

The loading effect of an AC rectifier voltmeter in both the cases (i.e. half wave diode rectifier and full wave diode rectifier) is high as compared to the loading effects of DC voltmeters as the sensitivity of the voltmeter either using in half wave or full wave rectification is less than the sensitivity of DC voltmeters.

Electrostatic Type Instruments

Working Principle of Electrostatic Type Instruments

As the name suggests the electrostatic type instrument use static electrical field to produce the deflecting torque. These types of instrument are generally used for the measurement of high voltages but in some cases they can be used in measuring the lower voltages and powers of a given circuit. Now there are two possible ways in which the electrostatic force can act. The two possible conditions are written below,

Construction of Electrostatic Type Instruments

1. When one of the plates is fixed and other plate is free to move, plates are oppositely charged in order to have attractive force between them. Now due this attractive force movable plate will move towards the stationary or fixed plate till the moving plate stored maximum electrostatic energy.
2. In other arrangement there may be force of attraction or repulsion or both, due to some rotary of plate.

Force and Torque Equation of Electrostatic Type Instrument

Now let us derive force equation for the linear electrostatic type instruments. Let us consider two plates as shown in the diagram given below.

Plate A is positively charged and plate B is negatively charged. As mentioned above as per the possible condition (a) we have linear motion between the plates. The plate A is fixed and plate B is free to move. Let us assume there exists some force F between the two plates at equilibrium when electrostatic force becomes equal to spring force. At this point, the electrostatic energy stored in the plates is

Now suppose we increase the applied voltage by an amount dV, due to this the plate B moves towards the plate A by a distance dx. The work done against the spring force due to displacement of the plate B be F.dx. The applied voltage is related to current as From this value of electric current the input energy can be calculated as

From this we can calculate the change in the stored energy and that comes out to be By neglecting the higher order terms that appears in the expression. Now applying the principle of energy conservation we have input energy to the system = increase in the stored energy of the system + mechanical work done by the system. From this we can write, From the above equation the force can be calculated as Now let us derive force and torque equation for the rotary electrostatic type instruments. Diagram is shown below, In order to find out the expression for deflecting torque in case of rotary type electrostatic instruments, just replace the in the equation (1) F by T_d and dx by dA. Now rewriting the modified equation we have deflecting torque is equals to Now at steady state we have controlling torque is given by the expression T_c = K × A. The deflection A can be written as From this expression we conclude that the deflection of the pointer is directly proportional to the square of the voltage to be measured hence the scale will be non uniform. Let us now discuss about Quadrant electrometer. This instrument is generally used in measuring the voltage ranging from 100V to 20 kilo volts. Again the deflecting torque obtained in the Quadrant electrometer is directly proportional to the square of the applied voltage; one advantage of this is that this instrument can used to measure both the AC and DC voltages. One advantage of using the electrostatic type instruments as voltmeters is that we can extend the range of voltage to be measured. Now there are two ways of extending the range of this instrument. We will discuss them one by one.

(a) By using resistance potential dividers: Given below is the circuit diagram of this type of configuration. The voltage which we want to measure is applied across the total resistance r and the electrostatic capacitor is connected across the portion of the total resistance which is marked as r. Now suppose the applied voltage is DC, then we should make one assumption that the capacitor which is connected is having infinite leakage resistance. In this case the multiplying factor is given by the ratio of electrical resistance r/R. The ac operation on this circuit can also be analyzed easily again in case of ac operation we multiplying factor equal to r/R. (b) By using capacitor multiplier technique: We can increase the range of voltage to be measured by placing a series of capacitors as shown in the given circuit. Let us derive the expression for multiplying factor for the circuit diagram 1. Let us mark the capacitance of the voltmeter be C₁ and series capacitor be C₂ as shown in the given circuit diagram. Now the series combination of these capacitor be equal to Which is the total capacitance of the circuit. Now the impedance of the voltmeter is equal to Z₁ = 1/jωC₁ and thus total impedance will be equal to Now the multiplying factor can be defined as the ratio of Z/Z₁ which is equal to 1 + C₂ / C₁. Similarly the multiplying factor can also be calculated. Hence by this way we can increase the range of voltage to be measure.

Advantages of Electrostatic Type Instruments

Now let us look at some advantages of electrostatic type instruments.

1. The first and the most important advantage is that we can measure both AC and DC voltage and the reason is very obvious the deflecting torque is directly proportional to the square of the voltage.
2. Power consumption is quite low in these types of instruments as the current drawn by these instruments is quite low.
3. We can measure high value of voltage.

Disadvantages of Electrostatic Type Instruments

Instead of various advantages, electrostatic instruments posses few disadvantages and these are written below.

1. These are quite costly as compared to other instruments and also these have large size.
2. The scale is not uniform.
3. The various operating forces involved are small in magnitude.