Operating principle of ammeters and voltmeters

In our day today life, many times we require to measure different electrical quantities like current, voltage, resistance, etc. While doing experiment, there is necessity of multimeter. As we have already discussed about multimeter, how it measures different electrical quantities like electrical current, voltage, resistance, etc. But the basic instruments for the measurement of electric current and voltage are ammeters and voltmeters respectively.

Let us discuss these instruments one by one, operating principle or working principle of ammeters and voltmeters, finally major differences between ammeters and voltmeters.

Operating Principle:

Analog ammeters and voltmeters are classed together as there are no fundamental differences in their operating principles. The action of all ammeters and voltmeters, with the exception of electrostatic type of instruments, depends upon a deflecting torque produced by an electric current. In an ammeter this torque is produced by a current to be measured or by a fraction of it. In a voltmeter this torque is produced by a current which is proportional to the voltage to be measured. Thus all analog voltmeters and ammeters are essentially current measuring devices.

See also: Study & use of Digital Multimeter

The essential requirement of measuring instruments are (i) that its introduction into the circuit, where measurements are to be made, does not alter the circuit conditions ;(ii)the power consumed by them for their operation is small.

Working principle of Ammeters:

Ammeter

Ammeter

Ammeters are connected in the series with the circuit whose current is to be measured. The power loss in an ammeter is (I^2.Ra) where I is the current to be measured Ra is the resistance of the ammeter therefore ammeter should have low electrical resistance so that they cause a small voltage drop and consequently absorb small power.

Working principle Voltmeters:

Voltmeter

Voltmeter

Voltmeters are connected in parallel with the circuit whose voltage is to be measured .the power loss in voltmeter is (V^2/Rv), where V is the voltage to be measured and Rv is the resistance of the voltmeter. Therefore voltmeters should have a high electrical resistance, in order that the current drawn by them is small and consequently the power consumed is small.

Difference between Ammeters and voltmeters:

 Parameters

Ammeter

Voltmeter

Connection It is to be connected in series mode It is to be connected in parallel mode
Resistance It has comparatively low resistance It has high resistance
Uses It is used to find the amount of current flowing in the circuit It is used to find the potential difference in the circuit
Circuit Circuit must be disconnected in order to attach the ammeter Circuit does not need to be disconnected
Accuracy Considered as less accurate Considered as more accurate compared to ammeter

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Hall Effect Multiplier – Construction, Working and Applications

Hi friends, in this article we are going to learn Hall Effect Multiplier which can be used in applications where power is to be controlled. Hall Effect Multiplier is also used where the multiplication of signals over a wide range of amplitudes is performed. Now let us see the construction and working of Hall Effect Multiplier.

Hall Effect:

“When a current carrying conductor is placed in a magnetic field, a transverse effect is noted. This effect is called Hall Effect.” It was discovered by scientist Hall in 1879. Hall found that:

“When a magnetic field is applied at right angles to the direction of electric currents, an electric field is setup which is perpendicular to both the direction of electric current and the applied magnetic field.”

In other words:

“When any specimen carrying a current ‘I’ is placed in the transverse magnetic field B, then an electric field ‘E’ is induced in the specimen in the direction perpendicular to both ‘I’ and ‘B’. The phenomenon is known as Hall effect.”

Applications of Hall Effect Multiplier:

  1. Determining whether a semiconductor is N-type or P-type.
  2. Determining the carrier concentration.
  3. Calculating the mobility, having measured the conductivity.
  4. Magnetic field meter: the hall voltage vH for a given current is proportional to B. hence measurement of vH measure the magnetic field B.
  5. Hall Effect multiplier: the instrument gives an output proportional to the product of two signals. Thus if current I made proportional to one input and if B is proportional to second input, then Hall voltage vH is proportional to the product of two signals.

Hall Effect multiplier:

In applications where the power is to be controlled or processed further Hall Effect Multipliers are used . A Hall Effect multiplier is shown in figure. The Hall Effect multiplier uses a Hall Effect element.

Hall effect multiplier wattmeter
Hall effect multiplier wattmeter

The current is passed through the current coil which produces a magnetic field proportional to the current I .this field is perpendicular to the Hall Effect element. A current Ip, proportional to the voltage, is passed through the Hall Effect element in a direction perpendicular to field as shown. The current is limited by the multiplier resistance Rs. The output voltage of the Hall Effect multiplier is:

vH=KH.Ip.B/t

Where KH =Hall co-efficient  m/A-Wb m^(-2),

B=flux density;Wb/m^2 and t= thickness of hall element ; m.

Now B is directly proportional to Vi

Therefore the output voltage of the Hall Effect multiplier is proportional to instantaneous power.

Hence the voltmeter connected at the output terminals can be calibrated in terms of power. The Hall Effect voltage which is of the power can be processed further for control and other purposes. This is the major advantage of Hall Effect multiplier over electrodynamometer wattmeters the output of the later being the deflection of a pointer which cannot be processed further.

I hope you understood the construction and working of Hall Effect Multiplier and its Applications. If you have any doubts please let us know. Share your doubts in the comments below. If you liked this article please share and like our facebook page.

Heterodyne Wave Analyzer

Heterodyne Wave Analyzer:

Introduction:

Analysis of the waveform means determination of the values of amplitude, frequency and sometime phase angle of the harmonic components.

A wave analyser is an instrument designed to measure relative amplitude of signal frequency components in a complex waveform .basically a wave instruments acts as a frequency selective voltmeter which is tuned to the frequency of one signal while rejecting all other signal components.

It is well known that any periodic waveform can be represented as the sum of a d.c. component and a series of sinusoidal harmonics. Analysis of a waveform consists of determination of the values of amplitude, frequency, and sometime phase angle of the harmonic components. Graphical and mathematical methods may be used for the purpose but these methods are quite laborious. The analysis of a complex waveform can be done by electrical means using a band pass filter network to single out the various harmonic components. Networks of these types pass a narrow band of frequency and provide a high degree of attenuation to all other frequencies.

A wave analyzer, in fact, is an instrument designed to measure relative amplitudes of single frequency components in a complex waveform. Basically, the instrument acts as a frequency selective voltmeter which is used to the frequency of one signal while rejecting all other signal components. The desired frequency is selected by a frequency calibrated dial to the point of maximum amplitude. The amplitude is indicated either by a suitable voltmeter or CRO.

This instrument is used in the MHz range. The input signal to be analysed is heterodyned to a higher IF by an internal local oscillator. Tuning the local oscillator shifts various signal frequency components into the pass band of the IF amplifier. The output of the IF amplifier is rectified and is applied to the metering circuit. The instrument using the heterodyning principle is called a heterodyning tuned voltmeter.

The block schematic of the wave analyser using the heterodyning principle is shown in fig. above. The operating frequency range of this instrument is from 10 kHz to 18 MHz in 18 overlapping bands selected by the frequency range control of the local oscillator. The bandwidth is controlled by an active filter and can be selected at 200, 1000, and 3000 Hz.

Block schematic of a heterodyne wave analyser
Block schematic of a heterodyne wave analyser

 

Applications of Wave Analyzers:

Wave analyzers have very important applications in the following fields:

1)      Electrical measurements

2)      Sound measurements and

3)      Vibration measurements.

The wave analyzers are applied industrially in the field of reduction of sound and vibrations generated by rotating electrical machines and apparatus. The source of noise and vibrations is first identified by wave analyzers before it can be reduced or eliminated. A fine spectrum analysis with the wave analyzer shows various discrete frequencies and resonances that can be related to the motion of machines. Once, these sources of sound and vibrations are detected with the help of wave analyzers, ways and means can be found to eliminate them.

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Signal waveform analysing instruments

Waveform Analyzing Instruments:

Introduction:

The analysis of electrical signals is used in different applications. The different instruments used for signal analysis are wave analyses, distortion analysers, spectrum analysers, audio analysers, and modulation analysers.

All signal analysis instruments measure the basic frequency properties, but they use different techniques to do so. A spectrum analyser sweeps the signal frequency band and displays a plot of amplitude versus frequency. It has operating range of 0.02 Hz to 250 GHz. A wave analyser is a voltmeter which measures the amplitude of a signal frequency within a band of about 10 Hz – 40MHz.

A) Wave Analysers :

Analysis of the waveform means determination of the values of amplitude, frequency and sometime phase angle of the harmonic components.

A wave analyser is an instrument designed to measure relative amplitude of signal frequency components in a complex waveform .basically a wave instruments acts as a frequency selective voltmeter which is tuned to the frequency of one signal while rejecting all other signal components.

Types of wave analysers:

There are two types of wave analysers, depending upon the frequency ranges used,

a) Frequency selecting wave analyser.

b) Heterodyne wave analyser.

B) Harmonic distortion analyser :

The application of the sinusoidal input signal to an electronic device , such as amplifier should result in generation of a sinusoidal output waveform .generally the output is not exactly as the input i.e. it is not exactly sinusoidal because of various types of distortion may occur. Non linear behavior of circuit elements introduces harmonics in the output waveform and the resultant distortion is often referred as harmonic distortion.

Types of distortion analyzer :

  1. Frequency distortion
  2. Phase distortion
  3. Amplitude distortion
  4. Intermodulation distortion
  5. Cross-over distortion.

C) Spectrum analyzers :

A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. The primary use is to measure the power of the spectrum of known and unknown signals. The input signal a spectrum analyzer measures is electrical, however, spectral compositions of other signals, such as acoustic pressure waves and optical light waves, can be considered through the use of an appropriate transducer.

Types of Spectrum analysers :

Spectrum analyzer types are dictated by the methods used to obtain the spectrum of a signal. There are 1)swept-tuned and

2) FFT based spectrum analyzers.

D) Audio Analyser

An Audio Analyser is a test and measurement instrument used to objectively quantify the audio performance of electronic and electro-acoustical devices. Audio quality metrics cover a wide variety of parameters, including level, gain, noise, harmonic and intermodulation distortion, frequency response, relative phase of signals, interchange crosstalk, and more. In addition, many manufacturers have requirements for behavior and connectivity of audio devices that require specific tests and confirmations.

Audio analysis requires that the device under test receive a stimulus signal of known characteristics, with which the output signal (response) may be compared by the analyzer in order to determine differences expressed in the specific measurements. This signal may be generated or controlled by the analyzer itself or may come from another source (e.g., a recording) as long as characteristics relative to the desired measurement are defined.

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Series type ohmmeter and Shunt type ohmmeter

Hello friends, in this article we are going to learn about Series type ohmmeter along with basic working principle and circuit diagram.

Ohmmeters

Introduction:

The ohmmeter is one of the conventional devices which measures the resistance with direct readings.

Ohmmeters has a low degree of accuracy. This does not represent its negative side. Instead, this instrument is widely used in the applications where there is need to determine the approximate value of resistance.

We can use ohmmeters to measure the approximate resistance of electronic components like

  • heater elements
  • machine field coils
  • checking of semiconductor diodes
  • checking for continuity of the electic circuit.

Ohmmeters can be used as a precision bridge, it quickly measures the approximate value of resistance which can save a lot of time in balancing the bridge. There are ttwo types of Ohmmetes as 1) Series type ohmmeter and 2) Shunt type ohmmeter. Lets see some more details about these ohmmeters one by one.

Series type ohmmeter circuit diagram:

Below circuit shows a series tye ohmmter.

Basic series type ohmmeter
Basic series type ohmmeter

In the figure,

  • Rm is the internal resistance of d’Arsonval movement
  • E denotes an emf of internal battery
  • R1  is the current limiting resistor
  • R2 is a zero adjusting resistor

Series type ohmmeter working:

In figure, if you can see that a basic d’Arsonval movement  Rm is connected in parallel with a shunting resistor R2. This parallel circuit is connected in series with a battery of emf E and resistance R1. This series circuit is connected to the terminals A and B of resistor Rx. Here Rx is an unknown resistor which we are going to calculate in next steps :).

We know that when any conductor has zero resistance, the current flow will be maximum through that conductor. Similarly, when the unknown resistance Rx is zero (that means there will be no resistor in the circuit, terminals A and B shorted)  maximum current will flow through the meter. For this condition, adjust the resistor R2 so that the basic movement meter indicates full-scale current Ifs. The full-scale current position of the pointer is marked “0” on the scale.

On the other hand, we know that when resistance of a conductor is infinity (that means the two ends of conductor kept open), there will be no flow of current. In this case when Rx is removed from the circuit i.e. Rx =  (that means when terminal A and B are kept open), the current in the meter will be zero and the basic movement indicates zero current which is the marked “”.

In this way our ohmmeter will show infinite resistance at the zero current position and zero resistance at a full scale current position. Here zero resistance indicates that the current in the meter is the maximum and hence the pointer goes to the top mark.

When Rx is inserted at terminal A, B the current through the meter will be less and therefore the pointer will drop lower on the scale. Therefore the meter has “0” at the extreme right and “” at the extreme left position.

Now as we have the range for 0 and infinity resistance on our scale, we can mark intermediate scale by different known values of the resistance Rx to the ohmmeter.

When the pointer of a meter shows the resistance point exactly in the middle of 0 and inifinity, this value of the resistance across terminals A and B is defined as the half scale position resistance Rh.

You may also like:

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D’Arsonval meter movement and its principle

D’Arsonval movement principle :

An action caused by electromagnetic deflection, using a coil of wire and a magnetized field. When current passes through the coil, a needle is deflected.

Whenever electrons flow through a conductor, a magnetic field proportional to the current is created. This effect is useful for measuring current and is employed in many practical meters.
Since most of the meters in use have D’Arsonval movements, which operate because of the magnetic effect, only this type will be discussed in detail. The basic dc meter movement is known as the D’Arsonval meter movement because it was first employed by the French scientist, D’Arsonval, in making electrical measurement.

This type of meter movement is a current measuring device which is used in the ammeter, voltmeter, and ohmmeter. Basically, both the ammeter and the voltmeter are current measuring instruments, the principal difference being the method in which they are connected in a circuit. While an ohmmeter is also basically a current measuring instrument, it differs from the ammeter and voltmeter in that it provides its own source of power and contains other auxiliary circuits.

D’Arsonval Galvanometer :

This instrument is very commonly used in various methods of resistance measurement and also in d.c. potentiometer work.

Construction of D’Arsonval galvanometer:

The construction of D’Arsonval galvanometer is shown in figure below. Let us discuss different parts of D’Arsonval galvanometer.

D'Arsonval Galvanometer

1) Moving coil:

It is the current carrying element. It is either rectangular or circular in shape and consists of number of turns of fine wire. This coil is suspended so that it is free to turn about its vertical axis of symmetry. It is arranged in a uniform, radial, horizontal magnetic field in the air gap between pole pieces of a permanent magnet and iron core. The iron core is spherical in shape if the coil is circular but is cylindrical if the coil is rectangular. The iron core is used to provide a flux path of low reluctance and therefore to provide strong magnetic field for the coil to move in. this increases the deflecting torque and hence the sensitivity of the galvanometer. The length of air gap is about 1.5mm. In some galvanometers the iron core is omitted resulting in of decreased value of flux density and the coil is made narrower to decrease the air gap. Such a galvanometer is less sensitive, but its moment of inertia is smaller on account of its reduced radius and consequently a short periodic time.

2) Damping:

There is a damping torque present owing to production of eddy currents in the metal former on which the coil is mounted. Damping is also obtained by connecting a low resistance across the galvanometer terminals. Damping torque depends upon the resistance and we can obtain critical damping by adjusting the value of resistance.

3) Suspension:

The coil is supported by a flat ribbon suspension which also carries current to the coil. The other current connection in a sensitive galvanometer is a coiled wire. This is called the lower suspension and has a negligible torque effect. This type of galvanometer must be leveled carefully so that the coil hangs straight and centrally without rubbing the poles or the soft iron cylinder. Some portable galvanometers which do not require exact leveling have ” taut suspensions” consisting of straight flat strips kept under tension for at the both top and at the bottom.

The upper suspension consists of gold or copper wire of nearly 0.012-5 or 0.02-5 mm diameter rolled into the form of a ribbon. This is not very strong mechanically; so that the galvanometers must he handled carefully without jerks. Sensitive galvanometers are provided with coil clamps to the strain from suspension, while the galvanometer is being moved.

4)  Indication:

The suspension carries a small mirror upon which a beam of light is cast. The beam of light is reflected on a scale upon which the deflection is measured. This scale is usually about 1 meter away from the instrument, although ½ meter may be used for greater compactness.

5) Zero setting:

A torsion head is provided for adjusting the position of the coil and also for zero setting.

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Wheatstone Bridge Method for Measurement of Resistance

Another method of measuring the value of a resistor is the Wheatstone bridge. This device sets up a parallel resistor system that measures the differences in voltage between two legs of a circuit. If there is a difference of voltage between the branches it will be detected by the galvanometer. A special type of Wheatstone bridge is a slide wire bridge. The setup for a slide wire bridge is shown in Figure 1. In this case, there is a wire of constant resistivity with a contact that can move over the entire length.

Wheatstone bridge method for measurement of resistance
Wheatstone bridge method for measurement of resistance

The resistance of the lengths of wire created by moving the contact is proportional to the length of the wire. So, expressing the resistance in terms of the wire length we have,
R2/R1 = L2/L1
If you are trying to determine the resistance of an unknown resistor and you have a known resistance available, the value of the unknown resistor is,
R2 = (L2/L1) R1
Where R1 is a specified resistance.

Procedure

Ammeter – Voltmeter Method

  1. Measure the resistance of an unknown resistor using an ohmmeter.
  2. With the help of your instructor set up a circuit using the unknown resistor as
    shown in Figure 1. Turn on the power.
  3. Measure the current flowing through the resistor.
  4. Measure the voltage across the resistor
  5. Calculate the value of the unknown resistor using Ohm’s Law.
  6. Find the % difference between your two values.

Wheatstone Bridge Method

  1. Set up a slidewire bridge circuit as shown in Figure.
  2. The wires connecting the resistances and the bridge should be as short as practically possible. Use a decade box with a known resistance as R1. This should be set to a value about equal to R2 (as measured by an ohmmeter). You can test this value by changing the decade box resistance and testing the bridge balance point until it is near the center of the bridge. Contact is made to the wire by sliding contact key C. Do not slide the key along with the wire while it is pressed down. This will scrape the wire causing it to be nonuniform. Have the instructor check your wiring before activating the circuit.
  3. Activate the circuit by closing the switch S, and balance the bridge by moving the slide wire contact. Open the switch and record R1, L1, and L2. Leave the switch open unless actually making measurements.
  4. Repeat procedure 1 for R1 settings of (a) R1 ??3 R2 and (b) R1 ??0.3 R2 .
  5. Compute the value of R2 for each case and find the average value. Compare this value to the directly measured value of R2 by finding the percent difference.
  6. Repeat the previous procedures with a large known resistance R1 and record your findings.

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Megohm Bridge Method for Measurement of Resistances

Megohm bridge is another important method for measurement of high resistances. It has one three terminal high resistance located in one arm of the bridge.

Fig a) shows the very high resistance with terminals A and B, and a guard terminal, which is put on the insulation. So it forms a three terminal resistance.

Megohm bridge three terminal resistance
Megohm bridge three terminal resistance

Let us consider take the hypothetical case of a 100 Mohm resistance .let we assume that this resistance is measured by an ordinary Wheatstone bridge. It is clear that Wheatstone will measure a resistance of 100*200/(100+200)=67Mohm instead of 100Mohm thus the error is 33 percent.

Megohm Bridge method for measurement of resistances
Megohm Bridge method for measurement of resistances

However, if the same resistance is measured by a modified Wheatstone bridge as shown in fig b)with the guard connection G connected as indicated, the error in measurement will be reduced and this modified Wheatstone bridge is called megohm bridge.

The arrangement of above figure illustrated the operation of Megohm Bridge.

The figure shows the circuit of the completely self-contained Megohm Bridge which includes power supplies, bridge members, amplifiers, and indicating instrument. It has ranged from 0.1MO to 10^6MO. The accuracy is within 3% for the lower part of the range to possible 10% above 10000MO.

The sensitivity of balancing against high resistance is obtained by use of adjustable high voltage supplies of 500V or 1000V and the use of a sensitive null indicating arrangements such as a high gain amplifier with an electronic voltmeter or a C.R.O. The dial on Q is calibrated 1-10-100-1000 MO, with main decade 1-10 occupying greater part of the dial space. Since unknown resistance R=PS/Q, the arm Q is made, tapered, so that the dial calibration is approximately logarithmic in the main decade, 1-10. Arm S give five multipliers, 0.1,1,10,100 and 1000.

The junction of ratio arms P and Q is brought on the main panel and is designated as ‘Guard’ terminal.

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Loss of Charge Method for Measurement of Resistances

loss of charge method
Loss of charge method for measurement of resistances

In the loss of charge method unknown resistance is connected in parallel with the capacitor and electrostatic voltmeter. The capacitor is initially charged to some suitable voltage by means of a battery of voltage V and then allowed to discharge through the resistance. The terminal voltage is observed during discharge and it is given by,

OR

Or insulation resistance is given by,

The variation of voltage v with time is shown in figure,

Loss of Charge Method Plot
Loss of Charge Method Plot

From above equation, it follows that if V, v, C, and t are known the value of R can be computed.

If the resistance R is very large the time for an appreciable fall in voltage is very large and thus this process may become time-consuming. Also the voltage-time curve will thus be very flat and unless great care is taken in measuring voltages at the beginning and at the end of time t, a serious error may be made in the ratio V/v causing the considerable corresponding error in the measured value of R. more accurate results may be obtained by change in the voltage V-v directly and calling this change as e, the expression for R becomes:

123

This change in voltage may be measured by a galvanometer.

However, from the experimental point of view, it may be advisable to determine the time t from the discharge curve of the capacitor by plotting the curve of log v against time t. this curve is linear as shown in the second figure and thus the determination of time t from this curve for the voltage to fall from V to v yields more accurate results.

Loss of charge method is applicable to some high resistances, but it requires a capacitor of very high leakage resistance as high as resistance being measured. The method is very attractive if the resistance being measured is the leakage resistance of a capacitor as in this case auxiliary R and C units are not required.

Actually, in this method, we do not measure the true value of resistance since we assume here that the value of resistance of electrostatic voltmeter and the leakage resistance of the capacitor have infinite value. But in practice corrections must be applied to take into consideration the above two resistances. Let R1 be the leakage resistance of the capacitor. Also R’ be the equivalent resistance of the parallel resistances R and R1.

loss of charge method with leakage resistance of capacitor
loss of charge method with leakage resistance of the capacitor

Then discharge equation of capacitor gives,

R’=0.4343 t / (C log V/v)

The test is then repeated with the unknown resistance R disconnected and the capacitor discharging through R1. The value of R1 obtained from this second test and substituted into the expression,

R’=(R R1) / (R+R1)

In order to get the value of R.

The leakage resistance of the voltmeter, unless very high should also be taken into consideration.

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Substitution Method For Measurement Of Medium Resistance

The connection diagram for substitution method is shown in fig. below. R is the unknown resistance to be measured, S is the standard variable resistance, ‘r’ is the regulating resistance and ‘A’ is an ammeter. There is a switch for putting S and R in a circuit.

Substitution Method For Measurement Of Medium Resistance
Substitution Method For Measurement Of Medium Resistance

Firstly switch is at position 1 and R is connected in a circuit. The regulating resistance ‘r’ is adjusted till ammeter pointer is at chosen scale. Now switch is thrown to position ‘2’ and now ‘S’ is in a circuit. The value of standard variable resistance ‘S’ is varied till the same deflection as was obtained with R in the circuit is obtained. When the same deflection obtained it means same current flow for both the resistances. It means resistances must be equal. Thus we can measure the value of unknown resistance ‘R’ by substituting another standard variable resistance ‘S’. Therefore this method is called Substitution method.

This is more accurate method than ammeter-voltmeter method. The accuracy of this method is greatly affected if the emf of the battery changes during the time of readings. Thus in order to avoid errors on this account, a battery of ample capacity should be used so that the emf remains constant.

See also: Ammeter voltmeter method.

The accuracy of the measurement naturally depends upon the constancy of the battery emf and of the resistance of the circuit excluding R and S, upon the sensitivity of the instrument, and upon the accuracy with which standard resistance S is known.

This method is not widely used for simple resistance measurements and is used in a modified form for the measurement of high resistances. The substitution principle, however, is very important and finds many applications in bridge methods and in high-frequency a.c. measurements.

Example

In a measurement of resistance by substitution method, a standard 0.5MO resistor is used. The galvanometer has a resistance of 10KO and gives deflections as follows:

1) With standard resistor, 41 divisions,

2) with unknown resistance, 51 divisions.

Find the unknown resistance.

Solution

The deflection of the galvanometer is directly proportional to the current passing through the circuit and hence is inversely proportional to the total resistance of the circuit. Let S, R, and G be respectively the resistances of the standard resistor, unknown resistor and galvanometer. Also, let ?1 be the deflection with a standard resistor in the circuit and ?2 with an unknown resistor in the circuit.

Hence unknown resistance R=(S+G)*?1/?2-G

= (0.5*10^6+10*1000)*(41/51)-10*1000

=0.4*10^6O

=0.4 MO.

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