Transcript Measurement Of Resistance

Measurement Of Resistance
• In electrical circuit, basically three are three
elements viz, resistance, conductance and
capacitance.
• In fact each electrical equipment is consisting of
these elements or combination of these
elements.
• This chapter mainly concentrates on the method
of measurements of different resistances (i.e.
low, medium or high value resistance) and
inductance and capacitance.
Classification of resistance
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By definition : Resistance is the property of a material by virtue of which it opposes the flow of current.
According to nature of supply resistance can be denoted as DC resistance and AC resistance.
DC resistance :
Ohms Law : The law states that the direct current flowing in a conductor is
directly proportional to
the potential difference
between its ends
provided that all the physical states of conductor
remain same (dimensions
and temperature).
It is usually formulated as I α V, where V is the potential difference,
or voltage and I is the current.
The constant of proportionality is called the “resistance”, R
measured in Ω (ohms).
Then the ohm’s law is given by:
I = V/R ....Amps
....(3.2.1)
The resistance R of a conductor of uniform cross section can be
computed as,
R = ρl/a
.....(3.2.2)
Where,
l is the length of the conductor, it is measured in meters [m]
a is the cross-sectional area of the current flow, measured in square m2
ρ (Greek : rho) is the electrical resistivity of the material, it is measured in ohm-meters (Ω m). Resistivity is a
measure of the material’s ability to oppose electric current.
For practical reasons, any connections to a real conductor will almost
certainly mean the current density is
not totally uniform. However,
this formula still provides a good approximation for long thin
conductor
such as wires.
AC resistance :
If a wire conducts high-frequency alternating
current then the effective cross sectional area of
the wire is reduced because of the skin effect.
If several conductors are together, then due to
proximity effect, the effective resistance of each is
higher than that if conductors were alone.
According to the value of resistance : According
to the value, resistances are mainly categorized
into three parts viz low, medium and high
resistance.
• Low resistance :
• When the value of resistance is below one ohm. (R<1 Ω) then it is called as
low resistance e.g. resistance of armature winding of generator, resistance
of series field winding of DC series generator, resistance of transformer
winding, bus bar resistance, earth wire resistance etc.
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• Medium resistance:
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• When the value of resistance lies between 1Ω to 0.1 mega ohm (i.e. 1
Ω<R<0.1 M Ω) then it is called as medium resistance. e.g. resistance of
field winding of DC shunt generator, resistance of long transmission line
etc.
• High resistance:
• When the value of resistance ia greater than 0.1 mega
ohm (i.e. R>0.1
MΩ).Then it is called as high resistance. e.g. resistance of cable insulation,
resistance of insulatodisc of transmission line etc.
Generally the resistance of conductor is under the category of low
range and that of insulator is treated
as high range.
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Measurement of Medium Resistance
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This method(a and b) is very common as voltmeter and ammeter is available in all labs.
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There are two methods of connecting voltmeter and ammeter for measurement of resistances as shown in Fig. 3.3.1(a)
and(b). In both the cases the measured value of the unknown resistances is equal to the reading of voltmeter divided by
reading of ammeter.
Let the reading of voltmeter is 'V and ammeter I. hence measured value of the resistances = Rm =V/I.
For connection in Fig. 3-3.1(a) the reading of ammeter is equal to I = the current flowing through resistances.
Reading of voltmeter = the voltage across resistance + voltage across ammeter
Hence V=Va + Vr .let the resistance of ammeter be Ra Ω s.
V=IRa+IR
=I(Ra+R)
Hence
Rm=V/I=I(Ra+R)/I
Rm=Ra+R
True value of resistance ’R’ is equal to Rm-Ra.
R=Rm-Ra
R=Rm(1-Ra/Rm)
……….(3.3.1)
From this expression R=Rm only when Ra=0.
This is the ideal case but practically ammeter has low resistance.
Hence to reduce the error in measurement this method is used for measurement medium and high resistances. As in this
case ‘Rm’ will be very much greater than ‘Ra’ Hence the ratio ‘Ra/Rm’ is approximately = 0
From Fig. 3.3.1( b),
Rm =V/I.
The reading of ammeter is equal to I= Iv+ Ir let the resistance of voltmeter by ‘Rv’.
I=V/Rv+V/R
Reading of voltmeter V=Voltage across resistance VR.
Rm=V/(V/R+V/Rv )
=1/(1/R+1/Rv )=1/R+1/Rv
1/R=1/Rm-1/Rv
1/R=Rv-Rm/RmRv
R=RmRv/Rv-Rm
Divided by Rv;
R=Rm/(1-Rm/Rv)…………….(3.3.2)
From this expression the true value of resistance R=Rm only, when resistance of voltmeter is ‘∞’(This is the ideal case).
But practically Rv is in medium or high resistance class.
Hence this method is used for measurement of low resistance as in this case Rm /Rv will be approximately equal to zero.
In both cases % error in measurement of resistance is equal to true value R-measured value of Rm divided by product of R
and Rm.
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Whetstone's Bridge Method
• Resistance ‘R’ is the unknown resistances
under measurement.
• It is connected in the whetstone's bridge
circuit, formed by the ratio arm resistances
‘P’ and ‘Q’
standard arm resistances 'S'.
• This bridge operates on D.C supply and the
detector
used to detect the balanced
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condition of bridge is a sensitive
galvanometer ‘G’.
• Working :
• The bridge is said to be under balanced condition when galvanometer
shows zero deflection.
• By using the equations of voltage drops at balanced condition, we can find
the magnitude of unknown resistance, ‘R’.
• connect the source and galvanometer as shown in Fig 3.3.3 and unknown
resistance between points ‘B’ and ‘C’ of the bridge.
• Put ‘ON’ key ‘K1’ then put 'ON' key ‘K2'. Keep resistance ‘N’ at minimum
position.
• Observe the reading of galvanometer. If it is vary the resistance ‘N’ to the
maximum resistance position. If there is a small deflection on G adjust
ratio P/Q and variable resistance ‘S’ to get zero reading on the
galvanometer.
• Now the bridge is completely balanced hence points A and C are at same
potential.
• Hence voltage across AB is equal voltage across BC.
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VAB = VBC
I1P = I2R ………………(3.3.5)
VAD = VCD
I3Q = I4S
I1 = I3 and I2 = I4 (bridge is at balance condition i.e. Ia=0)
I1Q = I2S
………………..(3.3.6)
Dividing Equation (3.3.5) by Equation (3.3.6),
I1P/I1Q=I2R/I2S
P/Q = R/S
R=PS/Q
Thus we can find magnitude of unknown resistance by ratio
P/Q and the
standard resistance ‘S’.
This method is accurate than voltmeter ammeter method
because we are
calculating the value of ‘R’ by method of
comparison.
Hence this value is independent of error in the
galvanometer.
This method is used for measurement of medium
resistances very accurately.
Kelvin double Bridge Method
• The Kelvin double bridge is the modification of
the bridge and provides
• greatly increased accuracy measurement of low
resistance.
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• An understanding of the Kelvin bridge
arrangement may be obtained by the study of the
difficulties that arise in a Wheatstone bridge on
account of the resistance of the leads and the
contact resistances while measuring low valued
resistance
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Figure shows the schematic diagram of the Kelvin bridge. The first of ratio arms is P and Q. the second set of ratio arms, p
and q is used to connect the galvanometer to a point d at the appropriate potential between points m and n to eliminate
effect of connecting lead of resistance r between the unknown resistance, R, and the standard resistance, S.
The ratio p/q is made equal to P/Q. under balance conditions there is no current through the galvanometer, which means
that the voltage drop between a and b, Eab is equal to the voltage drop Eamd.
When the bridge is under balanced condition Ig = 0. Hence points H and F are at the same potential.
Hence voltage across ‘P’ = voltage across resistance R + voltage across’P’.
Hence,
VP = VR+ VP
I1 .P =I.R +I2 .p
……….. (3.4.3)
Similarly voltage across Q = voltage across ‘S’ + voltage across ’q’.
VQ=VS+Vq
I1.Q = I.S +I2.q
………(3.4.4)
As ’r’ is in parallel with p + q, voltage across ‘r’ = voltage across (p+q)
Vr = V(p+q)
(I-I2)r =I2 (p +q)
Ir = I2 (p+q+r)
I2
Subatituting This value of I in Equation (3.4.3) and (3.4.4)
I1P = IR +()p
I1Q =IS +()q
…..…..(3.4.5)
…….(3.4.6)
Divide Equation (3.4.5) by Equation (3.4.6)
• Usually while balancing the bridge ratio P/Q is adjusted
equal to p/q . Hence expression for resistance ‘R’ is
simplified as
• Above equation is the usual working equation for the
Kelvin Bridge. It indicates that the resistance of
connecting lead, r, has no effect on the measurement,
provided that the two sets of ratio arms have equal
ratio. The former equation is useful, however, as it
shows the error that is introduced in case the ratios are
not exactly equal. It indicates that it is desirable to
keep r as small as possible in order to minimize the
errors in case there is a difference between ratios P/Q
and p/Q.