Chapter 4 Resistance Precise Measurements

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• The ohmmeter can be used to measure the resistance, but it is not very accurate instrument.

• For precise measurements of resistance, the used instruments differ for different resistance ranges.

• Also the voltmeter and ammeter can be used in different ways according to the range of the unknown resistors.

• The Voltmeter and Ammeter Methods will be discussed in the following slides:

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1-Low Value Resistance Measurements

• The connection of voltmeter and ammeter in the figure shows that: the voltmeter is connected directly across the unknown resistor.

•

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• The measured resistor value :

• The actual resistor value:

• In this connection the current read by the ammeter does not equal to the current through the resistor (R), and the voltage read by the voltmeter equals the voltage drop across R.

• Therefore the error of this method is due to the current passing in the voltmeter (IV).

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• This will be greatly reduced if the resistance R is much lower than the internal resistance of the voltmeter (i.e.: IV ≈ 0)

• Thus:

if Iv << I

or

• This means that: Rm = R , If RV >> R ( i.e. IV << I ).

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• So this method actually measures the parallel combination of the internal resistance of voltmeter and the unknown resistance.

• If the readings of the ammeter does not change before and after the connection of the voltmeter to the circuit (i.e. IV << I),

• Then this method will be the best one for measuring a small value resistor R.

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2-High Value Resistance Measurements

• The connection diagram in the figure shows that: the ammeter is connected directly to the unknown resistor.

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• The measured resistor value :

• The actual resistor value:

• In this connection the voltage read by the voltmeter does not equal to the voltage drop across (R), and the current read by the ammeter equals the current passing in (R).

• Therefore the error of this method is due to the voltage drop across the ammeter Va [due to its non zero value of its internal resistance (Ra)].

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• This error will be greatly reduced if the resistance R is much higher than the ammeter internal resistance (i.e: Va ≈ 0)

• Thus:

if Va << V

or

= Ra + R

• This means that: Rm = R , If Ra << R

( i.e. Va << V )

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• So this method actually measures the series combination of the internal resistance of the ammeter and the unknown resistance.

• If the readings of the voltmeter does not change before and after the connection of the ammeter to the circuit (i.e. Va << V),

• This method will be the best one for measuring the resistance R.

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Substitution Method

• If the current (I) drawn by the unknown resistor (R) is maintained the same as the current drawn by an adjustable decade resistor (Rd),

then the value of R equals to the value of Rd

• The accuracy of this method depends on the accuracy of the decade resistor Rd and the sensitivity of the ammeter (A)

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If I | s on pos. 1= I | s on pos. 2

Then R = Rd

• Discuss the disadvantages of this method:

o Number of Rd .

o Accuracy of Measurements.

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Null Methods

• One of these methods is the wheatstone bridge

• The resistors P, and Q are precise decade resistors.

• The resistor S is a variable precise (Standard) resistor, and R is the unknown resistor.

• When SW is on position 2 and the galvanometer (G) reads zero (null), the bridge is balanced, and

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R = (P/Q)S

• The ratio of P, and Q can be considered as a coarse adjustment of the balance, while resistor S is the fine adjustment.

• Protection of Galvanometer

The galvanometer must be protected from excessive currents by the following procedure:

– Initially SW must be put to position 1 (galvanometer is in series with a high value resistor RS), the balance is adjusted.

– Then the value of the series resistor RS is decreased, and the balance is readjusted.

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– This step is repeated until the value of the series resistor RS equals to zero (i.e. SW may be put to position 2).

– In this case the value of R is calculated as:

R = (P/Q) S

• Balance condition

• At balance ( when SW on position 2):

IG = 0 (i.e. I1 passes in P & R, and I2 passes in Q & S)

VG = 0 (i.e. VP = VQ , and VR = VS).

I1 R = I2 S (1)

and, I1 P = I2 Q (2)

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• Divide eq.(1) by eq.(2) we get:

R/P = S/Q i.e

Which is called the balance condition of the bridge.

• Bridge Accuracy

• If P, Q, and S have the percentage errors P%, Q%, and S% respectively,

• Then the percentage error of R (R%) will be as follows:

R% = ± (P% + Q%+S%)

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R = (P/Q)S

Range of Measurements

• The measured resistance by wheatstone bridge must be much higher than the contact and lead resistances so that the measurement will be accurate.

• If the measured resistance (Q) is very small ( 5) the measurement will not be accurate because the galvanometer can be considered to be connected to points (b or a), if you considered the contact or lead resistor Y is a part of Q (measured resistance) or not,

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• If G is connected to a, then: R = (S+Y)P / Q

• If G is connected to b, then: R = SP / (Q+Y)

• i.e. to have unambiguous measurements, Y must be much less than S or Q, if this condition is satisfied then value of R is the same and equal to: R = SP / Q

• Therefore, the resistance S or Q must be much higher than Y.

• Also, if the measured resistor is much high (> 1012 ) the bridge currents will be very small and a bad accuracy of measurement will be obtained (IG 0 although the bridge is not balanced).

• So, a special techniques must be used to measure the very high resistors.

As a Conclusion:

The measured range of the wheatstone bridge is :

(5 ÷ 1012) .

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Very Low Resistance Measurements

• Four Terminal Resistors

• The low resistance is represented by 4- terminals such that the errors due to contact (contact resistance and lead resistances) are avoided.

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• The outer terminals are large and called current terminals, the internal terminals are smaller and called the potential terminals.

• The potential terminals carry very small current (voltmeter or ohmmeter current).

• So, the contact and lead resistance errors are not affecting its value, and the exact value of resistor (without errors) is measured across the potential terminals.

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• Kelvin Bridge

• The low value resistors are best measured by the Kelvin bridge, since they are out of the range of resistance measurement by the wheatstone bridge.

• The Kelvin bridge is the same as wheatstone bridge with two additional resistors (b and r) included such that the effect of the contact and lead resistance (Y) is eliminated in the balance condition.

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Balance condition

• At balance: IG = 0 (i.e. i1 passes through P & R, I2 passes through Q & S, and I passes through b & r ).

• Also VG = 0

i.e. VP = Vb + VQ and VR = Vr + Vs

i1 P = I b + I2 Q and i1 R = I r + I2S

i.e. i2 Q = P (I1 - I b/P) and i2 S = R (I1 - I r/R)

Dividing these two equations by each other :

Then the Balance Condition is:

if

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Q = (P/R)S b/P = r/R

Notes:

(1) Q & S are low resistors, P, R , b & r are high value resistors.

(2) The Kelvin bridge may be called double Kelvin bridge because it has double Wheatston bridges.

(3) The error due to the contact and lead resistor (Y) is completely eliminated as seen in the balance condition.

(4) The Kelvin bridge can measure as small value as

0.1 resistors.

(5) Its accuracy depends on the accuracy of its components, as discussed in wheatstone bridge, its sensitivity depends on the resolution of the adjustable resistor, and on the current sensitivity of the galvanometer.

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Very High Resistance Measurements 1- Measurement of High value Discrete resistors

• Very high value resistor (R) can be measured using voltmeter and ammeter method, as shown above.

• A microammeter must be used because the current (I) passing through R is very small.

• Also a high value power supply (VS) is used (e.g.10000V).

• The microammeter is connected directly to R, while the voltmeter is connected across the power supply.

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• The voltmeter reading is equal to the voltage drop (V) across R, because the voltage drop across the microammeter is much lower than the voltage drop across R, then:

R = V/I

2- Measurement of the Insulation Resistance

• In case of the insulation materials , two insulation resistive parts can be measured: Surface leakage resistance (RS), and a Volume resistance (RV)

RS = V/ IS and RV = V/ IV

Where: IS is the surface leakage current , and IV is the volume current

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• If it is measured using ammeter–voltmeter method, the measured resistance will equal to the parallel combination of these two insulation resistive parts: R = RV //RS ,

because the microammeter reading is: Ia = IV + IS

• In order to separate these two parts, a guard wire or a guard ring is used (in case of cable sheath or disc-shaped insulation materials respectively).

• The guard wire or guard ring is simply a several turns of wire wrapped tightly around the insulation.

• It separates the surface leakage current such that it passes through the guard wire or ring, not through the microammeter ( Ia = IV only), therefore the measured resistance Rm is :

Rm = V/ Ia = V/ IV = RV 3/26/2018 26

• Steps of measurement of Rs and RV : 1. Connect the guard wire or ring , by closing switch S, therefore : (Ia = IV), then measure RV:

RV = V/ Ia = V/ IV

2. Disconnect the guard wire or ring , by opening switch S, therefore : (Ia = IV + IS) , then measure RV // RS:

Rv // Rs = V/ Ia = V/ (IV+ IS) 3. Calculate Rs from steps (1) and (2):

RS = V/ ((IV+ IS)- IV) = V/ IS 3/26/2018 27

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