Measurement of Resistance by Voltmeter and Ammeter Method.

Experiment No.: 11

Aim of the Experiment:

Measurement of resistance by voltmeter and ammeter method.

Apparatus Required:

Sl. No.	Name	Specification	Quantity
01	Single Phase Auto-Transformer	5KVA, (0-270)V	1 no.
02	Rheostat	(0-80)Ω, 5A	1 no.
03	Voltmeter	(0-250)V, MI	1 no.
04	Ammeter	(0-5)A, MI	1 no.
05	Multimeter	Digital type	1 no.
06	Connecting Wire	PVC Insulated Copper	As per required

Table No. 11.1

Circuit Diagram: –

Theory: –

This method is used for measurement of medium resistance. Two types of connection is used for an ammeter voltmeter methods are shown in fig.11.2 & fig. 11.3. In both the case, the reading of ammeter and voltmeter are taken, then the measured value of resistance is

\[\color\magenta{R_m = \;\dfrac{Voltmeter\; Reading}{Ammeter\; Reading}\; =\;\dfrac{V}{I}}\]

The measured Value of resistance R_m, could be equal to the true value R, if the ammeter resistance is zero and the voltmeter resistance is infinite. So that the conditions in the circuit are not distributed. But in practical this is not possible and hence, both the methods gives inaccurate results.

Case: I

In this circuit the ammeter measures the true value of the current through the resistance but the voltmeter does not measure the true value of voltage across the resistance. The Voltmeter indicates the sum of the voltage across the ammeter and resistance.

So, the voltage across the ammeter is

\[\color\red{V_a \;= I\times{R_a}}\]

Now, measured value of resistance,

\[\color\red{R_{m1} =\;\dfrac{V}{I} =\;\dfrac{V_R+V_a}{I}\; =\dfrac{IR+IR_a}{I} = \;R+R_a}\]

So, the value of resistance

\[ \color\red{R =\;R_{m1}-R_a \;= \;R_{m1}\Big(1-\dfrac{R_a}{R_{m1}}\Big)}\]

Thus, the measured value of resistance is higher than the true value. It is also clear from above (case-I) that true value is equal to the measured value, only if the ammeter resistance (R) is zero.

\[\color\red{E_r = \dfrac{(R_{m1}-R)}{R}}\] \[ \color\red{= \dfrac{R_a}{R}}\]

Relative Error: –

The relative error would be small if the value of resistance under measurement is large as compared to the internal resistance of the ammeter. Therefore the circuit in fig. 11.2 is suitable for measuring high resistance.

Case: – II

The voltmeter measures the true value of voltage across the resistance but ammeter measures the sum of current through the resistance and the voltmeter.

So, the current through the voltmeter is

\[\color\red{I_V = \dfrac{V}{R_V}}\]

Measured value of resistance is

\[\color\red{R_{m2} =\;\dfrac{ V}{I} = \;\dfrac{V}{(I_R+I_V)}}\] \[\color\magenta{ = \;\dfrac{V}{\dfrac{V}{R}+\dfrac{V}{R_V}}}\] \[\color\red{= \dfrac{R}{ 1+\dfrac{R}{R_V}}}\]

True value of resistance is

\[ \color\red{R = \dfrac{R_{m2}R_V}{R_V-R_{m2}}}\] \[\color\red{= R_{m2}\Big(\dfrac{1}{1-\dfrac{R_{m2}}{R_V}}\Big)}\]

It is clear that from the above equations the true value of resistance is equal to the measured value if the resistance of voltmeter (Rv) is infinite. If the resistance of voltmeter is very high as compared to the resistance under measurement.

\[ \color\red{R_V\;>\;R_{m2}}\]

and therefore

\[ \color\red{\dfrac{R_{m2}}{R_V}}\] \[\color\green{is \;very \;small}\]

therefore we have

\[ \color\red{R = 1+\dfrac{R_{m2}}{R_V}}\]

thus the measured value of resistance is smaller then the true value.

The relative Error,

\[ \color\red{E_r = \dfrac{-R}{R_V}}\]

The relative error for the two case are equal,

when,

\[ \color\red{\dfrac{R_a}{R} =\;\dfrac{R}{R_v}}\] \[ \color\green{or \; when\; true\;value\;of\;resistance\; is\;}\] \[\color\red{R =\sqrt{(R_aR_v)}}\]

Procedures: –

Connect all the apparatus as per circuit diagram (fig. 11.1).
Keep the output of auto-transformer is zero.
Make sure that all the connections should be tight.
Now, slowly increased the applied voltage to the circuit using auto-transformer.
Take all the instruments readings as per observation table.

Observation Table: –

Sl. No,	Current (I) in Amps.	Voltage (V) in Volts.	Calculated unknown resistance (R) in Ω
1			R₁
2			R₂
3			R₃
4			R₄
5			R₅
6			R₆
7			R₇
8			R₈
9			R₉
10			R₁₀

Table No. 11.2

Calculation: –

\[ \color\red{R = \dfrac{V}{I}}\] \[\color\red{R_{ave} = \dfrac{(R_1+R_2+……+R_{10})}{10}}\]

Actual value of Resistance (R_A) = ____Ω (measure using multimeter)

Measured value of Resistance (R_m) = ______Ω

\[ % Error = ((R_A-R_m)/R_A)*100\]

Precaution: –

Connection should be proper.
Take reading carefully from the instruments.

Conclusion: –

To be written by student.

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