Measurement of self-inductance and quality factor using Anderson Bridge.

Experiment No.: 01

Aim of the Experiment: –

Measurement of self-inductance and quality factor using Anderson Bridge.

Apparatus Required: –

Sl. No.NameSpecificationQuantity
1Anderson Bridge Board1 no.
2Standard resistance and capacitance1 no.
3Head Phone1 no.
41000Hz oscillator1 no.
5Connecting wirePVC insulated copperas per required
Table No. 1.1

Circuit Diagram: –

Theory: –

Anderson Bridge is used to measure inductance of a coil and it requires a standard capacitor in terms of which self inductance is expressed. This method can be used for the precise measurement of the inductance over a wide range of values. The connection of the bridge is shown in fig. 1.

As shown by Stephen Butterworth, the greatest sensitivity to a change in r is obtained if,

\[\color{red}{ R_1 = R_2}\] \[\color{red}{R_3 = R_4 = {\frac{1}{2}}\times R_1}\] \[\color{green}{and}\] \[\color{red}{\frac{L_1}{C} = 2R_1^2}\]

The current distribution of the bridge under balanced condition is as shown in fig 1.

Hence under balanced conditions,

\[\color{red}{i_1\times (R_1+jwL_1) = i_2R_4+ir ………………..(1)}\] \[\color{red}{i{r+(\frac{1}{jwc})} = (i_2-i)\times R_3 } \] \[\color{red}{i{r+R_3+(\frac{1}{jwc})} = i_2R_3 ………………..(2)}\] \[\color{Green}{also,}\] \[\color{red}{i_1R_2 = (\frac{1}{jwc})}\] \[\color{red}{ i = ji_1wcR_2 ………………..(3)}\]

After substituting value of i from equation 3, equation 1 and 2 are reduced as equations 4 and 5 respectively.

\[\color{red}{i_1(R_1+jwL_1-jwcR_2r) = i_2R_4 ……………………(4) }\] \[\color{red}{i_1(R_2+jwcR_2r+jwcR_2R_3) = i_2R_3 ………………… (5)} \] \[\color{Green}{Hence,}\] \[\color{red}{\frac{(R_1+jwL_1-jwcR_2r)}{(R_2+jwcR_2r+jwcR_2R_3)} = (\frac{R_4}{R_3})}\] \[\color{Black}{or,}\] \[\color{red}{R_1+jwl_1 = {(\frac{R_4}{R_3})\times (R_2+jwcR_2r+jwcR_2R_3)+jwcR_2r}}\] \[\color{red}{ = \frac{(R_2R_4)}{R_3+{(\frac{jwc}{R_3})\times (R_2R_4r+R_2R_3R_4+R_2R_3r)}}}\]

Equating the real and imaginary parts the resistance of the coil is,

\[\color{red}{R = \frac{(R_2R_4)}{R_3}}\]

and the inductance of the coil

\[\color{red}{L_1 = (\frac{cR_2}{R_3})[{r(R_3+R_4)+R_3R_4}]}\]

In the present case, i.e according to circuit given in the bridge, L1=L, R1=S, R2=R, R3=P, R4=Q, r=m, c=c.

Thus, the self inductance to be

\[\color{red}{L = c[RQ+(R+S)m]}\]

Procedures: –

  1. The D.C. resistance of the coil is obtained. For that oscillator is replaced by a D.C. source and headphone by a D.C. galvanometer. The bridge is balanced by adjusting the resistance R, P and Q. The D.C. resistance is given by,
\[ \color{red}{S = \frac{QR}{P}}\]

2. The arm resistance are chosen for maximum sensitivity for the bridge as follows:

P = Q, Q = R = 1ks

For these value of resistance, bridge is shown in Fig.2

3. The D.C source and the galvanometer are replaced by the oscillator and the headphone respectively. For a particular value of C, the bridge is balanced for minimum sound by adjusting resistance r. The resistance S and inductance L are calculated as,

\[\color{red}{S = \frac{QR}{P}}\] \[\color{Green}{and}\] \[\color{red}{L = c[RQ+(R+S)m]}\]

4. Step-3 is repeated for at-least five different values of C.

5. The average value of L is calculated and for the average value of L, the value of C is calculated from,

\[\color{red}{\frac{L}{C} = 2S^2}\]

6. For this calculated value of C the bridge is again balanced and the value of L is calculated as shown in step 3. This value is the inductance of the coil.

7. For a given frequency the Q-factor of the coil is calculated as

\[\color{red}{Q = \frac{wL}{S}}\]

Observation Table: –

Sl. No.RP=QCrS=(QR/P)L=C[RQ+(R+S)m]
1
2
3
4
5
6
Table No. 1.2

Average value of L

For maximum bridge sensitivity, we calculate it based on our previous calculations,

\[\color{red}{C = \frac{L_1}{2R_1^2}}\]

The final value of self inductance,

\[\color{red}{ L_1 = (\frac{cR_2}{R_3})\times [r(R_3+R_4)+R_3R_4]}\]

and the Q-factor,

\[ \color{red}{Q = \frac{wL_1}{R_1}}\]

Precautions: –

1. Connecting wire used in the circuit should be as small as possible.

2. There should not be any external field nearby.

3. Supply should be pure sinusoidal.

Conclusion: –

To be written by student.

Next

For Your Reference: –

The Anderson Bridge is a specific type of bridge circuit used for measuring the self-inductance ( L ) and quality factor ( Q ) of an inductor. It’s particularly useful for inductors with high ( Q ) values. The Anderson Bridge balances the inductor’s reactance with a known resistance and capacitance, allowing for accurate measurements.

Here’s how you can measure ( L ) and ( Q ) using the Anderson Bridge:

  1. Setup the Circuit: Construct the Anderson Bridge circuit. It consists of a bridge with four arms: one arm contains the inductor ( L ), one contains a resistor ( R ), one contains a capacitor ( C ), and one contains a variable resistor ( R1 ). The bridge is supplied with an alternating current source.
  2. Balance the Bridge: Adjust the variable resistor (R1 ) until the bridge is balanced. At balance, the null detector (typically a galvanometer or voltmeter) connected across the diagonal of the bridge shows zero deflection.
  3. Measurements:
  • Resistance ( R ): The value of the known resistor ( R ) is known. It is used for balancing the inductor’s reactance. Measure its resistance using a multimeter.
  • Variable Resistor ( R1 ): Note down the value of the variable resistor ( R1 ) when the bridge is balanced. This value is crucial for calculations.
  • Capacitance ( C ): The value of the capacitor ( C ) is known. Measure its capacitance using a capacitance meter.
  • Frequency ( f ): The frequency of the alternating current source should be known. Measure it using a frequency counter or waveform generator.
  1. Calculate ( L ) and ( Q ):
  • Inductance ( L ): Use the formula
\[\color{red}{( L =R \cdot C \cdot R_1 )}\]
  • Quality Factor ( Q ): Calculate ( Q ) using the formula
\[\color{red}{( Q = \frac{1}{R_1} \cdot \sqrt{\frac{L}{C}} )}\]
  1. Additional Considerations:
  • Ensure the frequency of the AC source is sufficiently high to minimize any stray capacitance and resistance effects.
  • Be mindful of the limitations of the bridge, particularly at higher frequencies where stray capacitance and inductance might affect the accuracy of the measurements.

By following these steps and performing the necessary calculations, you can accurately determine the self-inductance ( L ) and quality factor ( Q ) of the inductor using the Anderson Bridge circuit.

Viva-voce Questions:

  1. What is the Anderson Bridge and how is it used to measure the self-inductance and quality factor of an inductor ?

Answer:

The Anderson Bridge is a specialized bridge circuit used for measuring the self-inductance (L) and quality factor (Q) of an inductor. It’s particularly effective for inductors with high (Q) values. The bridge achieves this by balancing the reactance of the inductor with known resistance and capacitance components.

To measure the self-inductance and quality factor using the Anderson Bridge:

  1. Setup: Construct the Anderson Bridge circuit, which consists of four arms: one with the inductor (L), one with a known resistance (R), one with a known capacitance (C), and one with a variable resistor (R1).
  2. Balancing: Adjust the variable resistor (R1) until the bridge is balanced, indicated by zero voltage across the bridge.
  3. Measurements:
  • Resistance (R): Measure the known resistance.
  • Variable Resistor (R1): Note down the value of the variable resistor when the bridge is balanced.
  • Capacitance (C): Measure the known capacitance.
  • Frequency (f): Know the frequency of the AC source.
  1. Calculations:
  • Inductance ( L ): Use the formula
\[\color{red}{( L =R \cdot C \cdot R_1 )}\]
  • Quality Factor ( Q ): Calculate ( Q ) using the formula
\[\color{red}{( Q = \frac{1}{R_1} \cdot \sqrt{\frac{L}{C}} )}\]

The Anderson Bridge provides a precise method for determining the self-inductance and quality factor of an inductor, particularly useful for applications where high (Q) values are involved.