A bridge for measuring the inductance L 1 and resistance R 1 of a coil is shown in Figure. R 2 and R 3 are variable resistors and C 3 and C 4 are fixed capacitors. Show that the conditions for the bridge to be balanced (no signal at the detector) are independent of the frequency of the alternating voltage source V.
The bridge is found to be balanced with the following values:
C 3 = 8 μ F, C 4 = 5 μ F, R 2 = 0.5 Ω , R 3 = 3.2 k Ω .
What are the values of L 1 and R 1 ?
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. When the circuit is balanced, the voltages at the two terminals of the detector are equal. If we write Z 1 for the total complex impedance of the inductor L 1 and the resistor R 1 , and define Z 2 , Z 3 and Z 4 similarly, the condition for balance is
= 
,
Which can be rewritten as
or Z 1 Z 4 = Z 2 Z 3 .
Now we have Z 1 = R 1 + j ω L 1 , Z 2 = R 2 ,
Z 3 = R 3 + 
, Z 4 = 
.
So 
+ 
= R 2 R 3 + 
.
Equating real and imaginary pats of this expression gives
R 1 = 
and L 1 = C 4 R 3 R 2 .
These conditions for balance are independent of the frequency ω as required.
Substituting the values given for C 3 , C 4 , R 2 and R 3 gives R 1 = 0.31 Ω and L 1 = 8.0 mH.
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