Alternating current bridge methods are important for measurement of inductance, capacitance, storage factor, dissipation factor etc.The a.c. bridge is natural out growth of Wheatstone Bridge.It's basic form consist of four arms, a source of excitation(ac source), a balance detector.
GENERAL EQUATION FOR BRIDGE BALANCE
For balance of bridge the condition required is no current through the detector
i.e. the potential difference between points b and d should be zero. This is possible when, e1=e2
or,i1z1=i2z2 ......................(1)
Also at balance,
i1=i3=ez1+z3 .......(2)
and i2=i4=ez2+z4 ........(3)
substitute eqn.(2) and eqn.(3) gives,
z1z4=z2z3
This is the basic equation for balanced ac bridge.
Consider the polar form, the impedance can be written as,z=z∠θ. z represents magnitude and θrepresents the phase angle of the complex impedance.
Hence, (z1∠θ1)(z4∠θ4)=(z2∠θ2)(z3∠θ3)
For balance,z1z4∠(θ1+θ4)=z2z3∠(θ2+θ3)
MEASUREMENT OF INDUCTANCE
MAXWELL'S INDUCTANCE BRIDGE:
This bridge is used to measure self-inductance by comparison with a variable standard self-inductance.
The connection of bridge and phasor diagram for balanced circuit is shown.
Here,
L1= inductance to be measured with resistance R1
L2=variable inductance
R2=variable resistance
R3,R4= pure resistance
The balanced condition is that, z1z4=z2z3
or, (R1+jωL1)R4=(R2+jωL2)R3
Equating real and imaginary part of both sides,
R1R4=R2R3R1=R2R3R4
and,
L1R4=L2R3L1=L2R3R4
Hence the unknown resistance can be measured in terms of known inductance and two resistors.And the whole process are independent of frequency. This bridge is used for measurement of iron losses of transformer at audio frequency.
MAXWELL,S INDUCTANCE CAPACITANCE BRIDGE:
Here, an inductance is measured by comparison with standard capacitance.
The connection of bridge and phasor diagram for balanced circuit is shown.
Here,
L1= inductance to be measured with resistance R1
R2,R3,R4= are known pure resistance.
C4=variable standard capacitor.
Now,
Z1=(R1+jωL1);Z2=R2;Z3=R3;Z4=R41+jωC4R4
The balanced condition is that,
z1z4=z2z3
(R1+jωL1)(R41+jωC4R4)=R2R3R1R4+jωL1R4=R2R3+jωR2R3C4R4
Equating real and imaginary part of both sides,
R1=R2R3R4L1=R2R3C4
Expression for Q factor, Q=ωL1R1=ωC4R4
HAY'S BRIDGE:
Hay's bridge is a modification of Maxwell's bridge with a series resistance connected to standard capacitor, instead of resistance parallel with capacitor.
Let,
L1= inductance to be measured with resistance R1
R2,R3,R4= are known pure resistance.
C4= Standard capacitor.
Now,
Z1=R1+jωL1;Z2=R2;Z3=R3;Z4=R4−jωC4
The balanced condition is that,
z1z4=z2z3
or,(R1+jωL1)(R4−jωC4)=R2R3
Separating real and imaginary terms,we get,
R1R4+L1C4=R2R3L1=R1ω2R4C4
Solving these simultaneous equations we get,
R1=ω2R2R3R4C241+ω2R24C24L1=R2R3C41+ω2R24C24
The expressions contain the frequency term which is same as the frequency of source of supply.
Hence for calculation of unknown inductance we should know the frequency of supply.
ANDERSON'S BRIDGE:
This bridge is modification of the maxwell's inductance capacitance bridge. Here self inductance is measured in terms of standard capacitor over a wide range of values.

L1= the self-inductance to be measured
R1=resistance connected in series with self-inductor
r,R2,R3,R4= are known pure resistance.
C= fixed standard capacitor.
At balance,
I1=I3I2=Ic+I4I1R3=Ic×1jωCIc=I1jωCR3
Writing the other balance eqations
I1(R1+jωL1)=I2R2+Icr
I1(R1+jωL1)=I2R2+I1jωCR3r
I1(R1+jωL1−jωCR3r)=I2R2 ..................................(1)
and
Ic(r+1jωC)=(I2−Ic)R4
I1jωCR3(r+1jωC)=(I2−I1jωCR3)R4
I1(jωCR3r+R3+jωCR3R4)=I2R4 ..........................(2)
From equation (1) and (2) we get
I1(jωCR3r+R3+jωCR3R4)×1R4=I1(R1+jωL1−jωCR3r)×1R2
Equating real and imaginary part,
R1=R2R3R4L1=CR3R4[r(R4+R2)+R2R4]
This method is used for measurement over a wide range of values from few micro_henrys to several henrys.
OWEN'S BRIDGE:
The circuit diagram of this bridge is shown. Here also unknown inductance is measured in terms of resistance and capacitance.
From figure,
Z1=−jωC1Z2=R2Z3=R3+jωL3Z4=R4−jωC4
Balance condition is,
Z1Z3=Z2Z4
−jωC1(R3+jωL3)=R2(R4−jωC4)
By separating real and imaginaries, we get
R3=R2C1C4L3=C1R2R4
Here is no need of frequency in final balance equation. Hence bridge is unaffected by frequency variation and wave form.
MEASUREMENT OF CAPACITANCE
De SAUTY'S BRIDGE:
In this bridge it is simplest to measure capacitance by comparing two capacitor. With respect to figure,
C2=capacitor to be measured
C3=a standard capacitor
R1,R4= non-inductive resistor
By varying either R4 or R1
Let I1 be current through R1 andC2
and I2 be current through R4andC3
At balanced condition,
I1R1=I2R4 .......................(1)
and, I1×−jωC2=I2×−jωC3 .......................(2)
From the two equations we get,
R1R4=C2C3
C2=C3×R1R4
If both the capacitors not free from dielectric loss this bridge method will goes offset. A perfect balance occur if air capacitors are used.
SCHERING BRIDGE:
C1=capacitor to be measured
C2=a standard capacitor
R3= non-inductive resistor
R4=variable non-inductive resistor
C2=variable capacitor
At balance,
(r1+1jωC1)(R41+jωC4R4)=R3jωC2(r1+1jωC1)R4=R3jωC2(1+jωC4R4)
Equating real and imaginary part,
r1=R3C4C2C1=C2(R4R3)
MEASUREMENT OF FREQUENCY
WIEN'S BRIDGE:
The bridge circuit is shown in figure. The usual balanced relationship gives,
R4(R1−jωC1)=R3(R21+jωC2R2)
Separating real and imaginary part,
R1R4+R2R4C2C1=R2R3C2C1=R3R4−R1R2
And,
ωC2R2R4−R4ωC1=0ω2=1C1C2R2R4
Wien's bridge also had applications on audio and HF oscillators as the frequency determining device.