A.C. Bridges for Measurement of Resistance, Inductance, Capacitance, Frequency etc.

Submitted by Sudeepta Pramanik on Tue, 05/08/2018 - 21:19

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.

 

ac bridgeGENERAL 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=zθ. z represents magnitude and θrepresents the phase angle of the complex impedance.

Hence,            (z1θ1)(z4θ4)=(z2θ2)(z3θ3)

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.

maxwell's inductance bridge

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.

maxwell's L/C bridge

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.

hay's bridge

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=R4jωC4

The balanced condition is that,      

                                                    z1z4=z2z3

                                                or,(R1+jωL1)(R4jωC4)=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.

 

anderson's bridge

 

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+jωL1jωCR3r)=I2R2     ..................................(1)

and

Ic(r+1jωC)=(I2Ic)R4

I1jωCR3(r+1jωC)=(I2I1jωCR3)R4

I1(jωCR3r+R3+jωCR3R4)=I2R4      ..........................(2)

From equation (1) and (2) we get

I1(jωCR3r+R3+jωCR3R4)×1R4=I1(R1+jωL1jωCR3r)×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:

owens 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=R4jωC4

Balance condition is,

Z1Z3=Z2Z4

jωC1(R3+jωL3)=R2(R4jωC4)

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:

De sauty bridgeIn 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:

Schering

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:

wien bridgeThe bridge circuit is shown in figure. The usual balanced relationship gives,

R4(R1jωC1)=R3(R21+jωC2R2)

Separating real and imaginary part,

R1R4+R2R4C2C1=R2R3C2C1=R3R4R1R2

 

 

And,

ωC2R2R4R4ωC1=0ω2=1C1C2R2R4

Wien's bridge also had applications on audio and HF oscillators as the frequency determining device.

 

 

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