Bird J. Electrical Circuit Theory and Technology

478 Electrical Circuit Theory and Technology

Types of detector used with a.c. bridges vary with the type of bridge

and with the frequency at which it is operated. Common detectors used

include:

(i) a C.R.O., which is suitable for use with a very wide range of

frequencies;

(ii) earphones (or telephone headsets), which are suitable for frequen-

cies up to about 10 kHz and are used often at about 1 kHz, in which

region the human ear is very sensitive;

(iii) various electronic detectors, which use tuned circuits to detect

current at the correct frequency; and

(iv) vibration galvanometers, which are usually used for mains-operated

bridges. This type of detector consists basically of a narrow moving

coil which is suspended on a ﬁne phosphor bronze wire between

the poles of a magnet. When a current of the correct frequency

ﬂows through the coil, it is set into vibration. This is because

the mechanical resonant frequency of the suspension is purposely

made equal to the electrical frequency of the coil current. A mirror

attached to the coil reﬂects a spot of light on to a scale, and

when the coil is vibrating the spot appears as an extended beam

of light. When the band reduces to a spot the bridge is balanced.

Vibration galvanometers are available in the frequency range 10 Hz

to 300 Hz.

27.3 Types of a.c. bridge

circuit

A large number of bridge circuits have been developed, each of which

has some particular advantage under certain conditions. Some of the most

important a.c. bridges include the Maxwell, Hay, Owen and Maxwell-

Wien bridges for measuring inductance, the De Sauty and Schering

bridges for measuring capacitance, and the Wien bridge for measuring

frequency. Obviously a large number of combinations of components in

bridges is possible.

In many bridges it is found that two of the balancing impedances will be

of the same nature, and often consist of standard non-inductive resistors.

For a bridge to balance quickly the requirement is either:

(i) the adjacent arms are both pure components (i.e. either both resis-

tors, or both pure capacitors, or one of each)—this type of bridge

being called a ratio-arm bridge (see, for example, paras (a), (c),

(e) and (g) below); or

(ii) a pair of opposite arms are pure components—this type of bridge

being called a product-arm bridge (see, for example, paras (b),

(d) and (f) below).

A ratio-arm bridge can only be used to measure reactive quantities of

the same type. When using a product-arm bridge the reactive component

of the balancing impedance must be of opposite sign to the unknown

reactive component.

A.c. bridges 479

Figure 27.2 Simple Maxwell

bridge

A commercial or universal bridge is available and can be used to

measure resistance, inductance or capacitance.

(a) The simple Maxwell bridge

This bridge is used to measure the resistance and inductance of a

coil having a high Q-factor (where Q-factor D ωL/R, see Chapters 15

and 28).

A coil having unknown resistance R

x

and inductance L

x

is shown in

the circuit diagram of a simple Maxwell bridge in Figure 27.2. R

4

and

L

4

represent a standard coil having known variable values. At balance,

expressions for R

x

and L

x

may be derived in terms of known components

R

2

, R

3

, R

4

and L

4

.

The procedure for determining the balance equations given in

Section 27.2 may be followed.

(i) From Figure 27.2, Z

x

D R

x

C jX

L

x

, Z

2

D R

2

, Z

3

D R

3

and

Z

4

D R

4

C jX

L

4

.

At balance,Z

x

Z

3

 D Z

2

Z

4

, from equation (27.3),

i.e., R

x

C jX

L

x

R

3

 D R

2

R

4

C jX

L

4



(ii) Isolating the unknown impedance on the left-hand side of the equa-

tion gives

R

x

C jX

L

x

 D

R

2

R

3

R

4

C jX

L

4



(iii) Manipulating the right-hand side of the equation into a C jb form

gives

R

x

C jX

L

x

 D

R

2

R

4

R

3

C j

R

2

X

L

4

R

3

(iv) Equating the real parts gives R

x

D

R

2

R

4

R

3

Equating the imaginary parts gives X

L

x

D

R

2

X

L

4

R

3

(v) Since X

L

D ωL, then

ωL

x

D

R

2

ωL

4



R

3

from which L

x

D

R

2

L

4

R

3

Thus at balance the unknown components in the simple Maxwell bridge

are given by

R

x

=

R

2

R

4

R

3

and L

x

=

R

2

L

4

R

3

These are known as the ‘balance equations’ for the bridge.

480 Electrical Circuit Theory and Technology

Figure 27.3 Hay bridge

(b) The Hay bridge

This bridge is used to measure the resistance and inductance of a coil

having a very high Q-factor. A coil having unknown resistance R

x

and inductance L

x

is shown in the circuit diagram of a Hay bridge in

Figure 27.3.

Following the procedure of Section 27.2 gives:

(i) From Figure 27.3, Z

x

D R

x

C jX

L

x

,Z

2

D R

2

,Z

3

D R

3

 jX

C

3

,and

Z

4

D R

4

.

At balance Z

x

Z

3

 D Z

2

Z

4

, from equation (27.3),

i.e., R

x

C jX

L

x

R

3

 jX

C

3

 D R

2

R

4



(ii) R

x

C jX

L

x

 D

R

2

R

4

R

3

 jX

C

3

(iii) Rationalizing the right-hand side gives

R

x

C jX

L

x

 D

R

2

R

4

R

3

C jX

C

3



R

3

 jX

C

3

R

3

C jX

C

3



D

R

2

R

4

R

3

C jX

C

3



R

2

3

C X

2

C

3

i.e. R

x

C jX

L

x

 D

R

2

R

3

R

4

R

2

3

C X

2

C

3

C j

R

2

R

4

X

C

3

R

2

3

C X

2

C

3

(iv) Equating the real parts gives R

x

D

R

2

R

3

R

4

R

2

3

C X

2

C

3

Equating the imaginary parts gives X

L

x

D

R

2

R

4

X

C

3

R

2

3

C X

2

C

3

(v) Since X

C

3

D

1

ωC

3

,

R

x

D

R

2

R

3

R

4

R

2

3

C 1/ω

2

C

2

3



D

R

2

R

3

R

4

ω

2

C

2

3

R

2

3

C 1/ω

2

C

2

3



i.e. R

x

D

ω

2

C

2

3

R

2

R

3

R

4

1 C ω

2

C

2

3

R

2

3

Since X

L

x

D ωL

x

,

ωL

x

D

R

2

R

4

1/ωC

3



ω

2

C

2

3

R

2

3

C 1/ω

2

C

2

3



D

ω

2

C

2

3

R

2

R

4

ωC

3

1 C ω

2

C

2

3

R

2

3



i.e. L

x

D

C

3

R

2

R

4

1 C ω

2

C

2

3

R

2

3



by cancelling.

Thus at balance the unknown components in the Hay bridge are given by

R

x

=

!

2

C

2

3

R

2

R

3

R

4

.1 Y !

2

C

2

3

R

2

3

/

and L

x

=

C

3

R

2

R

4

.1 Y !

2

C

2

3

R

2

3

/

A.c. bridges 481

Since ωD 2f appears in the balance equations, the bridge is

frequency-dependent.

(c) The Owen bridge

This bridge is used to measure the resistance and inductance of coils

possessing a large value of inductance. A coil having unknown resistance

R

x

and inductance L

x

is shown in the circuit diagram of an Owen bridge in

Figure 27.4, from which Z

x

D R

x

C jX

L

x

, Z

2

D R

2

 jX

C

2

, Z

3

DjX

C

3

and Z

4

D R

4

.

At balance Z

x

Z

3

 D Z

2

Z

4

, from equation (27.3), i.e.,

R

x

C jX

L

x

jX

C

3

 D R

2

 jX

C

2

R

4

.

Rearranging gives R

x

C jX

L

x

D

R

2

 jX

C

2

R

4

jX

C

3

By rationalizing and equating real and imaginary parts it may be shown

that at balance the unknown components in the Owen bridge are given by

R

x

=

R

4

C

3

C

2

and L

x

= R

2

R

4

C

3

(d) The Maxwell-Wien bridge

This bridge is used to measure the resistance and inductance of a coil

having a low Q-factor. A coil having unknown resistance R

x

and induc-

tance L

x

is shown in the circuit diagram of a Maxwell-Wien bridge in

Figure 27.5, from which Z

x

D R

x

C jX

L

x

,Z

2

D R

2

and Z

4

D R

4

.

Arm 3 consists of two parallel-connected components. The equivalent

impedance Z

3

, is given either

(i) by

product

sum

, i.e., Z

3

D

R

3

jX

C

3



R

3

 jX

C

3



,or

Figure 27.4 Owen bridge Figure 27.5 Maxwell-Wien bridge

482 Electrical Circuit Theory and Technology

(ii) by using the reciprocal impedance expression,

1

Z

3

D

1

R

3

C

1

jX

C

3

from which Z

3

D

1

1/R

3

 C 1/jX

C

3



D

1

1/R

3

 C j/X

C

3



or Z

3

D

1

R

3

C jωC

3

, since X

C

3

D

1

ωC

3

Whenever an arm of an a.c. bridge consists of two branches in

parallel, either method of obtaining the equivalent impedance may

be used.

For the Maxwell-Wien bridge of Figure 27.5, at balance

Z

x

Z

3

 D Z

2

Z

4

, from equation (27.3)

i.e., R

x

C jX

L

x



R

3

jX

C

3



R

3

 jX

C

3



D R

2

R

4

using method (i) for Z

3

. Hence

R

x

C jX

L

x

 D R

2

R

4

R

3

 jX

C

3



R

3

jX

C

3



By rationalizing and equating real and imaginary parts it may be

shown that at balance the unknown components in the Maxwell-

Wien bridge are given by

R

x

=

R

2

R

4

R

3

and L

x

= C

3

R

2

R

4

(e) The de Sauty bridge

This bridge provides a very simple method of measuring a capacitance

by comparison with another known capacitance. In the de Sauty bridge

shown in Figure 27.6, C

x

is an unknown capacitance and C

4

is a standard

capacitor.

At balance Z

x

Z

3

 D Z

2

Z

4



i.e. jX

C

x

R

3

 D R

2

jX

C

4



Hence X

C

x

R

3

 D R

2

X

C

4





1

ωC

x



R

3

 D R

2





1

ωC

4



Figure 27.6 De Sauty bridge

A.c. bridges 483

from which

R

3

C

x

D

R

2

C

4

or

C

x

=

R

3

C

4

R

2

This simple bridge is usually inadequate in most practical cases. The

power factor of the capacitor under test is signiﬁcant because of internal

dielectric losses—these losses being the dissipation within a dielectric

material when an alternating voltage is applied to a capacitor.

(f) The Schering bridge

This bridge is used to measure the capacitance and equivalent series resis-

tance of a capacitor. From the measured values the power factor of insu-

lating materials and dielectric losses may be determined. In the circuit

diagram of a Schering bridge shown in Figure 27.7, C

x

is the unknown

capacitance and R

x

its equivalent series resistance.

From Figure 27.7, Z

x

D R

x

 jX

C

x

,Z

2

DjX

C

2

Z

3

D

R

3

jX

C

3



R

3

 jX

C

3



and Z

4

D R

4

Figure 27.7 Schering bridge

At balance, Z

x

Z

3

 D Z

2

Z

4

 from equation (27.3), i.e.,

R

x

 jX

C

x



R

3

jX

C

3



R

3

 jX

C

3

D jX

C

2

R

4



from which R

x

 jX

C

x

 D

jX

C

2

R

4

R

3

 jX

C

3



jX

C

3

R

3

D

X

C

2

R

4

X

C

3

R

3

R

3

 jX

C

3



Equating the real parts gives

R

x

D

X

C

2

R

4

X

C

3

D

1/ωC

2

R

4

1/ωC

3



D

C

3

R

4

C

2

Equating the imaginary parts gives

X

C

x

D

X

C

2

R

4

R

3

i.e.

1

ωC

x

D

1/ωC

2

R

4

R

3

D

R

4

ωC

2

R

3

from which C

x

D

C

2

R

3

R

4

Thus at balance the unknown components in the Schering bridge are

given by

R

x

=

C

3

R

4

C

2

and C

x

=

C

2

R

3

R

4

484 Electrical Circuit Theory and Technology

Figure 27.8 Phasor diagram

for the unknown arm in the

Schering bridge

The loss in a dielectric may be represented by either (a) a resistance

in parallel with a capacitor, or (b) a lossless capacitor in series with a

resistor.

If the dielectric is represented by an R-C circuit, as shown by R

x

and

C

x

in Figure 27.7, the phasor diagram for the unknown arm is as shown

in Figure 27.8. Angle  is given by

 D arctan

V

C

x

V

R

x

D arctan

I

x

X

C

x

I

x

R

x

i.e.,  D arctan



1

ωC

x

R

x



The power factor of the unknown arm is given by cos .

The angle υ (D 90

°

 ) is called the loss angle and is given by

υ D arctan

V

R

x

V

C

x

= arctan!C

x

R

x

and

υ D arctan



ω



C

2

R

3

R

4



C

3

R

4

C

2



= arctan .!R

3

C

3

/

(See also Chapter 39, page 716)

(g) The Wien bridge

This bridge is used to measure frequency in terms of known components

(or, alternatively, to measure capacitance if the frequency is known). It

may also be used as a frequency-stabilizing network.

Figure 27.9 Wien bridge

A typical circuit diagram of a Wien bridge is shown in Figure 27.9,

from which

Z

1

D R

1

,Z

2

D

1

1/R

2

 C jωC

2

(see (ii), para (d), page 482),

Z

3

D R

3

 jX

C

3

and Z

4

D R

4

.

At balance, Z

1

Z

3

 D Z

2

Z

4

 from equation (27.3), i.e.,

R

1

R

3

 jX

C

3

 D



1

1/R

2

 C jωC

2



R

4



Rearranging gives



R

3



j

ωC

3



1

R

2

C jωC

2



D

R

4

R

1

R

3

R

2

C

2

C

3

 j



1

ωC

3

R

2



C jωC

2

R

3

D

R

4

R

1

A.c. bridges 485

Equating real parts gives

R

3

R

2

Y

C

2

C

3

=

R

4

R

1

27.4

Equating imaginary parts gives



1

ωC

3

R

2

C ωC

2

R

3

D 0

i.e. ωC

2

R

3

D

1

ωC

3

R

2

from which ω

2

D

1

C

2

C

3

R

2

R

3

Since ω D 2f,

frequency, f =

1

2p

p

.C

2

C

3

R

2

R

3

/

(27.5)

Note that if C

2

D C

3

D C and R

2

D R

3

D R,

frequency,fD

1

2

p

C

2

R

2



D

1

2CR

Problem 1. The a.c. bridge shown in Figure 27.10 is used to

measure the capacitance C

x

and resistance R

x

. (a) Derive the

balance equations of the bridge. (b) Given R

3

D R

4

, C

2

D 0.2 µF,

R

2

D 2.5k and the frequency of the supply is 1 kHz, determine

the values of R

x

and C

x

at balance.

Figure 27.10

(a) Since C

x

and R

x

are the unknown values and are connected in

parallel, it is easier to use the reciprocal impedance form for this

branch



rather than

product

sum



,

i.e.

1

Z

x

D

1

R

x

C

1

jX

C

x

D

1

R

x

C

j

X

C

x

from which Z

x

D

1

1/R

x

 C jωC

x

From Figure 27.10, Z

2

D R

2

 jX

C

2

, Z

3

D R

3

and Z

4

D R

4

.

At balance, Z

x

Z

3

 D Z

2

Z

4





1

1/R

x

 C jωC

x



R

3

 D R

2

 jωX

C

2

R

4



486 Electrical Circuit Theory and Technology

hence

R

3

R

4

R

2

 jX

C

2



D

1

R

x

C jωC

x

Rationalizing gives

R

3

R

2

C jX

C

2



R

4

R

2

C X

2

C

2



D

1

R

x

C jωC

x

Hence

1

R

x

C jωC

x

D

R

3

R

2

R

4

R

2

C 1/ω

2

C

2



C

jR

3

1/ωC

2



R

3

R

2

C 1/ω

2

C

2



Equating the real parts gives

1

R

x

D

R

3

R

2

R

4

R

2

C 1/ω

2

C

2



i.e. R

x

D

R

4

R

2

R

3



R

2

ω

2

C

2

C 1

ω

2

C

2



and R

x

=

R

4

.1 Y !

2

C

2

R

2

/

R

2

R

3

!

2

C

2

Equating the imaginary parts gives

ωC

x

D

R

3

1/ωC

2



R

4

R

2

C 1/ω

2

C

2



D

R

3

ωC

2

R

4

R

2

ω

2

C

2

C 1/ω

2

C

2



i.e. ωC

x

D

R

3

ω

2

C

2

ωC

2

R

4

1 C ω

2

C

2

R

2



and C

x

=

R

3

C

2

R

4

.1 Y !

2

C

2

R

2

/

(b) Substituting the given values gives

R

x

D

1 C ω

2

C

2

R

2



R

2

ω

2

C

2

since R

3

D R

4

i.e. R

x

D

1 C 21000

2

0.2 ð 10

6



2

2.5 ð 10

3



2

2.5 ð 10

3

21000

2

0.2 ð 10

6



2

D

1 C 9.8696

3.9478 ð 10

3

 2.75 k

C

x

D

C

2

1 C ω

2

C

2

R

2



since R

3

D R

4

D

0.2 ð 10

6



1 C 9.8696

µF D 0.01840 µFor18.40 nF

Hence at balance R

x

= 2.75 kZ and C

x

= 18.40 nF

A.c. bridges 487

Problem 2. For the Wien bridge shown in Figure 27.9, R

2

D

R

3

D 30 k, R

4

D 1k and C

2

D C

3

D 1 nF. Determine, when

the bridge is balanced, (a) the value of resistance R

1

, and (b) the

frequency of the bridge.

(a) From equation (27.4)

R

3

R

2

C

2

C

3

D

R

4

R

1

i.e., 1 C 1 D 1000/R

1

, since R

2

D R

3

and C

2

D C

3

, from which

resistance R

1

D

1000

2

D 500 Z

(b) From equation (27.5),

frequency, f D

1

2

p

C

2

C

3

R

2

R

3



D

1

2



[10

9



2

30 ð 10

3



2

]

D

1

210

9

30 ð 10

3



 5.305 kHz

Problem 3. A Schering bridge network is as shown in Figure 27.7,

page 480. Given C

2

D 0.2 µF, R

4

D 200 , R

3

D 600 , C

3

D

4000 pF and the supply frequency is 1.5 kHz, determine, when

the bridge is balanced, (a) the value of resistance R

x

, (b) the value

of capacitance C

x

, (c) the phase angle of the unknown arm, (d) the

power factor of the unknown arm and (e) its loss angle.

From para (f), the equations for R

x

and C

x

at balance are given by

R

x

D

R

4

C

3

C

2

and C

x

D

C

2

R

3

R

4

(a) Resistance, R

x

D

R

4

C

3

C

2

D

2004000 ð 10

12



0.2 ð 10

6

D 4 Z

(b) Capacitance, C

x

D

C

2

R

3

R

4

D

0.2 ð 10

6

600

200

F D 0.6 mF

(c) The phasor diagram for R

x

and C

x

in series is shown in Figure 27.11.

Phase angle,  D arctan

V

C

x

V

R

x

D arctan

I

x

X

C

x

I

x

R

x

D arctan

1

ωC

x

R

x

i.e.  D arctan



1

215000.6 ð 10

6

4



D arctan44.21 D 88.7

°

lead

Figure 27.11

Bird J. Electrical Circuit Theory and Technology

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