AC Bridge

 A

 A

C

C

B

B

R

R

I

I

D

D

GE

GE

Molla Shahadat Hossain Lipu

Molla Shahadat Hossain Lipu

 Assistant Professor

 Assistant Professor

Department of EEE

Department of EEE

University of Asia

University of Asia

Pacific

Pacific

Email:

Email:

lipu.eee@uap

lipu.eee@uap

-bd.edu

-bd.edu

EEE 457:



 AC Bridges consist of a source,

balance detector and four arms.



In AC bridges, all the four arms

consists of impedance.



The AC bridges are formed by

replacing the DC battery with an

AC source and galvanometer by

detector of Wheatstone bridge.



They are highly useful to find out

inductance, capacitance, quality

factor, dissipation factor etc.

Detector:

(i) Headphones (250 Hz -3~4 kHz) (most sensitive)

Four Arms Z₁, Z₂, Z₃ & Z

4

Detector

Source

(ii) Vibration Galvanometer (5-1000 Hz) (commonly used below 200 Hz) (iii) Tuneable Amplifier (100 Hz-1000 Hz) (respond to narrow bandwidth)

General Expression for an AC Bridge

Now let us derive general expression for an AC bridge balance

Balance Condition

The potential difference between b and d must be zero. Therefore, voltage drop from a to d equals to drop from a to b both in

magnitude and phase.

Thus, we have from figure e₁ = e₂

From equation 1, 2 and 3 we have

Z

₁

.Z

₄

= Z

₂

.Z

₃

Now consider the basic form of an AC bridge. Suppose we have bridge circuit as shown below.

In this circuit R₃ and R₄ are pure electrical

resistances. Putting the value of Z₁, Z₂, Z₃ and Z₄ in the equation that we have derived above for AC bridge.

��1��4 = ��2��3

1 + ����1 ��4 = 2 + ����2 ��3 1��4 + ����1��4 = 2��3 + ����2��3

Now equating the real and imaginary parts we get, Real Part: 1��4 = 2��3 Imaginary Part: ��1��4 = ��2��3 1

=

��

3

��

2 and

��

1

=

��

3

��

2

Following are the important conclusions that can be drawn

from the above equations:

(i) We get two balanced equations that are obtained by

equating real and imaginary parts

(ii) Two balanced equations give two unknown quantities

(iii) Both the equations are said to be independent if and

only if both equation contain single variable element.

This variable can be inductor or resistor.

(iv) The above equations are independent of frequency that

means we do not require exact frequency of the source

voltage and also the applied source voltage waveform

need not to be perfectly sinusoidal.

In this bridge the arms bc and cd are purely resistive while the phase balance depends on the arms ab and ad.

Here, l₁ = unknown inductance of r ₁.

l₂ = variable inductance of resistance R₂. r ₂ = variable electrical resistance.

 According to balance condition, we have at balance point,

Z

₁

.Z

₄

= Z

₂

.Z

₃ => 1

+

 

����

1 4

= (

2

+

��

2

+

 

��

2

)

3 3

Real Part:

14

=

23

+

��

23 => 1

=

3 4

(

2

+

��

2

)

Imaginary Part:

��

14

=

23 =>

��

1

=

 4 ₂

Maxwell's Inductance Capacitance

Bridge

In this Maxwell Bridge, the unknown inductance is measured by the standard variable capacitor.

Here, l₁ is unknown inductance, C₄ is standard capacitor.

Now under balance conditions we have from ac bridge that

Z

₁

.Z

₄

= Z

2

.Z

3

Let us separate the real and imaginary parts, the we have,

Real Part: Imaginary Part: Now the quality factor is given by,

Maxwell's Bridge

Advantages

(1) The frequency does not appear in the final expression of both

equations, hence it is independent of frequency.

(2) Independent equation if chosen r 

_4,

C

₄

as variable elements.

(3) Simple expression for unknown inductor and resistor.

(4) Maxwell's inductance capacitance bridge is very useful for the wide

range of measurement of inductance at audio frequencies.

Disadvantages

(1) The variable standard capacitor is very expensive.

(2) The bridge is limited to measurement of low quality coils (1 <

Q < 10) and it is also unsuitable for low value of Q (i.e. Q < 1) from

this we conclude that a Maxwell bridge is used suitable only for

Hay

’

s Bridge

Hay's bridge is modified Maxwell bridge, now question arises here in our mind that where we need to do modification. In order to understand this, let us

consider the connection diagram given below: In this bridge the electrical resistance,

r 

₄ is connected in series with the standard capacitor,

c

4

Here

l

₁ is unknown inductance connected in series with resistance

r 

₁.

c

₄ is standard capacitor and

r 

₂

, r 

₃

, r 

₄ are pure electrical resistance forming other arms of the bridge. From the theory of ac bridge, we can write at balance point,

4

�� ��

−

��

1 1 4

��

₄

+

 

��



��

+

����

1 1 4

��

₄

=

23 ��1

Real Part:

14

+



=

23

………..(2)

₁

Imaginary Part:

��

1

=

�� 2_�� 4��4

………(3)

On solving equation (2) and (3), we have

frequency hence, in order to find the accurate value of

l

₁

and r 

₁

we should know the correct value of source

frequency.

Let us rewrite the expression for l

₁

,

Now if we substitute Q >10 then 1/Q

2

_{= 1 / 100 and hence}

we can neglect this value, thus neglecting 1/Q

2

_{we get}

r 

₂

r 

₃

c

₄

which is same as we have obtained in Maxwell

bridge hence Hay's bridge circuit is most suitable for high

inductance measurement.

factor 

,

Bridge

Advantages



The bridge gives very simple expression for the calculation of 

unknown inductance of high value (Q>10).



The Hay's bridge require low value of r 

₄

while Maxwell

bridge requires high value of r 

₄

. Now let us analyze why should put

low value of r 

₄

in this bridge: Consider the expression of quality

 As r 

₄

presents in the denominator hence for high quality

factor, r 

4

must be small.

Disadvantages



Equation are not independent (c

₄

r 

₄

are chosen as variable elements).



Complex expression for

��



and

r  ₁ 

.



Frequency dependent.