CHAPTER 5

Bridge circuits (DC & AC) are an instrument to measure resistance, inductance, capacitance and impedance.

Operate on a null-indication principle. This means the indication is independent of the calibration of the indicating device or any characteristics of it.

# Very high degrees of accuracy can be achieved using the bridges. Used in control circuits. # One arm of the bridge contains a resistive element that is sensitive to the physical parameter (temperature, pressure, etc.) being controlled.

TWO (2) TYPES of bridge circuits are used in measurement:

1) DC bridge:a) Wheatstone Bridgeb) Kelvin Bridge

2) AC bridge:a) Similar Angle Bridgeb) Opposite Angle Bridge/Hay Bridgec) Maxwell Bridged) Wein Bridgee) Radio Frequency Bridgef) Schering Bridge

Figure 5.1: Wheatstone Bridge Circuit

The Wheatstone bridge is an

electrical bridge circuit used

to measure resistance.

It consists of a voltage source

and a galvanometer that

connects two parallel branches,

containing four resistors.

One parallel branch contains one known resistance and one unknown; the other parallel branch contains resistors of known resistances.

Figure 5.1: Wheatstone Bridge Circuit

In the circuit at right, R4 is the unknown resistance; R1, R2 and R3 are resistors of known resistance where the resistance of R3 is adjustable.

How to determine the resistance

of the unknown resistor, R4?

“The resistances of the other three

are adjusted and balanced until

the current passing through the

galvanometer decreases to zero”.

R3 is varied until voltage between the two midpoints (B and D) will be

zero and no current will flow through the galvanometer.

Figure 5.1: Wheatstone Bridge Circuit

A

B

C

D

Figure 5.2: A variable resistor; the amount of resistance between the connection terminals could be varied.

Figure 5.1: Wheatstone Bridge Circuit

A

B

C

D

When the bridge is in balance

condition (no current flows through

galvanometer G), we obtain; voltage drop across R1 and R2 is equal,

I1R1 = I2R2

voltage drop across R3 and R4 is equal,

I3R3 = I4R4

Figure 5.1: Wheatstone Bridge Circuit

A

B

C

D

2

4

1

3

R

R

R

R

1

234 R

RRR

In this point of balance, we also obtain;

I1 = I3 and I2 = I4

Therefore, the ratio of two resistances

in the known leg is equal to the ratio

of the two in the unknown leg;

Example 1

Find Rx?

Figure 5.3

Sensitivity of the Wheatstone Bridge

When the pointer of a bridge galvanometer deflects to right or to left direction, this means that current is flowing through the galvanometer and the bridge is called in an unbalanced condition.

The amount of deflection is a function of the sensitivity of the galvanometer. For the same current, greater deflection of pointer indicates more sensitive a galvanometer.

Figure 5.4.

Sensitivity of the Wheatstone Bridge (Cont…) Sensitivity S can be expressed in units of:

A

radiansS

orA

reesS

orA

etersmilS

I

D

Current

DeflectionS

;deg

;lim

How to find the current value?

Figure 5.4.

Fig. 5.5: Thevenin’s equivalent voltage

Thevenin’s Theorem Thevenin’s theorem is a approach used to determine the current flowing through the galvanometer.

Thevenin’s equivalent voltage is found by removing the galvanometer from the bridge circuit and computing the open-circuit voltage between terminals a and b.

31

3

RR

REVa

Applying the voltage divider equation, we express the voltage at point a and b, respectively, as

42

4

RR

REVb

42

4

31

3

RR

R

RR

REVVV baTh

Thevenin’s Theorem (Cont…) The difference in Va and Vb represents Thevenin’s equivalent voltage. That is,

Fig. 5.5: Wheatstone bridge with the galvanometer removed

Fig. 5.6: Thevenin’s resistance

Thevenin’s equivalent resistance is found by replacing the voltage source with its internal resistance, Rb. Since Rb is assumed to be very low (Rb ≈ 0 Ω), we can redraw the bridge as shown in Fig. 5.6 to facilitate computation of the equivalent resistance as follows:

Thevenin’s Theorem (Cont…)

Fig. 5.6: Thevenin’s resistance

4231 //// RRRRRTh

42

42

31

31

RR

RR

RR

RRRTh

If the values of Thevenin’s equivalent voltage and resistance have been known, the Wheatstone bridge circuit in Fig. 5.5 can be changed with Thevenin’s equivalent circuit as shown in Fig. 5.7,

Fig. 5.5: Wheatstone bridge circuit Fig. 5.7: Thevenin’s equivalent circuit

Thevenin’s Theorem (Cont…)

Fig. 5.7: Thevenin’s equivalent circuit

If a galvanometer is connected to terminal a and b, the deflection current in the galvanometer is

gTh

Thg RR

VI

where Rg = the internal resistance in the galvanometer

R2 = 1.5 kΩR1 = 1.5 kΩ

R3 = 3 kΩR4 = 7.8 kΩ

Rg = 150 Ω

GE= 6 V

Figure 5.8 : Unbalance Wheatstone Bridge

Calculate the current through the galvanometer ?

Example 2

Slightly Unbalanced Wheatstone Bridge

22

E

R

RE

RR

REVa

If three of the four resistors in a bridge are equal to R and the fourth differs by 5% or less, we can develop an approximate but accurate expression for Thevenin’s equivalent voltage and resistance. Consider the circuit in Fig- 5.9, the voltage at point a is given as

The voltage at point b is expressed as

rRR

rREVb

Figure 5.9: Wheatstone Bridge with three equal arms

Thevenin’s equivalent voltage is the difference in this voltage

rR

rE

rRR

rREVVV abth 242

1

If ∆r is 5% of R or less, Thevenin equivalent voltage can be simplified to be

R

rEVth 4

Slightly Unbalanced Wheatstone Bridge (Cont…)

rRR

rRRRRTh

))((

2

If ∆r is small compared to R, the equation simplifies to

22

RRRth RRth or

Thevenin’s equivalent resistance can be calculated by replacing the voltage source with its internal resistance and redrawing the circuit as shown in Figure 5.10. Thevenin’s equivalent resistance is now given as

o o

R

R

R

R + Δr

Slightly Unbalanced Wheatstone Bridge (Cont…)

Figure 5.10: Resistance of a Wheatstone.

We can draw the Thevenin equivalent circuit as shown in Figure 5.11

Figure 5.11: Approximate Thevenin’s equivalent circuit for a Wheatstone bridge containing three equal resistors and a fourth resistor differing by 5% or less

Slightly Unbalanced Wheatstone Bridge (Cont…)

Kelvin bridge is a modified

version of the Wheatstone bridge.

The purpose of the modification is

to eliminate the effects of contact

and lead resistance when

measuring unknown low

resistances.

The measurement with a high

degree of accuracy can be done

using the Kelvin bridge for

resistors in the range of 1 Ω to approximately 1 µΩ.

Fig. 5.12: Basic Kelvin Bridge showing a second set of ratio arms

Since the Kelvin bridge uses a second set of ratio arms (Ra and Rb, it is sometimes referred to as the Kelvin double bridge.

Fig. 5.12: Basic Kelvin Bridge showing a second set of ratio arms

The resistor Rlc represents the lead and contact resistance present in the Wheatstone bridge.

The second set of ratio arms (Ra and Rb in figure) compensates for this relatively low lead-contact resistance.

When a null exists, the value for Rx is the same as that for the

Wheatstone bridge, which is

At balance the ratio of Rb to Ra must be equal to the ratio of R3 to R1. Therefore,

1

32

R

RRRx

1

3

2 R

R

R

Rx or

a

bx

R

R

R

R

R

R

1

3

2