The Wheatstone Bridge

The Wheatstone Bridge consists of a dc voltage source, four resistors and a detector. The
detector is a type of ammeter called a galvanometer.



The galvanometer is used to detect the condition
g
0 i = . When the circuit satisfies the condition
we say that the bridge is balanced.
g
0 i =

Because the galvanometer is a type of ammeter,
g
0 v = . (Its always true that , whether
the bridge is balanced or not. When the bridge is balanced it is also true that .) Apply KVL
to the top mesh of the bridge to get
g
0 v =
g
0 i =


2 2 1 1 1 1 2 2 g
0 R i v R i R i R i = = (1)

Apply KVL to the bottom mesh of the bridge to get

(2)
3 3 3 3 g m m m
0 v R i R i R i R i + = =
m
1 3

When the bridge is balanced . Apply KCL to node b of the balanced bridge to get
g
0 i =


1 g 3
0 i i i i i = + = = (3)

Apply KCL to node c of the balanced bridge to get


2 g m m
i i i i i
2
+ = = (4)

Using equations 3 and 4 to substitute for the currents in equation 2 gives


3 1 m 2
R i R i = (5)

1
Dividing equation 5 by equation 1 gives


1 2
3
R
m
R
R R
= (6)
Now and solving for
m
R we get

3
1
2
m
R
R R
R
= (6)

Typically,
1
R and
2
R are fixed resistors and
3
R is a variable resistor.
m
R is the resistance that
is being measured.
3
R is adjusted until the detector indicates that the bridge is balanced. Then
the value of
m
R is determined using equation 6.

Example
Consider using a Wheatstone bridge having
1
200 R = and
2
2000 R = to measure a
resistance
m
R . The bridge is balanced by adjusting
3
R until
3
250 R = . What is the value of
m
R ?

Solution
From equation 6
3
1
2
m
2000
250 2500
200
R
R R
R
= = =

Example
Consider using a Wheatstone bridge having
1
200 R = and
2
2000 R = to measure a
resistance,
m
R , of a temperature sensor. Suppose the resistance of the temperature sensor,
m
R ,
in , is related to the temperature T, in C, by the equation


m
1500 25 R T = +

The bridge is balanced by adjusting
3
R until
3
250 R = . What is the value of the temperature?

Solution
From equation 6
3
1
2
m
2000
250 2500
200
R
R R
R
= = =
Next, the temperature in C is given by

m
1500
2500 1500 1000
40 C
25 25 25
R
T


= = = =
2
Example
Consider using a Wheatstone bridge having
1
200 R = and
2
2000 R = to measure a
resistance,
m
R , of a temperature sensor. Suppose the resistance of the temperature sensor,
m
R ,
in , is related to the temperature T, in C, by the equation


m
1500 25 R T = +

The temperature is expected to vary over the range 0 to 100 C. Over what range must
3
R vary
in order for the bridge to measure temperature over the range 0 to 100 C?


Solution:
Solve equation 6 for
3
R :

1
3
2
m
R
R R
R
= (7)

When T =0 C, and
m
1500 R =
1
3
2
m
200
1500 150
2000
R
R R
R
= = =

When T =100 C, ( )
m
1500 25 100 4000 R = + = and

1
3
2
m
200
4000 400
2000
R
R R
R
= = =

3
R could be implemented as a 150 resistor in series with a 250 potentiometer:



3