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 i g = 0 . When the circuit satisfies the condition
i g = 0 we say that “the bridge is balanced”.
Because the galvanometer is a type of ammeter, v g = 0 . (It’s always true that v g = 0 , whether
the bridge is balanced or not. When the bridge is balanced it is also true that i g = 0 .) Apply KVL
to the top mesh of the bridge to get
R 2 i 2 − v g − R1 i1 = 0 ⇒ R1 i1 = R 2 i 2
(1)
Apply KVL to the bottom mesh of the bridge to get
v g + R m i m − R3 i 3 = 0 ⇒ R3 i 3 = R m i m
(2)
When the bridge is balanced i g = 0 . Apply KCL to node b of the balanced bridge to get
i 1 = i g + i 3 = 0 ⇒ i1 = i 3
(3)
Apply KCL to node c of the balanced bridge to get
i2 + ig = im
⇒ i2 = im
(4)
Using equations 3 and 4 to substitute for the currents in equation 2 gives
R 3 i1 = R m i 2
(5)
1
Dividing equation 5 by equation 1 gives
R3
R1
=
Rm
R2
(6)
Now and solving for R m we get
Rm =
R2
R1
R3
(6)
Typically, R1 and R 2 are fixed resistors and R 3 is a variable resistor. R m is the resistance that
is being measured. R 3 is adjusted until the detector indicates that the bridge is balanced. Then
the value of R m is determined using equation 6.
Example
Consider using a Wheatstone bridge having R1 = 200 Ω and R 2 = 2000 Ω to measure a
resistance R m . The bridge is balanced by adjusting R 3 until R 3 = 250 Ω . What is the value of
Rm ?
Solution
From equation 6
Rm =
R2
R1
R3 =
2000
250 = 2500 Ω
200
Example
Consider using a Wheatstone bridge having R1 = 200 Ω and R 2 = 2000 Ω to measure a
resistance, R m , of a temperature sensor. Suppose the resistance of the temperature sensor, R m ,
in Ω, is related to the temperature T, in °C, by the equation
R m = 1500 + 25 T
The bridge is balanced by adjusting R 3 until R 3 = 250 Ω . What is the value of the temperature?
Solution
From equation 6
Rm =
R2
R1
R3 =
2000
250 = 2500 Ω
200
Next, the temperature in °C is given by
T=
R m − 1500
25
=
2500 − 1500 1000
=
= 40 °C
25
25
2
Example
Consider using a Wheatstone bridge having R1 = 200 Ω and R 2 = 2000 Ω to measure a
resistance, R m , of a temperature sensor. Suppose the resistance of the temperature sensor, R m ,
in Ω, is related to the temperature T, in °C, by the equation
R m = 1500 + 25 T
The temperature is expected to vary over the range 0 to 100 °C. Over what range must R 3 vary
in order for the bridge to measure temperature over the range 0 to 100 °C?
Solution:
Solve equation 6 for R 3 :
R3 =
R1
R2
Rm
(7)
When T = 0 °C, R m = 1500 Ω and
R3 =
R1
R2
Rm =
200
1500 = 150 Ω
2000
When T = 100 °C, R m = 1500 + 25 (100 ) = 4000 Ω and
R3 =
R1
R2
Rm =
200
4000 = 400 Ω
2000
R 3 could be implemented as a 150 Ω resistor in series with a 250 Ω potentiometer:
3