Wheatstone Bridges
Introduction

In earlier studies you will have been introduced to two resistors connected in series with a power supply to form a voltage divider such as that shown below. These are frequently used in sensing circuits where changes in the environment surrounding the sensor alter its resistance.

You should already have met the idea that the voltage at the midpoint of a voltage divider can be calculated using the equation:

The voltage at the midpoint depends upon the individual resistor values and the power supply voltage. The theory of voltage dividers can be extended to cover other circuits.

Balanced bridge formula
One important type of circuit based on
voltage
The voltage across a component is the electrical energy transferred by 1 coulomb of charge passing through the component.
voltage
dividers, called a Wheatstone bridge, is shown in Fig.1. It consists of four resistors arranged as two voltage dividers connected in parallel with the same
power
The power of system is a measurement of the rate at which energy is transferred from one form to another. The scientific unit of power is the watt.
power
supply. The 10 kΩ and 20 kΩ resistors are described as forming one arm of the
Wheatstone bridge
A Wheatstone bridge is a circuit made of two voltage dividers connected in parallel with the same power supply. The midpoints of the two voltage dividers are connected with a voltmeter.
Wheatstone bridge
, while the variable resistor and the 6 kΩ fixed resistor make up the other arm of the bridge circuit.

Alter the value of the variable resistor in Fig.1 and observe the reading on the voltmeter.

Click on the figure below to interact with the model.

 Figure 1.  A Wheatstone bridge.

When the variable resistor in Fig.1 is altered through its full range of values, the voltmeter reading is …
What is the
resistance
The opposition to the flow of current provided by a circuit is called resistance. Resistance is measured in units called Ohms.
resistance
of the variable resistor when the voltmeter reading is 0 V?

The bridge is said to be balanced when the voltages at the midpoints of each arm have the same value. When the bridge is balanced a voltmeter connected between the two midpoints will therefore show a potential difference of 0 V.

Adjust the variable resistor in Fig.2 to balance the Wheatstone bridge.

Click on the figure below to interact with the model.

 Figure 2.  A Wheatstone bridge.

Complete the following sentences to describe the voltages in the balanced bridge circuit of Fig.2.

• The two 10 kΩ resistors set the voltage at point . The voltage at point X is  V.

The variable resistor and 5 kΩ resistors set the voltage at point . To balance the bridge by making the voltage at point Y 4.5 V, the variable resistor must have a value of  kΩ.

The example in Fig.2 is simple because the resistors in each arm have the same value. However, this need not always be the case.

 Figure 3. Resistor ratios for a balanced Wheatstone bridge.

For the bridge to be balanced, the resistors in each arm must have the same ratios. This can be expressed mathematically as:

This equation describes the condition needed for a Wheatstone bridge to be balanced. The resistor positions are called A, B, C, and D. The resistors at these locations are labelled as shown in Fig.4
 Figure 4. Formula nomenclature.

Enter values for RA, RB, and RC into the bridge calculator of Fig.5 and note that the value of RD needed to balance the bridge depends upon the values chosen for RA, RB, and RC.

 Figure 5. Bridge balance calculator.

Set the variable resistor values in Fig.6 to those shown in the table below and record the value of resistor RD required for balance.

Click on the figure below to interact with the model.

 Figure 6.  Bridge balancing example.

 RA / kΩ RB / kΩ RC / kΩ RD / kΩ 60 40 90 50 30 75 40 30 60 80 30 40

Now check the values you have determined experimentally by entering each set of values for RA, RB, and RC from this table into the bridge calculator of Fig.2. Note the calculated value that RD must have for the bridge to be balanced in each case. Compare the calculated values with the values of RD determined experimentally and recorded in the table above.

Using Wheatstone bridges
Wheatstone bridge circuits such as that shown in Fig.2 were first developed to allow very precise measurement of the resistances of samples of materials. The method for finding an unknown resistance involves having two resistors whose values are known precisely, and using a resistance substitution box such as is shown in Fig.7 to balance a bridge circuit. In this way, three of the resistances in the bridge are known accurately and the fourth can be calculated using:

Click on specific digits on the resistance substitution box of Fig.7 to alter its resistance.

 Figure 7. Balancing the bridge.
What value is the resistance of the substitution box when the bridge in Fig.7 is balanced?

If the precise values of the resistors A and B in the circuit of Fig.7 are known, the resistance of the substitution box can be adjusted until the bridge is balanced. The value of the unknown resistance can then be found using the balanced bridge formula.

In one situation the bridge balances when the resistance substitution box setting is 259 Ω. The resistance beside the substitution box is 690 Ω and the resistance vertically above the substitution box is 2.2 kΩ. What is the value of the other resistor in the bridge?
Determining temperature
In Fig.8 a
thermistor
A thermistor is an electronic component whose resistance changes when its temperature alters. The resistance of a 'negative temperature coefficient' (n.t.c.) thermistor reduces as the temperature increases.
thermistor
immersed in water forms one arm of a bridge circuit. Turn on the Bunsen to warm the water briefly. Then adjust the value of the resistance substitution box until the bridge balances.

Since the fixed resistors in the bridge have the same value, balance is achieved when the resistance of the substitution box equals that of the thermistor. Therefore when balance is achieved we know the resistance of the thermistor and can use its calibration graph to determine its temperature.

Heat the water further and note the effect on the resistance of the thermistor.

When the thermistor warms, its resistance …
 Figure 8. Measuring temperature.

As the water is heated the thermistor's resistance decreases, so different values of the variable resistor are needed to balance the bridge at different temperatures. Balancing the bridge at any specific temperature allows the thermistor's resistance to be determined at that temperature. The actual temperature can then be found from the thermistor's resistance using the thermistor's calibration graph.

Complete the following statements to summarize the steps in determining the temperature of the water.

• When the water is heated, the thermistor warms to the same temperature as the water and its changes. Adjusting the resistance substitution box the bridge. When the bridge is balanced the voltmeter reads volts. By referring to the graph the at which the thermistor has the calculated resistance can be determined.

The principle behind this method could be used to determine the light level in a room. The thermistor would be replaced with an
LDR
The resistance of a light dependent resistor reduces as the light intensity increases. This feature makes LDRs ideal for use in light sensing circuits.
LDR
and the light level determined from an appropriate LDR calibration graph. This basic principle governs the operation of light meters such as that shown in Fig.9.

 Figure 9. A light meter.

In fact, any
sensor
Sensors used in electronics produce a change in their resistance when some feature of their surrounding environment changes. The resistance of a thermistor changes as the surrounding temperature alters.
sensor
whose resistance depends upon some physical property of the surrounding environment can be used with a Wheatstone bridge in this way to measure the value of that property, if an appropriate calibration graph is available.

The
strain
The strain ε produced by a particular force is defined as the ratio of the extension to the original length.
strain
gauge is another example of a resistive sensor. The resistance of the strain gauge depends upon the forces acting on it. Forces acting on a beam can be monitored with a strain gauge. In
normal
The normal to a surface at a given point is a line drawn at right angles to the surface at that point.
normal
operation the circuit would be arranged so that the bridge was balanced and the voltmeter reading would be 0 V. Any changes in the forces acting on the
strain
The strain ε produced by a particular force is defined as the ratio of the extension to the original length.
strain
gauge would be indicated by a non-zero reading on the voltmeter.

Out-of-balance bridges
When a Wheatstone bridge is balanced, the ratio of the resistance values in each arm is the same. If the ratios are not equal, the bridge is described as being 'out of balance' and a p.d. is created between the midpoints of each arm.
What value should the resistance substitution box have to balance the bridge in Fig.10?

Click the appropriate digits on the resistance substitution box in the circuit of Fig.10 to check that your answer is correct.

When a Wheatstone bridge is out of balance, the size of the p.d. ΔV between the midpoints increases as the value of the resistance substitution box deviates further from the balance condition. The difference between the setting on the resistance substitution box and the value of resistance required for balance is referred to as ΔR.

Alter the resistance substitution box in Fig.10 between 420 Ω and 520 Ω in steps of 10 Ω and plot readings of the out-of-balance voltage ΔV and out-of-balance resistance ΔR for each setting.

 Figure 10. Out-of-balance bridge.

The graph in Fig.10 is a straight line for small values of ΔR. This shows that for small changes around the balance condition the out-of-balance voltage ΔV is directly proportional to the out-of-balance resistance ΔR. The relationship ΔV is proportional to ΔR is true for resistance changes within approximately 5 per cent of the resistance needed to balance the bridge. When the changes from the balance condition are larger, ΔV still increases as ΔR increases but there is no longer a direct proportionality.

Before the days of the microprocessor, out-of-balance bridges were used extensively in electronic instrumentation where thermistors, light dependent resistors or strain gauges were used in the bridge. When the monitored property changes by a small amount, the magnitude of the change can be indicated by the out-of-balance voltage.

The potential differences produced by sensing circuits using out-of-balance bridges are small. These small voltages can be amplified by the type of differential amplifier circuit shown in Fig.11.

Click on the figure below to interact with the model.

 Figure 11.  Sensing circuit with difference amplifier.

In addition, whether the resistance of the sensor is increasing or decreasing can be determined from the sign of the out-of-balance voltage.

Summary

Two voltage dividers can be made into a bridge circuit, commonly called a Wheatstone bridge. The Wheatstone bridge is balanced when there is no p.d. between the midpoints of the voltage dividers. For this to happen, the ratio of the resistors in each voltage divider must be the same.

The value of resistance needed to balance a Wheatstone bridge circuit can be determined using the equation:

Wheatstone bridges can be used in circuits to measure environmental properties such as temperature or light level.

Exercises
 Figure 12.
1. When the circuit shown in Fig.12 is set up, the ammeter reads zero. What is the value of resistor X?
• Ω.   (to the nearest whole number)
 Figure 13.
2. In the circuit shown in Fig.13 an ammeter connected between points X and Y reads zero. This means that …
3. Calculate the resistance of the variable resistor required to balance the bridge in Fig.14.
• kΩ.   (to 1 d.p.)
 Figure 14.
4. The student now connects another 39 kΩ resistor in parallel with the 39 kΩ resistor already in the bridge of the circuit in Fig.14. Is it now possible to balance the bridge circuit?
5. The student then interchanges the positions of the 100 kΩ resistor and the variable resistor in Fig.14. What value should he set the variable resistor to in order to balance the bridge now
•  kΩ   (to 1 d.p.)

6. A student investigating the behaviour of a light dependent resistor (LDR) finds that its resistances in bright daylight and darkness are 400 Ω and 100 kΩ respectively. The LDR is added into a circuit as shown in Fig.15. The battery has negligible
internal resistance
All batteries or power supplies have internal resistance. This resistance has the effect of reducing the output p.d. as the current supplied increases.
internal resistance
. Calculate the reading on the voltmeter when the LDR is in bright daylight.
•  V   (to 2 d.p.)
 Figure 15.
7. Calculate the reading on the voltmeter in Fig.15 when the LDR is placed in darkness.
•  V   (to 2 d.p.)
 Figure 16.
8. The LDR of Fig.15 is now used as one of the resistors in the Wheatstone bridge circuit shown in Fig.16. Calculate the resistance value needed to balance the bridge when the LDR is placed in bright daylight
• .
Ω   (to the nearest whole number)
9. In the circuit shown in Fig.17 the values of R1 and R2 are as indicated and the voltmeter reads zero. Which of the following shows a set of possible values for R3 and R4?
 Figure 17.
10. Calculate the value of the variable resistor in Fig.18 when the bridge is balanced.
• kΩ    (to the nearest whole number)
 Figure 18.
11. What will happen to the balanced bridge if the 6 V battery in Fig.18 is replaced with a 9 V battery.
 Figure 19.

12. The circuit in Fig.19 is set up to investigate the action of the thermistor (TH3). The variable resistor is adjusted until the voltmeter reads 0.00 V. The thermistor's calibration graph is as shown in Fig.19. Classify the following statements as true or false.
•  When the voltmeter reads zero, the variable resistor and the thermistor have equal resistances. False True If the thermistor is warmed the voltage at point Y increases. False True The voltage at X is 12 V. False True
 Figure 20.
13. In an experiment to investigate unknown resistors X and Y, a student sets up the Wheatstone bridge circuit shown in Fig.20 and the voltmeter reads 0.00 V. When he is told that resistors X and Y have the same resistance he makes the following statements. Classify these statements as true or false.
•  As drawn the bridge circuit is not balanced. False True The resistance of X must be 200 kΩ. False True The bridge would also be balanced if X were removed from its current position and used to replace R3. False True
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