Why a Wheatstone Circuit is so Awesome at Measuring Small Changes in Resistance

A Wheatstone Bridge circuit is a great method for measuring small changes in resistance. They are often used with resistance based sensors like strain gauges, Resistance Temperature Detectors (RTDs), Piezoresistive pressure, and Magnetoresistive. The following are some key benefits of using a Wheatstone Bridge circuit:

  • Its configuration has the benefit of cancelling out error terms with the ADC circuit used to measure it
  • They produce a differential voltage output centered around 0V so they can easily be interfaced with precision high gain amplifiers, like instrumentation amplifiers, enable the detection of very small resistance changes

In this tutorial we will explain how a Wheatstone bridge works — mathematically and practically — and demonstrate the benefits, mentioned above, of using a Wheatstone bridge. We will compare it to another resistance measurement technique for comparison to better highlight its benefits.

Overview of a Wheatstone Bridge Circuit

Below is a schematic of a Wheatstone bridge circuit. R1 in the circuit represents the resistance we want to measure any changes in from its nominal value, it could represent a  RTD sensor or a strain gauge. The circuit is configured as a Quarter Wheatstone Bridge circuit since 3 of the 4 resistance values in the bridge are known and the other (R1) is not known, but we know it has a value close to the other resistors. There are half and full Wheatstone bridge configurations where two and four of the resistance values are unknown. We will just focus on the quarter bridge configuration in this tutorial, we will have a link at the end if you want more information on the other configurations. 

Let's walk through the math for how a Quarter Wheatstone Bridge works, then we will use a PT100 RTD temperature sense to walk through a real world example. 

From the bridge circuit schematic we can see that:

R2 = R3 = R4 and since they are equal we can represent them all as R0

and we can represent R1: R0 + ΔR

A voltage source is applied across the bridge that we will refer to as the excitation voltage or Vex. To calculate ΔR, we make a differential measurement of the voltage in between the two parallel pairs of resistors, as shown in the schematic. 

Voltage measurement Vo = Vl - Vr, if all resistance values are equal Vo = 0V

The two midpoint voltage values can be represented mathematically as:

Vl = Vex ⋅ (R0 / (2R0 + ΔR) 

Vr = Vex / 2

We can combine these two expressions to calculate Vo:

Vo = Vl - Vr so Vo = Vex ⋅ ( R_0 / (2R0 + ΔR) − 1/2 ) 

We can use algebra to rearrange the expression:

Vo = −Vex ⋅ ΔR / [ 2 ⋅ (2R0 + ΔR) ]

We can now apply a simplification step if |ΔR| ≪R0:

Vo ≈ −Vex ⋅ ΔR / (4R0) which can be rearranged to:
Vo / Vex ≈ −(1/4) · (ΔR / R0)

The final equation for getting ΔR (the only unknown if we measure Vo) does not look special but there are a couple of key points to keep in mind that tie back to the advantages of a Wheatstone Bridge circuit mentioned in the beginning of this tutorial. 

  • Cancels error factors: the term "Vo / Vex" shows that this is a ratio metric measurement. Do you know what else makes a ratio metric measurements, an ADC! If you use the voltage reference for the ADC as the excitation voltage source for the bridge, any error or drift in the voltage reference is applied to both the ADC's ratio measurement and the Wheatstone Bridge's ratio measurement which means it cancels out. 
  • Centered around 0V: The polarity of the output voltage V0 is based on whether R1 is greater or less than R2 so it is centered around 0V so that it can work well with a high gain precision amplifier. It is also a differential measurement, which cancels out any common mode noise in the measurement.

Next let's do a real world example of a Wheatstone bridge measurement on a 3 wire RTD sensor and compare it to the precision current source resistance measurement technique. 

Application example using a Quarter Wheatstone Bridge to Measure an RTD Sensor

Let's say we have a PT100 3-wire RTD sensor. The "100" in PT100 means the RTD sensor's nominal resistance at 0 °C is 100ohms. With that in mind we would want the 3 known resistor values in our Wheatstone to be 100 ohms. For this example, we will compare the Wheatstone Bridge approach to another resistance measurement approach. The second approach uses a precision current source and an ADC voltage measurement. The known current flows through the unknown resistance of the RTD and the measured voltage drop is used to calculate the resistance (V / I = R). Now this is a common way to measure an unknown resistance value and Anabit has a detailed tutorial on using this method to make a resistance measurement. The issue is this method is not good at measuring small changes in a resistance like the bridge approach as we will see.

The following schematic shows how we connect a 3 wire RTD sensor to a Quarter Wheatstone Bridge setup

For a PT100 RTD sensor the nominal resistance is: R₀ = 100 Ω at 0 °C. The resistance change from the nominal value is 0.385 ohms per degree Celsius. For our example let's assume the temperature being measured is 2 °C so the resulting resistance value of the RTD is 100.77 Ω. Let's set our ADC voltage reference and Wheatstone Bridge excitation voltage to 2.048V. They are both using the same 2.048V voltage source to cancel out any error or drift, as mentioned earlier. 

For our example situation let's calculate what the measured differential voltage will be from the Quarter Wheatstone Bridge:

Vo = −Vex ⋅ ΔR / (4R0) = -2.048V ⋅ (0.77Ohms / (4 ⋅ 100Ohms)) =  -3.94mV

To measure -3.94mV requires an accurate high resolution ADC and remember to resolve a change of a single degree it would be half that voltage. There are two important points that this example calculation show:
  • The result can be positive or negative depending on the temperature value. But it is always centered around 0V so that makes it easy to amplify with a high gain instrumentation amplifier, which essentially increases the resolution of the ADC. 
  • The sign of the result depends on which side of the bridge the sensor is on and how you connect the ADC to the differential measurement points. In this example a negative voltage means a temperature that is greater than 0 °C

Now let's look at an example of the same measurement done with an precision current source and an ADC. For this example the current source is 1mA.

At 0 °C the ADC will measure a voltage of: Vmeas = I x R = 0.001A ⋅ 100Ohms = 100mV

At 2 °C the ADC will measure a voltage of: Vmeas = 0.001A ⋅ 100.77Ohms = 100.77mV

That's a change of only 0.77mV which is about 5x smaller than the output of the Wheatstone bridge for the same measurement. But why not use a larger current source than 1mA, how about we use 5mA? You can do that and then the result is about the same voltage change per degree as the Wheatstone Bridge approach. But that means the nominal voltage across the RTD will be 500mV! If you wanted to amplify the bridge output by a gain of 10x the resulting nominal output voltage from the amplifier would be 5V. You are not going to find an instrumentation amplifier that can handle output voltages > 5V so you will have to get a higher voltage op amp that will have more error contributors compared to an instrumentation amplifier. You could not amplify the signal and then you will need a high accuracy and high resolution ADC to resolve milli volt or sub milli volt voltage changes. Finally keep in mind that you are not getting the benefits of the ADC's voltage reference cancelling out error when it is used as the excitation voltage source for the bridge. With the current source approach any error in the current source signal and the ADC's voltage reference both contribute to the resulting error in the measured voltage. 

How to handle resistance due to wiring when working with a Wheatstone Bridge

Chances are that if you are using a Wheatstone Bridge to monitor a sensor like an RTD or strain gauge, it is not going to be right next to the Wheatstone Bridge. It could be located a meter away or 5 meters away. That wiring connected to the senor and to the Wheatstone bridge will have some resistance that will add error to the output voltage of the bridge. Compensating for wiring resistance is the reason resistor based sensors typically have more than two wires. The PT100 RTD sensor that we used for our example calculation has 3-wires. If you refer back to the diagram of the Wheatstone Bridge with the RTD sensor attached, notice that all three wire are connected to the bridge. Also notice that the wire that represents the left side differential measurement point essentially splits the two other wires so that one is on the top of the measurement point and one is on the bottom of the measurement point. This approach slits the wire resistance that the current from the excitation voltage flows through between the sensor and the bottom resistor in the bridge R2. If we assume the resistance of the two wires are the same, we could represent the resistance on the top and bottom of that measurement node as:

R1 = R0 + ΔR + Rw and R2 = R0 + Rw, where Rw is the resistance of the RTD wire

If we combine the terms R0 and Rw into R0w, we can see that the expression to calculate the left side voltage of the Wheatstone Bridge does not really change:

Vl = Vex ⋅ (R0w / (2R0w + ΔR)

Now the only difference in our bridge circuit is the left side has higher resistance values than the right side, but if you remember the expression for the voltage on the right simplified to an expression that did not even include resistance values because R3 = R4:

Vr = Vex / 2

Therefore we can simply ignore the wiring resistance if it is evenly split between the left measurement node. Because the value of Vl is dependent on the ratio of the resistance values, the only thing that affects Vl is a mismatch in resistance between the top and bottom resistance values which is only caused by ΔR or a temperature change away from 0 °C. Of course in practice chances are the two wire paths will have slightly different resistance values that will introduce some very small error to the measurement. 

Calibrating your Wheatstone Bridge setup for greater measurement accuracy

Throughout this tutorial we have been mainly looking at the resistance values in the Wheatstone Bridge as ideal. When we consider the resistors to be ideal and if we are using an RTD sensor that is at exactly 0 °C the output voltage of the bridge Vo would be 0V. But in practice we know that the resistor values are not going to be ideal. Also if we are measuring the output of the bridge with a amplifier and an ADC, they will also add error to the measurement in the form of offset voltages. For a lot of use cases these error factors maybe trivial and can be ignored, but if your use case requires high accuracy you could eliminate these error factors via calibration measurements. 

One approach to calibrating your Wheatstone Bridge setup is as follows:

  1. Put your Wheatstone Bridge and your measurement circuit in a state where the bridge's differential output voltage should be 0V.
  2. Measure the actual output voltage which comes from various error sources in your setup. Ideally measure it multiple times concurrently and average the readings together to cancel noise in the measurement.
  3. You would then take the measured voltage value and put it in your design's software as a correction factor. From there you would then subtract that correction factor from each measured value.

Setting your Wheatstone bridge circuit for a 0V differential output state can be easier said then done. One way to do it would be to replace the sensor with a precision resistor along with wiring that matches the wiring of the sensor. If you are working with an RTD sensor you could use an ice bath to get the sensor to its 0 °C value. 

Conclusion and other resources related to resistance measurements and Wheatstone Bridges

In this tutorial we went through in detail how Quarter Wheatstone Bridge circuit works and its advantages for measuring small changes in resistance. The ability to measure small changes in resistance makes it a common fixture for precision circuits measuring sensors such as RTD, Piezoresistive pressure, Magnetoresistive, and strain gauges. If you are looking for a precision ADC reference design to make measurements from a Wheatstone bridge, check out Anabit's Precision Logger. It features a 32 bit ADC with a precision programmable gain amplifier. It can support up to 5 differential measurement channels. Click to learn more about the Precision Logger.

If you have any questions or comments from this tutorial, please use the Anabit Forum

Video tutorial from Anabit on resistance measurement techniques

Tutorial from Transducer Techniques on Half and Full Wheatstone Bridge configurations

 

 

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