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Lock-in Amplifier - Wheatstone Bridge Application Note

Overview

 Wheatstone bridges are commonly used in strain gauges to allow the small changes in resistance to be more easily measured.

 This application note shows how a Wheatstone bridge can be driven by AC rather than DC to enable changes of 2 parts per million to be seen and measured.

Standard DC Wheatstone Bridges

DC Wheatstone Bridge Wheatstone Bridges are made from four similar value resistors connected together as shown on the right. Typically the wheatstone bridge is powered by 10V DC at Vin. As the resistor values are nominally the same, the voltage across the output is normally small. A change in the value of any one of the resistors will cause a change in the output voltage (Vout).

 This circuit is commonly seen in strain gauges where all four resistors are strain gauges. The circuit is so designed that Ra and Rd will increase in value with applied strain and Rb and Rc will decrease in value with applied strain. The main advantage of this circuit over using a single strain gauge is that some of the error sources (such as temperature dependence) cancel each other out.

 The accuracy of measurements taken with this arrangement of resistors depends on the accuracy with which Vout can be measured. Unfortunately, if Vout is in the µV region there are a number of difficulties in making this measurement accurately. The input impedance of a measuring op-amp needs to be very high to avoid loading the circuit and the input offset voltage needs to be very small. Also the Common Mode Rejection Ratio (CMRR) needs to be very high as both inputs will be at approximately 5V (for 10V drive). Noise will also be a problem and care will have to be taken that all junctions between dissimilar metals cancel each other out to avoid voltages being introduced due to the thermocouple effect.

Detailed Example of Taking Measurements Using a Wheatstone Bridge and a Scitec 420 Lock-in Amplifier

Ra = Rb = Rc = Rd = 4k99 ohms 0.1% tolerance.

Re = Rf = 100 ohms.

Rg = 20k ohms 10 turn.

Ca = Cc = 10pF.

Cb = Cd = 5 to 75pF.

 The idealised Wheatstone bridge shown above needs a few additions to be able to adjust the errors within it. A modified circuit is shown to the right. The first thing to notice is the symmetry. The symmetry of the circuit helps cancel out error sources down each side and the symmetry should be followed when laying out and building the circuit.

 The variable resistor, Rg, together with Re and Rf, is used to zero the output voltage by modifying the resistance down either branch by a small amount. (Please note that the relationship between the wiper position and the output voltage is non-linear.) The variable capacitors, Cb and Cd, are used to compensate for any mismatches in the capacitance or inductance down either branch. (The capacitors Ca and Cc are used to reduce the effective value of Cb and Cd. You don't need much adjustment.)

 This modified Wheatstone bridge is connected to a dual phase lock-in amplifier as shown above. If a Scitec Instruments 420 lock-in amplifier is used then the input needs to be set to differential mode as explained in the instruction manual. Care should be taken in keeping the output signal wires separate from the drive wires so that the high amplitude signals on the drive are not picked up on the output wires. Screened or twisted pair wires are recommended.

420 Maximum Input Voltage
(either input when in differential mode)

Gain Setting

Maximum Voltage

1V

10V

300mV

10V

100mV

10V

30mV

10V

10mV

10V

3mV

10V

1mV

10V

300µV

10V

100µV

1V

30µV

1V

10µV

1V

3µV

300mV

 The maximum drive voltage that can be used to drive the Wheatstone bridge is dictated by the maximum input voltage levels that are allowed by the 420 instrument. This value is dependant on gain setting as shown on the left.

 As the amplitude of the signals seen at the input of the lock-in will be half of the drive voltage it is possible drive the Wheatstone bridge with a maximum of 20V peak signal (14.1V RMS) for gain settings down to 300µV. For this example a 10V RMS 1kHz stimulating signal will be used with the lock-in amplifier gain setting set to 300µV. With these values a 0.01% mismatch in any of the resistors Ra, Rb, Rc or Rd, relative to the others will cause a 250µV signal to be generated across the input to the lock-in. This will be amplified by the lock-in to produce a 0.83V DC output signal. The potentiometer Rg can be used to correct for this mismatch and could, in an ideal world, bring the output from the lock-in to zero. Unfortunately, it will be found that although either the X or the Y outputs of the lock-in can be made to reach zero, it is not possible to bring both of them to zero at the same time. This can most easily be seen by looking at the R output which calculates the amplitude (or modulus) of the input signal. Adjusting Rg will decrease the output R to a minimum but will not bring it to zero. Due to the very small parasitic capacitance associated with the circuit a small phase shift is seen down either branch of the Wheatstone bridge. This small phase shift stops the two signals generated down either branch from cancelling out each other completely.

 By adjusting Cb and Cd (and Rg) it is possible to compensate for the minor phase shift and reduce the output of the lock-in close to zero. In practise the output R value has been reduced to below 50mV. This value corresponds to an input signal level of 15µV. To test the performance of the system a 1M ohm resistor was connected in parallel with Re, reducing its value by 0.01 ohm. This corresponds to a 2 parts per million change in Rc. Theory states that this will produce a 4.8µV change in the input voltage to the lock-in which corresponds to a 24.6mV change in the output. In practise a 20mV change was seen although there were problems with taking a reading due to output voltage drift.

Drift

 A major problem with the circuit, as outlined above, is the amount of drift seen on the output. There are two major causes of voltage drift, resistor value change and capacitance change.

 As a 2 parts per million (ppm) change in one of the resistors Ra, Rb, Rc or Rd causes a 25mV change in the output it is easy to see how temperature mismatches will cause drift. The 0.1% resistors used in the above circuit have a temperature coefficient of 15 ppm/°C so it only takes a 0.13°C temperature difference to produce a 25mV change in output. Great care must be taken to keep the temperature of the resistors the same.

 Very small phase shifts in the network can produce major changes in output.. These phase shifts are produced by very small capacitors and these are unfortunately found everywhere. Problems have been seen with a Wheatstone bridge circuit placed too close to a grounded static work bench. Lifting the circuit by about 10cm above the bench produced a 50mV change in output due to the capacitance between the bench and the circuit.

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