EE3901/EE5901 Sensor Technologies
Week 6 Tutorial

College of Science and Engineering, James Cook University

Last updated 4 February 2022

Question 1

A four-wire resistance measurement is performed using the circuit shown in Figure Q1.

The wires connecting the interface circuit to the sensor have resistance Ω. The device is a voltmeter, which has an input impedance of The circuit is designed to measure resistances in the range .

(a) Find the excitation current . Assume that the op-amp is ideal.

(b) Notice that some current will be diverted through the voltmeter, so the measured voltage will not be the expected value of . If the current flowing into the meter is then the measured voltage is actually . Find the minimum value of such that the relative measurement error is no worse than 0.01%.

(c) In next week’s practical, you will use a Texas Instruments INA826 instrumentation amplifier. This device has an input impedance of approximately 20 GΩ. Would this device be a suitable buffer for the analog front end of the voltage meter?

Answer

(a) Recognise that the op-amp has negative feedback, and so it will drive its output in such a way that a voltage of appears across . Consequently, the excitation current is simply

(b) We need to analyse the circuit to find the voltage . Perhaps the easiest approach is to use a current divider formula to find how much of the current flows into the meter. Let the current entering the meter be denoted . Since the wire resistance is a total of 2 Ω, we have

Hence the measured voltage is

The ideal value (with no loading due to the meter) is , so the relative error is

We require to meet the specification given in the question. Hence, the question arises: will the worst case error occur for small or large ? Which end of the measurement range should we use when calculating the required ?

In the case of small :

Hence we conclude that small is the “easy case” to measure. Intuitively, we want current to flow through the sensor instead of the meter, and this is achieved when the sensor has lower resistance.

We can also consider the case of large :

This means there is a relative error of -100%. Hence we conclude that the most difficult case to measure accurately will be when the sensor resistance is at the top end of the measurement span ( Ω).

Substituting this limiting case:

Given that the error is negative, substitute

Question 2

The Wheatstone Bridge can be used to compensate for changes in temperature. Suppose that you have two identical strain gauges. The “active gauge” is glued to the stressed material, and the “dummy gauge” is kept away from any mechanical stress. If the gauges are at equal temperature, then the effects of temperature will be the same on both.

Suppose that the strain gauges have a nominal resistance of at a temperature Also, they have a temperature coefficient of resistance (TCR) of and a gauge factor of .

(a) Write down the transfer function for the gauges as a function of strain () and temperature ().

Hint: a TCR represents a relative change per Kelvin. In other words, to account for temperature, multiply the entire resistance by a correction factor .

(b) Suppose that these gauges are placed in a half-bridge configuration as shown below:

Figure Q2:
The circuit for Question 2.
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Sensor is the dummy gauge (with zero strain), and sensor is the active gauge (with strain of ).

Analyse this circuit and show that is independent of temperature.

Answer

(a) Assuming that the gauge factor is temperature independent, we have

Using voltage divider formulas, the output voltage is given by

Since we are asked for the sensitivity for “small strain” we can consider the sensitivity in the limit as . Hence our result simplifies to

Question 3

A quarter-bridge circuit is used to interface with a resistive sensor as shown below:

Figure Q3:
The circuit for Question 3.
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The sensor is placed at position and is described by a transfer function of the form

where is a large fixed constant, and is the sensor response that we seek to measure.

(a) Analyse each side of the voltage divider and prove that the bridge is balanced (i.e. ) when

(b) Suppose that the resistances satisfy a relationship

Find an expression for the output voltage in terms of and .

(d) Find the value of that maximises the sensitivity at .

Answer

Using voltage divider formulas,

(a) Let and solve to obtain the result.

(b) The definition of suggests that we should rearrange to this form:

Hence, we can substitute and

(d) At the maximum sensitivity is obtained by solving

We can exclude the negative square root because resistances must be positive.

Question 4

The circuit below is used to perform a resistance measurement. Balanced current sources inject power into the circuit. There is a sensor with transfer function

where is a small relative change in resistance that we seek to measure. The circuit also has a fixed resistance .

Figure Q4:
The circuit for Question 4.
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(a) Assuming perfectly balanced current sources (), find the value of such that when .

(b) Write the transfer function for this circuit (i.e. the relationship between and ).

Answer

(a) Choose to cancel out the voltage caused by the fixed resistance in the sensor.

(b) The transfer function is where .

The requirement for matching current sources . If this condition is not met then the effect of will not perfectly cancel out.
The requirement that be matched to the sensor . Manufacturing tolerances will limit how well these resistors can be matched.
The TCR of and may be different, in which case the condition will not be maintained at different temperatures.
The circuit that measures must have a very large input impedance compared with so that the current predominantly flows through the sensor.