EE3901/EE5901 Sensor Technologies
Chapter 3 Tutorial

Written by A/Prof Bronson Philippa and Laurance Papale

College of Science and Engineering, James Cook University

Last updated 24 February 2025

Question 1

A 1 kΩ linear potentiometer is used to deliver a voltage to a load , as shown in Figure 1.

Figure 1:
A potentiometer is used to control a voltage . The potentiometer has the equivalent circuit shown in the circle, where is the potentiometer’s resistance and is the fractional position of the potentiometer.
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Suppose that the potentiometer is at its halfway point (). When there is no external load connected ( open circuit), the potentiometer forms a simple voltage divider, and the output voltage is obviously half of the supply voltage.

However, in practical situations, the potentiometer must be connected to some downstream circuit. If that circuit has an input resistance of , then the resistive loading of the potentiometer will affect the voltage .

Given , what is the minimum value of such that the voltage is maintained within 5% of its unloaded value?

Answer

Applying the voltage divider formula, the loaded voltage is

where means “in parallel with”.

After some algebraic simplification:

The unloaded voltage (when is disconnected) is half the supply, i.e. V. The required threshold is 5%, i.e.

Solving for we obtain

Question 2

The power rating of a potentiometer can affect the maximum allowed driving voltage. The worse case for power dissipation in the potentiometer will occur in the limit of , i.e. when the maximum current flows in the load. If the voltage source is modelled by a Thevenin equivalent then the following circuit is obtained:

Figure 2:
A potentiometer with its external load shorted.
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(a) Calculate the power dissipated in the potentiometer when the wiper is at a fractional position (measured from the top), i.e. means the wiper is at the top and means the wiper is at the bottom.

(b) Find the value of that maximises the dissipated power.

Answer

(a) The dissipated power is

(b) We have

Setting the derivative to zero

For the supplied values

Question 3

A Ω strain gauge with gauge factor is attached to a steel beam that is supporting a load of 3 kN. The steel beam has a cross-sectional area of 250 and a Young’s modulus of 190 GPa. What is the total strain gauge resistance when the beam is loaded?

Answer

The beam is supporting a strain of , which results in a stress of $μ ε$ Consequently there is a change in resistance

The total resistance of the gauge will be

Question 4

A 120 Ω strain gauge with is used to measure a strain of . What is the resistance change from the unloaded to the loaded state?

Answer

Question 5

The resistance of a strain gauge changes by 0.5% when a strain of 25 με is applied. Calculate the gauge factor.

Answer

From the equation

we have

Question 6

A nickel RTD has a transfer function

The device has a resistance of 500 Ω at 0 °C, and a temperature coefficient of .

(a) Calculate the sensitivity of the temperature sensor.

(b) Determine its resistance at 100 °C.

Answer

Substituting the known values into the transfer function:

Here it is OK to use temperature in °C because is the same regardless of whether °C or K are used.

(a) The sensitivity is Ω/K.

(b) The resistance at 100 °C is Ω.

Question 7

A coil of wire is used to activate a relay. However, the coil’s resistance changes with temperature, resulting in a different activation threshold at different temperatures. It is proposed to use an NTC thermistor to compensate for the temperature dependence in the coil, such that the total resistance of the circuit remains constant as the temperature varies. The circuit diagram is shown below:

Figure 3:
A series connection of a thermistor and a relay coil.
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The coil has a resistance of 5 kΩ at 25 °C, and a temperature coefficient of resistance (TCR) of 0.0069 . The thermistor also has a resistance of 5 kΩ at 25 °C. Its temperature dependence is given by

Find a value of such that has the opposite TCR to the coil at 25 °C. In other words, find such that the series combination of the two devices has no temperature dependence (within the range over which the linear TCR is a valid approximation).

Hint: TCR is defined as

Answer

The TCR for the thermistor is

Solving for , we obtain

Question 8

Building upon the previous question, suppose that a thermistor with the required value of is not available. The only available thermistor has too large a temperature dependence. Hence, to reduce the temperature sensitivity of the compensation circuit, you shunt the thermistor with a fixed resistance :

Figure 4:
Modified temperature compensation circuit.
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The goal in this question is to choose the value of such that the TCR of the compensation circuit is -0.0069 .

Define to be the parallel combination of the resistor and the thermistor:

Assume that resistor has no temperature dependence.

(a) Show that the partial derivative of with respect to temperature is

(b) Show that the TCR of is

find a value of such that the TCR of is -0.0069 at 25 °C.

Answer

(a) Use the quotient rule to evaluate this derivative, noting that is a constant because it is assumed to be temperature independent. Let and . Then and . This gives:

(b) The TCR of the parallel combination is

Further substituting K and and solving,

Question 9

A four-wire resistance measurement is performed using the circuit shown in Figure 5.

Figure 5:
A circuit intended for resistance measurement. The op-amp works as a constant current source, as you will prove in part (a). The component is a voltage meter, measuring the voltage drop across the sensor . The resistors represent the unwanted resistances in the wires that connect to the sensor.
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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 10

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 6:
The circuit for Question 10.
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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 11

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

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 12

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 8:
The circuit for Question 12.
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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.