EE3901/EE5901 Sensor Technologies
Week 5 Tutorial

Written by A/Prof Bronson Philippa
College of Science and Engineering, James Cook University
Last updated 3 February 2022

Question 1

A 1 kΩ linear potentiometer is used to deliver a voltage to a load , as shown in Figure Q1.

Figure Q1
Figure Q1:

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 Q2
Figure Q2:

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.

(c) Using your result from part (b), find the worst case dissipated power for Ω, Ω, and V.

Answer

(a) The dissipated power is

(b) We have

Setting the derivative to zero

(c) The worse case dissipated power is

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 με 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 Q7
Figure Q7:

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 Q8
Figure Q8:

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

(c) Assume that the available thermistor has resistance

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

(c) Substituting into the TCR and simplifying

Further substituting K and and solving,