# EE3901/EE5901 Sensor TechnologiesWeek 6 Tutorial

**Dr. Bronson Philippa**

## 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 $R_{w_{1}}=R_{w_{2}}=R_{w_{3}}=R_{w_{4}}=1$ Ω. The device $V$ is a voltmeter, which has an input impedance of $Z_{meter}.$ The circuit is designed to measure resistances in the range $10\ \text{Ω}\leq R\leq500\ \text{kΩ}$.

(a) Find the excitation current $i_{excite}$. Assume that the op-amp is ideal.

(b) Notice that some current will be diverted through the voltmeter, so the measured voltage $V$ will not be the expected value of $V=Ri_{excite}$. If the current flowing into the meter is $i_{sense}$ then the measured voltage is actually $V=i_{sense}Z_{meter}$. Find the minimum value of $Z_{meter}$ such that the relative measurement error is no worse than $\pm$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?

## 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 $R_{0}$ at a temperature $T_{0}$. Also, they have a temperature coefficient of resistance (TCR) of $\alpha$ and a gauge factor of $G$.

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

Hint: a TCR represents a **relative** change per Kelvin. In
other words, to account for temperature, multiply the entire resistance
by a correction factor $\left(1+\alpha(T-T_{0})\right)$.

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

Sensor $R_{1}$ is the dummy gauge (with zero strain), and sensor $R_{2}$ is the active gauge (with strain of $\epsilon$).

Analyse this circuit and show that $V_{0}$ is independent of temperature.

(c) Find the sensitivity of this circuit for small values of strain.

## Question 3

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

The sensor is placed at position $R_{3}$ and is described by a transfer function of the form

where $R_{0}$ is a large fixed constant, and $x$ 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. $V_{0}=0$) when

(b) Suppose that the resistances satisfy a relationship

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

(c) Find the sensitivity of $V_{0}$ to changes in $x$.

(d) Find the value of $k$ that maximises the sensitivity at $x=0$.

## Question 4

The circuit below is used to perform a resistance measurement. Balanced current sources $i_{1}=i_{2}$ inject power into the circuit. There is a sensor $R$ with transfer function

where $x$ is a small relative change in resistance that we seek to measure. The circuit also has a fixed resistance $R_{z}$.

(a) Assuming perfectly balanced current sources ($i_{1}=i_{2}$), find the value of $R_{z}$ such that $V_{0}=0$ when $x=0$.

(b) Write the transfer function for this circuit (i.e. the relationship between $V_{0}$ and $x$).

(c) Discuss the practical issues that would be associated with this design.