6.7: Temperature, Pressure, Flow, Current, and Voltage Sensing

Learning Objectives

After completing this section, you will be able to:

1. Describe the working principle of temperature sensors such as RTD (resistance temperature device), thermal couple, and air flow sensors.
2. Grasp the working principle of pressure sensors such as piezo-pressure sensors.
3. Explain electric current and voltage sensing principles.

In earlier sections, we reviewed a few mechatronic sensors including rotational hall-effect speed sensors, accelerometers, and rate gyros. In this section we will briefly review sensing principles of temperature, pressure, air flow, current, and voltage sensors. Our intension is to provide an overview and how to use these sensors in general.

First let’s look at temperature sensing technology. Generally, there are four main kinds of electrical temperature sensors. The first kind is the thermistor [45], a resistor made of semiconductor materials; its resistance is strongly dependent on temperature. Note that the word thermistor combines thermal and resistor. Negative temperature coefficient (NTC) thermistors exhibit a decrease in electrical resistance as their body temperature increases. By contrast, positive temperature coefficient (PTC) thermistors have an increase in electrical resistance as body temperature goes up.

The second kind is a thermocouple [46], also known as a thermoelectrical thermometer. It is an electrical junction consisting of two dissimilar electrical conductors. A thermocouple produces a temperature-dependent voltage as a result of the Seebeck effect, that is, the electromotive force developed across two points of an electrically conducting material when there is a temperature difference between them. Note that the generated voltage can be interpreted to measure temperature.

The third kind is a resistance thermometer [9], also called a resistance temperature detector (RTD). It is used to measure temperature. Most RTDs consist of a length of fine wire wrapped around a heat-resistant ceramic or glass core. The RTD wire is a pure material, typically platinum, nickel, or copper. These materials have an accurate resistance and temperature relationship that is used to provide an indication of temperature. Because RTD elements are fragile, they are often housed in protective probes. Due to RTDs’ high accuracy and repeatability, they are slowly replacing thermocouples in industrial applications below \(600~^\circ \text{C}\).

The last kind is a silicon bandgap temperature sensor in the form of thermometer. It is widely used in electronic equipment because it can be part of a silicon integrated circuit at very low cost. The principle of the sensor is that the forward voltage of a silicon diode (see Chapter 2), such as the base-emitter junction of a bipolar junction transistor, is temperature-dependent. It can be used to detect junction temperature [47].

We will concentrate on resistance temperature detector (RTD) for the rest of the temperature sensing discussion. Note that the resistance temperature detector is described by its temperature and resistance relationship. For most RTDs, their resistance increases as the temperature goes up within the measurement range. This is due to the fact that the atoms in the metal wire of the sensor vibrate more at higher temperatures; this causes more collisions with flowing electrons, thus increasing the resistance to electrical current flow.

Figure 6.33 shows a typical RTD sensor characteristic chart representing the relationship between RTD temperature and associated resistance value. One can see that RTD resistance increases as temperature rises. This phenomenon is known as a positive temperature coefficient, where the resistance of the material directly increases with temperature. To use the RTD to measure temperature, the RTD resistance value needs to be converted into a voltage signal that can be sampled into the information domain. One approach is to connect the RTD (\(R_{RTD}\)) to a reference voltage through a reference resistor \(R_{ref}\), as shown in Figure 6.34, where an operational amplifier is used to isolate the RTD circuit from the connected circuit.

The simple derivation, utilizing the techniques discussed in Chapter 2, leads to the following relationship between the RTD and \(v_{RTD}\).

\[v_{RTD}=\dfrac{R_{RTD}}{R_{RTD}+R_{ref}}v_{ref} \label{6.70}\]

Another method is to use an H-bridge circuit, shown in Figure 6.35, with a differential operational amplifier circuit because the RTD voltage signal is referenced as being half of \(v_{ref}\). The advantage of this circuit will be addressed later in this section.

Utilizing the techniques discussed in Chapter 2, the following relationship between RTD resistance \(R_{RTD}\) and circuit output voltage \(v_{RTD}\) can be derived.

\[v_{RTD}=\dfrac{R_2}{2R_1}\dfrac{\left(R_{RTD}-R_{ref}\right)}{\left(R_{RTD}+R_{ref}\right)}v_{ref} \label{6.71}\]

Comparing equations \ref{6.70} and \ref{6.71}, we find that using the H-bridge circuit leads to more freedom for tuning the output voltage \(v_{ref}\) in terms of the output voltage range and output voltage for a given temperature by adjusting \(R_1\), \(R_2\), and \(R_{ref}\). We will use Example 6.11 and Problem 6.24 to illustrate.

Example 6.11

Let \(R_1=5\ kΩ\), \(R_2=10\ kΩ\), and \(R_{ref}=75\ Ω\). Find the relationship from \(R_{RTD}\) to \(v_{RTD}\) and using the RTD relationship in Figure 6.33 to plot the relationship from measured voltage to RTD temperature, given \(v_{ref}=5\ V.\ \)Substituting given parameters into equation \ref{6.71} yields the following.

\[v_{RTD}=\dfrac{R_2}{2R_1}\dfrac{\left(R_{RTD}-R_{ref}\right)}{\left(R_{RTD}+R_{ref}\right)}v_{ref}=5\dfrac{R_{RTD}-75}{R_{RTD}+75} \label{6.72}\]

Using the formulae above, we can calculate the RTD resistance \(R_{RTD}\) for a given output voltage \(v_{RTD}\). Then use the relationship shown in Figure 6.33 to find the associated RTD temperature. The relationship from the output voltage \(v_{RTD}\) to RTD temperature is shown in Figure 6.36. Note that the relationship is no longer linear.

Note that the map shown in Figure 6.36 can be used to find the RTD temperature based upon the H-bridge circuit output voltage \(v_{RTD}\) directly.

Now we will discuss the working principles and different types of pressure transducers and pressure sensors. Pressure is defined as the force applied by a liquid or gas on a surface. It is usually measured in units of force per unit of surface area, commonly in Pascal (Pa), Bar (bar), \(N/{mm}^2\), or PSI (pounds per square inch). A pressure sensor is a device that measures the physical quantity (pressure) and translates it to an electrical or an optical signal. A pressure sensor consists of a pressure sensitive element to determine the actual pressure applied to the sensor and some components to convert this information into an output signal. As a result, pressure sensors are also known as pressure transmitters.

A variety of properties can be used to classify pressure sensors, including absolute, gauge, and differential pressure sensors. Absolute pressure sensors measure the pressure relative to a reference chamber (nearly vacuum). Gauge pressure sensors (also called relative pressure sensors) determine the pressure relative to the present atmospheric pressure. Differential pressure sensors find the difference between two pressure chambers for measuring pressure drops, fluid levels, and flow rates.

Pressure sensors can also be categorized by their sensing element types mainly as strain gauge [48], piezoelectric [49], and capacitive pressure [50] sensors. A strain gauge pressure sensor measures change in the ohmic resistance of a strain gauge with its deflection magnitude caused by applied pressure proportional to the change in resistance. A piezoelectric pressure sensor determines the electric charge due to applied stress resulted from pressure, bending, twisting, or vibrations. The electric charge is used to determine the amount of pressure. Capacitive pressure sensors use a diaphragm and pressure cavity to create a variable capacitor as a function of pressure applied to the diaphragm.

We will focus on the working principle of the strain gauge, piezoelectric, and capacitive pressure sensors. First, for the strain gauge pressure sensor, the resistance of the strain resistance absorbed on the base material changes with the mechanical deformation. Figure 6.37 illustrates this effect, which is commonly known as the resistance strain effect.

The top plots in Figure 6.37 show an unstretched strain gauge strip with resistance of \(R_{su}\) (top left) and a stretched strain gauge strip with resistance of \(R_{ss}\), where \(R_{ss}>R_{su}\) (top right). The bottom plots in Figure 6.37 show unstretched and stretched strain gauge strips installed on a pressure sensor diaphragm. When the pressure is applied, the sensor diaphragm is deformed downward (bottom right). The strain gauge is stretched so that its resistance is increased. Note that the resistance of the strain gauge also changes under different environments (e.g., temperature and electromagnetic field). If the resistance is too high, resulting in high impedance, the ability to resist external electromagnetic interference is poor. For most of the strain gauges, the resistance is around tens of ohms to tens of thousands of ohms.

Figure 6.38 shows a strain gauge pressure sensor with four strain gauges evenly distributed on the diaphragm. The associated resistance values are denoted by \(R_{vh}\), \(R_{vl}\), \(R_{hl}\), and \(R_{hr}\), marked on the left plot of Figure 6.38. We can define the following resistance values marked on the circuit part of Figure 6.38.

\[R_v=R_{vh}+R_{vl},\ \ R_h=R_{hl}+R_{hr}\ \ \ \ \label{6.73}\]

Using the circuit on the right of Figure 6.38, we can find the change of resistance values due to applied pressure. Note that the formula for calculating the output voltage as a function of these resistance values is found in Problem P6.25 at the end of the section.

Next, we will discuss the working principle of a piezoelectric pressure sensor. The piezoelectric materials used in piezoelectric pressure sensors include quartz, potassium sodium tartrate, and dihydrogen phosphate. Among them, quartz is a natural crystal. The piezoelectric effect is found in this crystal. Within a certain temperature range, the piezoelectric property always exists. After the temperature exceeds this range, the piezoelectric property completely disappears. When certain dielectrics inside the piezomaterial are deformed by external forces in a specific direction, polarization will occur inside them. Positive and negative charges appear on their two opposing surfaces, generating a positive voltage \(v_{ps}\) (see the right plot of Figure 6.39 for the opposition surfaces). When the external force is removed, it will return to the uncharged state with \(v_{ps}=0\). This phenomenon is called a positive piezoelectric effect.

When the direction of the applied force changes, the polarity of the charge changes accordingly. Conversely, when an electric field is applied in the polarization direction of the dielectric, these dielectrics will also deform. After the electric field is removed, the deformation of the dielectric will disappear. This phenomenon is called the inverse piezoelectric effect. The type of sensor developed based on the dielectric piezoelectric effect is called a piezoelectric sensor. The right plot of Figure 6.39 illustrates the sensor structure. The piezoelectric material is sandwiched between the fixed insulation (below the piezoelectric material) and the diaphragm on top of the piezoelectric material. As a result, when the diaphragm is deformed downward due to pressure, voltage \(v_{ps}\) (sensor signal) is produced. This signal is very weak. A charge amplifier is often used to obtain a signal with certain magnitude by isolating the piezoelectric material output signal and sensor output.

The next group are capacitive pressure sensors. They use a diaphragm and pressure cavity to create a variable capacitor to detect strain due to applied pressure (see the left plot of Figure 6.39), where capacitance increases as pressure deforms the diaphragm. The capacitance of the sensor is typically around 50 to 100 \(pF\) with the change being a few picofarads (\(pFs\)). The stiffness and strength of the diaphragm material can be chosen to provide a range of sensitivities and operating pressures. To get a large signal, the capacitive pressure sensor may need to be fairly large. This increases the capacitance, which can limit its operational frequency range. However, smaller diaphragms are more sensitive and have a faster response time. Common technologies use metal, ceramic, and silicon diaphragms. Capacitive pressure sensors can also be integrated into CMOS (complementary metal-oxide-semiconductor) technology [51].

Let’s look now at the fluid and gas flow sensors. A flow sensor is a component that measures the flow of a fluid such as gas or liquid. Flow sensors utilize both mechanical and electrical subsystems to measure changes in the fluid’s physical characteristics and calculate its flow. Gaseous and liquid fluids behave vastly different from one another. Thus, the methods we use to measure their flow must be different. We will mainly address the gear flow meter for measuring liquid flow and thermal mass flow meter for determining gases mass flow.

Figure 6.40 presents the principle of a fluid gear flow sensor. A pair of mechanical gears are used in the center of the plot. As the fluid mass flow into the gear flow meter, both gears rotate in an opposite direction with their speed as a function of the mass flow rate \(\dot{m}\). To convert the rotational speed to the mass flow rate, a rotational speed measurement device, such as the one introduced in Section 6.2, can be used to obtain the gear rotational speed \(ω\). A calibration lookup table can then be made to convert the measured speed \(ω\) to mass flow rate \(\dot{m}\). There is another type of gear flow meter utilizing the oval-shape gears with improved performance by replacing circular gears with oval-shape gears.

For the mass air flow measurement, the hot-wire mass air flow (MAF) sensor converts the amount of mass air flow into a voltage signal \(v_{maf}\) (see Figure 6.41). The primary components of the hot-wire MAF sensor are a thermistor (represented by \(R_{th}\)), a platinum hot wire (represented by \(R_{hot}\)), and an electronic control circuit consisting of the transistor \(T\) and temperature regulator. The thermistor measures the temperature of the incoming air, and the hot wire is maintained at a constant temperature in relation to the thermistor (normally higher than the thermistor temperature) by an electronic control circuit. An increment in air flow will increase the heat loss of the hot wire. The electronic control circuitry will compensate by increasing the current flown through the wire.

Note from Figure 6.41 that the electronic control circuit measures voltages \(v_{th}\) and \(v_{hot}\) while the temperature regulator controls the voltage \(v_s\) to make the voltage difference \(v_{hot}-v_{th}\) equal to zero (or a given difference). The goal is to maintain the temperature of the hot-wire resistor \(R_{hot}\) greater than the thermistor one by a designed value. Note that setting \(v_{hot}-v_{th}=0\) leads to the following.

\[v_{th}=\dfrac{R_1}{R_{th}+R_1}=\dfrac{R_2}{R_{hot}+R_2}{=v}_{hot} \label{6.74}\]

The above equation is equivalent to the following relationship.

\[\dfrac{R_{th}}{R_1}=\dfrac{R_{hot}}{R_2}\ \ \to \ \ {R_{hot}=\dfrac{R_2}{R_1}R}_{th} \label{6.75}\]

Therefore, by selecting the proper properties of \(R_{th}\) and \(R_{hot}\) and maintaining \(v_{hot}-v_{th}=0\), the hot-wire resistor temperature can be maintained at a level above air temperature. Also, mass air flow rate can be calculated based up output \(v_{maf}=v_{hot}\). Normally a calibration table is needed to convert the measured \(v_{maf}\) to the mass air flow rate. We will use Example 6.12 to show how to achieve the target hot-wire resistor temperature by maintaining \(v_{hot}-v_{th}=0\).

Example 6.12

Let the thermistor resistance and temperature have a relation of \(R_{th}=2\left(T_{th}+50\right)+1200\ Ω\) for temperature \(T\in \left[-50,\ 200\right]^\circ \complement\), the hotwire resistor \(R_{hot}=0.2\left(T_{hot}+40\right)+120\ Ω\) for temperature \(T∈\left[-50,\ 200\right]\ ^\circ \complement\), \(R_1=100\ Ω\), and \(R_2=10\ Ω\). Note that \(10\) times the difference between \(R_1\) and \(R_2\) is mainly due to the difference in \(R_{th}\) and \(R_{hot}\). Assume that for the hot-wire MAP shown on Figure 6.41, \(v_{hot}-v_{th}\) is maintained at zero and \(T_{th}=20\mathtt{^\circ\!{C}}\ \). Find the hot-wire resistor temperature \(T_{hot}\).

Figure 6.42 plots the temperature and resistance relations of both the thermistor and hot-wire resistors. First, we draw a vertical line at 20\(\mathtt{^\circ\!{C}}\) on the thermistor plot, map the intersection horizontally to the equivalent resistance hotwire plot, and draw the vertical line through the intersection on the right plot. We can then see that if \(v_{hot}-v_{th}=0\), the temperature of hot-wire resistor is 30\(\mathtt{^\circ\!{C}}\), which is 10\(\mathtt{^\circ\!{C}}\) higher than the thermistor’s temperature.

As a result, to make a hot-wire mass air flow sensor work, it is very important to properly match thermistor property with hot-wire resistor characteristics. Otherwise, the hot wire may not maintain at its temperature above the thermistor one. The hot-wire resistor temperature can also be calculated analytically. If the thermistor is held at 20\(\mathtt{^\circ\!{C}}\), the corresponding resistance is \(R_{th}=2\left(25+50\right)+1200=1340\ Ω\). We can calculate the hot-wire resistance as follows.

\[{R_{hot}=\dfrac{R_2}{R_1}R}_{th}=\dfrac{10}{100}1340=134\ Ω \label{6.76}\]

Solving the given equation \(R_{hot}=0.2\left(T_{hot}+40\right)+120\ Ω\) leads to a hot-wire resistor temperature of 30\(\mathtt{^\circ\!{C}}\).

Next up are techniques for current and voltage sensing. For a mechatronic system, the current sensor signal will eventually be converted into voltage signals. A/D converters can then sample them into the information domain. Let’s first address how to detect a voltage signal that is not referenced to the ground (also called floating voltage signal). One example is the H-bridge RTD circuit shown in Figure 6.35. The voltage difference over two sample points is the measuring voltage. An operational amplifier circuit is used to convert the voltage signal referenced to the ground. The A/D converter in the information domain is usually referenced to ground. As a result, the operational amplifier circuit in Figure 6.35 is widely used.

In certain applications, the distance from the sensor (or converted voltage) to the A/D converter in the information domain is fairly long. In these cases, the sensor ground and A/D converter ground may be slightly different due to ground noise or bias, especially for these high-power (current) applications. We recommended converting the voltage signal into a current source signal. Then use a burn resistor to convert the current signal back to voltage for the A/D converter in the information domain; see Figure 6.43.

For Figure 6.43, \(v_s\) is the sensor output voltage. An operational amplifier is used to isolate the input from the rest of the circuit and convert the voltage to current with \(i_s=v_s/R_2\) with the help of transistor \(T_3\). This is because the voltage drop between positive and negative inputs of the operational amplifier is zero. Inside the red-dashed box is a current mirror circuit, making the current flow through collectors of transistors \(T_1\) and \(T_2\) the same. In order to have an accurate current mirror, transistors \(T_1\) and \(T_2\) needs to be manufactured on the same silicon so that their characteristics are the same.

Finally, the sensor output voltage \(v_s\) is recovered at the microcontroller side with \(v_{s-A/D}\)\(=R_2i_s\). As indicated in Figure 6.43, the sensor and microcontroller ground are different. If \(v_s\) is directly connected to the A/D converter, there is significant ground noise, leading to noisy sensor signal sampling by the A/D converter. This is the main reason to use the current source signal to connect the sensor signal to the A/D converter of the microcontroller in the information domain.

Now we’ll address current measurement. Information regarding current flow is required in a wide range of mechatronic systems. Often, current information needs to be available in the digital form in the information domain for feedback control. This requires the current signal to be converted into a digital signal by an analog-to-digital (A/D) converter. For example, to measure current, we can use a shunt resistor (e.g., resistor \(R_1\) of the circuit in P6.27) connected to the load source. Shunt resistors are widely used in power electronics due to their low cost, small size, and simplicity with reasonable accuracy. This method converts the current measurement problem into a voltage measurement problem. A disadvantage of using the shunt resistor for current sensing is the high energy consumption for applications with a very large current. For example, if the current flow through the shunt resistor is \(i_{sh}=10\ A\) and the resistor is \(R_{sh}=0.05\) \(\Omega\), the power consumed by the shunt resistor is \(P=i^2_{sh}R_{sh}=5\ W.\)

Let’s discuss a current measurement approach based on the Rogowski coil principle [52], [53]. A Rogowski coil is a toroid of wire used to measure an alternating current \(i(t)\) through a cable encircled by the toroid. Figure 6.44 shows a Rogowski coil encircling a current-carrying cable. The output of the coil, \(v(t)\), is connected to a lossy integrator circuit to obtain a voltage \(v_{out}(t)\ \)that is proportional to \(i(t)\). The Rogowski coil displayed in Figure 6.44 is a typical application example using Faraday’s induction law.

By applying Faraday’s law, we can have the relationship shown in the following equation (see [53] for detailed derivation).

\[v\left(t\right)=-N\dfrac{Aμ_0}{2πr}\dfrac{di(t)}{dt} \label{6.77}\]

where \(N\) is the number of turns; \(A\) is the cross-sectional area of the coil body formed by the windings; \(r\) is the radius of the toroid, and \(i(t)\) is the current to be measured. Parameter \(μ_0\) [54] is the coefficient of permeability of free space, which is a measure of how much magnetic field is produced per unit current in vacuum with its value shown below.

\[μ_0=1.256\times {10}^{-6}\ \ (N/A^2) \label{6.78}\]

To recover the current flowing through the toroid \(i(t)\), an integration with integral constant (gain) \(k\) and with infinitely high input impedance can yield the exact result as follows.

\[v_{out}\left(t\right)=-kN\dfrac{Aμ_0}{2πr}\int^t_0{\left(\dfrac{di\left(t\right)}{dt}\right)dt}+v_{out}\left(0\right)=-kN\dfrac{Aμ_0}{2πr}i\left(t\right)+v_{out}\left(0\right) \label{6.79}\]

so that the current can be calculated by the following equation.

\[i\left(t\right)=\dfrac{2πr}{kNAμ_0}\left(v_{out}\left(0\right)-v_{out}\left(t\right)\right) \label{6.80}\]

Although equation \ref{6.80} indicated that the Rogowski coil can also be used to measure direct current (DC), its basic principle is based on detecting a flux change proportional to a current change. Without knowing current \(i(0)\) represented by \(v_{out}\left(0\right)\) in equation \ref{6.80}, it is impossible to reconstruct the DC current component. In practice, integration operation is always limited by the signal bias. A low-pass filter is always used to approximate the integral to get rid of the effect of bias (see Section 6.5 for the case of integrating a rate gyro signal to get angular position). The signal conditioning circuit presented in Figure 6.44 uses a low-pass filter with the following transfer function from \(v(t)\) to \(v_{out}(t)\) as shown in equation \ref{6.81}.

\[\dfrac{V_{out}(s)}{V(s)}=-\dfrac{1}{R_1}\dfrac{R_2}{R_2Cs+1} \label{6.81}\]

As resistance \(R_2\) goes to infinity, the transfer function reduces down to an integrator (\(1/s\)) with gain (\(1/R_1C\)). Normally, a fairly large \(R_2\) is used. To measure the DC current, it has been recently proposed to combine the Rogowski coil with an open-loop magnetic field sensor that provides the DC information [55].

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