2.1.1 Classification of quantities

Various attempts have been made to set up a consistent framework of quantities. Physical quantities can be divided into subgroups according to various criteria. This leads to subgroups with different characteristics.

Within one domain state and rate variables are related as follows:

$X_{\text{rate}}=\frac{\text{d}}{\text{d}t}{X}_{\text{state}}={\dot{X}}_{\text{state}}\text{or}{X}_{\text{state}}=\int X_{\text{rate}}\text{d}t$ (2.1)

(2.1)

Examples

The electrical domain

$I=\frac{\mathrm{d}}{\mathrm{d}t}Q\mathrm{or}Q=\int I\mathrm{d}t$ (2.2)

(2.2)

The mechanical (translation) domain

$v=\frac{\mathrm{d}}{\mathrm{d}t}x\mathrm{or}x=\int v\mathrm{d}t$ (2.3)

(2.3)

Table 2.2

Summary of relations between types of variables

Domain	State/Extensive	Rate/Flow	Effort/Intensive	Energy (J)	Power (W)
Mechanical (translation)	Position x (m)	$v=\dot{x}$	Force F (N)	F·dx	F·v
Mechanical (rotation)	Angle φ (rad)	$\omega =\dot{\phi }$	Torque T (Nm)	T·dφ	T·ω
Electrical	Charge Q (C)	$I=\dot{\mathit{Q}}$	Voltage V (V)	V·dQ	V·I
Magnetic	Flux Φ (Wb)	$V=\dot{\text{Φ}}$	Current I (A)	I·dΦ	I·V
Thermal	Entropy σ (J/K/m3)	$\dot{\sigma }$	Temperature Θ (K)	Θ·dσ	$\text{Θ}\text{·}\dot{\sigma }$

We repeat the list of pairs of the conjugate variables in Table 2.3, together with their symbols:

Comparing these pairs with the groups from other categories given previously, we can make the following observations. The quantities E, D, B, H, T, and S are vector variables, whereas σ and Θ are scalars (therefore often denoted as Δσ and ΔΘ indicating the difference between two values). Further, in the above groups of quantities, T, E, and Θ are across-variables. On the other hand, S, D, and σ are through-variables. Finally, note that the dimension of the product of each domain pair is always J/m³ (energy per unit volume), whereas the product of the pairs effort and flow variables in Table 2.1 has the dimension power (W).

2.1.2 Relations between quantities

The energy content of an infinitely small volume of an elastic dielectric material changes by adding or extracting thermal energy and the work exerted upon it by electrical and mechanical forces. If only through-variables affect the energy content, the change can be written as follows:

$\text{d}U=\mathit{T}\text{d}S+E\text{d}D+\text{Θ}\text{d}\sigma$ (2.4)

(2.4)

where we disregard the magnetic domain (in Appendix B this domain is included).

Obviously, the across-variables in this equation can be expressed as partial derivatives of the energy:

$\begin{array}{c}T(S,D,\sigma )={\left({\displaystyle \frac{\text{∂}U}{\text{∂}S}}\right)}_{D\text{,}\sigma }\\ E\left(S,D,\sigma \right)={\left({\displaystyle \frac{\text{∂}U}{\text{∂}D}}\right)}_{S,\sigma }\\ \text{Θ}\left(S,D,\sigma \right)={\left({\displaystyle \frac{\text{∂}U}{\text{∂}\text{σ}}}\right)}_{S\text{,}D}\end{array}$ (2.5)

(2.5)

Likewise, if only across-variables affect the energy content, the energy change is written as follows:

$\text{d}G=-S\text{d}T-D\text{d}E-\sigma \text{d}\text{Θ}$ (2.6)

(2.6)

G is called the Gibbs potential (see Appendix B). The through-variables can be written as follows:

$\begin{array}{c}S\left(T,E,\text{Θ}\right)=-{\left({\displaystyle \frac{\text{∂}G}{\text{∂}T}}\right)}_{\text{Θ},E}\\ D\left(T,E,\text{Θ}\right)=-{\left({\displaystyle \frac{\text{∂}G}{\text{∂}E}}\right)}_{T\text{,}\text{Θ}}\\ \sigma \left(T,E,\text{Θ}\right)=-{\left({\displaystyle \frac{\text{∂}G}{\text{∂}\text{Θ}}}\right)}_{T,E}\end{array}$ (2.7)

(2.7)

From these equations we can derive the various material properties. We will extend these equations only for Eq. (2.7) because the resulting parameters are more in agreement with experimental conditions (constant temperature, electrical field strength, and stress), as denoted by the subscripts in Eqs. (2.5) and (2.7).

The variables S, D, and Δσ are approximated by linear functions, so:

$\begin{array}{c}\text{d}S\left(T,E,\text{Θ}\right)={\left({\displaystyle \frac{\text{∂}S}{\text{∂}T}}\right)}_{E\text{,}\text{Θ}}\text{d}T\text{+}{\left({\displaystyle \frac{\text{∂}S}{\text{∂}E}}\right)}_{T\text{,}\text{Θ}}\text{d}E\text{+}{\left({\displaystyle \frac{\text{∂}S}{\text{∂}\text{Θ}}}\right)}_{T\text{,}E}\text{d}\text{Θ}\\ \text{d}D\left(T,E,\text{Θ}\right)={\left({\displaystyle \frac{\text{∂}D}{\text{∂}T}}\right)}_{E,\text{Θ}}\text{d}T+{\left({\displaystyle \frac{\text{∂}D}{\text{∂}E}}\right)}_{T,\text{Θ}}\text{d}E+{\left({\displaystyle \frac{\text{∂}D}{\text{∂}\text{Θ}}}\right)}_{T,E}\text{d}\text{Θ}\\ \text{d}\sigma \left(T,E,\text{Θ}\right)={\left({\displaystyle \frac{\text{∂}\sigma }{\text{∂}T}}\right)}_{E,\text{Θ}}\text{d}T+{\left({\displaystyle \frac{\text{∂}\sigma }{\text{∂}E}}\right)}_{T,\text{Θ}}\text{d}E+{\left({\displaystyle \frac{\text{∂}\sigma }{\text{∂}\text{Θ}}}\right)}_{T,E}\text{d}\text{Θ}\end{array}$ (2.8)

(2.8)

Combining Eqs. (2.6) and (2.7) results in:

$\begin{array}{c}\text{d}S=-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{{\text{∂}T}^2}}\right)}_{\text{Θ}\text{,}E}\text{d}T-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}T\text{∂}E}}\right)}_{\text{Θ}}\text{d}E-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}T\text{∂}\text{Θ}}}\right)}_E\text{d}\text{Θ}\\ \text{d}D=-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}E\text{∂}T}}\right)}_{\text{Θ}}\text{d}T-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}{E}^2}}\right)}_{\text{Θ}\text{,}T}\text{d}E-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}E\text{∂}\text{Θ}}}\right)}_T\text{d}\text{Θ}\\ \text{d}\sigma =-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}\text{Θ}\text{∂}T}}\right)}_E\text{d}T-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}\text{Θ}\text{∂}E}}\right)}_T\text{d}E-{\left({\displaystyle \frac{{\text{∂}}^{2}G}{\text{∂}{\text{Θ}}^2}}\right)}_{E,T}\text{d}\text{Θ}\end{array}$ (2.9)

(2.9)

Now we have a set of equations connecting the (dependent) through-variables S, D, and σ with the (independent) across-variables T, E, and Θ. The system configuration (or the material) couples the conjugate variables of each pair. The second-order derivatives in the diagonal represent properties in the respective domains: mechanical, electrical, and thermal. For example, the top left second derivative in Eq. (2.9) represents the elasticity (or compliance) of the material (actually Hooke’s law). All other derivatives represent cross effects. Note that these derivatives are pair-wise equal since (assuming linear equations) the order of differentiation is not relevant:

$\frac{\text{∂}}{\text{∂}x}\left(\frac{\text{∂}G}{\text{∂}y}\right)=\frac{\text{∂}}{\text{∂}y}\left(\frac{\text{∂}G}{\text{∂}x}\right)$

So the derivatives in Eq. (2.9) represent material properties; they have been given special symbols. The variables denoting constancy are put as superscripts, to make place for the subscripts denoting orientation.

$\begin{array}{c}S=s^{E,\text{Θ}}T+d^{\text{Θ}}E+{\text{α}}^E\text{Δ}\text{Θ}\\ D=d^{\text{Θ}}T+{\epsilon }^{\text{Θ}\text{,T}}E+p^T\text{Δ}\text{Θ}\\ \Delta \sigma ={\alpha }^{E}T+p^{\text{T}}E+{\frac{\rho }{T}c}^{E,T}\text{Δ}\text{Θ}\end{array}$ (2.10)

(2.10)

For instance s^E,Θ is the compliance at constant electric field E and constant temperature Θ. The nine associated effects are displayed in Table 2.4.

Table 2.5 shows the associated material properties. The parameters for just a single domain (ε, c_p, and s) correspond to those in Tables A.2, A.5, and A.8 of Appendix A. The other parameters denote “cross effects” and describe the conversion from one domain to another. The piezoelectric parameters p and d will be discussed in detail in the chapter on piezoelectric sensors.

Note that direct piezoelectricity and converse piezoelectricity have the same symbol (d) because the dimensions are equal (m/V and C/N). The same holds for the pair pyroelectricity and converse pyroelectricity as well as for thermal expansion and piezocaloric effect.

Eqs. (2.7) and (2.10) can be extended just by adding other couples of conjugate quantities, for instance from the chemical or the magnetic domain. Obviously, this introduces many other material parameters. With three couples we have nine parameters, as listed in Table 2.3. With four couples of intensive and extensive quantities we have 16 parameters, so seven more (for instance, the magnetocaloric effect, expressed as the partial derivative of entropy to magnetic field strength, see Appendix B). Further, Appendix B gives a visualization of these relations using Heckman diagrams.

2.2.1 Classification based on measurand and application field

Many books on sensors follow a classification according to the measurand because the designer who is interested in a particular quantity to be measured can quickly find an overview of methods for that quantity. The more experienced designer may also consult books that deal with just one quantity (for instance temperature or liquid flow). Much information on sensors can also be found in books focusing on a specific application area, for instance (mobile) robots [1], industrial inspection [2], buildings [3], manufacturing [4], mechatronics [5], automotive, biomedical, and many more. However, an application field provides no restricted set of sensors since in each field many types of sensors could be applied.

Fig. 2.1 presents a list of physical quantities (measurands) [6]. The list is certainly not exhaustive, but it shows the many possible measurands. For each of these quantities one or more measurement principles are available.

2.2.2 Classification based on port models

The distinguishing property in the classification based on port models is the need for auxiliary energy (Fig. 2.2). Sensors that need no auxiliary energy for their operation are called direct sensors or self-generating sensors. Sensors that need an additional energy source for their operation are called modulating sensors or interrogating sensors.

Direct sensors do not require additional energy for conversion. Since information transport cannot exist without energy transport, a direct sensor withdraws the output energy directly from the measurement object. As a consequence, loss of information about the original state of the object may occur. There even might be energy loss too—for instance, heat. An important advantage of a direct sensor is its freedom from offset: at zero input the output is essentially zero. Examples of direct sensors are the piezoelectric acceleration sensor and the thermocouple.

Modulating or interrogating sensors use an additional energy source that is modulated by the measurand; the sensor output energy mainly comes from this auxiliary source, and just a fraction of energy is withdrawn from the measurement object. The terms modulating and interrogating refer to the fact that the measurand affects a specific material property which in turn is interrogated by an auxiliary quantity. Most sensors belong to this group: all resistive, capacitive, and inductive sensors are based on a parameter change (e.g., resistance, capacitance, and inductance) caused by the measurand. Likewise, most displacement sensors are of the modulating type: displacement of an object modulates optical or acoustic properties (e.g., transmission, reflection, and interference), where light or sound is the interrogating quantity.

Energy (and thus information) enters or leaves the system through a pair of terminals making up a “port.” We distinguish input ports and output ports. A direct sensor can be described by a two-port model or four-terminal model (Fig. 2.3A).

The input port is connected to the measurand; the output port corresponds with the electrical connections of the sensor. Likewise, a modulating sensor can be conceived as a system with three ports: an input port, an output port, and a port through which the auxiliary energy is supplied (Fig. 2.3B). In these models the variables are indicated with across or effort variables $\mathcal{E}$ and through or flow variables , respectively. The subscripts x, y, and z are chosen in accordance with the sensor cube, to be discussed in Section 2.2.4.

Direct sensors provide the information about the measurand as an output signal, an energetic quantity. Modulating sensors contain the information as the value of a material property, or a geometric quantity, not an energetic signal. The information enters the system through the input port, where the measurand affects specific material or geometric parameters. To extract the information from such a sensor, it has to be interrogated using an auxiliary signal. The information stored in the sensor is available latently, in the latent information parameters or LIPs [7]. These parameters are modulated by the input signal and interrogated by the auxiliary or interrogating input.

At zero input the LIPs of a modulating sensor have initial values, set by the material and the construction. Generally, the input has only a small effect on these parameters, resulting in relatively small deviations from the initial values. Note that direct sensors too have LIPs, set by materials and construction. They determine the sensitivity and other transfer properties of the sensor. So the input port of all sensors can be denoted as the LIP input port. As a consequence, any sensor can be described with the three-port model of Fig. 2.3B. Only the functions of the ports may differ, notably the LIP input port and the interrogating input port.

According to the “unified transducer model” as introduced in Ref. [7], an input port can be controlled either by design (it has a fixed value) or by the environment (the measurand or some unwanted input variable). So we have four different cases (Fig. 2.4). The characteristics of these four transducer types are briefly reviewed:

It is important to note that any practical transducer shows all four types of responses. A strain gauge (a modulating transducer) produces, when interrogated, an output voltage related to the strain-induced change in resistance. But the circuit can also generate spurious voltages caused by capacitively or magnetically induced signals. A thermocouple (a direct transducer) produces an output voltage proportional to the measurand at the interrogating input. If, however, the material parameters change due to (for instance) strain or nuclear radiation (inputs at the LIP port), the measurement is corrupted.

Since just one response is desired, other responses should be minimized by a proper design. The universal approach helps to identify such interfering sensitivities.

2.2.3 Classification based on conversion principles

The classification according to conversion principles is often used for the reason that the sensor performance is mainly determined by the physics of the underlying principle of operation. However, a particular type of sensor might be suitable for a variety of physical quantities and in many different applications. For instance, a magnetic sensor of a particular type could be applied as displacement sensor, a velocity sensor, a tactile sensor, and so on. For all these applications the performance is limited by the physics of this magnetic sensor, but the limitations manifest in completely different ways. A closer look at the various conversion effects may lead to the observation that the electrical output of a sensor depends either on a material property or the geometry or a movement. Fig. 2.5 tabulates these three phenomena for various types of sensors.

2.2.4 Classification according to energy domain

A systematic representation of sensor effects based on energy domains involves a number of aspects. First, the energy domains have to be defined. Second, the energy domains should be allocated to both the sensor input and output. Finally, since many sensors are of the modulating type, the domain of the auxiliary quantity should also be considered. From a physical point of view, nine energy forms can be distinguished:

This classification is rather impractical for the description of sensors. Lion [8] has proposed only six domains and adopted the term signal domain. These six domains are: radiant, thermal, magnetic, mechanical, chemical, and electrical. The number of domains is a rather arbitrary choice, so for practical reasons we will continue with the system of six domains and call them energy domains.

Information contained in each of the six domains can be converted to any other domain. These conversions can be represented in a 6×6 matrix. Fig. 2.6 shows that matrix, including some of the conversion effects. An input transducer or sensor performs the conversion from a nonelectrical to the electrical domain (the shaded column), and an output transducer or actuator performs the conversion from the electrical to another domain (the shaded row). The cells on the diagonal of the matrix indicate effects within a single domain.

This two-dimensional representation can be extended to three dimensions, when the interrogating energy domain is included. This gives 216 energy triplets. To get a clear overview of all these possible combinations, they can be represented in a 3D Cartesian space, the “sensor cube” (Fig. 2.7). The three axes refer to the input energy domain, the output energy domain, and the interrogating input energy.

On each of the 216 elements of the 6×6×6 matrix a conversion effect is located. When restricting to electrical transducers, there are 5 direct input transducers, 5 direct output transducers, 25 modulating input transducers, and 25 modulating output transducers.

To facilitate notation, the transducers can be indicated by indices, like in crystallography, the so-called Miller indices: [x, y, z]. The x-index is the input domain, the y-index is the output domain, and the z-index is the domain of the interrogating quantity. With these three indices a transducer can be typified according to the energy domains involved. Some examples are as follows:

These transducers are also visualized in Fig. 2.7. The practical value of such a representation is rather limited. It may serve as the basis of a categorization for overviews or as a guide in the process of sensor selection.