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theory:sensor_technology:st13_lumped_element_models_advanced

Systems Theory

The theory known as “dynamical analogies” or “systems theory” is as old as the existence of electrodynamic transducers1) 2) 3). Systems Theory acknowledges the similarity between physical domains. Because the differential equations describing various systems across many domains look so similar, you may ask whether there is a uniform way of describing (modelling) all domains. Systems Theory gives you the unification with the advantage that models can be derived that include the full measurement, control and actuation chain. Especially to describe transducers (sensor and actuators) and their transduction principle, the unified Systems Theory is a very powerfull tool.

Definition

The Systems Theory, also referred to as System Dynamics, Dynamical Analogies, or Transduction Techniques, aims at the description and explanation of the behaviour of objects in time; it is based on a generalised and unified set of concepts from several domains in physics and engineering. The concepts are mainly based on mathematics, mechanics, thermodynamics, network analysis, measurement theory, control theory, and informatics.

An object that is being studied by means of the Systems Theory is referred to as a System, which is of course a circular definition. It is better to define a system as a set of entities being interconnected by relations. This is already a definition very close to the notion of models. And indeed: Systems Theory is mainly about models because the whole theory is intended to explain and to optimise physical phenomena in engineering projects.

Historical background

Taken from the reader of Breedveld.4) The mechanical laws of Newton can be seen as the first scientific theory that was based on generic physical principles, like the law of conservation of momentum. It was constructed using the mathematics of differential equations which is in fact a mathematical abstraction level needed for unification. The relation to other domains is introduced in the 19th century when Sadi Carnot added the notion of conservation of energy and applied it as Thermodynamics, linking mechanics to all other physical domains.
The first unifications came from the 1930's in the field of acoustical transducers, because there transduction from electronics to mechanics and subsequently to acoustics is observed. Electromechanical transducers using electrodynamics (a coil speaker) and electrostatics (condenser microphone) need appropriate modelling of their transduction function and all electrical, mechanical and acoustical phenomena influencing the behaviour5) 6) 7). Especially Chapter 3 in the book by Beranek8) is a very structured explanation of the method. The book can be found on the Internet as a pdf-file9).
The insights are later in the 1940's and 1950's formalized in the Control Theory where the benefits of feedback, stability, and adaptivity could be described at an abstract level. It was Norbert Wiener who introduced the concept of Cybernetics10) in 1948. Cybernetics is the science of “control and theory in biological systems and machines” and is derived from the Greek word $\kappa \mu \beta \epsilon \rho \nu \eta \tau \eta \rho$ = Kubernètèr = governor = steersman. Nowadays, the word cybernetics is normally replaced by generalised systems theory and expands now even to biology, economics and sociology.

Basic concepts

Physical meaning of lumped elements

Beranek describes four principal requirements to enable schematic representations11):

  1. The methods must permit the formation of schematic diagrams from visual inspection of devices;
  2. They must be capable of such manipulation as will make possible the combination of electrical, mechanical, and acoustical elements into one schematic diagram;
  3. They must preserve the identity of each element in combined circuits so that one can recognize immediately a force, voltage, mass, inductance, and so on;
  4. They must use the familiar symbols and the rules of manipulation for electrical circuits.

The reason for these four requirements is to guarantee that any lumped element in the circuit can be re-traced to the physical meaning of a single physical phenomenon.

State variables

What was seen on the page of Lumped Element Modelling is that in all domains we can define a set of three variables; the state variable, a flow and an effort. The flow is defined as the derivative in time of the state variable and the effort is defined as the cause of the flow. Combined with the section on quantities on the Sensor Theory page, this gives table 1.

Extensive variable State variable
$\partial / \partial t$ Extensive variable Rate variable = flow
Intensive variable effort
Tab. 1: Relation of quantities to lumped element terminology

The most appropriate quantities to choose as state variables are quantities that can be stored and buffered. These are the variables that have to be initialized to re-define the state of the system completely. For example, in an electromagnetic system, we have to know the charges $q$ on the capacitive elements and the magnetic fluxes $\Phi$ of the inductors to know the initial condition. These are the quantities that have to be set to initial values before doing a simulation in PSpice and in Simulink/MatLab. In other words, the state variables are the memory elements of the system. Once the state variables are known, the complete initial condition is known (for example, initial potentials and currents can be calculated).

The state variables to be chosen are:

  • Electrical → charge $q$ [$C$]
  • Magnetic → coupled magnetic flux $\Phi_{k}$ [$Vs$]
  • Mechanic potential (translation) → position $x$ [$m$], or
  • Mechanic kinetic (translation) → momentum $p$ [$Ns$]
  • Mechanic potential (rotation) → angle $\alpha$ [$rad$], or
  • Mechanic Kinetic (rotation) → angular momentum $b$ [$Nms$]
  • Hydraulic → volume $V$ [$m^{3}$]
  • Chemical mass → mole number $N_{i}$ [mole]
  • Thermal → entropy $S$ [$JK^{-1}$]

You may wonder why the mechanical domain is represented two times, and why the magnetic and electrical domain are not simply one electromechanical domain. The list above is very formalistic, in practice some combinations can be made:

  • The mechanic potential en mechanic kinetic domains can be combined with $\dot{x}=v$ and $\dot{p}=F$
  • The electric and magnetic domains can be combined with $\dot{q}=i$ and $\dot{u}=\Phi_{k}$

where $\dot{x}$ means the first derivative in time of variable $x$, so $\dot{x}=\partial x/ \partial t=v$. These combinations are legitimate because in technical systems and domain transitions (transduction) we need a consistency in energy. The energy balance in the mechanical domain equals $F \dot{x}=\dot{p} v$ and in the electromechanical domain $i \dot{\Phi_{k}}=u \dot{q}$ (static versions of the Maxwell equations). In fact, in this reasoning we are using already the property of power conjugation (next subsection). The formalistic description in either the state variables $x$ or $p$ for the mechanical domain and $q$ and $\Phi_{k}$ for the electromechanical domain, are in fact already the conjugated descriptions in the mobility type and impedance type analogies.

Nevertheless, in for a fundamental and formalistic approach, the long list of state variables above is preferred. With this we accept the consequence that there are always conjugated equivalents.

Flow and Effort

Once a state variable $q$ is defined, we can define a flow $f$ of the state variable by

\begin{equation} f=\frac{\partial q}{\partial t}. \end{equation}

Finally, we can define a third variable which is the cause of the flow $f$ and is referred to as the effort $e$. table 2 shows the resulting efforts and flows for the selected state variables. In the effort and flow, we can recognise electrical potential and current:

  • Kirchhoffs voltage law is applicable to the effort $e$: the sum of potential differences along a loop equals zero Volt;
  • Kirchhoffs current law applies to the flow $f$: the sum of electrical currents flowing to a node equals zero.

In the mechanical domain, similar laws apply and are defined in the motion laws of Newton.

Physical domain State variable Effort Flow
Mechanical Impedance translation $x [m]$ force $F [N]$ velocity $v [m/s]$
Mobility momentum $p [N/s]$ velocity $v [m/s]$ force $F [N]$
Mechanical rotation Impedance angle $\phi [rad]$ torque $F [Nm]$ angular velocity $v [rad/s]$
Mobility angular momentum $p [N/s]$ angular velocity $v [rad/s]$ torque $F [Nm]$
Electrical (magnetic) Impedance charge $q [C]$ potential diff. $u [V]=[Wb/s]$ current $i [A]$
Mobility flux $\Phi [Vs]$ mmf $i [A]$ potential diff. $u [V]=[Wb/s]$
Acoustical vol. displ. $x [m]$ pressure $p [Pa]$ vol. velocity $U [m^{3}/s]$
Thermal Heat $Q [Ws=J]$ temp. $T [K]$ heat flow $\partial Q/ \partial t=P [W]$
Thermo dynamical entropy $S [J/K]$ temp. $T [K]$ entr. flow $f_{S} [J/s \cdot K]$
Chemical $N [mole]$ chemical potential $\mu [J/mole]$ mass flow $\dot{N} [mole/s]$
Tab. 2: Summary of state variables and signals in several physical domains

Power conjugation

When the efforts and flows are defined well, there is a clear similarity across the domain boundaries. A proper definition is based on power conjugated efforts and flows, meaning that the product has the meaning of power:

\begin{equation} P=f \cdot e [Watt]. \end{equation}

In the SI system, the product of effort and flow is the power in Watts. In table 2, we can see that only for the thermal convention this does not work. For intuitive reasons we prefer to take heat flow as the flow (to make the heat the state variable), although it is for the uniformity of the models better to use entropy as the state variable. The mathematical damage of such a psuedo analogy is limited, because thermal systems are irreversible in most cases anyway.

Across and through variables, mobility and impedance type

Power conjugates always have one across-network variable (the voltage equivalent in Kirchhoff terminology) and one through-network variable (the current equivalent in Kirchhoff terminology). Being “conjugates” means they can be mathematically exchanged depending on the definition of the state variable. This means there are always two conventions - the mobility analogy and the impedance analogy12). In table 2 the impedance analogies are given where potential difference, force, pressure, and temperature are the across network variables (efforts). For electrical circuits, the impedance analogy is off course the most intuitive, but note that the magnetic domain is in fact the conjugate because there $u$ and $i$ are exchanged. In the mechanical domain the mobility type (also called inverse analogy) is also used, and even preferred by Beranek13). In that case, velocity $v$ is the effort and force $F$ the flow. This make impulse $p$ the state variable, and not displacement $x$.

On the across-variables, the efforts, Kirchhoffs voltage law applies: the sum of all voltages along a closed loop is zero. On the through-variables, Kirchhoffs current laws applies: the sum of all currents to a network node is zero. These two laws determine the network equations.

Awareness of the across and through varables gives us the ability to derive lumped equivalent models directly from real-world systems. For example, in the mechanical mobility analogy, the efforts are the velocities $v$. This means that objects that have the same speed are in parallel (potential differences the same). As a result they will have different forces $F$ (flows) acting through them. The sum of forces (flows) gives the resultant force equivalent to Kirchhoffs current law.

Inspection in the mechanical mobility analogy is done by observing:

  • points with same velocity are in parallel in the equivalent circuit (for exampe, a mass on a spring results in the same velocity for the mass as for one side of the spring)
  • points with the same force are in series in the equivalent circuit (for example, a motor driving a mass results in the same force on the mass as delivered by the motor)

Inspection in the mechanical impedance analogy is done by observing:

  • points with same force are in parallel in the equivalent circuit
  • points with the same velocity are in series in the equivalent circuit

Beranek describes a structured method to convert mobility type anologies into impedance type anologies in section 3.8.14)

Component equations

Energy can be stored in either a capacitive or an inertial buffer or be drained in a dissipative element, creating three passive components for each domain:

\begin{equation} \begin{aligned} f \quad & = \quad C\frac{\partial e }{\partial t}, & \quad \text{(Capacitive buffer)} \\[5pt] e \quad & = \quad L\frac{\partial f }{\partial t}, & \quad \text{(Inertial buffer)} \\[5pt] e \quad & = \quad R\cdot f, & \quad \text{(Dissipative element)} \\[5pt] \end{aligned} \end{equation}

with $e$ the effort and $f$ the flow. These are the one-port elements $C$, $L$ and $R$ respectively, and are summarized in table 3.

Physical domain Dissipative Inertial buffer Capacitive buffer
Mechanical Impedance lossy friction $[N \cdot s/m]$ mass $M [kg]$ compliance $[m/N]$
Mobility inverse of lossy friction $[m/(s \cdot N)]$ compliance $[m/N]$ mass $M [kg]$
Electrical (magnetic) Impedance resistor $R [\Omega]$ coil $L [H]$ capacitor $C [F]$
Mobility conductance $G [Siemens]$ capacitor $C [F]$ coil $L [H]$
Acoustical resistor $[Pa \cdot s/m^{3}]$ mass $[kg/m^{4}]$ compliance $[m^{3}/Pa]$
Thermal $R_{th}=\partial T/\partial{P} [K/W]$ heat capacity $c_{m}m [J/kg]$
Thermo dynamical
Tab. 3: Summary of components in several physical domains

Transduction

The ideal transformer

Besides the one-port elements $R$, $L$ and $C$ (and $e$ and $f$ sources), there are also two-port elements: the transformer and the gyrator. Both can be within a domain or across a domain. The electrical transformer is a single domain two-port, a gear box can be mechanical transformer element. We will not discuss the gyrator here: it can simply be seen as a transformer with a successive step to invert the output impedance into an admittance.

The generic transformer model is given in figure 1. The performance is described by conservation of energy

\begin{equation} e_{1}f_{1}=e_{2}f_{2} \end{equation}

which results into the ideal transformer equations assuming a symmetrical transformer constant $T$ \begin{equation} \begin{aligned} e_{1} \quad &= \quad T \cdot e_{2} \\ f_{1} \quad &= \quad T^{-1} \cdot f_{2}. \end{aligned} \end{equation}

For an electrical transformer, we know the transformer constant $T$ because this is equal to the ratio of windings $T = N_{1}/N_{2}$. For cross-domain transformers (called transduction) the transformation constant (referred to a transduction constant) may be less evident.

Fig. 1: Generic ideal transformer

Cross effects

In table 4 we can see some cross-effects of transduction.15) This table indicates the name of the physical effect related to the transduction phenomenon.

Output domain
Radiant Thermal Magnetic Mechanical Chemical Electrical
Input domain Radiant Luminescense Radiation heating Photomagnetism Radiant pressure Potohcemical process Photoconductivity
Thermal Incandescense Thermal conductivity Curie-Weiss law Thermal expansion
Magnetic Faraday effect Ettinghausen effect Magnetic induction Converse magnetostriction Hall effect
Mechanical Photo-elastic effect Friction heat Magnetostriction Gear Pressure-induced reaction Piezoelectricity, electrostatic transducer
Chemical Chemo luminiscense Exothermal reaction Explosive reaction Chemical reaction Volta effect
Electrical Injection luminiscense Peltier effect Ampère's law Converse piezoelectricity Electrolysis Ohm's law
Tab. 4: Physical domains and cross effects of transduction

How this phenomenon is implemented in a transduction effect depends on the geometry of the transducer, the material properties and the transduction principle.

Electrostatic transduction

Electrostatic transduction is a situation where an electric field is used to transform displacement or force (mechanical) into a voltage or current (electrical) or vice versa. The three seminal examples are the capacitive accelerometer, the capacitive pressure sensor and the condensor microphone. Actually, the condenser microphone transforms sound pressure into displacament first, but still the transduction from displacement to an electrical signal is still illustrative for the electrostatic transducer. The Accelerometer Model in Appendix B is a transducer from the mechanical to the electrical domain. The transduction is functionally not so different from a capacitive (or condenser) microphone or a capacitive pressure sensor.

The reasoning in Appendix B starts with noticing that the potential accross a parallel plate capacitor is given by

\begin{equation} u = \frac{q}{C} \rightarrow C= \frac{\epsilon_{0}A}{d}. \label{eq:ParallelCapacitor} \end{equation}

after biasing with a fixed charge $q$. On the bottom plate there is a charge $–q$, on the upper plate a charge $+q$. In fact, the upper plate is placed in the electric field of the bottom plate. The result is that the upper plate experiences an electrostatic force caused by the electric field of the magnitude $E/2$. This force is the mechanical force on the top plate as a result from the bottom plate and is equal to

\begin{equation} F_{el} = q \cdot \frac{E}{2} = \frac{q^{2}}{2 \epsilon_{0}A}. \end{equation}

If the upper plate is suspended from a spring with spring constant $k$, and so can move with respect to the fixed bottom, the linearised transduction equations are:

\begin{equation} \begin{aligned} \mathrm{d}u \quad & = \quad \frac{d_{0}}{\epsilon_{0}A}\mathrm{d}q + \frac{q_{0}}{\epsilon_{0}A}\mathrm{d}x \\ \mathrm{d}F \quad & = \quad \frac{q_{0}}{\epsilon_{0}A}\mathrm{d}q+k\mathrm{d}x \end{aligned} \label{eq_electrostatic_transduction_equations} \end{equation}

assuming small variations ($x<<d_{0}$) and a fixed bias $q_{0}$. This set of equations is known as the characteristic equations and can be written in a matrix form

\begin{equation} \begin{bmatrix} \mathrm{d}u\\ \mathrm{d}F \end{bmatrix} = \begin{bmatrix} \frac{d_{0}}{\epsilon_{0}A} & \frac{q_{0}}{\epsilon_{0}A}\\ \frac{q_{0}}{\epsilon_{0}A} & k \end{bmatrix} \begin{bmatrix} \mathrm{d}q\\ \mathrm{d}x \end{bmatrix} =M \cdot \begin{bmatrix} \mathrm{d}q\\ \mathrm{d}x \end{bmatrix} \end{equation}

using the transduction matrix $M$, not to be confused with the mass $M$. In practical microphones and accelerometers, we know all elements of the matrix $M$. The $d_{0}$ does not deviate so much from the gap size, $A$ is the plate area, $k$ the (effective) spring constant of the suspension and $q_{0}$ a direct consequence of the biasing voltage by means of

\begin{equation} q_{0}=C \cdot V_{bias} = \frac{\epsilon _{0}A}{d_{0}}V_{bias}. \end{equation}

The entry $M_{11}$ in the matrix represents the electrical capacitance of the transducer. Entry $M_{22}$ is simply the spring constant. Both elements represent single domain phenomana. However, the cross entries $M_{12}$ and $M_{21}$ in the transduction matrix indicate there is a coupling between the two domains when they are non-zero. These terms are equal and referred to as the transduction coefficient. When they are non-zero, a displacement $dx$ translates into a potential difference $du$ and a charge variation $dq$ into a force variation $dF$. This means we can relate the mechanical domain to the electrical domain and back.

An alternative way of writing \eqref{eq_electrostatic_transduction_equations} is in terms of the nominal capacitance $C_{0}$ and the bias voltage $V_{bias}$:

\begin{equation} \begin{aligned} \mathrm{d}u \quad & = \quad \frac{1}{C_{0}}\mathrm{d}q + \frac{V_{bias}}{d_{0}}\mathrm{d}x \\ \mathrm{d}F \quad & = \quad \frac{V_{bias}}{d_{0}}\mathrm{d}q+k\mathrm{d}x \end{aligned} \label{eq_electrostatic_transduction_equations2} \end{equation}

which is exactly the same, but shows a more direct relation to the lumped elements.

The coupling factor for electrostic trasnduction

The coupling factor is defined as

\begin{equation} K = \sqrt{\frac{M_{12}M_{21}}{M_{11}M_{22}}} \end{equation}

and is a measure for the quality of the transducer because it relates the cross-domain operation with the isolated-domain operation. It is a real number which is between $0$ and $1$ for normal, stable operation. In our case, it becomes equal to

\begin{equation} K = \frac{q_{0}}{\sqrt{\epsilon_{0}Ad_{0}k}}=\sqrt{\frac{\epsilon _{0} A}{d_{0}^{3}k}}V_{bias}=\sqrt{\frac{C_{0}}{k}} \frac{V_{bias}}{d_{0}}. \end{equation}

We can easily convert it to

\begin{equation} \begin{bmatrix} \frac{\mathrm{d} u}{\mathrm{d} t}\\ \frac{\mathrm{d} F}{\mathrm{d} t} \end{bmatrix} = M \cdot \begin{bmatrix} \frac{\mathrm{d} q}{\mathrm{d} t}\\ \frac{\mathrm{d} x}{\mathrm{d} t} \end{bmatrix} = M \cdot \begin{bmatrix} i\\v \end{bmatrix} \end{equation}

with $i$ an electrical current ($i=dq/dt$) and $v$ a mechanical velocity ($v=dx/dt$). From this we can step to the impedance equation

\begin{equation} \begin{bmatrix} j \omega \overline{u}\\ j \omega \overline{F} \end{bmatrix} = M \cdot \begin{bmatrix} \overline{i}\\ \overline{v} \end{bmatrix}. \label{eq_transduction} \end{equation}

The impedance equation is a powerful tool to find, for example, the electrical input impedance of the transducer. We find

\begin{equation} \overline{Z}\left ( j \omega \right ) = \frac{\overline{u}\left ( j \omega \right )}{\overline{i}\left ( j \omega \right )} = \frac{1}{j \omega} M_{11}\left ( 1-K^{2} \right ). \end{equation}

Earlier, it was said that the cross entries $M_{12}$ and $M_{21}$ in the transduction matrix represent the coupling between the two domains. So, we may think they are equal to the transformer constant $T$. Equation \eqref{eq_transduction} shows it is not: we miss a differentiation step when the in- and outputs of the transformer are formalised as efforts and flows. We find: \begin{equation} M_{12}=M_{21}=j \omega T. \end{equation}

Electrodynamic transduction

While electrostatic transduction is between an electric field and mechanical displacement (or force, or velocity), electrodynamic transduction is about a magnetic field and mechanical displacement (or force, or velocity). On the page about Actuators there is an example of an electrodynamic motor. This is in fact electrodynamic transduction and is quite similar for AC motors, solenoids, speakers, and electrodynamic microphones. The model derived there is

\begin{equation} \begin{aligned} u \quad & = \quad Blv , & \quad \text{(Back emf creation)} \\[5pt] F \quad & = \quad Bli , & \quad \text{(Biot and Savart force creation)} \\[5pt] \end{aligned} \label{eq_magnetic_coupling} \end{equation}

with $B$ and magnetic field somewhere in the motor (permanent magnet?), $l$ an effective wire length somewhere in the motor, $i$ and $u$ the electric inputs and $v$ and $F$ the mechanical outputs. What is input and what is output is normally not so relevant in these transducer principles because the transduction principles are reversible.

However, there is something confusing:

  • the transduction equation \eqref{eq_electrostatic_transduction_equations} for the electrostatic transducer expressed the efforts in terms of the state variables,
  • in the set of equations \eqref{eq_magnetic_coupling} we see the efforts expressed in terms of flows.

The reason is that (although they are correct), equations \eqref{eq_magnetic_coupling} are derived from linking the electrical domain to the mechanical domain, while it is more appropriate and formalistic to derive the relation from the mechanical and magnetic domain.

Using the mechanical impedance type and electrical mobility (= magnetic) type, we find the linearised transduction equations

\begin{equation} \begin{aligned} \mathrm{d}i \quad & = \quad \frac{1}{L}\mathrm{d}\Phi_{k} - \frac{W}{L}\mathrm{d}x \\ \mathrm{d}F \quad & = \quad -\frac{W}{L}\mathrm{d}\Phi_{k}+ \left ( \frac{W^{2}}{L} + D \right ) \mathrm{d}x \end{aligned} \label{eq_electrodynamic_transduction_equations} \end{equation}

with $L$ the self inductance of the coil, $W$ relates to the electromagnetic coupling, and $D$ a compliance $1/k$ for example from the suspension of the speaker cone. The coupling $W$ can be estimated by the product of the number of windings per meter, the flux $\Phi_{k}$ and the cross-sectional area of the coil.

Again, we can write the matrix form

\begin{equation} \begin{bmatrix} \mathrm{d}i\\ \mathrm{d}F \end{bmatrix} = \begin{bmatrix} \frac{1}{L} & - \frac{W}{L}\\ - \frac{W}{L} & \frac{W^{2}}{L} + D \end{bmatrix} \begin{bmatrix} \mathrm{d}\Phi_{k}\\ \mathrm{d}x \end{bmatrix} =M \cdot \begin{bmatrix} \mathrm{d}\Phi_{k}\\ \mathrm{d}x \end{bmatrix} \end{equation}

to recognise the transduction matrix $M$. The coupling factor for the electrodynamic system is

\begin{equation} K = \sqrt{\frac{M_{12}M_{21}}{M_{11}M_{22}}} = \sqrt{\frac{1}{1+\frac{DL}{W^{2}}}}, \end{equation}

This can be interpreted as

  • if $W=0$, $K$ becomes $0$, so there is no coupling,
  • if $D=0$ the $K$ will be $1$ which means the plate is hanging loose and the system is instable.

The sensitivity can be estimated by assuming a high impedance voltmeter. This means that $\mathrm{d}i=0$ in equation \eqref{eq_electrodynamic_transduction_equations} and so

\begin{equation} 0= \frac{1}{L}\mathrm{d}\Phi_{k} - \frac{W}{L}\mathrm{d}x \end{equation}

which gives us

\begin{equation} u = \frac{\mathrm{d}\Phi_{k}}{\mathrm{dt}}= W \frac{\mathrm{d}x}{\mathrm{d}t} \end{equation}

and so

\begin{equation} \overline{u} = j \omega W \overline{x}. \end{equation}

If we take $W$ the product of $10^{6}$ windings per meter, a flux of $1 Vs/m^{2}$, and a coil cross-sectional area of $10^{-3}m$, then the sensitivity $u/x$ is about $10 \mu /nm$ at a frequency of $20Hz$.

Unification

With the two previous examples, we have seen typical two-port elements. These examples where not just an ideal transformer, because there the transduction matrix elements $M_{11}$ and $M_{22}$ would have been $0$. The non-zero values in the matrix represent other elements like the inductance of the coil in the transducer, the spring constant of the suspension, or the capacitance of the electrostatic transducer.

The characteristic transduction equations define a relation between the sate variables $q_{m}$ and the efforts $e_{m}$ according to \begin{equation} \begin{aligned} \mathrm{d}e_{1} \quad & = \quad b_{11}\mathrm{d}q_{1} + \ldots + b_{1m}\mathrm{d}q_{m} \\ \mathrm{d}e_{2} \quad & = \quad b_{21}\mathrm{d}q_{1} + \ldots + b_{2m}\mathrm{d}q_{m} \\ \ldots \quad & = \quad \ldots + \ldots\ldots\ldots\ldots.. \\ \mathrm{d}e_{m} \quad & = \quad b_{m1}\mathrm{d}q_{1} + \ldots + b_{mm}\mathrm{d}q_{m} \end{aligned} \label{eq_linearised_transduction_equations} \end{equation}

which generalises of course equations \eqref{eq_electrostatic_transduction_equations} and \eqref{eq_electrodynamic_transduction_equations}. Every element $b_{ij}$ can be found by taking $\mathrm{d}e_{i}/\mathrm{d}q_{j}$ with all $q_{k}$'s constant for $k \neq j$. At any time we find $b_{ij}=b_{ji}$ which is referred to as the Maxwell symmetry. Regtien16) formalises all transduction matrix elements $b_{ij}$ for the mechanical, electrical, magnetic and thermal domain in appendix B.2.

Equation \eqref{eq_linearised_transduction_equations} can be written as

\begin{equation} \begin{aligned} \frac{\mathrm{d}e_{1}}{\mathrm{d}t} \quad & = \quad b_{11}f_{1} + \ldots + b_{12}f_{2} \\ \frac{\mathrm{d}e_{2}}{\mathrm{d}t} \quad & = \quad b_{21}f_{1} + \ldots + b_{22}f_{2} \end{aligned} \end{equation}

which is reduced to two domains for readability. In impedance notation this becomes the set of equations

\begin{equation} \begin{aligned} \overline{e}_{1} \quad & = \quad \frac{b_{11}}{j \omega} \overline{f}_{1} + \frac{b_{12}}{j \omega} \overline{f}_{2} \\ \overline{e}_{2} \quad & = \quad \frac{b_{21}}{j \omega} \overline{f}_{1} + \frac{b_{22}}{j \omega} \overline{f}_{2} \end{aligned} \end{equation}

also known as the generalised impedance equations. Furthermore

\begin{equation} \begin{bmatrix} \overline{e}_{1}\\ \overline{e}_{2} \end{bmatrix} = \begin{bmatrix} \frac{b_{11}}{j \omega} & \frac{b_{12}}{j \omega}\\ \frac{b_{21}}{j \omega} & \frac{b_{22}}{j \omega} \end{bmatrix} \begin{bmatrix} \overline{f}_{1}\\ \overline{f}_{2} \end{bmatrix} =M \cdot \begin{bmatrix} \overline{f}_{1}\\ \overline{f}_{2} \end{bmatrix} \end{equation}

and the coupling factor is \begin{equation} K = \sqrt{\frac{b_{12}b_{21}}{b_{11}b_{22}}} \end{equation} where the $j \omega$ denominators drop out. The calculation of the input impedance can be generalized as \begin{equation} \overline{Z}\left ( j \omega \right ) = \frac{\overline{e}\left ( j \omega \right )}{\overline{f}\left ( j \omega \right )} = \frac{1}{j \omega} b_{11}\left ( 1-K^{2} \right ). \end{equation}

Sensor Technology TOC

These are the chapters for the Sensor Technology course:

1) , 5)
F.A. Firestone, A new analogy between mechanical and electrical systems, J. Acoust. Soc. Amer., 4 (1933), pp. 249 - 267
2) , 6)
F.A. Firestone, The mobility method for computing the vibrations of linear mechanical and acoustical systems: mechanical-electrical analogies, J. Appl. Phys., 9 (1938), pp. 373 - 387
3) , 7) , 8) , 11) , 12) , 13) , 14)
Leo L. Beranek, Acoustics, McGraw-Hill, New York, 1954
4)
P.C. Breedveld, J.J. van Dixhoorn, dictaat Technische Systeemleer, Universiteit Twente, Faculteit der Elektrotechniek, studiejaar 1989/1990
10)
Norbert Wiener, Cybernetics: Or Control and Communication in the Animal and the Machine. Paris, (Hermann & Cie) & Camb. Mass. (MIT Press), 1948
15)
Middelhoek, Simon, and Sarah A. Audet. Silicon sensors. Academic Press, 1989.
16)
Paul P.L. Regtien, Sensors for Mechatronics, Elsevier, 2012
theory/sensor_technology/st13_lumped_element_models_advanced.txt · Last modified: 2017/10/10 18:40 by glangereis