# Sensor Systems

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theory:sensor_technology:st10_lumped_element_models

# Lumped element models

A generalized, cross domain method to start calculations on transducers, is to use lumped element models. With lumped element models, entities of all physical domains are represented in a single formalised network. This yields optimum transparency of transduction principles, where electrical-circuit theory can be applied to optimise and calculate performance parameters. The theory is known as “dynamical analogies” or “systems theory” and is as old as the existence of electrodynamic transducers1) 2) 3).

## An example

This method will be illustrated using the example of the mass on a cantilever as used on the page about scaling. There we have figure 1 which is a mass on a cantilever. To turn it into a two-domain physical transducer, an electrostatic actuator is added, and we end up with the configuration as shown in figure 1 below. Assume the mass and the cantilevers to be good conductors. The counter electrode is placed at a distance $y_{0}$ to create a parallel plate capacitor.

Fig. 1: An electrostatically actuated mass on a spring

## State variables, efforts and flows

The starting point is the definition of three variables in each physical domain: a 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. See also the section on quantities on the Sensor Theory page. In the electrical domain, the state variable is the charge $q$. This makes the flow to be the electrical current $i = \partial q/ \partial t$ and the potential the effort since it is the cause of an electrical current. In the mechanical domain we may consider translation as the state variable, resulting into force as the flow and speed as the effort4).

For a selection of physical domains the state variables, flows and efforts are given in table 1.

Physical domain State variable Effort Flow Dissipative Inertial buffer Capacitive buffer
Electrical charge $q [C]$ potential difference $u [V]$ current $i [A]$ resistor $R [\Omega]$ coil $L [H]$ capacitor $C [F]$
Mechanical translation $x [m]$ force $F [N]$ velocity $v [m/s]$ lossy friction $[N \cdot s/m]$ mass $M [kg]$ compliance $[m/N]$
Acoustical vol. displ. $x [m]$ pressure $p [Pa]$ vol. velocity $U [m^{3}/s]$ resistor $[Pa \cdot s/m^{3}]$ mass $[kg/m^{4}]$ compliance $[m^{3}/Pa]$
Magnetic flux $\Phi [Vs]$ mmf $M [A]$ potential difference $u [V]$
Thermal Heat $Q [Ws=J]$ temp. $T [K]$ heat flow $\partial Q/ \partial t=P [W]$ $R_{th}=\partial T/\partial{P} [K/W]$ heat capacity $c_{m}m [J/kg]$
Thermo dynamical entropy $S [J/K]$ temp. $T [K]$ entr. flow $f_{S} [J/s \cdot K]$
Tab. 1: Summary of state variables, signals and components in several physical domains

## The network elements

Energy can be stored in either a capacitive or an inertial buffer or be drained in a dissipative element, satisfying the equations

$$f=C\frac{\partial e }{\partial t} , e=L\frac{\partial f }{\partial t} , \text{or } e=R\cdot f$$

respectively, with $e$ the effort and $f$ the flow. Besides the one-port elements $R$, $C$ and $L$, there are also two-port elements like transformers and gyrators.

Now we can derive an equivalent circuit for the electrostatically actuated mass on a spring of figure 1. The system is driven by an electric potential source $u \left ( t \right )$. The source has an electrical internal resistance $R_{e}$ is loaded by the impedance of the capacitance $C_{e}$. This is shown at the electrical side of figure 2.

Transduction from the electrical to the mechanical domain is modelled by an ideal transformer. In a certain regime, where the system is biased by a voltage $V_{Bias}$, the transduction is linear and satisfies $u = T_{EM} \cdot F$ and $v = T_{EM} \cdot i$. The transduction coefficient equals

$$T_{EM}=y_{0}^{2}/\left (\epsilon_{0}R^{2}V_{Bias} \right).$$

On the mechanical side, the compliance of the beam $C_{M}$ (equal to $k^{-1}$ as given on the page about scaling), the mass $M_{M}$ and a certain mechanical friction with air $R_{M}$ are incorporated in the model. These components are placed in series since they are exposed to the same velocity $v$.

Fig. 2: Equivalent model for the electrostatically actuated mass on a spring

From a circuit analysis we can understand the behaviour of the system. The voltage source is not only loaded by the impedance due to the electrical parts $R_{e}$ and $C_{e}$, but also by the mechanical parts $R_{M}$, $C_{M}$ and $M_{M}$. This is due to (electro-mechanical) coupling acting in two directions, back and forth. The idealised part of the coupling is modelled by the ideal transformer. The frequency response can be derived from the model. There will be a resonance due to the mass and the spring at

$$f_{res} = \frac{1}{2 \pi \sqrt{C_{M}M_{M}}}.$$

At lower frequencies where the effect of $C_{e}$ is negligible, the transduction from input voltage $\overline{u} \left ( \omega \right )$ to mass velocity $\overline{v} \left ( \omega \right )$ is given by:

$$\overline{v}\left ( \omega \right ) = T_{EM}^{-1} \left ( \frac{R_{e}}{T_{EM}^{2}} + \frac{1}{j \omega C_{M}} + R_{M} + j \omega M_{M} \right )^{-1} \overline{u}\left ( \omega \right )$$

which shows how the method of lumped elements easily helps us to develop mathematical models. When the model of figure 2 shows too much disagreement with observations, we have to refine the model by adding more components. For example, oscillating modes in the cantilever are not implemented in the model, but could be included by adding $LC$ circuits. At any time, the lumped element model is linked phenomenologically to reality.

# Sensor Technology TOC

These are the chapters for the Sensor Technology course:

1)
F.A. Firestone, A new analogy between mechanical and electrical systems, J. Acoust. Soc. Amer., 4 (1933), pp. 249 - 267
2)
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)
Leo L. Beranek, Acoustics, McGraw-Hill, New York, 1954
4)
This is the impedance equivalence which opposes the more commonly used mobility analogy where it is just the opposite way (speed is flow and is force is effort) since mechanical impulse is considered as the state variable. These conventions are called dual and result in mathematical interchangeability of the associated buffers.