# Sensor Systems

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theory:sensor_technology:st20_actuators

# Actuators: An Example of an Electrodynamic Motor

Now we will consider some actuators as a lumped element system. We will use the same electro-mechanical analogies as introduced in the lumped element page and on the systems theory page. The analogy concept is based on the work of Firestone1) 2) and Beranek3)

## Electrodynamic transduction

Let us now restrict to two domains: the electrical and the mechanical. The state variables, flows and efforts are given in table 1. One new row is added: the mobility type equivalent for the mechanical domain. The mobility type analogy is the conjugate of the impedance type analogy. The concept of conjugates is briefly explained in the section about across and through varables on the lumped element pages. The only reason to introduce it, is that the derivation of the model becomes easier when using the impedance type for the electrical domain and the mobility type of analogy for the mechanical domain (see chapter 2.17 in the book of Borwick4) ).

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 (Impedance type) translation $x [m]$ force $F [N]$ velocity $v [m/s]$ lossy friction $[N \cdot s/m]$ mass $M [kg]$ compliance $[m/N]$
Mechanical (Mobility type) impulse $p [N/s]$ velocity $v [m/s]$ force $F [N]$ inverse of lossy friction $[m/s / N]$ compliance $[m/N]$ mass $M [kg]$
Tab. 1: Summary of state variables, signals and components in the electrical and mechanical domain

The meaning of compliance can be understood as the inverse of a spring constant $c_{m}^{-1}=k [N/m]$.

Electromechanical transducers, like speakers, solenoids and motors, are in fact linking the mechanical domain to the electrical domain. This is a bidirectional effect: electrical current will result into the rotatiopn of a motor, but moving the shaft of the motor will also result into an electric voltage on the electrical connectors of the motor.

The physical phenomena describing the link from the electrical domain to the mechinical are the Lorentz force and the back emf as descibed by the Biot and Savart law. The Lorenz force law has two terms constituting a force, beging (a) the motion of charge (an electric current) in a magnetic field gives a force and (b) a static charge in an electrical field results in a force. These two phenomena are captured in the equation \begin{equation} \mathbf{F}=q\left ( \mathbf{E}+\mathbf{v}\times\mathbf{B} \right ). \label{eq_lorenz} \end{equation} The Biot and Savart law describes the resultant magnetic field $\mathbf{B}$ at position $\mathbf{r}$ generated by a steady current $I$ (for example due to a wire): \begin{equation} \mathbf{B}\left ( \mathbf{r} \right )=\frac{\mu_{0}}{4 \pi}\int_{l}\frac{I \mathrm{d}\mathbf{l}\times \mathbf{r}}{\left | \mathbf{r} \right | ^{3}}. \label{eq_biotsavart} \end{equation} Equations \eqref{eq_lorenz} and \eqref{eq_biotsavart} can be simplified into the coupled interaction

which is true for simple electromechanical transducers. The first equation of \eqref{eq_coupling} describes the interaction between the two efforts, while the second equation describes the interaction between the two flows. The coupling is given by the $Bl$ product, representing a magnetic field times a certain length of a conducting wire in the transducer. These two equations constitte a transformer element as shown in figure 1. Fig. 1: The electromagnetic coupling in an electrodynamic motor can be seen as a transformer

## Modelling an electromagnetic motor with a load

The transformer of figure 1 can be seen as the element that does the bidirectional transduction from the electronic to the mechanical domain (and vice versa) using the electrodynamic principle. This means, it is using electromagnetic effects (a coil) to make displacement form an electric signal.

When adding some other lumped elements we can complete the model for motors, speakers and solenoids:

• $R_{e}$ : the electrical resistance of the transducer in $[\Omega]$
• $C_{m}$ : the mechanical compliance (inverse of the spring constant) of the actuated object (speaker cone, rubber, tissue, or whatever) in $[m/N]$
• $M_{m}$ : the mass of the actuated object (cone mass, effective mass of moved tissue, or whatever) in $[kg]$
• $G_{m}$ : the dissipation in the mechanical domain (a resistive element) in $[m/s/N]$

the total model figure 2 can be formed. Note that the mechanical components $C_{m}$, $M_{m}$ and $G_{m}$ are in parallel because the sum of forces is equal to the spring force plus the force on the mass plus the lossy force (equivalent to Kirchhoffs currrent law). Fig. 2: The total model of an electrodynamic motor

Finally, in figure 3 a simplification is made, by removing the transformer element. In that case, the mechanical components will get scaled values

\begin{equation} \begin{aligned} C'_{m} \quad & = \quad \left ( Bl \right )^{2} C_{m} \\ M'_{m} \quad & = \quad \frac{M_{m}}{\left ( Bl \right )^{2}} \\ G'_{m} \quad & = \quad \left ( Bl \right )^{2} G_{m} = \frac{\left ( Bl \right )^{2}}{R_{m}}\\ \end{aligned} \label{eq_conversions} \end{equation}

and $R_{e}$ will stay the same. With this simplified model we can easily do some simulations. The derived equations will be in the electrical domain (currents and voltages) but these are easily converted into forces and velocities using equations \eqref{eq_coupling}. Fig. 3: The total model of an electrodynamic motor, after elimination of the transformer

## The dynamic behaviour

The total model can be written as:

\begin{equation} \bar{u}\left (j \omega \right )=\bar{u}_{in}\left (j \omega \right )\frac{1}{1+R_{e}\left ( \frac{1}{G'_{m}} + j \omega M'_{m} +\frac{1}{j \omega C'_{m}}\right )} \end{equation}

which is the transfer function in the frequency domain from the input voltage $\bar{u}_{in}$ to the output voltage $\bar{u}$. In case we are interested in the output motion $\bar{v}$ we can easily convert

\begin{equation} \bar{v}\left ( j \omega \right )=\frac{\bar{u}\left ( j \omega \right )}{Bl} \end{equation} which follows from equation \eqref{eq_coupling}. In case we are not interested in the motion $\bar{v}$ of the actuator but in the excursion $\bar{x}$, an integration step is needed. This is described in the frequency domain as \begin{equation} \bar{x}\left ( j \omega \right )=\frac{\bar{v}\left ( j \omega \right )}{j \omega}=\frac{\bar{u}\left ( j \omega \right )}{j \omega Bl}. \end{equation}

In total, the electromechanical actuator as modelled in figure 2, can be characterized by two transfer functions:

\begin{equation} \begin{aligned} \bar{x}\left ( j \omega \right )\quad & = \quad \bar{u}_{in}\left (j \omega \right )\frac{1}{j \omega Bl\left ( 1 + R_{e}\left ( \frac{1}{G'_{m}} + j \omega M'_{m} +\frac{1}{j \omega C'_{m}}\right ) \right )} \\ \bar{Z}\left ( j \omega \right ) \quad & = \quad \frac{\bar{u}_{in}\left (j \omega \right )}{\bar{i}\left (j \omega \right )} = R_{e} + \frac{1}{\frac{1}{G'_{m}} + j \omega M'_{m} +\frac{1}{j \omega C'_{m}}} \end{aligned} \end{equation}

where the first one shows the excursion $\bar{x}(j \omega)$ as a result of the input voltage $u_{in}(j \omega)$ and the second one the electrical input impedance $\bar{Z}(j \omega)$. For understanding the behaviour (like resonances and damping factors) it is easier to write

\begin{equation} \begin{aligned} \bar{x}\left ( j \omega \right ) \quad & = \quad \bar{u}_{in}\left (j \omega \right )\frac{Bl \frac{C_{m}}{R_{e}}}{\left ( j \omega \right )^{2} C_{m}M_{m} + j \omega \frac{C_{m}}{G_{m}}\left ( 1+ \left ( Bl \right )^{2}\frac{G_{m}}{R_{e}} \right )+1} \\ \bar{Z}\left ( j \omega \right ) \quad & = \quad R_{e} \frac{\left ( j \omega \right )^{2} C_{m}M_{m} + j \omega \frac{C_{m}}{G_{m}}\left ( 1+ \left ( Bl \right )^{2}\frac{G_{m}}{R_{e}} \right )+1}{\left ( j \omega \right )^{2} C_{m}M_{m} + j \omega \frac{C_{m}}{G_{m}}+1} \end{aligned} \end{equation}

where the conversions \eqref{eq_conversions} are used. The corresponding curves are plotted in figure 5 for arbitrary component values. Fig. 4: The transfer function of an electrodynamic actuator Fig. 5: The impedance of an electrodynamic actuator

## Interpretation of the dynamic behaviour

What can we see?

• At DC input, the electrical input impedance is $R_e$ and the excursion of the mechanical output is $u_{in}Bl\cdot C_{m} /R_{e}$
• For very high frequencies, the electrical input impedance is $R_e$ and the mechanical output excursion is $0m$
• There is a resonance at $f_{res}=1/ (2 \pi \sqrt {C_{m}M_{m}})$, so independent of the coupling factor $Bl$.

This means that based the resonance frequency, we can determine the mechanical compliance $C_{m}$ from the electrical impedance assuming a constant effective mass $M_{m}$.

## Other Types of actuators

The example above is for an electrodynamic transducer, like speakers, motors and solenoids. The physical principle to drive a motor is the Biot and Savart force creation (that a current through a wire results into a magnetic field that exerts a force on a permanent magnet in the field as described by equation \eqref{eq_biotsavart}). We may wonder whether there are other principles to drive machinery from an electric source. An interesting summary of actuation principles from the field of MEMS micro-valves can be found in table 2. This table is based on a literature search for micro valve principles5), so the devices are optimised for valve purposes in the submillimeter range.

Actuation principle Pressure [$10^{5} Pa$] Stroke [$\mu m$] Voltage [$V$] Power consumption [$mW$]
Electrostatic $< 0.5$ $< 10$ $100 – 200$ $< 1$
Thermopneumatic $1 < P < 100$ $30 < d < 100$ $5 – 20$ $2000 – 5000$
Electromagnetic $< 0.5$ $> 100$ $20 – 50$ $50 – 300$
Piezoelectric $< 1$ $10 < d < 50$ $100 – 200$ $< 5$
Tab. 2: Properties of micro valves made with MEMS technology[Cristina Neagu, 1998]

The thermopneumatic principle (thermophone) clearly produces the largest pressure, however the power consumption is huge. The electrodynamic principle is the proper choice for a large stroke. For small applications we can benefit from the low power electrostatic principle. In practical scales in the centimeter range and above, other physical principles may be beneficial. See also the page on the principle of scaling. From our experience we may understand that electromagnetic priciples are beneficial over electrostatic principles in the region of centimeters and metres. Thermopneumatic effects can be huge in the macroscopic world, although it takes time and a significant amount of energy to heat larger volumes.

The models for other actuation principles from the perspective of miniature speakers (recievers) is described in Appendix C: Actuator principles for small speakers.

# 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)
John Borwick (Editor), Loudspeaker and Headphone Handbook, 3rd Edition, Focal Press, 2001
5)
Cristina Neagu, A medical microactuator based on an electrochemical principle, Thesis University of Twente, 1998 