theory:sensor_technology:st21_actuator_models

*The following paragraphs are written from the perspective of small acoustical transducers that are intended to be used on the ear (mini speakers, also called *recievers*). However, the insights are applicable for all actuator methods.*

An electrostatic speaker uses the electrostatic charge across a parallel plate capacitor to move air. The equivalent circuit of an electrostatic transducer as shown in figure 1 is based on a systems theory lumped element view on the transduction path^{1)}^{2)}.

The resistor $R_{e}$ is the low-ohmic electrical resistance of the connection wire between voltage source $V(t)$ and the transducer. The electrical capacitance of the transducer is represented by $C_{e}$. Transduction is modelled by an ideal transformer with a coefficient

\begin{equation} T_{EM}=\frac{d}{C_{e}V_{Bias}}=\frac{d^{2}}{\epsilon _{0}A^{2}V_{Bias}} \label{eq_Tem} \end{equation}

with $d$ the nominal air gap distance and $A$ the parallel plate size. This shows that the model is only valid for small membrane displacements (which is in practice not the case for high sound levels). In the mechanical domain we can find the compliance of the membrane $C_{M}$, the acoustical resistance of the membrane RM and the mass of the membrane $M_{M}$. The acoustical radiation is represented by front and backside radiation impedance $Z_{MR}$.

To calculate the membrane movement $u$, which is the flow in the mechanical domain, it is easier to remove the transformer from the circuit of figure 1. This is done in figure 2. The function of the transformer is now included in the electrical components $V’(t)$, $R’_{e}$ and $C’_{e}$.

Extremely high frequencies are filtered out in the electrical domain by $R_{e}$ and $C_{e}$. This happens at frequencies above $(2R_{e}C_{e})^{-1}$. Since the mechanical branch will be much more important at lower frequencies, we can remove $C_{e}$ for simplicity. In that case, the velocity $u$ of the membrane is given by

\begin{equation} \bar{u}\left ( \omega \right )=\frac{1}{T_{EM}}\left ( \frac{R_{e}}{T_{EM}^{2}} +\frac{1}{j \omega C_{M}}+R_{M}+j \omega M_{M}+2Z_{MR}\left ( \omega \right )\right )^{-1}\bar{v}\left ( \omega \right ) \label{eq_membrane_velocity} \end{equation}

There are two masses: the mass of the membrane and the mass of the air into which is radiated (included in $Z_{MR}$). A quick estimate of a $1 \times 1 mm^{2}$ silicon-nitride membrane with a thickness of $1 m$ resulted into a radiation mass which is ten times larger than the mass of the membrane itself.

The critical frequencies are:

- An electrical low pass cut-off frequency at $(2 \pi R_{e}C_{e})^{-1}$. When the wire resistance $R_{e}$ is in the region of Ohms, this cut-off frequency can be very high ($MHz$);
- The more important high frequency limit is determined by $f_{high} = R_{M}/2 \pi (M_{M}+M_{MR})$ with $M_{MR} = Im(2Z_{MR})/ \omega$;
- The compliance of the membrane and the air flow resistance of the speaker determine the lower frequency bound: $f_{low} = (2 \pi R_{M}C_{M})^{-1}$;
- Resonance frequencies are determined by compliance/mass systems. The compliance of the membrane $C_{M}$ and the masses in the system result into the resonance frequency $f_{res} = (2 \pi \sqrt{ C_{M}(M_{MR}+M_{M}) } )^{-1}$;

Transduction in the middle frequency range is found by removing all frequency dependent elements from equation \eqref{eq_membrane_velocity} resulting into

\begin{equation} \bar{u}\left ( \omega \right )=\frac{T_{EM}^{-1}}{R_{M}+\frac{1}{T_{EM}^{2}}R_{e}}\bar{v}\left ( \omega \right ) \label{eq_membrane_velocity2} \end{equation}

which reduces to

\begin{equation} \bar{u}\left ( \omega \right )=\left ( R_{M}T_{EM} \right )^{-1}\bar{v}\left ( \omega \right ) \label{eq_membrane_velocity3} \end{equation}

in practical cases because $R_{e}/T_{EM}^{2} << R_{M}$. Note that both $R_{M}$ and $T_{EM}$ must be small for a large transduction effect^{3)}.

The results are summarised in table 1.

According to a report from the University of Twente^{4)} concerning new actuator principles for hearing aid receivers, the fundamental problem of electrostatically driven microspeakers is not in the frequency response. For a small speaker, a large diaphragm displacement is needed. This means for an electrostatically driven speaker that the air gap is large as well. This will result into huge voltages, probably even close to the breakdown electrical field in air.

The force between the plates is equal to
\begin{equation}
F=\frac{q^{2}}{2 \epsilon_{0}A}
\label{eq_membrane_force}
\end{equation}
where the charge $q$ can be expressed in terms of voltage by using the definition of capacitance $C = q/V$:
\begin{equation}
F=\frac{C^{2}V^{2}}{2 \epsilon_{0}A}.
\label{eq_membrane_force2}
\end{equation}
Then the relation between sound pressure and voltage in a closed volume with cross section $A_{ear}$ is found from
\begin{equation}
p=\frac{F}{A_{ear}}=\frac{C^{2}V^{2}}{2 \epsilon_{0}A A_{ear}}=\frac{\left (\frac{\epsilon_{0}A}{d} \right )^{2}V^{2}}{2 \epsilon_{0}A A_{ear}}=\frac{\epsilon_{0}A V^{2}}{2 d^{2} A_{ear}}.
\label{eq_membrane_pressure}
\end{equation}
For an ear diameter of 8 mm and a receiver of $2 mm$, the voltages are well below 100 Volts when the air gap is only $10 m$. The breakthrough voltage of air is in the range of $50 V/m$. So, for a receiver the situation is probably not as bad as claimed in the report^{5)}.

The electrodynamic transducer is based on a coil and a permanent magnet. In figure 3, the system representation is given^{6)}^{7)}. This figure is of the *mobility type* where force is represented as mechanical flow (current) and movement $u$ as mechanical effort (voltage). In figure 1, we used the dual form, referred to as *impedance type* where force is the mechanical effort and movement $u$ is the flow^{8)}.

The transducer is actuated by a source with voltage $V(t)$ and an internal resistance $R_{S}$. This source faces the electrical resistance $R_{E}$ of the coil and the electrical inductance $L$ at zero-movement. Transduction is linear from the driving current to the movement $u$ of the diaphragm, given by $u = Bli$. In figure 3 the central element is an ideal transformer with transduction coefficient $T_{EM} = Bl$.

In the mechanical domain, the mass, compliance and friction of the diaphragm are modelled by $M_{M}$, $C_{M}$ and $R_{M}$ respectively. The acoustical impedance of the air, as seen by one side of the diaphragm, is given by $Z_{MR}$.

The membrane mass $M_{M}$ and the membrane compliance $C_{M}$ are comparable to the case of the electrostatic transducer. The resistance $R_{M}$ however, is not limited by the air gap since there is not a back electrode at a small distance from the membrane needed to actuate the membrane. So, $R_{M}$ can be much lower in the electrodynamic case than in the electrostatic case.

For convenience, the mobility-type schematic of figure 3 is re-arranged into the impedance-type schematic of figure 4. Transformation to the dual representation exchanges node efforts (movement $u$) with branch flows (force or pressure)^{9)}. In addition, the transformer is removed.

The diaphragm movement u is given by the source effort divided by the total impedance

\begin{equation} \bar{u}\left ( \omega \right )=\frac{\frac{\bar{v}\left ( \omega \right ) T_{EM}}{\left ( R_{S}+R_{E} \right )+j \omega L}}{\left (\frac{1}{j \omega L / T_{EM}^{2}} \right )//\left ( R_{S}/T_{EM}^{2} +R_{E}/T_{EM}^{2} \right )+j \omega M_{M}+R_{M}+\frac{1}{j \omega C_{M}}+2Z_{MR}} \label{eq_membrane_movement_electrodynamic} \end{equation}

in which $//$ represents the parallel configuration of two impedances: $A//B = AB/(A+B)$.

Just like with the electrostatic transducer, there are two masses: the mass of the membrane and the mass of the air into which is radiated.

Now we can indicate the critical frequencies:

- The inductance of the coil determines a high frequency limit at the frequency of $(R_{S}+R_{E})/2 \pi L$;
- The more dominating high frequency limit is given by $f_{high} = R_{M}/2 \pi (M_{MR}+M_{M})$ with $M_{MR} = Im(2Z_{MR})$;
- The compliance of the membrane and the air flow resistance of the speaker determine the lower frequency bound: $f_{low} = (2 \pi R_{M}C_{M})^{-1}$;
- The resonance frequency is $f_{res} = (2 \pi \sqrt{C_{M}(M_{MR}+M_{M})})^{-1}$;

The transduction in the middle range is given by

\begin{equation} \bar{u}\left ( \omega \right )=\frac{T_{EM}}{T_{EM}^{2} + R_{M}\left ( R_{S} +R_{E} \right )}\bar{v}\left ( \omega \right ) \label{eq_membrane_movement_electrodynamic2} \end{equation}

which reduces to

\begin{equation} \bar{u}\left ( \omega \right )=\frac{T_{EM}}{ R_{M}\left ( R_{S} +R_{E} \right )}\bar{v}\left ( \omega \right ) \label{eq_membrane_movement_electrodynamic3} \end{equation}

in practical cases because $T_{EM}^{2} << R_{M}(R_{S}+R_{E})$. So, in contrast to the electrostatic transducer, $T_{EM}$ must be large (just a matter of definition).

So, with the exception of the electrical front end (which has it’s impact probably outside the frequency range of interest), the electrostatic and electrodynamic configuration result into the same design constraints. The reason is that the acoustical performance is mainly determined by the interface between radiation and membrane. For practical reasons we might prefer one of the driving mechanisms. For example, with electrodynamic actuation, the mechanical resistance of the membrane can be lower, although the fabrication of microcoils and permanent magnets is more problematic.

Using a block of electrostrictive material as a sound generating body will not give the desired effect. The volume displacement of piezo materials is that small that a huge volume of material would be needed to move a sufficient amount of air.
It is better to use piezo bars^{10)} to move a plate. In that case, the mechanical and acoustical parts of figure 2 or figure 4 can be re-used to model the actuator. However, new oscillation frequencies are introduced due to the (rather complex) modes of the piezo bars. This is the reason that we will see multiple resonance frequencies in the high frequency areas of such devices. The high resonance frequencies shift the main part of the response curve towards the high frequency region above $10kHz$ which make the piezo-based transducers less suitable for audio-range actuation.
Despite the frequency problems, piezo actuators may result into simple devices because there is only a single vibrating plate. There is no need for air gaps and piezoelectric materials can be created relatively easy in micromachining processes.
A comprehensive mathematical model for piezoelectric sources is given by Lihoreau^{11)} and a fundamental description of piezo materials by Liu^{12)}.

A thermally driven speaker is referred to as thermophone^{13)}. A thermophone consists of a heater element in a closed chamber. By heating the gas in the chamber, the pressure increases which modulates one flexible wall of the chamber: the membrane. The other side of the membrane emits sound into the environment. In the classical implementation of the thermophone, the chamber is filled with hydrogen in order to shift the standing waves to a more useful response range.

Thermal actuation in the sense of heating the enclosed air in a chamber benefits from the gas law in which pressure divided by temperature is constant. So when increasing temperature, pressure increases linearly. The advantage of miniaturised devices is that cooling and heating small volumes can be much faster than heating larger volumes, especially for arrays of small heaters. So the frequency issue can probably be solved. A fundamental problem is the relation between input voltage and sound pressure. An increase in temperature is the result of the dissipated heat in the heating element. Therefore, the pressure is related to the square of the input voltage. So, we must first take the square root of the input signal before supplying it to the heater which is an undesired operation.

Moving membranes in MEMS devices using a heater is reported for the use in micropumps^{14)}^{15)}. The effect is referred to as thermopneumatic actuation. The membrane for this specific pump is configured for relaxation times of five seconds which is obviously too slow for acoustic actuation. In air we will see that acoustic frequencies can be theoretically reached.

Simple calculations of miniature heaters using equivalent circuits can be found in literature^{16)}. Consider a heater in an enclosed volume of air as shown in figure 5. The silicon walls of the chamber act as a heat sink, in this case assumed to be a single block on one side of the chamber. This heat sink transports the heat to the infinite environment consisting of air.

The equivalent model is shown in figure 6. In this equivalence, the heat source is a current source while temperatures are node “voltages”. We are interested in the temperature in the chamber which determines the sound pressure level.

The thermal resistance of the air chamber is

\begin{equation} R_{Chamber}=\frac{d_{wall}}{A \lambda_{air}} \label{eq_thermophone_resistance_air_chamber} \end{equation}

with A the cross section of the chamber, dchamber the thickness of the chamber and air the thermal conductivity of air. The heat capacitance of the chamber is found from

\begin{equation} C_{Chamber}=c_{m.air}\rho_{air}d_{chamber}A \label{eq_thermophone_capacitance_air_chamber} \end{equation}

with $c_{m.air}$ the specific heat capacity of air and $\rho_{air}$ the density of air. For the silicon walls we can use similar equations
\begin{equation}
R_{wall}=\frac{d_{wall}}{A \lambda_{Si}}
\label{eq_thermophone_resistance_wall}
\end{equation}
and
\begin{equation}
C_{Chamber}=c_{m.Si}\rho_{Si}d_{wall}A
\label{eq_thermophone_capacitance_wall}
\end{equation}
where the surface $A$ is chosen equal to the cross section of the chamber as drawn in figure 5. The infinite environment of air is modelled by a transmission line with thermal resistances per meter $ \lambda _{air}A$ and thermal capacitances per meter $c_{m.air} \rho_{air}A$. The total impedance of such a transmission line as seen from the walls is equal to^{17)}^{18)}
\begin{equation}
\bar{Z}\left ( \omega \right )=\sqrt{\frac{R_{air}}{j \omega C_{air}}}=\sqrt{\frac{\left ( \lambda_{air}A \right )^{-1}}{j \omega c_{m.air}\rho_{air}A}}
\label{eq_thermophone_impedance_air}
\end{equation}

with the radiating surface equal to $A$.

The transfer function from heat source to the chamber temperature is equal to \begin{equation} \bar{H}\left ( \omega \right )=\frac{\bar{T} \left ( \omega \right )}{\frac{dQ}{dt}\left ( \omega \right )}=\frac{1}{j \omega C_{air}}//\left (R_{wall}+ \frac{1}{j \omega C_{wall}} //\bar{Z}_{air}\left ( \omega \right )\right ) \label{eq_thermophone_transfer_function} \end{equation}

with $//$ a symbol used for parallel configurations. The material constants^{19)} can be found in table 3.

^{*)} Thermally grown

Equation (2.21) is plotted in figure 7. In the lower curve, walls of silicon are assumed. The resulting impedance is $3.4 K/Watt$. When using silicon dioxide walls, the thermal impedance is a factor of ten higher. The reason is that, in the horizontal region from $10Hz$ to $10kHz$, the temperature in the chamber is mainly determined by the a heat drain to the environmental air. This heat drain is ten times lower for silicon dioxide than for silicon.

The sound pressure level corresponding to a certain temperature difference is found by \begin{equation} SPL=20\cdot log \left (\frac{\Delta T}{T} \frac{P_{0}}{P_{ref}} \right ) \label{eq_thermophone_SPL} \end{equation}

with $P_{ref} = 20 Pa$ assuming that the small displacement of the membrane does not affect the volume significantly. An SPL level of more than $150 dB$ SPL for only $1^{\circ} C$ temperature increase is found at the location of the membrane. This is in the intended range for miniature speakers (receivers).

Besides thermopneumatics, another option for thermal actuation is to use a bimorph^{20)}. This method is used for resonating mass flow sensors and actuating other MEMS devices. Some helpful numerical values can be found in literature^{21)}. A bimorph as shown in figure 8 has a displacement $h$ at the tip which is proportional to $(T-T_{0})(\alpha _{2}-\alpha_{1})/w$ with $\alpha _{1}$ and $\alpha_{2}$ the thermal expansion coefficients. For beams with a length of $100 m$, displacements of $50 m$ are possible for temperature differences of $70^{\circ} C$.

The resonance frequency depends on the used temperature difference as shown in figure 9^{22)}. This is an undesired effect for speakers. What can be seen is that the highest speed of actuation can be shifted to the $kHz$ range.

An electrochemically driven microvalve is described in literature^{23)}. Such devices are based on the electrochemical removal or generation of gas from a liquid. Just like with the thermophone, the increase in pressure in a closed chamber is used to move a membrane. One major difference is that the chamber is inflated actively and deflated actively while thermal actuation only has to “inflate” the chamber because pressure is lost anyway by leakage of heat.

Electrochemical actuation has a close to linear response due to the Butler-Volmer equation \begin{equation} j=j_{0}\left [ e^{\frac{\left ( 1-\alpha \right )nF}{RT}\eta}-e^{\frac{\alpha nF}{RT}\eta} \right ] \label{eq_buttler_volmer} \end{equation}

which couples electrode overpotential $\eta$ to the electrode current $j$. The current is the direct cause of the gas formation \begin{equation} 2H_{2}O\rightarrow O_{2}\left ( g \right )+4H^{+}+4e^{-}. \label{eq_gas_formation} \end{equation}

From the thesis of Neagu^{24)} we can obtain some indicative numbers. The production of $O_{2}$ from water is in the range of $5 \cdot 10^{-12} mol/s$ for an electrode current of $2 \mu A$. We can see membrane deflections in the order of microns. The largest problem is the slow response time in the order of seconds. However, this microvalve was not optimised for speed. It uses a Nafion film to protect the electrode which reduces the response time by means of diffusion. The response is limited by saturation at the electrode. There are options to reduce this phenomenon in order to make the process faster (nano electrodes, smaller volumes or stirring).

The ionophone is the ultimate speaker because it converts the movement of a plasma directly to sound^{25)}. There are no moving parts involved. For portable consumer applications it is less suitable because it has a plasma in the open air, ozone is generated and non-linearities are easily introduced.

The driving mechanism left out of consideration, a speaker based on a vibrating membrane must move a sufficient amount of air to create a certain audible SPL level. This means that either the membrane is sufficiently large or the displacement is large. As a speaker (in open air), a small device will definitely not generate a satisfying sound level, especially not in the low frequency range. As a receiver (closed volume application), a rough estimation showed that we might reach the intended region.

With conventional configurations using planar membranes the microspeaker will not be dramatically smaller than the currently used models. So we must look for new configurations for sound generation or find a very cheap process to make $5-8 mm$ “large” MEMS speakers. Referring to the pulls for MEMS devices in cellular phones as mentioned in chapter 1, five out of six benefits can be accomplished with large MEMS speakers.

Each driving mechanism has its own advantages and disadvantages These are summarised in table 4.

Note that miniaturisation itself does not have benefits for the performance. There is one exception. Thermopneumatic actuation is not easily achieved with large speakers since the frequency is determined by the speed of heating and cooling the compression chamber. For small configurations however, cooling down can be that fast that audible sounds can be reached by heating and cooling.

These are the chapters for the Sensor Technology course:

- Chapter 1: Measurement Theory
- Chapter 2: Measurement Errors
- Chapter 3: Measurement Domains
- Chapter 4: Circuits, Graphs, Tables, Pictures and Code
- Chapter 5: Basic Sensor Theory
- Chapter 6: Sensor-Actuator Systems
- Chapter 7: Modelling
- Chapter 8: Modelling: The Accelerometer - example of a second order system
- Chapter 9: Modelling: Scaling - why small things appear to be stiffer
- Chapter 10: Modelling: Lumped Element Models
- Chapter 11: Modelling: Finite Element Models
- Chapter 13: Modelling: Systems Theory
- Chapter 14: Modelling: Numerical Integration
- Chapter 15: Signal Conditioning and Sensor Read-out
- Chapter 16: Resistive Sensors
- Chapter 17: Capacitive Sensors
- Chapter 18: Magnetic Sensors
- Chapter 19: Optical Sensors
- Chapter 20: Actuators - an example of an electrodynamic motor
- Chapter 21: Actuator principles for small speakers
- Chapter 22: ADC and DAC ← Next
- Chapter 23: Bus Interfaces - SPI, I
^{2}C, IO-Link, Ethernet based - Appendix A: Systematic unit conversion
- Appendix B: Common Mode Rejection Ratio (CMRR)
- Appendix C: A Schmitt Trigger for sensor level detection

Leo L. Beranek, Acoustics, McGraw-Fill Inc., New York 1954

John Borwick (editor), Loudspeaker and headphone handbook, Focal Press, 1994

For a condenser microphone, sensitivity depends on the compliance of the membrane. We do not see a dependency on the compliance in the reverse system (a speaker). This is the result of the transduction in the microphone from pressure to diaphragm movements, while in the speaker transduction is from voltage to acoustical pressure and membrane velocity. Since membrane displacement is the integrated membrane velocity, we get a different sensitvity. For the speaker, the compliance determines the lower frequency boundary in the acoustical power output.

T.A.J. Cats, Alternatieve actuatorprincipes t.b.v. hoortoestellen (in Dutch), Masters thesis BIO 90/20, University of Twente, 1990

Harry F. Olson, Acoustical engineering, D. van Nostrand company, Inc., Princeton, 1957

Don L. DeVoe and Albert P. Pisano, Modeling and optimal design of piezoelectric cantilever structures, Journal of Microelectromechanical Systems, Vol. 6, no. 3, September 1997

B. Lihoreau, P. Lotton, M. Bruneau and V. Gusev, (Maine Univ. Le Mans), Piezoelectric source exciting thermoacoustic resonator: analytical modelling and experiment, Acta Acustica united with Acustica, Vol 88 (2002), pp 986 – 997

Yuan Liu, (University Newfoundland, Institute of acoustics Beijing), Surface sources of piezoelectric transduction, Appl. Phys. Lett. 66 (5) jan 1995

F.C.M. van de Pol, H.T.G. van Lintel, M. Elwenspoek and J.H.J Fluitman, A thermopneumatic micropump based on micro-engineering techniques, Sensors and Actuators, A21 (1990), pp. 198-202

M. Elwenspoek, T.S.J. Lammerink, R. Miyake and J.H.J. Fluitman, Towards integrated microliquid handling systems, J. Micromech. Microeng. 4 (1994). pp. 227 – 245

G.R. Langereis, An integrated sensor array for monitoring washing processes, Chapter 6: Thermoresistive heating, Thesis University of Twente 1999

D.K. Cheng, Field and wave electromagnetics, 2nd edition, Addison-Wesley Publishing Company, 1989

D.R. Lide, Handbook of chemistry and physics, 74th edition 1993 - 1994, CRC Press, Boca-Raton, Florida

T.S.J. Lammerink, M. Elwenspoek and J.H.J. Fluitman, Thermal actuation of clamped silicon microbeams, Sensors and Materials A, vol.3, pp 217 – 238, 1992

M. Helmbrecht and R. Muller, Center for Adaptive Optics, University of California, MEMS for adaptive optics, Powerpoint presentation from website, November 15, 1999, http://cfao.ucolick.org/past_meetings/Mems/

Cristina Neagu, A medical microactuator based on an electrochemical principle, Thesis University of Twente, 1998

Roger Russell, Ionophones, http://home.earthlink.net/~rogerr7/ionovac.htm, Webpage with overview on ionophones

theory/sensor_technology/st21_actuator_models.txt · Last modified: 2017/10/10 18:45 by glangereis