PAYNTER’S VERIDICAL STATE EQUATION IN INTEGRAL CAUSAL FORM

PETER BREEDVELD

University of Twente, Control Laboratory, EL/TN 8250P.O Box 217, 7500 AE Enschede, Netherlands

phone: +31 53 489 2792, fax: +31 53 489 2223, e-mail: P.C.Breedveld@el.utwente.nl

Keywords: van-der-Waals gas, thermodynamics, port-baseddescription, initialization, integral causality

AbstractCompared to the ideal gas law, the van-der-Waals state equation

considerably improves the thermodynamic description of asubstance at a qualitative level. However, the quantitative accuracyis in many cases not sufficient to obtain reasonably realisticsimulation results. Many state equations with a more satisfyingnumerical accuracy have been proposed. However, most do notsatisfy the conditions derived in previous work [Breedveld, 2000]for symbolically deriving the preferred integral causal form of thethree-port storage element that describes the energy storage in agas. Paynter [1985] has introduced a modification of the van-der-Waals equation that may be considered the necessary quantitativeimprovement in order to obtain realistic simulation results, whichhe called the veridical state equation. However, like the commonform of the van-der-Waals equation, his state equation is not in thefull integral causal form required for constitutive relations ofpower ports of an energy-based dynamic submodel and thus notoptimally suited for simulation. Consequently, it has been hardlyused in port-based, ‘plug-&-play’ modeling and simulation. Thispaper shows that the integral causal form of the correspondingconstitutive relations can be symbolically derived and introduces asuitable initialization strategy. Some examples of its use insimulation are shown.

INTRODUCTIONThe van-der-Waals state equation considerably improved the

thermodynamic description of a substance at a qualitative levelwith respect to the ideal gas law, as it shows phase transition(intrinsically unstable region, cf. [Breedveld, 1991; 2000]) and acritical point. Although many close quantitative approximations tothe measurement data for the state equation p(v,T) of puresubstances exist (e.g. [Çengel and Boles, 1989] or [Gyftopoulosand Beretta, 1991]), these relations commonly require specificationof quite a few parameters and, more importantly, they require thetemperature as a dynamic input variable, i.e. differential causalityof the thermal port in bond graph terminology. Althoughspecifying the state of matter by its temperature makes much moresense than specifying its initial entropy, one should only concludefrom this that initialization of storage elements in dynamicsimulations is better done in terms of the temperature, not theentropy. However, the dynamic state itself should be the entropy inorder to prevent numerical differentiation. Previous workdemonstrated that if cv is assumed constant, the preferred integralform (‘causality’) of the thermal port can be found in case of theideal and the van-der-Waals gas, but not for the more accuratecurve fits of the measurement data commonly used in simulation,

as they do not depend linearly on the temperature. Karnopp andRosenberg [1975] solved this causality problem for the ideal gaslaw by formulating the constitutive relations of the two-port Cdescribing a fixed amount of gas in integral causal form. Theauthor showed how this can be extended to a three-port Cdescribing a variable amount of gas [Breedveld, 1984] and laterhow a substance characterized by the van-der-Waals equation canbe described by a three-port C element with constitutive relationsin integral causal form [Breedveld, 1991, 2000]. This paper takesPaynter’s veridical (literally: ‘truth-speaking’) state equation as astarting point to find the constitutive relations of a three-port C-type storage element in integral causal form that describes a puresubstance also quantitatively in a reasonably accurate manner,even though Paynter considered his results as an intermediate step,predicting that further modifications along the lines of hisextension with a non-analytical term would give even moreaccurate results. If such other extensions are found, it will have tobe tested in each case whether or not these forms satisfy theconditions for derivation of the integral causal form. Naturally, analternative path would be to try to improve the numerical accuracyof the integral causal form. However, this would go beyond thescope of this paper.

Some typical simulation results are shown and compared withthe results based on the ideal gas law and the van-der-Waalsequation, in order to demonstrate that the results lead to afunctional implementation. Although the essence of Paynter’sextension is that it contains a non-analytic term, it is still such thatthe pressure depends linearly on the temperature and that it leads toanalytically integrable expressions. These are two crucialconditions on a state equation in order to be able to find an integralcausal form [Breedveld, 2000].

The models will not include the dynamics of phase transitions,i.e. the (saturated) vapor state is assumed below the criticaltemperature and the results for the region of co-existing phases willnot be accurate. Hence this approximation is mainly intended toget good results for temperatures close to, but above, the criticaltemperature, or for low densities at temperatures below the criticalpoint.

Before a simulation is possible, initial conditions should be set.At this point an additional reason for the commonly useddifferential causality of the thermal port becomes clear: thereference value for the entropy just plays an arbitrary role in therelations, but a non-zero reference temperature should be provided,due to the nature of the constitutive form. This means that relationsshould be found between common sets of initial values like (T0,p0, V0), (T0, p0, N0) or even (T0, V0, N0), and the required initialstates (S0, V0, N0). These relations are not trivial and will be given

special attention, in particular (T0, p0, N0), which will be used inthe simulations.

PAYNTER’S MODIFICATIONPaynter [1985] proposed a much closer approximation of the

state equation of pure substances than the one by van-der-Waals bymaking a non-analytic extension. However, his equation is not inoptimal form for numerical simulation either, as the thermal port ofthe multiport capacitor characterized by this equation is still indifferential causal form. This severely restricts dynamic simulationto implicit numerical integration schemes and it often leads tomodels in which thermal damping is ignored without justification,as demonstrated by Pourmovahed and Otis, 1984]. In this paper theintegral causal form is derived assuming cv to be constant, viz.

12( )Vc n R= + , where R is the universal gas constant and n the

number of atoms per molecule (e.g. n =2 for nitrogen (N2), used inthe example simulations). The relations are written such that theycharacterize a proper three-port C-element, by allowing variationof the amount of gas (addition of a material port) in principle.

Paynter’s structural extension consists of adding a term 1rρ −

to the van-der-Waals equation, where rρ is the reduced density. He

also adapted the reduced van-der-Waals parameters a and b fromtheir common values, viz. 3 and 1/3 respectively, to 7 and 1/4respectively, although he clearly states that these are parameters tobe fitted to actual data. In order to support that, a and b areexpressed herein in terms of the critical values of the temperature

Tc, pressure pc and the compressibility factor Zc, thus allowing theuser to obtain an even closer approximation for the pure substancebeing simulated (note that the subscript c will be used to indicatethe critical point). The example simulations will be performed withthe values used by Paynter, i.e. 7 and 1/4 respectively, althoughsome multiple run tests were performed varying a and b, whichshowed that a=6 gives a result that much closer approximatesfigure 21.1 in [Gyftopoulos and Beretta, 1991]. The relation givenby Paynter is [1985] using the same symbols:

3 21 ( 1) (1 )v x

r r r r rp M T A Tρ ρ

= + − − − (1)

with 10 (1 )

v

rM M bρ −= = − for the van-der-Waals’ form,

1 01 1 (1 )v

r r rM M M bρ ρ ρ= = − = − − for Paynter’s veridical

form and 2x

c cA a a p v′= = where a′ is the common van-der-Waals

parameter. The other common van-der-Waals parameter b′ isrelated to b by cb bv′ = . For the reduced specific volume holds

1r rv ρ−= , such that

4

3 2

4

2 3 2

( 1) sgn( 1)1 (1 )

( )

( 1)1

( )

r r cc r r

r r r

r cc r

r r r r

v v app p T T

v v b v

a v app T

v v v b v

ξ

− − = − − − = − − = + − − −

(2)

using the non-analytical term sgn( 1) 1 ( 1)r r rv v vξ = − = − − .

Also note that Paynter used reduced variables throughout his 1985

paper, even though he omitted the common subscript r. Thesereduced, i.e. dimensionless, variables are not the required portvariables for port-based submodels. Herein, in order to keepnotation compact, we will take the following strategy. We firstassume that after integration of the flows of the ports of thesubmodel into the extensive states S (entropy), V (volume) and N(amount of moles), first the intensive states s and v are computedby dividing by N and next vr is computed by dividing by vc. It thensuffices to derive the temperature T(s, vr), the pressure p(s, vr) and

the total material potential µ tot(s, vr) in order to obtain the effortsof the corresponding ports in integral causality. It is assumed thatcv is independent of T, such that

( , , ) ( , , ) ( , )v v rU V S N Nc T V S N Nc T s v= = resulting in the required

expression for the effort of the material port (cf. [Breedveld,2000]):

tot ( , ) ( )

( ) ( , ) ( , )r v

v r c r r

s v u pv Ts c s T pv

c s T s v v v p s v

µ = + − = − + == − +

(3)

Note that the relation for µ tot may be omitted if the amount ofmoles N is kept fixed at all times which means that the C-elementreduces to a two-port. Equation (2) already provides the desiredrelation p (T, vr), however for later use it is already written in

separately integrable terms as follows (Note that rv b> always

holds as rv b≤ would mean the presence of a solid phase for

which this relation is not adequate):

( ) ( )2

21c c r

r r cr r

a ab Tp p ap v

v v b Tv v bξ −

= − Ω + + − − − −

(4)

where

( ) ( ) ( )( )( )3 2

41 4 6+1

r r rr r r r

b

v v b v bv v b v v b

− Ω = − + −

− − − − The compressibility factor Z, which is a measure for the deviationfrom the ideal gas law for which Z is always equal to 1, is definedas /Z pv RT= . Its critical value /c c c cZ p v RT= is often used as

one of the parameters to specify pure substances. The van-der-Waals equation in reduced form can thus be written:

2( ( ))r r c r rp T Z v b av−= − − (5)

For large values of vr, Paynter’s relation should converge to the

van-der-Waals form, hence 1(4 )cZ b −= − and consequently b is

found in terms of Zc: 14 cb Z −= − . The Riedel parameter at the

critical point ac becomes [Paynter, 1985]: ,

1

c c

rc

r v T

pa a

T

∆ ∂= = +∂

.

Hence 1ca a= − , such that the reduced van der Waals constants a

and b can be expressed in terms of the parameters ac and Zccommonly known for real substances. In the example simulationswe will follow Paynter and choose ac = 8 and Zc = 4/15 = 0.27,corresponding to a = 7 and b =0.25, to obtain values close to theaverage for many substances. Gyftopoulos and Beretta [1991]show in their Table 20.1 that most of the listed pure substanceshave values that lie within 10% of this value for Zc. However, theplug & play port-based submodel that is proposed will require only

specification of pc, Tc, ac and Zc from which vc, a and b arederived in an initial section of the model (only computed at thefirst time step), such that the results can be easily adapted to thepure substance at hand. Naturally, the plug & play submodel canbe easily modified to allow specification of a substance by anydesired parameter set.

DERIVATION OF THE INTEGRAL CAUSAL FORMFOR THE THERMAL PORT

Expressions for p (T, vr) = p(T(s, vr), vr), viz. (2) and µ(T(s, vr) ,p(T, vr), s, vr), viz. (3), have been found already, which meansthat, if T is known, the mechanical and material port are in integralcausality. What rests is to find T(s, vr), which is the key issue if thethermal port has to be in integral causality too. In previous work[Breedveld, 1987, 2000] it has been shown that if p(v, T) dependslinearly on T, and if the resulting expression for ds is integrable,the required T(v, s) can be found. It is obvious that Paynter’srelation satisfies the first condition and consequently

d 0d d dv

v

p cs v T

T T

∂∂ =

= +

or

d 0

d 1d d

r

c rr

v v r v

s Z R pv T

c c T T

∂∂ =

= +

(6)

needs to be integrated (cf. [Breedveld, 2000]). The result of theintegration depends on the two possible integration intervals,which are most easily handled if the arbitrary reference is chosenin the critical point: 0 0 0 0 so 1, , c r c cv v v T T s s= = = = . Note

that this reference is not necessarily equal to the initial equilibriumstate from which a simulation can be started. From (4) follows

d 0

3 2

2

1 4 6 4+1

( ) ( )( ) ( )

1( )( )

r

r

r v

r r rr r r r

r rr r

p

T

b

v v b v bv v b v v b

ab a

v v bv v b

∂∂

ξ

==

−= − − + − − − −− −

− − − −−

(7)

and thus

( )

1 4

d 01

2

(1 )d ( ) ( 1) ln

1 1 1 7 4 ln ( )

2 2

r

r

vr r

r rr rv

rr r r

r r

p v bv v b b

T v b

vb v v v a

v bv bb

∂ξ λ∂

ξ

−

=

− = = − + −

−+ − + + − − + +

∫(8)

such that an expression for s is found that can be used forinitialization

d 01

d ln lnr

r

vr

c r v vr c cv

p T Ts Z R v c c

T T T

∂ ξ∂ =

= + = Λ +

∫ (9)

with 1 4 (1 )( ) ( 1) ln lnr

c r c rr

v bZ R v Z Rb b R v

v bλ − −Λ = = − + +

−

2

( 1) 1 1 7( )

2 2c r

r rr r

Z R vv v a

v bv bbξ

−+ + − − + +

and finally the required form for T is obtained:

(1 )expr

c rr

v bT T v

v b

βγ − = Ξ − (10)

where

2

1 (1 ) 1 7 1( )

2 2c r

r rv r r

RZ vv v a s

c v b bvbξ

−Ξ = − + − + + +

(11)

with 1 4 4 3( 1) (1 3 ) (4 1)c c cb b Z Z Zγ − −= − = − − and 1vRcβ ξ −= − .

INITIALIZATIONAs the choice for the initial value of the entropy is arbitrary,

some attention should be paid to the proper initialization of the 3-port C-element characterized by the expressions derived above.The available volume V0, the initial temperature T0, and the initialpressure p0 commonly specify the initial state of a substance in acontainer. It is thus essential to derive the initial amount of molesN0 and the initial entropy S0 before a simulation is started, e.g. inan initial section of the submodel. The initial amount of moles N0can be found from the initial (reduced) density, which is a functionof pressure and temperature. However, in case of the van-der-Waals equation this is a third order polynomial and in case ofPaynter’s extension a fourth order polynomial in the density:

1 2 14( 1) ( 1) ( 1) 1 =0r r r r r rb T a p Tρ ξ ρ ρ− − − + − − + − (12)

Apart from using iteration in the initial section of the model,which is numerically somewhat costly, the ideal gas law can beused to approximate the initial amount of moles: 0 0 0 0N p V RT= .

This initial guess can be further improved by using thecompressibility factor for initial states close to the critical point,e.g.: 0 0 0 0 cN p V RT Z= for 1.2, 0.5 < 3r rT p< < . Another

solution, especially if the simulation tool does not allow initialiteration, is to set the volume flow to zero, connect sources withthe desired initial temperature and total material potential to thethermal and material port via relatively small resistors and next torun a dynamic simulation in order to find the equilibrium statesthat are the initial states to be computed.

As soon as N0 is known, S0 can be found using (taking sc againas the reference)

10 0 0 0 0ln( )v cS N c T Tξ − = Λ + (13)

with

00 0

0

00 0 02

0 0

(1 )ln ln

( 1) 1 1 7 ( )

2 2

rr

r

c rr r

r r

v bR R v

v b

Z R vv v a

v bv bb

γ

ξ

−Λ = + +−

−+ + − − + +

or approximated by the much simpler expression for the van-der-

Waals case: 110 0 0 02

ln (3 1) ln( )r v cS N R v c T T − = − + .

Now that we have expressions for all efforts as a function of theintegrated flows (extensive states), viz. equations 2, (3), and (11)and a way to initialize the storage element ((13)or (14)), it is readyfor simulation.

SIMULATIONS RESULTSA test-bed model in a plug & play modeling environment [20-

sim, 2000] in which the three port C (van-der-Waals or Paynter)can be easily replaced is used to generate some simple example

env.temp.

thermalport

Paynters gas

heat conduction

mech.port material

port

C

RS

MSf Directsum

0

Se

SfSf1

WaveGenerator1

p0

Figure 1: Test bed model for the three port C representing energy-storage in a Paynter gas (direct graphical model input; signals onlyfor initialization)

0.01 0.1 1 10pr

Z

0.251189

0.398107

0.630957

1

1.58489

2.51189

3.98107

Figure 3: Z- pr-plot (log-scales), critical isotherm for nitrogen, usingintegral causality, explicit integration and high heat conduction(computed points are shown)

simulation runs (Figure 1). A zero flow source (Sf) is attached tothe material port in order to restrict the example experiments to afixed number of moles. The thermal port is connected to theenvironmental temperature via a Fourier type heat conductionrepresented by an RS two-port element. The container is assumedto have a piston to be able to vary the available volume, driven bya velocity source at the mechanical port which imposes either asinusoidal signal to demonstrate loop behavior and thermaldamping or a constant negative velocity representing compression,followed by a constant positive velocity representing expansion.Combined with a high thermal conduction to the environment toguarantee isothermal behavior, the latter mechanical input is usedto create the commonly used log Z – log pr – plots, which may beused to judge the accuracy of the model. A version of the three-port with differential causality at the causal port provides the

env.temp.

thermalport

Paynters gas

heat conduction

mech.port material

port

CMSf Directsum

0

Se

SfSf1WaveGenerator1

p0

Figure 2: Alternative test bed model with thermal port indifferential causality

0.001 0.01 0.1 1 10log pr

log

Z

0.251189

0.398107

0.630957

1

1.58489

2.51189

3.98107

Figure 4: like figure 5, but using the test bed in Figure 2(thermal port in differential causality) and implicit numericalintegration (computed points are shown)

possibility to eliminate the heat conduction (RS) completely toprovide exact isothermal results (Figure 2), but requires use ofimplicit integration schemes like the BDF-method (DASSL). Thelog Z – log pr – plots in figures 3 and 4 demonstrate that thesimulation results are identical. Although this may seem to makethis whole exercise of deriving a symbolic integration superfluous,the reader should keep in mind that implicit numerical integrationis not always a solution, even if it is available, for instance whenthe submodel is embedded in a model with certain discontinuitiesor nonlinearities that do not match well with implicit integration.

Figure 5 shows cycles in the p-v-plane and T-v-plane near thecritical point to demonstrate that the behavior is as expected.Figure 6 demonstrates in a multiple run for n = 1, 2, 3 that thethermal damping (area of the cycle) decreases when the heatcapacity and thus the time constant increase, i.e. for a higher

0.8 0.9 1 1.05 1.15vr

pr Tr

0.5

1

1.5

2

2.5

3

3.5

0.89675

0.945137

0.993525

1.04191

1.0903

1.13869

1.18707

Figure 5: pr-vr- (o) and Tr-vr-cycles (x) for nitrogen showingthermal damping for a much lower heat conduction than in Figure 3.

0.5 1 1.5 2 2.5 3 3.5vr

pr

0.5

1

1.5

2

2.5

3

3.5

4

Figure 7: pr-vr-plot for nitrogen: isotherms for various temperatures(119 – 139 K in steps of 2 K) below and above the critical point (127K)

number of atoms per molecule. Figure 7 gives the results of

isotherms in the pr-vr-plot for nitrogen below and above thecritical point (127 K). In Figure 8 the value for a was reduced from

7 to 6 in order to match the resulting Z-pr-plot (log-scales) for

nitrogen with isotherms for T = 120,127 (Tc),165, 210, 255, 300K,

pc=3.3 Mpa with the experimental version in figure 21.1 in[Gyftopoulos and Beretta, 1991]. Next some comparisons are madebetween a van der Waals gas and Paynter’s modification. Figure 9shows that the compressibility factor Z takes off to infinity earlier

for increasing density ρ . Finally Figure 10 shows pr-vr-cycles forthe van-der-Waals and Paynter cases.

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2vr

pr

Pn

tr

0.6

0.8

1

1.2

1.4

1.6

Figure 6: pr-vr- cycles for n = 1, 2, 3, showing a decreasingenclosed area (thermal damping) for increasing n (Paynter gas)

a=6

0.01 0.1 1 10

log pr

co

mpr

es

sib

ility

fa

cto

r Z

0

0.5

1

1.5

2

Figure 8: Z-pr-plot (log-scale), for nitrogen with a=6; isothermsfor T = 120,127 (Tc),165, 210, 255, 300K, pc=3.3 MPa

CONCLUSIONIn this paper the preferred integral causal form of Paynter’s

veridical state equation [1985] has been derived in order to make itavailable for port-based modeling and simulation, in which storageelements in integral causality are preferred and sometimes evenrequired. This is not only done in order to be able to use explicit

numerical simulation schemes (other nonlinear elements mayobstruct the use of implicit integration schemes for instance), butalso in order to prevent omission of the thermal time constant dueto heat exchange with the environment, as the differential causalityof the port may mislead the modeler to just choose theenvironmental temperature. Pourmovahead and Otis [1984] haveshown that this thermal damping often has a considerable dynamiceffect, such that it should not be omitted from a model a priori.

As Paynter’s extension is non-analytic, the expression was firstseparated in analytical expressions on separate intervals,substituted in an expression for the total differential of the entropyafter which integration could be performed term by term as toobtain the desired expression for the temperature. After rewritingthe expression could be collected again using the non-analyticterms ( )sgn 1rvξ = − in order to compensate sign changes and

( )12 1 ξ− in order to allow additional terms. The resulting

expressions were written in terms of the critical pressure,temperature, compressibility factor and Riedel parameter toprovide for a generic submodel.

1 10log rho

log

Z P

ntr

log

Z v

dW

1

10

Figure 9: compressibility factor versus density for Paynter (o) andvan der Waals gas (x; van der Waals curve taking off to infinityearlier for increasing ρ)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2vr

pr P

ntr

pr v

dW

1

2

3

4

5

Figure 10: pr-vr-cycles for van der Waals (x) and Paynter (o)case

An important aspect that is often not well understood is that thefact that initial conditions have to be specified in terms of thetemperature, due to the nature of the relations, does not imply thatthe temperature should be the independent variable duringsimulation. As soon as the initial temperature (and pressure) can beused to find the proper initial conditions for the extensive states,simulation can take place in an optimal way, viz. in integralcausality, thus allowing robust explicit integration schemes likeRunge-Kutta 4th order. Herein, expressions were derived andstrategies proposed to find the initial extensive states in an initialsection of the model (only computed at the first time step).

REFERENCES20-sim, 2000, port-based modeling and simulation software, see:

www.20sim.com for more information and free demo (noimplicit integration); contact author for models used herein.

Breedveld, P.C., 1984, ‘Physical systems theory in terms of bondgraphs’, Ph.D. Thesis, Electrical Engineering, University ofTwente, Netherlands, ISBN 90-9000599-4.

Breedveld, P.C., 1991, ‘An alternative formulation of the stateequations of a gas’, Entropie, énergétique et dynamique dessystèmes complexes, Vol. 164/165, pp. 135-138, ISSN 00139084.

Breedveld, P.C., 2000, ‘Constitutive relations of energy storage ina gas in preferred integral causality’, to be published inProceedings IEEE IECON 2000, Nagoya, Japan, October 22-27,2000.

Çengel, Y.A. and Boles, 1989, M.A., ‘Thermodynamics, anengineering approach’, McGraw-Hill, N.Y.

Gyftopoulos, E.P. and Berretta, 1991, G.P., ‘Thermodynamics –Foundations and Applications’, MacMillan, N.Y.

Karnopp, D.C. and Rosenberg, R.C., 1974, ‘System Dynamics: AUnified Approach’, Wiley, N.Y.

Paynter, H.M., 1985 ‘Simple Veridical State Equations forThermofluid Simulation: Generalization and ImprovementsUpon Van der Waals’, ASME Journal of Dynamic Systems,Meas. & Control, Vol. 107, No. 4, pp. 233-234.

Pourmovahed, A. and Otis, D.R., 1984, ‘Effects of thermaldamping on the dynamic response of a hydraulic motor-

accumulator system’, ASME Journal of Dynamic Systems, Meas.& Control, Vol. 106, No. 1, pp. 21-26.

About the authorPeter Breedveld is an associate professor

with tenure at the University of Twente,Netherlands, where he received a B.Sc. in1976, an M.Sc. in 1979 and a Ph.D. in1984. He has been a visiting professor atthe University of Texas at Austin in 1985and at the Massachusetts Institute ofTechnology in 1992-1993. He is or hasbeen an industrial consultant. He initiatedthe development of the modeling andsimulation tool that is now commercially available under the name20-sim. In 1990 he received a Ford Research grant for his work inthe area of physical system modeling and the design of computeraids for this purpose.

He is an associate editor of the ‘Journal of the Franklin Institute’,SCS ‘Simulation’ and ‘Mathematical and Computer Modeling ofDynamical Systems’. His scientific interests are: Integratedmodeling, control and design of physical systems; graphical modelrepresentations (bond graphs); generalized thermodynamics;computer-aided modeling, simulation, analysis and design;dynamics of spatial mechanisms; mechatronics; generalizednetworks; numerical methods; applied fluid mechanics; appliedelectromagnetism; qualitative physics; surface acoustic waves inpiezo-electric sensors and actuators.