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Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
Simplifying SAW calculations
Adam Kawalec
Military University of Technology,
E-mail: mega@\vic. wat. waw.pl
Abstract
Numerical analysis of acoustic wave propagation in YZ lithium niobate covered
by ideal periodic metal strips is presented. A Mathematica package has been
developed for the analysis. Graphical results are presented for the surface
acoustic wave velocity and wave number for wide range of metallization ratios
and frequencies.
1
Introduction
Interdigital transducers may consist of thousands of metal strips. This allows us
to apply an approximation such that the surface acoustic wave (SAW) is
excited, propagates and is detected in an infinite periodic system of strips. This
key approximation, and the known B10tekjaer's theory [1] are the foundation of
a spectral theory for the interdigital transducer (IDT).
Spectral theory is a field theory, where we start from fundamental relations,
admit some well defined simplifications to get closed analytical expressions for
currents of strips contributing to one or another IDT bus-bar.
The theory has been further developed by E.Danicki [2-4] by solving an
inhomogeneous problem.
It is known that the ratio of metal strip width to the strip period (the
metallization ratio) influences the interdigital transducer frequency response,
especially of higher overtones. This phenomenon, rarely exploited in practice,
can be applied in casting the frequency characteristics of simple delay lines with
applications in inexpensive feedback oscillators working in the microwave
range[5].
Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
300 Innovation In Mathematics
In this paper we neglect all second-order effects in modeling periodic IDTs by
applying the spectral theory with bulk waves and Bragg reflections neglected.
This problem can be solved by numerically only for higher overtones.
Mathematica makes it possible to simplify solutions and graph results.
2
Fundamentals of periodic strips
For harmonics SAWs of angular frequency v and wave number r (a variable),
the Ingebrigsen's effective surface permittivity (EPS) is [6]
(1)
where <p is the electric potential on the surface of the substrate, AD is
the surface electric charge, k^^v/v^,
Vv,o are the SAW velocities of free (v), and metallized (o) surface of the
substrate and % is the effective permittivity.
Spectral theory of periodic strips [2] yields the relationship for the current /„
flowing to n* grounded strip resulting from the applied voltage K% to the m^
strip (Fig. 1).
Fig. 1: Metal strips on a substrate
K
/„ = jtos.
(2)
0
where K = 2jc/A, A is the strip period (in computation we applied K = 1), and
and Ajcr are determinants of the systems of equations for a*.
,=0
(3)
Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
Innovation In Mathematics 301
taken for n = 0, 1, 2, 3, ... N, appended by
(4)
for open strips or
(5)
for short-circuited strips, respectively. In (2)-(5), r e (0, K), and P denotes a
Legendre function, A = ATw/2, w is the strip width, S»= 1 for n > 0 and -1 otherwise, N is chosen larger then Nx=(k/K). k = fa ik,)/2.
In the case of a fundamental band, simple analytical expression for wave
numbers have been obtained from (3)-(5):
(7)
The influence of metallization ratio on wave number is important for higher
value of Av/v (fig. 2). This is characteristic of a piezoelectric substrate with
electromechanical coupling (l/2kc= Av/v).
r/K
Fig. 2: Wave numbers in fundamental band for open (TV) and short-circuited (r<,) strips
respectively.
Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
302 Innovation In Mathematics
1.01
1.005
0.04
r/K=0.4
0.02
dv/v Q.03
0.04
0.04
0.01
0.02
dv/v
0.04
0.04
0.04
0.04
Fig. 2a: Wave numbers in fundamental band for open (TV) and short-circuited (ro) strips
respectively for other r/K value (dv/v denotes Av/v).
Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
Innovation In Mathematics 303
Note that there are much more complicated expressions for higher overtones.
The zero of A^r), and the zero of AjcKr), which are at /v and r*, are the wave
numbers of SAWs propagating in free and short-circuited strips, reduced to the
Brillouin zone (0, K), respectively for fundamental band and higher overtones.
The corresponding SAW velocities are s^v/fa+NxK) and So^v/fa+NxK). Those
values are closely related to the SAW generation efficiency by strips, and are
defined as the residue of the integral (2) at the pole, r<> (AV, the zero of7f(r), is
close to TO).
3
The MATHEMATICA Package
Simple code is shown below for evaluation TV and r<,, which are the most
important in the numerical analysis (DK means K, v is frequency, d means cos(A)
and other notations are obvious).
v=2.;DK=l.;vo=3.4037;w=3.4878;NM2=2;
NM:=N[Floor[2.*v/(DK*(w+vo))]]
ss:=N[2.*v/(DK*(w+vo))-Floor[2.*v/(DK*(w+vo))]];s:=ss;
(*Ingebrigsten's effective surface permittivity*)
f[kfJ:=N[Sign[kf]*((kf-v/w)/(kf-v/vo))]
g:=N[(l-f[k-H-s]*Sign[k-j-f.5])*LegendreP[k-j,d]]
(*equation (3)*)
A:=Table[g,{k,NM+NM2-l},(j,NM-fNM2}]
(* equation (4)*)
B:=Table[N[LegendreP[-ss-l+l,d]*(s+NM-v/vo)],{l,NM+lSfM2}]
(^equation (5)*)
BB:=Table[N[(-l)^l-l)*LegendreP[-ss-l+l,-d]*
(s+NM-v/vo)],{l,NM+NM2}]
(*system of equations (3),(4)*)
X:=Table[If[ii<=lSM+NM2-l,A[[ii]],B],{ii,NM+NM2}]
(*system of equations (3),(5)*)
XX:=Table[Iftii<=NM+NM2-l,A[[ii]],BB],{ii,NM+NM2}]
sv: =N[v/w-Floor [v/w] ]
so:=N[v/vo-Floor[v/vo]]
(*calculations of system of equations (3),(4) *)
datn=Table[{d,s}/.FindRoot
pet[X]=0,{s,{.993*sv,so*1.007}}],
{d,-0.999,0.999,0.01}];
(*calculations of system of equations (3),(5) *)
dato=Table[{d,s}/.FindRoot
[Det[XX]=0,{s,{.993*sv,so*1.007}}],
{d,-0.999,0.999,0.01}];
Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
304 Innovation In Mathematics
4
Numerical results
4.1 Wave numbers r for fundamental band and higher overtone
Fig. 3 shows the dependence of wave numbers on metallization ratio and on
normalized frequency defined as /=2n/£/(Vv+Vo) for open and short-circuited
system of strips.
0.5875
-0-5 Metallization ratio, d 0.5
_o 5
Metallization ratio, d 0.5
Fig. 3: Wave numbers in the first, second and third frequency band, respectively. In the first
band and the second one there are differences between wave numbers for open and shortcircuited strips.
Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
Innovation In Mathematics 305
Curves are presented for other normalized frequencies /, which shows
fundamental band and higher overtone.
Note that at the fundamental band, we can only observe the difference between
the wave numbers of open and short-circuited system if the metallization-ratio
(d) is zero.
Wave number is increasing near the point d = -l(wide strips), for a shortcircuited system. The wave number is decreasing for d = 1 ( narrow strips) in
the case of open-strips.
4.2 Phase velocity S
for open strips:
.5
.5
for short-circuited strips:
Fig. 4: Dependence of phase velocity in wide band on normalized frequency.
Fig.4 shows the dependence of SAW phase velocity for open and shortcircuited strips.
Note that the phase velocity will change between the SAW velocity for a free
and for a metallized surface in the case of a piezoelectric substrate.
As we can see , wide strips (d = -0.9) and narrow strips (d = 0.9) have a
significant influence on SAW phase velocity.
Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533
306 Innovation In Mathematics
5
Conclusion
The above simple notebook is an example of using Mathematica for SAW
calculations.
We have studied how the metallization ratio, from extreme low values, to
extreme high values, affects the wave phase velocity and SAW number at
fundamental pass-band and at higher overtones.
Mathematica gives us simplicity in numerical calculations and has the following
advantages in SAW calculations:
• calculation of complicated system of equations
• simple graphics
• simple calculations because Mathematica can manipulate Legendre functions
which are used in SAW calculations
• interpolation of functions such as SAW wave number for other metallizationratios, used to obtain frequency characteristics of SAW filters.
References
1. K. B10tekjaer, K. Ingebrigsten, H.Skeie-A method for analysing waves in
structures consisting of metal strip on dispersive media, IEEE Trans. 1973,
ED-20, 1133-1138.
2. E Danicki, Theory of periodic interdigital transducer of surface acoustic
waves, J.Tech.Phys, 1978, 19, 89-102.
3. E Danicki, Unified theory of interdigital transducers and SAW reflectors,
J.Tech. Phys., 1980, 21, 387-403.
4. E Danicki, Spectral theory of IDT's, Ultras. Symp. Proc., 1994, 213-222.
5. E.Danicki, A.Kawalec Metallization-ratio dependent higher overtone
operation of IDTs, IEEE Ultras. Symp. Proc., 1996
6. K.A.Ingebrigsten, Surface waves in piezoelectrics, J. Appl. Phys.1969, 40,
2681-2686.