Surface Acoustic Wave (SAW) technology is low cost, rugged, lightweight, extremely low power and can be used to develop passive wireless sensors. For these reasons, NASA is investigating the use of SAW technology for Integrated Vehicle Health Monitoring (IVHM) of aerospace structures. To facilitate rapid prototyping of passive SAW sensors for aerospace applications, SAW models have been developed. This paper reports on the comparison of three methods of modeling SAWs. The three models are the Impulse Response Method (a first order model), and two second order matrix methods; the conventional matrix approach, and a modified matrix approach that is extended to include internal finger reflections. The second order models are based upon matrices that were originally developed for analyzing microwave circuits using transmission line theory. Results from the models are presented with measured data from devices. Keywords: Surface Acoustic Wave, SAW, transmission line models, Impulse Response Method

Atkinson, Gary

Wilson, William

NASA Technical Reports Server

 1
 
 
 
 
 
 
 
 
 
 
Comparison of Transmission Line Methods  
for Surface Acoustic Wave Modeling 
 
1William WILSON, 2Gary ATKINSON 
1NASA Langley Research Center, Hampton, VA, USA,  
2Virginia Commonwealth University, Richmond, VA, USA,  
Phone: +1 (804) 827-0185, Email: GMAtkins@vcu.edu  
 
 
 
 
 
 
Abstract: Surface Acoustic Wave (SAW) technology is low cost, rugged, lightweight, extremely 
low power and can be used to develop passive wireless sensors.  For these reasons, NASA is 
investigating the use of SAW technology for Integrated Vehicle Health Monitoring (IVHM) of 
aerospace structures.  To facilitate rapid prototyping of passive SAW sensors for aerospace 
applications, SAW models have been developed.   
This paper reports on the comparison of three methods of modeling SAWs.  The three models are 
the Impulse Response Method (a first order model), and two second order matrix methods; the 
conventional matrix approach, and a modified matrix approach that is extended to include internal 
finger reflections.  The second order models are based upon matrices that were originally developed 
for analyzing microwave circuits using transmission line theory.  Results from the models are 
presented with measured data from devices. 
 
Keywords: Surface Acoustic Wave, SAW, transmission line models, Impulse Response Method. 
 
 
 
1. Introduction 
 
First order models of SAW devices are based upon the Impulse Response[1, 2].  These models do 
not take into account second order effects such as internal reflections, frequency shifts, or allow for 
any physical arrangement other than equal electrode widths and spaces.  For more accurate results, a 
matrix based approach was developed [3].  This approach has been further refined and modified to 
include internal finger reflections [4].  The reflections occur when the thickness of the metallization is 
sizeable enough to result in significant reflections.  The extensions are based upon matrices that were 
https://ntrs.nasa.gov/search.jsp?R=20110016371 2019-08-30T17:36:48+00:00Z
 2
originally developed for analyzing microwave circuits using transmission line theory.  The 
modifications are accomplished by breaking up the SAW device into zones, where the area under a 
metalized region is treated as one zone, and the area without metallization is treated as another zone.  
The impedance discontinuities that occur at the edges of the metal fingers enable the simulation of the 
internal reflections of the mechanical acoustic wave.  The modifications also enable incorporation of 
the different velocities for each region, which produces a more accurate characterization of the 
frequency response of the device.  The modifications also allow for unequal finger widths and spacing.   
In this paper, the three modeling methods are briefly discussed, then a comparison of the three 
models with results from prototype devices are presented.  
 
 
2. First Order Modeling using the Impulse Response Method 
 
The Impulse Response method [1] was used as the baseline for modeling the SAW device.  This 
method is valid only for transducers where at least one of the two Inter-Digitated Transducers (IDTs) 
has a constant aperture or finger overlap [3].  This first order model includes both the mechanical and 
electrical behavior of SAW devices.  It calculates the frequency response, the loss of the system, the 
admittance, and parameters for circuit simulators.  This model assumes constant and equal spacing and 
finger widths.  A simple circuit model (Fig. 1) can be used to convey the basic elements of the Impulse 
Response Model.  The figure shows the source voltage and both the source and load impedances which 
are not part of the model.  In the circuit model CT is the total capacitance, Ba(f) is the acoustic 
susceptance and Ga(f) is the radiation conductance.  
 
 
  (a)        (b) 
 
Fig. 1. (a) Basic SAW delay line and (b) the circuit model used in the Impulse Response Modeling.  CT is the 
total capacitance, Ba(f) is the acoustic susceptance, and Ga(f) is the radiation conductance. 
The frequency response of a SAW device can be calculated by using the Impulse Response Model 
and is given by [5]: 
 
  
0
2
2 2
0
sin
( ) 20log 4
pN Dj
f
s a r p
X
H f k C H f Z N e
X
     
        
. (1) 
 
Where k is the piezoelectric coupling coefficient, Cs is the capacitance per finger pair and unit 
length, Ha is the aperture or overlap height of the fingers, f0 is the center or synchronous frequency, Np 
is the number of finger pairs, f is the frequency, D is the delay length between the IDTs, and Zr is the 
wave impendence and X is given by:   
 
 
0
0 )(
f
ffNX p
  . (2) 
Ba(f) CT 
Ga(f) 
IDT 
Ba(f)CT 
Ga(f) 
ZS 
IDT
VS 
V2V1 ZL
Piezoelectric Substrate 
IDT IDT 
Ha 
Wf 
Surface Acoustic Wave 
Ht 
 3
3. Conventional Matrix Method 
 
The matrices discussed in this work were originally developed for analyzing microwave circuits 
using transmission line theory.  The methodology utilizing transmission matrices was modeled on the 
approach given by Campbell [3].  This method is based upon the Mason equivalent circuit using the 
crossed field technique (Fig. 1b) [4].  In this method, for modeling purposes, a SAW device is modeled 
as two IDTs with two electrical ports, and two acoustic ports.  The acoustic ports represent mechanical 
waves traveling into and out of the IDT.  The electrical ports represent the current and voltage of the 
IDT.  The matrix for a SAW delay line is simply the multiplication of the matrices for the two IDTs 
and a matrix for the delay in between (Fig. 2). 
 
 
Fig. 2. Transmission line matrix model of a complete SAW delay line.   
 
 
The transmission matrix is often used because it can be cascaded easily.  The complete SAW device 
matrix is given by: 
 
        1 1 2( )SAW f T f D f T f   . (3) 
 
Where f is the frequency, Ti(f) is the transmission matrix for an IDT, and D(f)is the delay matrix.  
The delay matrix (D(f)) is modeled after an acoustic transmission line and is given by: 
 
 
 
2
2
0
( )
0
fj d
v
fj d
v
e
D f
e


     
 , (4) 
 
where λ is the wavelength at the center or synchronous frequency and d is the distance between the 
reference planes, or in this case the center of the two IDTs.   
 
 
 
 
w3+ 
w3- 
w0+ 
w0- 
t1 t3 D2 
W0 W1 W2 W3 
a1 a3 b1 b3 
w2+ 
w1-
w1+ 
w2-
 4
Each IDT is modeled separately as a single transmission matrix (Ti(f)) with one electrical port, and 
two acoustic ports (Fig. 3).  
 
Fig. 3. Transmission matrix model of an IDT.   
This allows the acoustic waves (Wi) and electrical parameters (ai and bi) to be related through the 
use of transmission matrix Ti(f) in: 
 
 
1
1 ( )
i i
i i i
i i
W W
W T f W
b a
 

 

               
 . (5) 
 
The transmission matrix is decomposed into sub-elements, given by 
 
 
11 12 13
12 22 23
13 23 33
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
i
t f t f t f
T f t f t f t f
st f st f t f
      
 . (6) 
 
The elements of the transmission matrix are given by:  
 
 
0
( )( )( )
(1 ( ))
a s e
e
G f R Zt f
j f
   , (7) 
 
 ( )
11 0( ) (1 ( )) t
j ft f t f e    , (8) 
 
 12 0( ) ( )t f t f  , (9) 
 
 ( )
2
13
(2 ( ) )
( )
1 ( )
j ft
a e
e
G f Z
t f e
j f

   , (10)
 
 ( )
22 0( ) (1 ( )) t
j ft f t f e    , (11)
 
 ( )
23 13( ) ( ) t
j ft f t f e    , (12)
 
 
33
2 ( )( )
1 ( )
c
e
j ft f
j f

   . (13)
w+i-1 
ai bi 
w-i-1 
w+i 
w-i 
Electrical Port 
Acoustic 
Port 
Acoustic 
Port 
 5
The symmetry factor s is based upon the number of fingers (Nt): 
 
 1 tNs    . (14)
 
The transit angles θe , θc, and θt,are given by: 
 
 ( ) ( )( )e t a s ef C B R Z     , (15)
 
 ( ) ( )c t s ef C R Z    , (16)
 
 
( )
2t t
f N    . (17)
 
The combined load or source impedance is Ze, and Rs is the combined IDT metal and lead 
resistance.  The frequency detuning parameter (δ) is given by  
 
 0
11
2 ( )f f k
v
    , (18)
 
where v is the velocity, and k11 is the mutual-coupling coefficient and is given by: 
 
 2
11 11 11 11p m s
h hk k k k 
        . (19)
 
where h is the height of the metal fingers.  The mutual coupling coefficient parameters k11p, k11m, 
and k11s, are material dependent, for quartz they are 0.0004, 0.02, and 7.9 [3].  The total capacitance for 
the IDT is given by: 
 
 ( 1)
2
t
t s
NC C  , (20)
 
where Cs is the capacitance per unit length for a pair of fingers.  The conductance (Ga) is given by [3]: 
 
 2
2 2
0
sin
28 ( 1)
2
t
a s t
t
G k C f N


           
. (21)
 
And finally, the susceptance (Ba) is given by [3]: 
 
  2 2
0 2
sin
16 ( 1) t ta s t
t
B k C f N
 

       , 
(22)
 
 
 
 
 
 6
4. Modified Matrix Method 
 
For more accurate results than are given by the first order approach or the conventional method, the 
matrix approach was extended to include second order effects such as the internal finger reflections [3, 
6].  The model divides an IDT into ½ wavelength sections.  These sections are further divided into 
zones.  Two of the zones are un-metalized areas (1/8 of a wavelength) around one zone that is 
comprised of a metal finger (1/4 of a wavelength).  Each zone is modeled by a transmission line matrix 
equivalent circuit (Fig. 4).  Two identical circuits model the un-metalized areas, while the middle 
circuit models the area under the metal finger.  The transmission matrix relates the SAW voltages V1 
and V2, to the currents I1 to I2 respectively.  The acoustic wave is assumed to have entered from the left 
and travels through the element towards the right.  In this model Zu and Zm are the acoustic 
impedances for the metalized and un-metalized areas, C0 is the capacitance for a single finger, u and 
m are the acoustic angles of the substrate, and the turns ratio of the transformer is assumed to be 1:1.   
 
 
 
Fig. 4.  Crossfield Model using Mason Equivalent Circuit for a ½ wavelength section of an IDT. 
 
The transmission matrix representing the middle area shown in Fig. 4 is the circuit for a metalized 
region that is assumed to be lossless and is given by: 
 
 
 
cosh( ( )) sinh( ( ))
( ) 1 sinh( ( )) cosh( ( ))
m m m
m
m m
m
j f Z j f
R f
j f j f
Z
 
 
     
. (23)
 
The transmission matrix (23) is determined by the acoustic angle m and the metalized region’s 
impedance Zm.  The impedance Zm is calculated with: 
 
 
2
1( )m
s a m
Z f
k C H f
 , (24)
V2 
I2 
Piezoelectric Substrate 
tan
2
m
mjZ
   
 cscm mjZ 
1:1 
C0 
Metal Finger 
I3 
V3 
tan
2
u
ujZ
   
 cscu ujZ 
I1 
1:1 1:1 
tan
2
u
ujZ
   
 cscu ujZ 
V1 
0
8
 0
4
 0
8

 7
 
where k2 is the piezoelectric coefficient, Cs is the capacitance for a single pair of electrodes per unit 
length, and Ha is the aperture width.  The acoustic angle of the substrate m, is given by 
 
 ( )
2m m
ff
f
   , (25)
 
where f is the frequency, and fm is the frequency shift caused by the velocity change as the acoustic 
wave travels under the metalized area: 
 
 m
m
vf   . (26)
 
Where vm is the acoustic wave velocity under the metalized area and λ is the wavelength of the 
device.  The metalized velocity (vm) is 3134 m/s for ST cut Quartz.  
The matrix (23) calculates the parameters for the metalized area, but cannot be used for the un-
metalized sections. This leads to the transmission matrix (Ru(f)) for the un-metalized region as is given 
by: 
 
 
 
cosh( ( )) sinh( ( ))
( ) 1 sinh( ( )) cosh( ( ))
u u u
u
u u
u
j f Z j f
R f
j f j f
Z
 
 
     
 . (27)
 
The transmission matrix (27) is determined by the acoustic angle u and the un-metalized region’s 
impedance Zu.  The impedance Zu is calculated with: 
 
 
2
0
1( )u
s a
Z f
k C H f
  . (28)
 
where f0 is the synchronous frequency of the acoustic wave under for the un-metalized area.  The 
acoustic angle of the substrate u, is given by 
 
 
0
( )
4u
ff
f
   . (29)
 
where f0 is the synchronous frequency of the acoustic wave, which is calculated using the acoustic 
wave velocity under the un-metalized area.  The un-metalized velocity (v) is 3158 m/s for ST cut 
Quartz. 
To find the transmission matrix for the ½ wavelength periodic element (RT(f)) one must multiply the 
three matrices together for both metalized region and the un-metalized regions adjacent to it: 
 
      ( ) ( ) ( ) ( )t u m uR f R f R f R f  . (30)
 
To find the transmission matrix (Q(F)) for an entire IDT one simply raises the (RT(f)) matrix to the 
power of 2Np: 
 
    21 ( ) pNtIDT f R f    . (31)
 
Where Np is the number of electrode pairs, so 2Np is the total number of electrodes in the IDT.   
 8
The matrix for a SAW delay line is simply the multiplication of the matrices for the two IDTs and 
the delay or space between the IDTs.  The SAW matrix is given by: 
 
        1 1 2SAW f IDT f D f IDT f                . (32)
 
5. Prototype Design 
 
A simple SAW delay line that consists of two identical un-apodized IDTs was chosen as a 
prototype to illustrate the capability of the models.  Each IDT has 63 finger pairs that are 17 μm wide.  
The spacing between the fingers is 17 μm also. The center or synchronous frequency is 46.44 MHz, or 
a wavelength of 68 μm.  The aperture height is 2730 μm.  The delay length between the IDTs is 10 
wavelengths or 680 μm.  The design was fabricated on two different ST cut quartz wafers.  One with a 
single side polished, and one wafer with both sides polished.  The aluminum thickness is 50 nm for the 
wafer with a single side polished and 250 nm for the wafer with both sides polished.  Also, a 112.78 
MHz device was fabricated on the single side polish wafer.  It has 90 pairs of 7 μm fingers, a wavelength 
of 28 μm, and an aperture height of 1105 μm.  The delay length is 104 wavelengths or 2.904 mm.   
 
6. Results 
 
All three methods adequately model the frequency response and amplitude for the main lobe and 
the first two side lobes for cases without any mass loading due to the metal fingers (Fig. 5).  The 
modified matrix more accurately captures the frequency shift due to the mass loading of the metal 
fingers.  In Figure 6, the ideal first order model (impulse response model) and the conventional matrix 
results are both centered about the synchronous frequency.  The measured results and the Modified 
matrix results are both shifted down in frequency due to velocity changes from mass loading effects. 
When comparing the measured data from the two figures, it is noticeable that the main lobe peak of 
Fig. 5 does not have the same artifacts as are seen on the peak of the main lobe in Fig. 6.  These 
artifacts are due in part from bulk waves that are reflected from the polished bottom surface of the 
wafer.  The roughness of the non-polished surface disperses the bulk waves which results in 
diminished artifacts in the main lobe peak (Fig. 5).  Also note that the peak in Fig 6 is shifted farther 
away by mass loading than the peak in Fig. 5. 
 
44.3 45 45.7 46.4 47.1 47.8 48.5
60
50
40
30
20
Impulse Response
Conventional Matrix
Modified Matrix
Measured
Frequency (MHz)
In
se
rti
on
 L
os
s (
dB
)
 
Fig. 5.  Comparison of model results with data from a single side polish wafer, with 50 nm of aluminum. 
 9
44.3 45 45.7 46.4 47.1 47.8 48.5
60
50
40
30
20
Impulse Response
Conventional Matrix
Modified Matrix
Measured
Frequency (MHz)
In
se
rti
on
 L
os
s (
dB
)
 
Fig. 6.  Comparison of model results with data from a double side polish wafer, with 250 nm of aluminum. 
 
While the models work well for the side lobes of low frequency devices at 46.4 MHz, as the frequency 
increases the accuracy goes down for the side lobes (Fig. 7).  The measured data shows higher order 
effects and noise at 112.8 MHz that is not present in the lower frequency device data.  These results 
show that opportunities exist for improvements to both the models and the test setup.  The inclusion of 
higher order terms to the models will increase their fidelity.  The fidelity of the test setup can be 
improved by dicing the wafer or by the inclusion of gating functions in the time domain.  Time gating 
will allow the removal of unwanted reflections from adjacent devices on the test wafer without having 
to dice the wafer and add absorber material to the devices.  Essentially the addition of the gating 
functions will allow us to take data on the wafer, while achieving the results as if the devices had been 
packaged in an ideal manner. 
 
109 109.5 110 110.5 111 111.5 112 112.5 113 113.5 114 114.5 115 115.5 116 116.5
60
50
40
30
20 Impulse Response
Conventional Matrix
Modified Matrix
Measured
Frequency (MHz)
In
se
rti
on
 L
os
s (
dB
)
 
Fig. 7.  Comparison of model results with data from a single side polish wafer, with 50 nm of aluminum. 
 
 
 10
7. Conclusion 
 
In this paper, results from prototype devices are compared to the Impulse Response model, a 
conventional matrix model, and a modified matrix model.  In the absence of mass loading from the 
metal fingers and the associated frequency shift, all three methods model the main lobe and next two 
side lobes fairly well.  However, the results show that the modified matrix methods more accurately 
modeled second order effects such as frequency shift due to the metal thickness of the IDT fingers.  
Neither the Impulse response method nor the conventional matrix method model the frequency shift.  
Therefore, the modified matrix method is the most accurate for a wider range of parameters such as 
finger metal thickness.  Future work will include a comparison the Coupling of Modes, the Impulse 
Response model, and the modified matrix model.  Also, to achieve higher fidelity our future models 
will include higher order terms.  To improve our wafer level test setup, time gating in the time domain 
will be added to remove artifacts that would not be present if the devices were packaged in an ideal 
manner with absorber materials. 
 
 
References 
 
 
[1]. C. S. Hartmann, Jr., D. T. Bell and R. C. Rosenfeld, "Impulse Model Design of Acoustic Surface-Wave 
Filters," Microwave Theory and Techniques, IEEE Transactions on, vol. 21, pp. 162-175, 1973. 
[2]. W. C. Wilson and G. M. Atkinson, "Frequency Domain Modeling of SAW Devices for Aerospace Sensors," 
Sensors & Transducers vol. 2, pp. 42-50, October 8 2007. 
[3]. C. Campbell, Surface Acoustic Wave Devices for Mobile and Wireless Communications, New York, NY, 
Academic Press, 1998. 
[4]. W. P. Mason, Physical acoustics vol. 1A, New York and London, Academic Press, 1964. 
[5]. V. M. Ristic, Principles of acoustic devices, New York, NY, Wiley, 1983. 
[6]. H. Y. Tung, "The Design of Surface Acoustic Devices using the Transmission Matrix Method," I-Shou 
University, 2005. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


Comparison of Transmission Line Methods for Surface Acoustic Wave Modeling

Surface Acoustic Wave (SAW) technology is low cost, rugged, lightweight, extremely low power and can be used to develop passive wireless sensors. For these reasons, NASA is investigating the use of SAW technology for Integrated Vehicle Health Monitoring (IVHM) of aerospace structures. To facilitate rapid prototyping of passive SAW sensors for aerospace applications, SAW models have been developed.

 

This paper reports on the comparison of three methods of modeling SAWs. The three models are the Impulse Response Method (a first order model), and two second order matrix methods; the conventional matrix approach, and a modified matrix approach that is extended to include internal finger reflections. The second order models are based upon matrices that were originally developed for analyzing microwave circuits using transmission line theory. Results from the models are presented with measured data from devices

Comparison of Transmission Line Methods for Surface Acoustic Wave Modeling

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