Appendix A
Coupled k and q Coefficients For Ladder Filter Design
This Appendix is referenced in Section 5.5.3 of the text. Coupling coefficients relating
the impedance values of neighboring ladder filter stages can be generated to aid in filter
design. An excellent explanation and definition of the k and q coefficients is found in
[25]. This section will only summarize the equations necessary to generate these
constants as found in [26].
The calculation begins by finding a series of values known as g-codes. These determine
the element values of a low-pass prototype filter. For a Chebychev filter of order n and
passband ripple LAr (in dB), the g-codes g0,1,…,n,n+1 are given by the following equations:
L ln 10 

B  ln  coth Ar

40 

 B
c  sinh  n 
2 


 2k  1  
ak  sin 
 , k = 1, 2, … , n
n
 2

 k 
bk  c 2  sin 2 
 , k = 1, 2, … , n-1
 n 
g0  1
g1 
2a1
c
gk 
4a k a k 1
, k = 2, … , n
bk 1 g k 1
g n1  1 for n odd
 B
 coth 2   for n even.
4
95
For the special case of LAr = 0 (a Butterworth response), the equations are:
g 0  g n1  1
 (2 k  1) 
g k  2 sin 
 , k = 1, 2, … , n
n
2


In both bases, the codes are normalized such that g0 = 1.
These equations give normalized k and q values as can be found in many published
tables. The index refers to the ladder filter stages, as explained in the references.
K j , j 1 
% BW
g j g j 1
QSOURCE 
QLOAD 
g 0 g1
% BW
g n g n1
%BW
where the fractional bandwidth is defined as the 3dB bandwidth divided by the center
frequency:
%BW 
BW3dB
f0
To get normalized k and q values, set the fractional bandwidth to one.
96
Appendix B
Material Parameters of Aluminum Nitride
  3.3  10 3 kg/m 3
e31  e32  0.58 C/m 2
e33  1.55
C/m 2
e14  1.13 C/m 2
e15  0.48 C/m 2
c11  3.45  1011 N/m 2
c13  1.20  1011 N/m 2
c33  3.95  1011 N/m 2
c44  1.18  1011 N/m 2
c66  1.10  10 8 N/m 2
 o  8.85  10 12 F/m
 33  9.5   o
97
References
[1] Lutsky, J., “A sealed cavity thin-film acoustic resonator process for RF bandpass
filters.” Ph.D. diss., Massachusetts Institute of Technology, 1997.
[2] Meier, H., Baier, T., and Riha, G., “Miniaturization and advanced functionalities of
SAW devices,” IEEE Transactions on Microwave Theory and Techniques, vol. 49,
no. 2, pp. 743-748, Apr. 2001.
[3] Morkner, H., Ruby, R., Frank, M., and Figueredo, D., “An integrated FBAR filter
and PHEMT switched-amp for wireless applications,” IEEE MTT-S International
Microwave Symposium Digest, vol. 4, pp. 1393-1396, 1999.
[4] King, D., and Gopani, S., “SAW filters in CDMA mobile communication networks,”
Wireless Technologies China ‘99 Conference Proceedings, Shanghai, China, pp.
104-107, Sept. 1999.
[5] Freisleben, S., Bergmann, A., Bauernschmitt, U., Ruppel, C., and Franz, J., “A
highly miniaturized recursive Z-path SAW filter,” Ultrasonics Symposium 1999
Proceedings, Caesars Tahoe, NV, vol. 1, pp. 347-350, Oct. 1999.
[6] Dubois, M. A., Muralt, P., Matsumoto, H., and Plessky, V., “Solidly mounted
resonator based on aluminum nitride thin film,” Ultrasonics Symposium 1998
Proceedings, Sendai, Japan, vol. 1, pp. 909-912, Oct. 1998.
[7] Lakin, K. M., Kline, G. R., and McCarron, K. T., “High-Q microwave acoustic
resonators and filters,” IEEE Transactions on Microwave Theory and Techniques,
vol. 41, no. 12, pp. 2139-2146, Dec. 1993.
[8] Ruby, R. C., Bradley, P., Oshmyansky, Y., Chien, A., and Larson, J. D., III, “Thin
film bulk wave acoustic resonators (FBAR) for wireless applications,” Ultrasonics
Symposium 2001, Atlanta, GA, vol. 1, pp. 813-821, Oct. 2001.
[9] Lakin, K. M., “Thin film resonators and filters,” Ultrasonics Symposium 1999
Proceedings, Caesars Tahoe, NV, vol. 1, pp. 895-906, Oct. 1999.
[10] Lakin, K. M., McCarron, K. T., Belsick, J., and McDonald, J. F., “Thin film bulk
acoustic wave resonator and filter technology,” Radio and Wireless Conference
(RAWCON) 2001, pp. 89-92, Aug. 2001.
[11] Ruby, R., and Merchant, P., “Micromachined thin film bulk acoustic resonators,”
Proceedings of the 1994 IEEE International Frequency Control Symposium, Boston,
MA, pp. 135-138, June 1994.
98
[12] Cleland, A. N., “Single-crystal aluminum nitride nanomechanical resonators,”
Applied Physics Letters, vol. 79, no. 13, pp. 2070-2072, Sept. 2001.
[13] Wang, K., Wong, A. C., and Nguyen, C. T. C., “VHF free-free beam high-Q
micromechanical resonators,” Journal of Microelectromechanical Systems, vol. 9,
no. 3, pp. 347-360, Sept. 2000.
[14] Antkowiak, B., Gorman, J. P., Varghese, M., Carter, D. J. D., and Duwel, A. E.,
“Design of a high-Q, low-impedance, GHz-range piezoelectric MEMS resonator,”
The 12th International Conference on Solid-State Sensors, Actuators, and
Microsystems (Transducers), Boston, MA, pp. 841-846, June 2003.
[15] ANSI/IEEE Std 176-1987, IEEE Standard on Piezoelectricity.
[16] Zelenka, J., Piezoelectric Resonators and their Applications, New York, NY:
Elsevier Science Publishing Co. Inc., 1986.
[17] Gerber, E. A., and Ballato, A., Precision Frequency Control, Volume 1: Acoustic
Resonators and Filters, New York, NY: Academic Press, Inc., 1985.
[18] Auld, B. A., Acoustic Fields and Waves in Solids, Vol. I, Malabar, FL: Robert E.
Krieger Publishing Co, Inc., 1990.
[19] Lakin, K. M., “Modeling of thin film resonators and filters,” IEEE MTT-S
International Microwave Symposium Digest, Albuquerque, NM, pp. 149-152, June
1992.
[20] Pozar, D. M., Microwave Engineering, Reading, MA: Addison-Wesley, 1990.
[21] Kinsman, R. G., Crystal Filters, New York, NY: John Wiley and Sons, 1987.
[22] Lakin, K. M., McCarron, K. T., Belsick, J., and Rose, R., “Filter banks implemented
with integrated thin film resonators,” 2000 IEEE Ultrasonics Symposium, San Juan,
Puerto Rico, vol. 1, pp. 851-854, Oct. 2000.
[23] Mang, L., Hickernell, E., Pennell, R., and Hickernell, T., “Thin-film resonator ladder
filter,” IEEE MTT-S International Microwave Symposium Digest, Orlando, FL, vol.
2, pp. 887-890, May 1995.
[24] Mang, L., and Hickernell, F., “ZnO thin film resonator lattice filters,” Proceedings
of the 1996 IEEE International Frequency Control Symposium, Honolulu, HI, pp.
363-365, June 1996.
[25] Zverev, A. I., Handbook of Filter Synthesis, New York, NY: John Wiley and Sons,
1967.
99
[26] Matthaei, G. L., Young, L., and Jones, E. M. T., Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Norwood, MA: Artech House, Inc.,
1980.
[27] Duwel, A. E., Charles Stark Draper Laboratories, Inc., personal communication.
100