Acoustic Waves From Microdevices to Helioseismology Part 13 pot

Acoustic Waves – From Microdevices to Helioseismology 468 Applying circuit theory, definitions of [y] matrix elements are presented: 21 33 11 22 11 11 12 00 12 00 13 13 33 00 33 00 ; ; ee ee ee ee ii yy ee ii yy ee == == == == == == (Appendix.4) And using trigonometric functions as follows: () 2 cot sin(2 ) 11 cot sin(2 ) 2 3cos(2 ) 1 sin(4 ) sin(2 ) (4 ) sin(4 ) sin(2 ) cos(2 ) tg g tg tg g tg tg tg tg tg tg αα α ααα α ααααα αα αααα α −=− −=− +− − =− +− (Appendix.5) The [y] matrix can be obtained for 2 models as follows: - for the “crossed-field” model: 11 0 0 12 13 0 33 0 0 cot (4 ) sin(4 ) (2 4 ) yjGg jG y yjGtg y jC Gtg α α α ωα =− = =− =+ (Appendix.6) - for the “in-line field” model: 2 0 0 0 11 0 2 0 0 0 2 0 0 12 0 00 00 13 0 0 0 0 33 0 1 sin(2 ) cot cot (2 ) 2 cot (2 ) 1 cot sin(2 ) 2 2cot cot(2) 2 1 2 2 1 G C G yjGg g C G g C G g C yjG GG gg CC tg yjG G tg C jC y G ωα αα ω α ω α ωα αα ωω α α ω ω ω      −      =− − −        −        −   =   −−     =− − = − 0 tg C α (Appendix.7) SAW Parameters Analysis and Equivalent Circuit of SAW Device 469 In IDT including N periodic sections, the N periodic sections are connected acoustically in cascade and electrically in parallel as Figure Appendix.6. Fig. Appendix.6. IDT including the N periodic sections connected acoustically in cascade and electrically in parallel Because the symmetric properties of the IDT including N section like these of one periodic section, and from (Appendix.2), (Appendix.3), Figure Appendix.4 and Figure Appendix.5, the [Y] matrices of N-section IDT are represented as follows: Fig. Appendix.7. The [Y] matrices and the model corresponsive models Since the periodic sections are identical, the recursion relation as follows can be obtained: e 1 m =e 2 m-1 (Appendix.8) e 3 N = e 3 N-1 = e 3 N-2 = = e 3 2 = e 3 1 =E 3 (Appendix.9) i 1 m =i 2 m-1 (Appendix.10) With m is integer number, m=1,2, …, N-1, N The total transducer current is the sum of currents flowing into the N sections. ()() ()() 33 1 3 2 3 N1 3 N 131 1 132 1 333 1 131 2 132 2 333 2 13 1 N 1 13 2 N 1 33 3 N 1 13 1 N 13 2 N 33 3 N Ii i . i i ye ye ye ye ye ye ye ye ye ye ye ye − −− − =+ +…+ + =−+ + −+ + +−+ +−+ (Appendix.11) Acoustic Waves – From Microdevices to Helioseismology 470 By applying (Appendix.8), (Appendix.9) and boundary conditions (e 11 = E 1 , e 2N =E 2 ), (Appendix.11) becomes: I 3 = y 13 e 1 1 -y 13 e 2 N + Ny 33 E 3 = y 13 E 1 -y 13 E 2 + Ny 33 E 3 (Appendix.12) From Figure Appendix.7, the Y 13 and Y 33 can be expressed as: Y 13 =y 13 (Appendix.13) Y 33 = Ny 33 (Appendix.14) Because the N periodic sections are connected acoustically in cascade and electrically in parallel, the model as in Figure Appendix.5 should be used to obtain the [Y] matrix of N- section IDT. From (Appendix.3) for one section, the i 1 and i 2 can be expressed i 1 = y 11 e 1 +y 12 e 2 + y 13 e 3 , i 2 = -y 12 e 1 -y 12 e 2 + y 13 e 3 (Appendix.15) Equations (Appendix.15) can be represented in matrix form like [ABCD] form in electrical theory as follows: [] 21 3 21 [] ee KLe ii   =+     (Appendix.16) Where [] 11 12 12 22 11 12 11 12 12 1 y yy K yy y yy   −     =   − −       (Appendix.17) [] 13 12 11 13 12 13 12 y y L yy yy y   −     =   +     (Appendix.18) By applying (Appendix.16) into N-section IDT as in Figure Appendix.6 and using (Appendix.9), the second recursion relation is obtained as follows: [] 1 3 1 [] mm mm ee KLE ii − −    =+       (Appendix.19) Where m is integer number, m=1,2, …, N-1, N Starting (Appendix.19)(Appendix.19) by using with m=N, and reducing m until m=1 gives the expression: [] [] 0 3 0 N N ee QXE ii   =+     (Appendix.20) SAW Parameters Analysis and Equivalent Circuit of SAW Device 471 Where [][] N QK= (Appendix.21) [] [] 1 1 0 2 [] N n n X XKL X − =  ==    (Appendix.22) Solving (Appendix.20) and using the boundary conditions (e 0 = E 1 , i 0 =I 1 ) gives: 11 1 1123 12 12 12 1QX IEEE QQQ =− + − (Appendix.23) Consequently, 11 11 12 Q Y Q =− (Appendix.24) 12 12 1 Y Q = (Appendix.25) 1 13 12 X Y Q =− (Appendix.26) The Y 13 is known by (Appendix.13), so (Appendix.26) and matrix [X] don’t need to be solved. To solve (Appendix.24) and (Appendix.25), matrix [Q] should be solved. In “crossed-field” model, matrix [Q] can be represented in a simple form as follows: [] 0 0 cos(4 ) sin(4 ) sin(4 ) cos(4 ) jR K jG αα αα −   =   −   (Appendix.27) [] 2 0 0 cos(8 ) sin(8 ) sin(8 ) cos(8 ) jR K jG αα αα −   =   −   (Appendix.28) [] 3 0 0 cos(12 ) sin(12 ) sin(12 ) cos(12 ) jR K jG αα αα −   =   −   (Appendix.29) . . . . . . etc. Consequently, matrix [Q] will be given: [][] 0 0 cos( 4 ) sin( 4 ) sin( 4 ) cos( 4 ) N NjRN QK jG N N αα αα −   ==   −   (Appendix.30) From (Appendix.24) and (Appendix.35), Y 11 and Y 12 in “cross-field” model can be expressed: 11 0 cot ( 4 )YjGgN α =− (Appendix.31) 0 12 sin( 4 ) jG Y N α = (Appendix.32) Acoustic Waves – From Microdevices to Helioseismology 472 In conclusion, matrix [Y] representation of N-section IDT is: - In "crossed-field" model, from (Appendix.6), (Appendix.13), (Appendix.14), (Appendix.31) and (Appendix.32): 11 0 0 12 13 0 33 0 0 cot (4 ) sin(4 ) (2 4 ) YjGgN jG Y N YjGtg YjNC Gtg α α α ωα =− = =− =+ (Appendix.33) - In "in-line field" model, from (Appendix.7), (Appendix.13), (Appendix.14), (Appendix.24) and (Appendix.25): 11 11 12 12 12 13 0 0 0 0 33 0 0 1 2 1 2 2 1 Q Y Q Y Q tg YjG G tg C jNC Y G tg C α α ω ω α ω =− = =− − = − (Appendix.34) Where [Q] can be calculated from (Appendix.17) and (Appendix.21). 7.2 Appendix 2: Equivqlent circuit for “N+1/2” model IDT In case IDT includes N periodic sections (like in section 3.2 plus one finger (in color red) as shown in Figure Appendix.8 that we call “N+1/2” model IDT. Fig. Appendix.8. “N+1/2” model IDT The equivalent circuit for this model is shown in Figure Appendix.9 and the matrix [Yd] representation is shown as in Figure Appendix.10 (letter “d” stands for different from model [Y] in section 3.2. SAW Parameters Analysis and Equivalent Circuit of SAW Device 473 Fig. Appendix.9. Equivalent circuit of “N+1/2” model IDT Fig. Appendix.10. [Yd] matrix representation of “N+1/2” model IDT The form of matrix [Yd] is: [] 11 12 13 21 22 23 31 32 33 Yd Yd Yd Yd Yd Yd Yd Yd Yd Yd     =       (Appendix.35) The elements of [Yd] matrix for “crossed-field” model are given as follows: 11 0 2 1 cot (4 ) sin (4 )(cot (2 ) cot (4 )) Yd jG g N Ng gN α αα α     =−   +    (Appendix.36) 0 12 sin(2 )[cot (4 )cos(2 ) sin(2 )] cos(2 ) sin(4 ) cos(2 ) cot (4 )sin(2 ) jG g N Yd NgN αααα α αααα   −   =−   +    (Appendix.37) 2 2 13 0 ( 2cot (4 )sin sin(2 ))sin(2 ) 2sin sin(4 )(cos(2 ) cot (4 )sin(2 )) sin(4 ) tg g N Yd j Gt g NgN N ααααα α α αα αα α  −+ +  =+−  +   (Appendix.38) 21 0 1 sin(4 )(cos(2 ) cot (4 )sin(2 )) Yd jG NgN αα αα =− + (Appendix.39) 22 0 cot (4 )cos(2 ) sin(2 ) cos(2 ) cot (4 )sin(2 ) gN Yd jG gN αα α ααα − = + (Appendix.40) Acoustic Waves – From Microdevices to Helioseismology 474 2 23 0 2cot (4 )sin (2 ) sin(2 ) cos(2 ) cot (4 )sin(2 ) tg g N Yd jG gN αααα ααα −+ + = + (Appendix.41) 31 0 Yd jG tg α =− (Appendix.42) 32 0 sin(2 )Yd jG α =− (Appendix.43) { } 33 0 0 (2 1) sin(2 ) (4 1)Yd j CN j GNt g ωαα =−+ ++ (Appendix.44) 7.3 Appendix 3: Scattering matrix [S] for IDT The scattering matrix [S] of a three-port network characterized by its admittance matrix [Y] is given by [3]: 1 33 2( )SYY − =Π − Π + (Appendix.45) Where 3 Π is the 3x3 identity matrix. After expanding this equation, the scattering matrix elements for a general three-port network are given by the following expressions: { } 11 33 11 22 11 22 12 21 13 31 22 21 32 23 32 11 12 31 1 (1 )(1 ) [ (1 ) ] [ ( 1) ] SYYYYYYY M YY Y YY YYY YY =+−+−+ + ++−+ −− (Appendix.46) [] 12 12 33 13 32 2 (1 )SYYYY M =− + − (Appendix.47) [] 13 13 22 12 23 2 (1 )SYYYY M =− + − (Appendix.48) [] 21 21 33 23 31 2 (1 )SYYYY M =− + − (Appendix.49) { } 22 33 11 22 11 22 12 21 13 31 22 21 32 23 32 11 12 31 1 (1 )(1 ) [ ( 1) ] [ ( 1) ] SYYYYYYY M YYY YY YYY YY =++−−+ + +−−++− (Appendix.50) [] 23 23 11 13 21 2 (1 )SYYYY M =− + − (Appendix.51) [] 31 31 22 21 32 2 (1 )SYYYY M =− + − (Appendix.52) [] 32 32 11 12 31 2 (1 )SYYYY M =− + − (Appendix.53) { } 33 33 11 22 11 22 12 21 13 31 22 21 32 23 32 11 12 31 1 (1 )(1 ) [ ( 1) ] [ ( 1) ] SYYYYYYY M YYY YY YYY YY =−+++− + ++−++− (Appendix.54) SAW Parameters Analysis and Equivalent Circuit of SAW Device 475 where 3 33 11 22 12 21 23 32 11 12 31 13 31 22 21 32 det( ) (1 )[(1 )(1 ) ] [ (1 ) ] [ (1 ) ] MY YYYYYYYYYY YY Y YY =Π+ =+ + + − − + − − −−− (Appendix.55) For model IDT including N identical sections, these equations can be further simplified. In case of Figure Appendix.7 (b): 11 22 21 12 31 13 23 32 13 YY YY YY YY Y = = = ==− (Appendix.56) Therefore, S ij ’s take the following form () {} 22 2 11 22 33 11 12 13 11 12 1 (1 )(1 ) 2SS Y YY YYY M == + −+ + + (Appendix.57) 2 12 21 12 33 13 2 (1 )SS Y Y Y M   ==− + +   (Appendix.58) 13 31 13 11 12 2 (1 )SS Y YY M ==− ++ (Appendix.59) 23 32 13 SS S==− (Appendix.60) {} 22 2 33 33 11 12 13 11 12 1 (1 )[(1 ) ] 2 (1 )SYYYYYY M =− +−+++ (Appendix.61) Where 22 2 33 11 12 13 12 (1 )[(1 ) ] 2 (1 ) M YYYYY=+ + − − + (Appendix.62) 7.4 Appendix 4: Equivalent circuit for SAW device base on Mason model, [ABCD] Matrix representation 7.4.1 Appendix 4.1: [ABCD] Matrix representation of IDT In SAW device, each input and output IDTs have one terminal connected to admittance G 0 . Therefore, one IDT can be represented as two-port network. [ABCD] matrix (as in Figure Appendix.11) is used to represent each IDT, because [ABCD] matrix representation has one interesting property that in cascaded network, the [ABCD] matrix of total network can be obtained easily by multiplying the matrices of elemental networks. Fig. Appendix.11. [ABCD] representation of two-port network for one IDT Acoustic Waves – From Microdevices to Helioseismology 476 To find the [ABCD] matrix for one IDT in SAW device, the condition that no reflected wave at one terminal of IDT, and the current-voltage relations by [Y] matrix in section are used as follows: Fig. Appendix.12. Two-port network for one IDT 11112131 21211132 31313333 IYYYV IYYYV IYYYV       =−       −    (Appendix.63) And I 1 =-G 0 V 1 (Appendix.64) From these current-voltage relations, the V 3 and I 3 are given: 22 11 12 11 0 0 11 322 12 13 11 13 13 0 12 13 11 13 13 0 YYYG GY VVI YY YY YG YY YY YG −+ + =− ++ ++ (Appendix.65) 2222 13 12 13 11 13 0 11 33 13 33 0 11 12 11 0 3 2 01112131113130 2 11 33 13 33 0 2 12 13 11 13 13 0 ()()() ()( ) YY YY YG YY Y YG Y Y YG IV GYYY YY YG YY Y YG I YY YY YG −++ +−+ −+ =− +++ −+ − ++ (Appendix.66) From (Appendix.65) and (Appendix.66), equivalence between port 3 in Figure Appendix.12 equals to port 1 in Figure Appendix.11, and consideration of direction of current I 2 in Figure Appendix.11 and Figure Appendix.12, [ABCD] matrix representation for two-port network of IDT in obtained: 22 11 12 11 0 12 13 11 13 13 0 YYYG A YY YY YG −+ = ++ (Appendix.67) 011 12 13 11 13 13 0 GY B YY YY YG + = ++ (Appendix.68) 2222 13 12 13 11 13 0 11 33 13 33 0 11 12 11 0 01112131113130 ()()() ()( ) YY YY YG YY Y YG Y Y YG C GYYY YY YG −++ +−+ −+ = +++ (Appendix.69) 2 11 33 13 33 0 12 13 11 13 13 0 YY Y YG D YY YY YG −+ = ++ (Appendix.70) In case of “crossed-field” model, the [ABCD] can be further simplified: [...]... of the modeled device as shown in Fig 5 As done for the viscosity, the dissipation in each resistor is coupled to the thermal model as a heat source In 490 Acoustic Waves – From Microdevices to Helioseismology fact, dissipation in the input and output resistors is coupled to the correspondent top and bottom thermal sections that model the electrodes The complete model can be seen in Fig 5, where a... state-of-the-art rectangular Solidly-Mounted Resonators (SMR) from a commercial manufacturer, with different areas summarized in Table 1, have been measured The resonators have a 1.25 μm thick aluminum nitride layer and a W - SiO2 Bragg mirror (alternating layers of W and SiO2), and show quality factors around 1800 492 Acoustic Waves – From Microdevices to Helioseismology From (8) it is clear that several sources,... closed-form expressions derived from the 484 Acoustic Waves – From Microdevices to Helioseismology circuit model and validates the model with extensive measurements that confirm the necessity to include dynamic self-heating to accurately predict the generation of spurious signals in BAW devices 2 Nonlinear generation mechanisms The origin of nonlinearities in BAW resonators has been controversial and... 498 Acoustic Waves – From Microdevices to Helioseismology c → c +η ∂ ∂t (20) The inverse damping coefficient can also be understood as the conductance per unit length Gd=η-1 With that, the acoustic telegrapher equations, making use of the analogy between the acoustic and electric domains, can be written as: ∂V = −Ld jω I ∂z (21) ∂I 1 =− D jωV ∂z Ac + jω Aη (22) and The shunt admittance of the acoustic. .. the acoustic distributed inductance Ld = ρ·A·Δz Thermal Domain Extension Top Layers (Thermal Domain) Piezoelectric Layer ∆z Rrad Rth,1/2 Rth,1/2 Tamb R th,Si/2 Rconv Cth,Si Cth,1 Top Layers (Acoustic Domain) Bottom Layers (Thermal Domain) Zair Rth,Si/2 Tamb Bottom Layers (Acoustic Domain) Zair Electro -Acoustic Conversion Nonlinear KLM model Rin Rout Fig 5 Complete circuit model with thermo -acoustic. .. ΔcD2 contributions to the 3IMD (2ω1-ω2 and 2ω2-ω1 overlap), for several separations between tones The intrinsic contributions cannot reproduce the envelope frequency-dependent 3IMD level 494 Acoustic Waves – From Microdevices to Helioseismology The parameter ΔcD1 is responsible for second harmonic generation, which in turn mixes with the fundamental frequencies ω1 and ω2, and gives rise to 3IMD On the... Vol.36, No.3, May 1989 [7] K.Hashimoto, and M.Yamaguchi, Precise simulation of surface transverse wave devices by discrete Green function theory, IEEE Ultrasonics Symposium, 1994, pp.253-258 480 Acoustic Waves – From Microdevices to Helioseismology [8] K.Hashimoto, G.Endoh, and M.Yamaguchi, Coupling-of-modes modelling for fast and precise simulation of leaky surface acoustic wave devices, IEEE Ultrasonics... possibility to predict 3IMD in BAW resonators, given their materials stack and geometry With such information one can use the resonator model to accurately predict 3IMD in filters Further research will be performed to investigate the relation between the electric-field contribution to 3IMD and the cancellation shown in the measurements The development of a 3D equivalent thermal model, to take into account... elasticity in (1) D D D D c D (T , K ) = c 0 + Δc1 T + Δc 2 T 2 + Δc K K (8) 486 Acoustic Waves – From Microdevices to Helioseismology where K represents the temperature, the equivalent capacitance is C d ( v , K ) = C d ,0 + ΔC 1 v + ΔC 2 v 2 + ΔC K K , (9) where each of the nonlinear terms ΔC1, ΔC2 and ΔCK are related to their counterparts Δc1D , Δc2D, ΔcKD respectively, as detailed in Appendix I The term... 60 years: A new formula for computing quality factor is warranted 2008 IEEE International Ultrasonics Symposium, pp 431-436, 2-5 Nov 2008 500 Acoustic Waves – From Microdevices to Helioseismology Ivira B., Benech P., Fillit R., Ndagijimana F., Ancey P., Parat G 2008 Modeling for temperature compensation and temperature characterizations of BAW resonators at GHz frequencies IEEE Transactions on Ultrasonics, . 11 322 12 13 11 13 13 0 12 13 11 13 13 0 YYYG GY VVI YY YY YG YY YY YG −+ + =− ++ ++ (Appendix.65) 2222 13 12 13 11 13 0 11 33 13 33 0 11 12 11 0 3 2 0111 2131 1131 30 2 11 33 13 33 0 2 12 13 11 13 13. 12 11 0 12 13 11 13 13 0 YYYG A YY YY YG −+ = ++ (Appendix.67) 011 12 13 11 13 13 0 GY B YY YY YG + = ++ (Appendix.68) 2222 13 12 13 11 13 0 11 33 13 33 0 11 12 11 0 0111 2131 1131 30 ()()() ()(. N The total transducer current is the sum of currents flowing into the N sections. ()() ()() 33 1 3 2 3 N1 3 N 131 1 132 1 333 1 131 2 132 2 333 2 13 1 N 1 13 2 N 1 33 3 N 1 13 1 N 13 2 N