E ve F devreleriyle band geçiren aktif SC filtre tasarımı ve duyarlık analizi

Abstract

Tümdevre tasarımında önemli bir yer tutan SC filtre yapılarının gerçekleştirilmesin de, E ve F olmak üzere iki ayrı devre topolojisinden yararlanılmaktadır, n. dereceden bir filtre, ikinci dereceden alt bloklara ayrılmakta ve her bir blok E veya F devresi ile gerçeMeştirilmektedir. Band geçiren SC filtre yapıları göz önüne alındığından blokla ra ayırma işlemi ise kaskad ve ya paralel tasarım yöntemlerinden birine uygun olarak yapılmaktadır. Bu amaçla, band geçiren filtre transfer fonksiyonu ya ikinci dereceden blokların çarpımı ya da bu blokların toplamı şeklinde düşünülmektedir. Öncelikle, s-domeni transfer fonksiyonu göz önüne alındığından uygun bir dönü şümle tasarımın yapılacağı z-domenine geçilmektedir. Bu amaçla, SC filtre sentezi için en uygun yöntem olan bilineer dönüşümden yararlanılmaktadır. Bu dönüşümün sebep olduğu frekans bozulmasını önlemek amacıyla da s-domeni. transfer fonksiyonuna ön- çarpıtma işlemi uygulanmaktadır. E ve F devrelerine ait tasarım eşitliklerinden devre- lerdeki kapasite değerleri bulunduktan sonra devre dinamiğini artırmak amacıyla bîr normalizasyon yapılmaktadır, ikinci bir normalizasyon ise minimum kapasite birim o- lacak şekilde gerçekleştirilmektedir. Tasarını aşamasından soma duyarlık analizine geçilmektedir. Duyarlık integrand- laıı ve duyarlık ölçütleri E ve F devreleri için ayrı ayrı hesaplanmaktadır. Duyarlık öl çütlerinin hesaplanmasında integrasyon işlemi için Simpson Yöntemi kullanılmaktadır. Analiz gerçekleştirilirken belli bir frekans bölgesi Simpson yöntemine uygun olarak (2m) parçaya ayrılmaktadır. Bunlara karşı düşen (2m+l) noktadaki frekans değeri nin her biri için duyarlık integıandı bulunmakta ve frakans-duyarlık integrandı eğrisi çizilmektedir. Bütün bu işlemler, kaskad ve paralel tasarımın her ikisi için de yapılmak tadır. Bahsedilen tasarım işlemleri ve duyarlık analizleri C-dilinde yazılmış bir program ile gerçekleştirilmektedir. E ve F devrelerinin her İkisi için ayrı ayn bulunan toplam kapasite değerleri ile duyarlık ölçütleri bu iki devrenin karşılaştırılması için önemli parametreler olmaktadırlar. Programın birkaç örneğe uygulanması sonucunda F devre sinin E devresine göre üstün olduğu ortaya çıkmaktadır. Bu sonucu desteklemek amacı ile ele alınan bir örnek Spice Monte Carlo simülasyon programıyla incelenmektedir. Varılan sonuç yine aynı olmaktadır.

The MOS-integrated-circuit technology has found wider usage in industry than the BIPOLAR technology. Millions of MO S transistors can be placed on a single chip, so the cost of digital or analog integrated circuits has decreased very much. Therefore, analog-sampled data filters are realised as switched-capacitor circuits. By this way, high Q's and flat passbands can be realized efficiently, much like the active-RC realizations. SC-filters take full advantages of the MOS-integrated circuit technology. Coeffici ents of the designed transfer function are determined as ratios between the capacities and as precise crystal-controlled clock frequency. A continious time analog signal has to be passed through a continious antialiasing (low-pass) filter in order to operate on the analog signal by SC-filters. Then, the band-limited signal is applied to input of a sample-and-hold circuit, it is sampled at intervals of l/fci and passes through an SC-filter. After the sample-and-hold circuit in the out of SC-filter has sampled the output signal, a reconstruction filter in the last block changes the signal to the analog signal again. Switches used in SC-filters are controlled by two phased antialiasing clock pulses. Even and odd clock phases are represented by respectively ee. and 30, 0,, and e0 have equal times and are equal to the 50% of the main period. The OP- AMP' s in the filter have been designed which settle to within 0.1% of final value in 2u sec and achive dc gains greater than 60 dB. The z-transform method is used in analysis and synthesis of SC-filters. Between the z-domain and the s-domain, there is a relation as z =- e*\ ( s: analog frequency, x- 2T: clock period ). This relation makes the frequency-domain analysis very easy. Because the operating principle of SC-filters depend on the charge-transfer ope rations between the capacities, discrete-time voltages Vj(kT) and discrete-time charge variations or transfers Aq,(kT) are used as port variables. At the switching times, charges are instantaneously redistributed, with the principle of charge conservation vimaintained at every node in the network. In general, due to the biphase switching operation, two distinct, but coupled, nodal charge equations are required to characterize the charge conservation condition at a particular node for all time instants, kT. Equations (1). (2), (3) and (4) arc written where qp*, qpi°, Afcp Mep A q« (kT) = Z q p i (kT) - 2 q°, [(k - 1 )TJ ; k even integer ( 1 ) i=l i=l Mop Mop Aq;(kT)= Zqpi(kT) - ZqPi [(k-l)T] ;k odd integer (2) M 1=1 Mep Mep AQUz)= SQPi(z) - z'"2 ZQ°pi(z) (3) i=l i=l Mop Mop ^q;cz)= Zq;i(z) - z'^Zq^z) h> and Qpi*, Qpi0 denote, respectively, the instantaneous charges stored on the i th capacitor connected to node p for the even and odd kT time instants and their z-transforms. Also. Mep and Mot> denote the total number of capacitors connected to node p during the even, and odd clock phases. hi SC-circuit design, first of all, it must be passed from the s-domain transfer function to the z-domain function. For this reason, the method used have to: i -) change the stable s-domain transfer functions to the stable z-domain functions. ii-) change the imaginary jw axis of s -plane to the unit-circle of z-plane. So, the methods which are used: ! -) Backward Difference (ED): (?.ai (5.b) Although the first condition is satisfied by this transform, the second one is not satisfied. VII2-) Forward Difference (FD): 1 T.z"1 1-z-1 (6.a) 8 1-Z"'1 ' " ~ T.Z"'1 z--l-r-sl (6,b) FD does not satisfy the two conditions, 3~) Bilinear Transform; 1 T 1+z"1 -=- 7-^T (7.a) s 2 1 - z (2/T) + s z= - - - - a.b) Stable-region corresponds to the inner of unit circle and s := jw corresponds to unit ;ifcle, So, both conditions are satisfied. 4-) Lossless Discrete Integrator (LDI): 1 z - = T. - - (8.a; s 1-z"1 z=- (2 + sV -)± v's2T2.(4 + s2T2) (8.b) LDI satisfies the two conditions as the Bilinear transform. Bilinear transform and LDI are the most suitable methods for synthesis. But bili near transform is chosen for realising SC-filters. When this method is used, there is a frequency warping effect between the analog frequency w, which is defined as s=jw, and sampled data domain frequency w in the z ~ eirot equation. This effect given by (9), loses its importance under the condition of wx « 1. If this operation is not done, the cutoff and stop band edge frequencies must be pre-warped according to (9). Apart from this problem, a zero effect occurs at z = -1 and a high-frequency warping effect destroys the band-pass character ( at ws /2 ). VIII2 i m'T ) -.t«j --i (9) After obtaining the z-domain function, SC-circuit topology must be defined. For this reason, two distinct circuits which realize a biquad z-domain function given by (10), have been used as E and F. v +e.z * -ffS.z'^ H(z)=; "IT"- T fi0) 1 + a.z + p.z Corresponding T and T' transfer functions for E and F are: İ£ '" ~ 1 +' (C + E '-"2). z"r+ (1. ~E)Tzl2" ( * 1;, (IC + IE - G ) + ( H + G - JC - JE IE ). z" ' + ( El - K I z~2 T- =- - - - - " -- - (12i 1-(C + E 2).z~' + (l-E).z~' ' ' Î't(Ğ-Î Jlz^-M J-H).z"2 e (F+l) + (C-F-2).z"1 + z-2 (-ÎC + ĞF - G *i + r-FH - H - Ğ + JC). z- + ( H \ z"2 T =: _ ^ _^_i_____ ^_ - (1 4 ? fF + lH(C-F-2Yz~1 + z~* There are also three capacitors (A, B. D) and they are chosen as unity to simplify the synthesis. The design equations, which are found after applying the bilinear transform, for E and F. are: D(s)=s2+a.s + b (15) a. X (16) l + a.(x/2) + b.(ti/4) b.x2 ç n 7\ l + a.(T/2)+b.(x2/4) a. x F = t ( 1 8) ]-a.(x 2)+b.(TV4) IXc = b.x2 la, (T, 2)+b.(Xi/4) (19) G, FL I, J and G, H, I, J capacitor values are calculated from the numerator of the z-transfer function. There-fore, a table is given in the following: Table 1. Zero-Placement Formulas For TE and TP ( For Band-Pass Filter ). The capacitor values for F are found by using (20). x = ( 1 + F).x x = G, H L J (20) Capacitor values, which are calculated at the end of the synthesis, are not nor malized values. A and D are scaled so that the first OP- AMP in the circuit does not overload the second OP- AMP. T function is not effected from, this normalization. T'-»u.T'. ( A.D)-»! --A. -D V ti ' u (21) 11 it is wished to be determined the output of the second OP- AMP, B, C, E and F have to be scaled. In this manner, T is not effected because the main function is T'., ! ' 1 1 1 1 T -» v.T İ B.CE.F ! -» I -B, -C. -E, F v V V V V (22)A second scaling is done to make the minimum capacitor unity. Then, (C, D, E, G, H) and (A, B, F, I, J) capacitor groups can be scaled seperately. Sensitivity analysis must be done for SC-filters after designing sensitivity measures for gain and phase functions which are written as: M"= 2c:.Kff.Ci.crx (23) k M* = Zc!,L,Cii 1 3 1 i=l (24) K. = J E, do (25) K, = I E,<, dec (2' z-domain function which is studied, is defined as: H(z, x) - ^7-t = - 7 ( 2 / ) D(z,x) z +a1,z4-a0 then, A and B matrices can be seperated into their real.,2 - IT b2.Z H-bpZ-r bD l J 1 i IT A = --. z 1 J (29) and imaginary components such as (30) and (31). After that, B = B" + j.Bim (30) A = A« + j.Aiin (31) XIEa and Ep matrices in the sensitivity measure formulas, can be written as: E" -\ B' 4 4l i (32) b:.b! b;.a! ! *^1 *"*J -A- B' A- 4* I (33) Q vectors in the sensitivity measure equations are determined by the capacitors of E and F circuits. There are nine C vectors for each network, because every circuits needs nine capacitors. ux\ represents the standard deviation for each capacitor in the circuit. 0 AG AH AfC+E) AE DB DB DB DB! (34) C, I (AG-DI-DJ) (DJ-AH) B DB DB A(C+_E) AE j" - DB ~~ DIi j (35) AC (36) (37) (38) (39) (40) (41) (42) XIIAG AH AC DfF+B) D(F+B) DİF+Bİ O ! (43) c, = S -. î (ÂĞ-DÎ-DJ) (DJ-ÂH) (ÂC+DFİ F (F+B)^ DİF+B)2 D(F+b)2 D(f+BJ2 (F+B)2J = \l I ^ (44) C, =1 0 0 0 AC D(F+Bl (45) C" =! 0 AG AH AC DfF+Bj D(F+B) D(F + B) İTT 0 (46) a = f, I (ÂG-DÎ-DJ) (DJ-ÂH) (DB-ÂC) B (F+BT D(F-B) DİF+B)' D(F+bT (f+ i) (47) '\ *? ' *-*) - ! AG ' C, =! 0 ---r^- ^r O O O i 0 I EtfF+B! ! (48) C,=i 0 0 AH DfF+B) 0 0 (49) C I I !f+b) (f+bi 0 0 0 (50) C9=j 0 :F-B! İF+B) 0 0 (51) If the order of the transfer function is n (n>2), only two of the designing methods are chosen: Parallel and Cascade Designing Methods. i-) Cascade Designing Method: Transfer function is considered as the product of biquad functions and each function is designed by using synthesis methods. Output of each block is connected to the other's input. XUlIn the cascade design, an. important problem occurs for circuit dynamics. Because the output of a block is connected to the other's input the first block can overload the output of the second one. Sensitivity for cascade design is calculated from the sum of each block's sensitivity. ii-) Parallel Designing Method; This turn, transfer function is considered as the sum of biquad functions. Output of each block is added by an OP- AMP adder and the main output is found. The circuit dynamics is not as bad as in the cascade design. This is an advantage of parallel design. Additionally, sensitivity is determined by (52). The main sen sitivity can be found by multiplying sensitivity of each block with a coefficient, This coefficient is defined as the square of the ratio between the gain of each block's transfer function and the gain of main function. M- ' i) l MT.?= J j 77i E 1 AT._ According to the SC-fîlter design methods and sensitivity analysis equations, a computer program, which designs Band-Pass SC -filters by parallel and cascade methods and does the sensitivity analysis, is written in C-programming language. 2. and 6. order examples have been applied to the program and then it is realized that F circuit has more advantage than E circuit, because sum of the values of capacitors in E circuit are greater than in F circuit and sensitivity of E circuits is greater than F circuits. Also, a second order band-pass SC-filter is analysed by SPICE-Monte-Carlo Simulation Program. It has been found that sensitivity of F circuit is less than E circuit. Finally, it can be said that F circuit is a better circuit than E.