Low-Sensitivity , Low-Power 4 th -Order Low-Pass Active-RC Allpole Filter Using Impedance Tapering Dražen Jurišić * , George S. Moschytz ** and Neven Mijat * * University of Zagreb/Faculty of Electrical Engineering and Computing, Unska 3, Zagreb HR-10000, Croatia ** Swiss Federal Institute of Technology Zürich (ETH), Sternwartstrasse 7, CH-8092 Zürich, Switzerland Abstract—The analytical design procedure of low- sensitivity, low-power, low-pass (LP) 2 nd -and 3 rd -order class-4 active-RC allpole filters, using impedance tapering, has already been published [1][2]. In this paper the desensitisation using impedance tapering is applied to the design of LP 4 th -order filters. The numerical design procedure was performed by Newton’s iterative method. Analytically designed unity-gain LP 4 th -order filters [3] can provide initial values for Newton’s method. The sensitivities of a filter transfer function to passive component tolerances, as well as active gain variations are examined by the Schoeffler sensitivity and Monte Carlo PSpice simulation. Butterworth and Chebyshev 0.5dB filter examples illustrate the design method. I. INTRODUCTION A procedure for the analytical design of low-sensitivity class-4 2 nd -and 3 rd -order Sallen-and-Key [4] active resistance-capacitance (RC) low-pass (LP) allpole filters was presented in [1], with the realizability constraints in [2]. It was shown in [1] that by the use of “impedance tapering”, in which L-sections of the RC network are successively impedance scaled upwards, from the driving source to the positive amplifier input, the sensitivity of the filter characteristics to passive component tolerances can be significantly decreased. In this paper, the design method based on “impedance tapering” is extended to the design of 4 th -order LP active- RC filters each with a single operational amplifier (opamp). It is also demonstrated that obtaining an analytical solution is possible only for lower-than-4 th - order filters and for a special case of 4 th -order unity-gain filter (β=1) [3]. The examples of Butterworth and Chebyshev filters with 0.5dB pass-band ripple illustrate optimal filter design with minimum passive and active sensitivities. II. FOURTH-ORDER ALLPOLE FILTER Consider the 4 th -order single-amplifier LP filter shown in Fig. 1. It is a low-power circuit, insofar as it uses only one opamp. Its voltage transfer function T(s), is given by: 0 1 2 2 3 3 4 0 1 2 ) ( a s a s a s a s a V V s T + + + + β = = , (1) where coefficients a i (i=0,…,3) as a function of components of the circuit are given by (2). Transfer function T(s) in (1), can be written in terms of pole Q-factors, q pi , and pole frequencies ω pi ; (i=1, 2) as [ ][ ] 2 2 2 2 2 2 1 1 1 2 2 2 2 1 ) / ( ) / ( ) ( p p p p p p p p s q s s q s s T ω + ω + ω + ω + ω βω = .(3) Note that the gain for the class-4 circuit is given by: 1 / 1 ≥ + = β G F R R . (4) V 2 R G R =R ( -1) F G β A→∞ V 1 R 1 C= / 2 2 ρ C 1 C= / 3 3 ρ C 1 C= / 4 4 ρ C 1 C 1 R 2 2 =r R 1 R 3 3 =r R 1 R 4 4 =r R 1 Fig. 1. 4 th -order LP filter with single opamp and impedance scaling factors r i and ρ i ; (i=2,3,4). Introducing the design frequency ω 0 and impedance scaling factors r i and ρ i as in Fig. 1, defined by 1 1 1 0 ) ( − = ω C R , R i =r i R 1 , C i =C 1 /ρ i ; i=2, 3, 4; (5) into (2), and using i n i i a − ω = α 0 / ; i=0, 1, 2, …, n-1; n=4, (6) we obtain a system of four equations with eight unknowns. To design the 4 th -order LP filter we have to solve this system, and therefore we must choose four variables, and then calculate the remaining four. For example, we can calculate the resistive tapering factors r i (i=2, 3, 4) and gain β from given coefficients a i (i=0, …, 3), chosen capacitive factors ρ i (i=2, 3, 4) and the design frequency ω 0 . Capacitive scaling factors ρ i should geometrically progress, providing “capacitive tapering” in the filter design. Note that, alternatively, we could have started by choosing resistive scaling factors r i , thus providing a "resistive tapering" design procedure. We can express values of r 4 and β explicitely, but we obtain a nonlinear relation between r 2 and r 3 . The new system of four nonlinear equations is given by: ) /( 0 3 2 4 3 2 4 α r r ρ ρ ρ r = ; 0 2 3 2 3 3 2 2 3 2 2 3 2 2 = ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ r f r e r d r r +c r r +b r r a ; (7) 0 2 2 3 2 2 3 3 2 2 2 3 2 2 3 2 2 2 3 3 2 = ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ r m r r +l r k r r +j r r +i r r h r r g ; ] / ) 1 ( / ) ( 1 )[ / ( / 1 2 2 3 3 2 3 4 4 4 3 r ρ - r ρ ρ - - α ρ r - ρ + ρ β = + + , where the constants a to m in the 2 nd and 3 rd equations can readily be calculated from eqs. (2) to (6). The next step, 1 4 3 2 1 4 3 2 1 0 ) ( − = C C C C R R R R a , )]}, ( [ ) ( ) ( ) ( { 3 2 1 3 1 1 4 4 4 3 3 4 3 2 2 4 3 2 1 1 0 1 R R R C C R C R C C R C C C R C C C C R a a + + + β − + + + + + + + + + = ]}, ) ( ) [( ) ( ) ( ) ( ) ( ) ( ) ( { 3 1 3 2 1 3 2 3 2 1 4 3 4 3 4 3 2 4 2 4 3 2 3 2 3 2 1 4 4 1 2 1 4 3 1 2 1 3 3 1 4 3 2 1 2 1 0 2 C C R R R C C R R R C C R R C C C R R C C C R R C C C C R R C C C R R C C C R R C C C C R R a a + + + β − + + + + + + + + + + + + + + + = (2) }. ) ( ) ( ) ( { 3 2 1 3 2 1 4 3 2 4 3 2 2 1 4 3 4 3 1 3 2 4 1 4 2 1 4 3 2 1 3 2 1 0 3 C C C R R R C C C R R R C C C C R R R C C C C R R R C C C C R R R a a β − + + + + + + = IEEE MELECON 2004, May 12-15, 2004, Dubrovnik, Croatia 0-7803-8271-4/04/$20.00 ©2004 IEEE 107