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RF Electronics Chapter 7: RF Filters Page 210 2022, C. J. Kikkert, James Cook University, ISBN 978-0-6486803-9-0. Table 7.1b. Butterworth Response, K And Q Value filter table, From Zverev[1], pp 341. n q0 I. L. q 1 q n k 12 k 23 k 34 k 45 5 INF. 0.000 0.6180 0.6180 1.0000 0.5559 0.5559 1.0000 32.361 1.045 0.4001 1.5527 1.4542 0.6946 0.5285 0.6750 32.361 1.045 0.5662 0.7261 1.0947 0.5636 0.5800 0.8106 16.180 2.263 0.3990 1.8372 1.4414 0.6886 0.5200 0.6874 16.180 2.263 0.5777 0.7577 1.0711 0.5408 0.6160 0.7452 10.787 3.657 0.4036 2.0825 1.4088 0.6750 0.5080 0.7066 10.787 3.657 0.5927 0.7869 1.0408 0.5144 0.6520 0.6860 8.090 5.265 0.4111 2.3118 1.3670 0.6576 0.4927 0.7290 8.090 5.265 0.6100 0.8157 1.0075 0.4844 0.6887 0.6278 6.472 7.151 0.4206 2.5307 1.3195 0.6374 0.4732 0.7542 6.472 7.151 0.6293 0.8449 0.9722 0.4501 0.7267 0.5681 5.393 9.425 0.4321 2.7375 1.2675 0.6149 0.4479 0.7821 5.393 9.425 0.6508 0.8748 0.9355 0.4103 0.7663 0.5048 Butterworth Filters Butterworth filters have a maximally flat amplitude response. That means all the derivatives of the amplitude with respect to frequency are zero at DC. The Butterworth response is a good compromise between attenuation characteristic and group delay. The characteristics of the other filter types described in this chapter will be compared with the Butterworth filter to highlight the advantages and disadvantages of each filter type. For a Butterworth Lowpass filter of n th order, with a cut off frequency of f c and a source and load resistance of R, the filter component values are given by: n i Sin R f C c i 2 ) 1 2 ( 1 i=1,3,5…. Eqn. 7.1 n i Sin f R L c i 2 ) 1 2 ( i=2,4,6…. Eqn. 7.2 The first component (i=1) is a shunt capacitor, the second (i=2) component is a series inductor etc. The normalised q values and coupling coefficients are related to the normalised lowpass impedances for a filter, using the following equations: 1 0 Z q n n Z q j i ij Z Z k 1 Eqn. 7.3 Where Z i is the normalised impedance as obtained from LC filter tables or from equations 7.1 and 7.2. Applying equation 7.3 to equations 7.1 and 7.2 gives: n Sin n Sin q q n 2 2 2 ) 1 2 ( 2 0 Eqn. 7.4 These equations will then allow the k and q values for IL = 0 and q 0 = as shown in table 7.1 to be calculated. RF Electronics: Design and Simulation 210 www.cadence.com/go/awr