Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part59

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part59

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 11 21 Copyright © Orchard Publications Low-Pass Analog Filter Prototypes Figure 11.18. VCVS low pass filter (Courtesy Reference Data for Engineers Handbook) The transfer function of the second order VCVS low pass filter of Figure 11.18 is given as (11.37) This is referred to as a second order all pole * approximation to the ideal low pass filter with cut- off frequency , where is the gain, and the coefficients and are provided by tables. For a non-inverting positive gain , the circuit of Figure 11.18 satisfies the transfer function of (11.37) with the conditions that (11.38) (11.39) (11.40) (11.41) From (11.40) and (11.41), we observe that . A fourth-order all pole low pass filter transfer function is a ratio of a constant to a fourth degree polynomial. A practical method of obtaining a fourth order transfer function, is to factor it into two second order transfer functions of the form of relation (11.37), i.e., *T h e t e r m i n o l o g y a l l pole” stems from the fact that the s plane contains poles only and the zeros are at , that is, the s plane is all poles and no zeros. C 1 R 3 R 4 R 1 R 2 C 2 v in v out Gs () Kb ω C 2 s 2 a ω C sb ω C 2 ++ ---------------------------------------- = ± ω C Ka b K R 1 2 aC 2 a 2 4b K 1 + [] C 2 2 4bC 1 C 2 +    ω C ----------------------------------------------------------------------------------------------------------------- = R 2 1 bC 1 C 2 R 1 ω C 2 ----------------------------- = R 3 KR 1 R 2 + K1 --------------------------- K 1 = R 4 1 R 2 + = R 4 R 3 + =
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Chapter 11 Analog and Digital Filters 11 22 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (11.42) Each factor in (11.42) can be realized by a stage (circuit). Then, the two stages can be cascaded as shown in Figure 11.19. Figure 11.19. Cascaded stages Table 11.3 lists the Butterworth low pass coefficients for second and fourth order designs, where and apply to the transfer functions of (11.37) and (11.42) respectively. For a practical design of a second order VCVS circuit, we select standard values for capacitors and for the circuit of Figure 11.18, we substitute the appropriate values for the coefficients and from Table 11.3, we choose desired values for the gain and cutoff frequency , and we substitute these in relations (11.38) through (11.41) to find the values of the resistors through . Example 11.6 Design a second order VCVS Butterworth low pass filter with gain and cutoff frequency . TABLE 11.3 Coefficients for Butterworth low pass filter designs Coefficients for Second and Fourth Order Butterworth Low Pass Filter Designs Order a1 . 4 1 4 2 1 2 b1 . 0 0 0 0
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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part59

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