Texa A&M Univerity Electrical Engineering Department ELEN Integrated Active Filter Deign Methodologie Alberto Valde-Garcia TAMU ID# 000 17 September 0, 001 HOMEWORK ASSIGNMENT # PROBLEM 1 Obtain at leat three different magnitude approximation that atify the following pecification: Figure 1 Four different approximation (Butterworth, Chebyhev, Elliptic and Invere Chebyhev) were obtained for the given pecification uing the CAD oftware FIESTA. Firt, the tranfer function and main performance plot will be hown for each of the approximation and then, a comparion between the four different approache will be preented upported by a table and comparion plot.
a) Butterworth Approximation Tranfer Function: 8 + 7.0e7 7 +.78e17 +1.97e.801e09 +.7e + 9.918e1 +1.17e1 +.181e8 + 1.9e7 Figure Figure
b) Chebyhev Approximation Tranfer Function: 8 +.1e7 7 +.8e17 +.1e 8.01e08 +.e +.78e1 + 1.1e1 + 7.e7 + 1.9e7 Figure Figure
c) Elliptic Approximation Tranfer Function: +.7e07.79e0 +1.99e017 +.701e08 +.e0 +1.e09 +1.1e0 +1.17e01 +.90e00 Figure Figure 7
d) Invere Chebyhev Approximation Tranfer Function: 8 + 7.e7 7 8 0.000719 + 1.e01 + 1.e01 +.e07 + 7.18e0 +.78e17 +1.0e +.7e + 9.9e1 +1.17e1 +.187e8 + 1.9e7 Figure 8 Figure 9
Table 1 how a comparion between the center frequency, bandwidth and quality factor of each econd order block of each one of the four implementation. A graphical comparion of the four different magnitude and group delay repone i hown in figure 10 and 11. Stage Butterworth Chebyhev ω 0 BW Q ω 0 BW Q 1.E+08.71E+07 9.70.E+08 9.E+0 8.8.E+08.0E+07 9.70.E+08 8.89E+0 8.8.71E+08 1.1E+07.9.9E+08.9E+0 8.87.E+08 1.0E+07.9.E+08.8E+0 8.87 Stage Elliptic Invere Chebyhev ω 0 BW Q ω 0 BW Q 1.7E+08 1.E+07 19.01.E+08.7E+07 9.80.9E+08.8E+0 9.1910.E+08.E+07 9.80.E+08.8E+0 9.1910.71E+08 1.1E+07.071 N/A N/A N/A.E+08 1.0E+07.0E+01 Table 1 Figure 10 Figure 11
Dicuion: The Chebyhev approximation yield the bet repone in term of harpne and out-of-band rejection but ha everal diadvantage: it require the highet quality factor in the econd order block, ha the larger ettling time and the larget group delay in the pa band. The Butterworth and Invere Chebyhev approximation are the eaiet to implement in term of the required quality factor. Their in-band group delay repone i very imilar and maller than the repone of the other two approache. The out of band rejection of the Invere Chebyhev approximation i better than the one obtained with the Butterworth but the ettling time of the former i lightly larger (around 10%) than the one of the later. The Elliptic approximation require one tage le ( in total) than the other. It in-band group delay i lightly greater than the one of the Butterworth approach and, both, it tep repone and out-of-band rejection are very imilar to the one of the Invere Chebyhev approximation. The required quality factor are not the mallet but not a high a the one demanded by the Chebyhev approach. PROBLEM Obtain the group delay approximation: Beel and Equal-Ripple that meet the following pecification: Ripple le or equal to 1% Group Delay le or equal to. nec Cutoff Frequency = 10MHz The approximation obtained with FIESTA are preented and compared below. a) Beel Approximation Tranfer Function: -.7e09 +.7e09 + e019 -.111e08 +.07e07 + e019 +.111e08 +.07e07
Figure 1 b) Equal Ripple Tranfer Function: -1.81e011 +1.81e011 +.98e01 +.98e01 -.e01 +.e01 + 9.e00 + 9.e00-1.e00 + 9.9e08 + 1.e00 + 9.9e08 Figure 1 The characteritic of the required econd order tage are hown in table. The Beel approximation appear to be better in thi cae ince it ue one tage le, ha lightly le ripple and there i practically no difference between the two tep repone. Stage Beel Equal Ripple ω 0 Q ω 0 Q 1.0E+09 0.19 1.9E+09 0.1179.E+09 0.808 7.09E+10 0.0008 N/A NA.E+09 0.9 Table
PROBLEM a) Deign a nd order LP Chebyhev for w 0 =p X10 rad/ and 1dB ripple Fieta yield the following approximation for the given pecification: DC GAIN 8.9109e-01 PASS/CENTER FREQUENCY.977e+0 (rad/) Q 0.90 POLE = -.8e+0+j(.9e+0) -.8e+0+j(-.9e+0) ZERO = Inf, Inf TRANSFER FUNCTION Table.879e007 + 897 +.e007 The correponding magnitude and group delay repone are hown in figure 1 Figure 1 b) Add a phae equalizer to the LP to yield a contant group delay in the range of 0 to 800Hz After a trial and error proce uing the Non-Conventional Group Delay Approximation feature of FIESTA, it wa determined that the following pecification yield convergence to an equalizer that atifactorily compenate the group delay repone howed in figure 1: Preditortion percentage: 0.000000 Relaxation factor: 0.100000 Contraint: Frequency: 100 Min: 0.0e-0 ec Max: 0.80e-0 ec Frequency: 800 Min: 0.00e-0 ec Max: 0.0e-0 ec A it can be oberved, the intention i to obtain a 0.1m difference in the delay at 800Hz with repect to the delay at 100Hz. In order to achieve convergence, all of the accuracy parameter had to be relaxed to the maximum and the Min-Max interval for the pecified point had to be broadened. The parameter of the obtained nd order equalizer are hown in table.
DC GAIN 1.000000e+00 PASS/CENTER FREQUENCY 7.7177e+0 (rad/) Q 0.001 POLE = -7.7987e+0+j(1.7178e+0) -7.7987e+0+j(-1.7178e+0) ZERO = 7.7987e+0+j(1.7178e+0) 7.7987e+0+j(-1.7178e+0) TRANSFER FUNCTION Table -1.8e00 +.99e007 +1.8e00 +.99e007 Both, the group delay repone of the equalizer and the group delay repone of the overall tranfer function (LP cacaded with equalizer) are hown in figure 1. Since the equalizer ha an (ideally) all pa magnitude repone, the overall magnitude repone i unaltered with repect of the tandalone LP. Figure 1 It i important to note on figure 1 (left) that the difference in the delay repone of the equalizer from 100Hz to 800Hz i cloe to 0.1m a intended. The overall group delay in the 0 to 800Hz range i around 0.7m with a ripple of le than 0.01 m, which correpond to only 1.%. It i worth to mention that by uing the ame approximation contraint, but targeting a higher contant group delay (i.e. 0.77m, 0.87m, etc.), convergence wa alo obtained but with a higher ripple (abolute and proportional) in the final repone and, in ome cae, a higher order for the equalizer. Reference: [1] Coure Note [] Fieta Manual