By Randall W. Rhea
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Additional resources for HF Filter Design and Computer Simulation
Next these results are substituted in the expression for the input impedance of the terminated network to be synthesized and we have Zi= s3+2s2+2s+1-s3 [ $+&2+2s+l+s3 ]=[ ,z~zi+1] (4% This expression for the impedance of the terminated network is used to find the network element values. First we rationalize the numerator Zi= 1 23+2s2+2s+l (43) 2s2+2s+1 Next we continually divide the lower order polynomial into the higher order polynomial and invert the remainder. 01 dB corner cutoff attenuation at a frequency of 1 radian per second.
I+ : . + . -12 + + ,L . -.. + + + t + Figure 2-7 Amplitude transmission and return loss for 5thorder (dashed) and 7th-order (solid) lowpass Butterworth (UL), Chebyshev (UR), Bessel (LL) and Cauer-Chebyshev (LR) filters. The Bessel transfer function has excellent group delay properties but poor selectivity. 17 Bessel Approximation Just as a maximally flat amplitude response is approximated by Butterworth, a maximally flat group-delay response is approximated by the Bessel transfer function.
13 Chebyshev Approximation If the poles of the Butterworth polynomial are moved toward the imaginary axis of the complex-frequency plane by multiplying their real parts by a constant factor iFz,< 1, the poles then lie on an ellipse and the maximally flat amplitude response of the Butterworth develops equal-attenuation ripples which increase with increased pole shifting. The resulting amplitude response becomes more zonal and the selectivity increases. Even a small amount of ripple can significantly improve the selectivity.
HF Filter Design and Computer Simulation by Randall W. Rhea