butterworth and chebyshev filter

[promach]

1) How to derive expression (3.11) for chebyshev filter ?

2) Why there is a factor of 2 in g_i expression (3.24) and n expression (3.25) for butterworth filter ?

 

[pancho_hideboo]

1) How to derive expression (3.11) for chebyshev filter ?

Definition of Chebyshev polynomials.

2) Why there is a factor of 2 in g_i expression (3.24) and n expression (3.25) for butterworth filter ?

(3.24) is due to complex conjugate pair of poles.

(3.25) is from (3.7).

 

[promach]

(3.24) is due to complex conjugate pair of poles.

I understand that butterworth function uses complex conjugate poles, but that does not justify the use of a factor of 2 in g_i expression.

 

[pancho_hideboo]

I understand that butterworth function uses complex conjugate poles,
but that does not justify the use of a factor of 2 in g_i expression.

Consider coefficients of polynomials.
Calculate S11 from S21.
Then calculate Zin.
Then evaluate g-parameters by continuation fraction expansion of Zin.
See https://www.edaboard.com/showthread.php?384998-Selection-of-Zeros-of-S11(s)

All zeros of S11 are located at origin for Butterworth filter.
So you can get Both S11 and Zin as close form.