With the previous gain function you calculated, design a Bessel fourth-order bandpass filter with center frequency \$f0 = 10 kHz\$ and bandwidth \$Bf = f0/10\$.
I did the calculations with the Mathematica program and even the factorization part is fine. Now I am trying to match the coefficients obtained, with the transfer function of the previous problem, to get the value of the capacitors and the resistors, but even assigning values to the capacitors I can not solve.
\$F(S)=\frac{1}{1+a_1S+b_1S^2}\rightarrow F(s)=\frac{1}{1+a_1\big(\frac{s^2+\omega_0^2}{Bs}\big)+b_1\big(\frac{s^2+\omega_0^2}{Bs}\big)^2}\$
\$F_{BP}(s)=\frac{1}{1+\frac{a_1s}{B}+\frac{a_1\omega_0^2}{Bs}+\frac{b_1s^2}{B^2}+\frac{b_12\omega_0^2}{B^2}+\frac{b_1\omega_0^4}{B^2s^2}}\Leftrightarrow\$
\$F_{BP}(s)=\frac{s^2}{s^2+\frac{a_1s^3}{B}+\frac{a_1\omega_0^2}{B}s+\frac{b_1}{B^2}s^4+\frac{b_12\omega_0^2}{B^2}s^2+\frac{b_1\omega_0^4}{B^2}}\Leftrightarrow\$
\$F_{BP}(s)=\frac{s^2}{\frac{b1}{B^2}s^4+\frac{a_1}{B}s^3+\big(\frac{b_12\omega_0^2}{B^2}+1\big)s^2+\frac{a_1\omega_0^2}{B}s+\frac{b_1\omega_0^4}{B^2}}\$
2th order Bessel coefficients:
\$a_1=1,3617\$
\$b_1=0,618\$
\$Q=0,58\$
Values of frequency and bandwidth:
\$f_0=10 KHz\$
\$\omega_0=2\pi\times 10^4 rad/s\$
\$B_f=1 KHz\$
\$B_{\omega}=B=2\pi\times10^3 rad/s\$
Substituting the values of the coefficients and frequencies in the transfer function:
\$F_{BP}=\frac{s^2}{1,565\times 10^{-8}s^4+2,167\times10^{-4}s^3+124,6s^2+855581,34s+2,44\times 10^{11}}\Leftrightarrow\$
Factoring in the Mathematica program:
\$F_{BP}(s)=\frac{6,38978\times 10^7s^2}{(3,69555\times10^9+6690,03s+s^2) (4,21887\times10^9+7156,62s+s^2)}\$
The transfer function of the previous problem is as follows (it is in this expression that I have to match the values):
\$H_{B}(s)=\frac{G_1(G_4+G_5)C_2s}{G_2G_3G_5+\frac{G_1G_4C_2}{G_2G_3G_5}s+\frac{G_4C_1C_2}{G_2G_3G_5}s^2}\$
My problem is now: I can not solve the system. Could you please help me?
System of equations:
\$G_1(G_4+G_5)C_2=\sqrt{6,38978\times 10^7}\$
\$G_2G_3G_5=3,69555\times 10^9\$
\$\frac{G_1G_4C_2}{G_2G_3G_5}=6690,03\$
\$\frac{G_4C_1C_2}{G_2G_3G_5}=1\$
I know I have to give values to both capacitors and to one of the resistors. G is the inverse of the resistance. But even giving values, I can not solve, gives me a mathematical incompatibility.