2nd order Sallen-Key Bandpass filter design in Matlab

Question

I'm trying to design a Sallen-Key BPF within Matlab. With the documentation found on ti: http://www.ti.com/lit/ml/sloa088/sloa088.pdf, I studied and understood the basic.

As for the design: The center frequency of my BPF is 40kHz. Q-factor and gain at center frequency must be 1. So in matlab I've made:

Fm = 40000;
R1 = 10000
R2 = 10000
C = 10*10^-9

R = 1/(2*pi*Fm*C);

G = 1+(R2/R1);
Q = 1/(3-G);
Am = G/(3-G);

w = 2*pi*Fm;

B=[G*R*C*w 0 ]
A=[(R^2)*(C^2)*(w^2) R*C*w*(3-G) 1]
sys=tf(B,A)

bode(sys,opts);

The issue is that my resulting bodeplot is indeed showing a BPF behaviour, but not at the desired frequency. Furthermore.

Can someone explain this to me? Thanks in advance

Answer 1

Your formulas do not take into account that the impedance of a capacitor is complex !

What you use is:

$$Zc(f) = 1 / 2 \pi f$$

but what you should use is:

$$Zc(s) = 1 / s$$

where $$s = 2 \pi f j $$ where j makes it imaginary remember: $$j^2 = -1$$

Have a look at this page on Wikipedia where they show how to determine the transfer function in the s domain.