The Sallen-Key filters

• Second order low-pass filter

  If we let $Z_1=R_1$, $Z_2=R_2$, $Z_3=1/j\omega C_1$, $Z_4=1/j\omega C_2$, the FRF becomes:

  
  

  $$H = \frac{1/(j\omega)^2C_1C_2} {R_1R_2+(R_1+R_2)/j\omega C_1+1/(j\omega)^2C_1C_2}$$

  $$= \frac{1/R_1R_2C_1C_2}{(j\omega)^2+j\omega(R_1+R_2)/R_1R_2C_1+1/R_1R_2C_1C_2}$$

  $$= \frac{1/R_1R_2C_1C_2}{(j\omega)^2+j\omega/R_pC_1+1/R_1R_2C_1C_2}$$

  $$= \frac{\omega_n^2}{(j\omega)^2+2\zeta\omega_n\;j\omega+\omega_n^2}... ...a+\omega_n^2} =\frac{\omega_n^2}{(j\omega)^2+\Delta\omega\;j\omega+\omega_n^2}$$

  
  

• Second order high-pass filter

  If we let $Z_1=1/j\omega C_1$, $Z_2=1/j\omega C_2$, $Z_3=R_1$, $Z_4=R_2$, the FRF becomes:
  

  $$H = \frac{R_1R_2} {1/(j\omega)^2C_1C_2+R_1(1/j\omega C_1+1/j\omega C_2)+R_1R_2}$$

  $$= \frac{(j\omega)^2}{1/R_1R_2C_1C_2+j\omega\;(C_1+C_2)/C_1C_2R_2+(j\omega)^2}$$

  $$= \frac{(j\omega)^2}{(j\omega)^2+j\omega\;/C_sR_2+1/R_1R_2C_1C_2}$$

  $$= \frac{(j\omega)^2}{(j\omega)^2+2\zeta\omega_n\;j\omega+\omega_n^2... ...mega_n^2} =\frac{(j\omega_n)^2}{(j\omega)^2+\Delta\omega\;j\omega+\omega_n^2}$$