Power Converters with Digital Filter Feedback Control

Appendix B

(1.17) to (1.19) and (1.22) to (1.23) Transform

Equation (1.17) has the form

$\frac{A_{1}s+A_0}{B_2{s}^2+B_{1}s+B_0}$

(B.1)

The Laplace operator s shall be replaced by

$s=C\frac{1-z^{-1}}{1+z^{-1}}$

(B.2)

(B.1) becomes

$\frac{A_{1}C\left((1-z^{-1})/(1+z^{-1})\right)+A_0}{B_2{C}^2{\left((1-z^{-1})/(1+z^{-1})\right)}^2+B_{1}C\left((1-z^{-1})/(1+z^{-1})\right)+B_0}$

(B.3)

It can be reduced to

$\frac{A_{1}C\left(1-z^{-1}\right)\left(1+z^{-1}\right)+A_0{\left(1+z^{-1}\right)}^2}{B_2{C}^2{\left(1-z^{-1}\right)}^2+B_{1}C\left(1-z^{-1}\right)\left(1+z^{-1}\right)+B_0{\left(1+z^{-1}\right)}^2}$

(B.4)

There are ...