Active filters

Abstract

The circuit of the simplest lowpass filter is shown in Fig. 3.1. The ratio of output voltage to input voltage can be expressed, for sinusoidal signals, as

$$ \underline A ({\text{j}}\omega ) = \frac{{{{\underline U }_0}}}{{\underline {{U_i}} }} = \frac{1}{{1 + {\text{j}}\omega RC}}, $$

and is called the frequency response of the circuit. Replacing j ω by j ω+σ = p gives the system function or generalized transfer function

$$ A(p) = \frac{{\text{L}\{ U_\text{o} (t)\} }}{{\text{L}\{ U_\text{i} (t)\} }} = \frac{1}{{1 + pRC}}. $$

This is the ratio of the Laplace transformed output and input voltage for signals of any time dependence. On the other hand, the transition from the transfer function A(p) to the frequency response $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} (\text{j}\omega )$ for sinusoidal input signals, is made by setting σ to zero.