Chapter 33, Problem 33.56P

Consider the Filter circuit shown in Figure P33.56. (a) Show that the ratio of the amplitude of the output voltage to that of the input voltage is to that of input voltage is

$\frac{\Delta V_{\text{out}}}{\Delta V_{\text{in}}}=\frac{1/\omega C}{\sqrt{R^2+{\left(\frac{1}{\omega C}\right)}^2}}$

(b) What value does this ratio approach as the frequency decreases toward zero? (c) What value does this ratio approach as the frequency increases without limit? (d) At what frequency is the ratio equal to one-half?