Chapter D2.5, Problem 24E

A high-pass filter Consider the LCR circuit shown in the figure.

a.    Explain why Kirchhoff’s voltage law for the circuit is

     $v_{\text{in}}-RI(t)-\frac{Q(t)}{C}-v_{\text{out}}=0.$

b.    Use the tacts that I(t) = Q′(t) and the voltage across the inductor is v_out = LI′(t) = LQ″(t) to show that the equation for the charge on the capacitor is

$Q^{″}+\frac{R}{L}{Q}^{\prime }+\frac{1}{LC}Q=\frac{1}{L}{v}_{\text{in}}=\frac{1}{L}f{e}^{i\omega t}.$

c.    Find the transfer function for the equation and compute the gain function.

d.    Show that the solution of the differential equation is $Q(t)=H(\omega )\frac{1}{L}f{e}^{i\omega t}$. Then conclude that Q″(t) = –ω²Q(t).

e.    Now we must relate v_out to Q. Use part (b) to show that v_out = –ω²LQ = –ω²H(_ω)v_in.

f.    The quantity of interest is the ratio of the magnitudes of the input and the output. Show that

     $\left|\frac{v_{\text{out}}}{v_{\text{in}}}\right|={\omega }^2\left|H(\omega )\right|={\omega }^{2}g(\omega )=\frac{{\omega }^2}{\sqrt{{\left({\omega }^2-\frac{1}{LC}\right)}^2+\frac{R^2}{L^2}{\omega }^2}}.$

g.    Show that tor $R>\sqrt{\frac{2L}{C}}$ the ratio in part (f) is monotonically increasing for ω > 0.

h.    Under what conditions, if any, does the ratio in part (f) have a local extremum for ω > 0?

i.    Explain why the circuit is called a high-pass filter.