Exercise 9.14 An harmonic oscillator?

(a)

(b)

An answer:

$$\begin{array}{llll}\hfill & H_1(\mathit{j\omega })=\frac{1+2\mathit{j\omega L}∕R}{\mathit{j\omega L}∕R}& \hfill & \\ \hfill & H_2(\mathit{j\omega })=\frac{1+\mathit{j\omega RC}}{\mathit{j\omega RC}}& \hfill & \\ \hfill & H_3(\mathit{j\omega })=-\frac{R_1}{R}& \hfill & \end{array}$$

(c)

An answer:

$$\begin{array}{llll}\hfill & -90^{\circ }\le \angle H_1(\mathit{j\omega })\le 0^{\circ }& \hfill & \\ \hfill & -90^{\circ }\le \angle H_2(\mathit{j\omega })\le 0^{\circ }& \hfill & \\ \hfill & \angle H_3(\mathit{j\omega })=\pm 180^{\circ }& \hfill & \end{array}$$

(d)

An answer:

The loop gain is $A_{\mathit{loop}}(\mathit{j\omega })=-\frac{1+2\mathit{j\omega L}∕R}{\mathit{j\omega L}∕R}\cdot \frac{1+\mathit{j\omega RC}}{\mathit{j\omega RC}}\cdot \frac{R_1}{R}$

It can be shown mathematically that $\angle A_{\mathit{loop}}$ can be $N\cdot 360^{\circ }$ only at $\omega =0$. This is not considered as a valid oscillation frequency.

(e)

An answer:

$$\begin{array}{llll}\hfill & H_1(\mathit{j\omega })=\frac{1+2\mathit{j\omega L}∕R}{\mathit{j\omega L}∕R}& \hfill & \\ \hfill & H_2(\mathit{j\omega })=\frac{1+\mathit{j\omega RC}}{\mathit{j\omega RC}}& \hfill & \\ \hfill & H_3(\mathit{j\omega })=-\frac{1}{\mathit{j\omega RC}}& \hfill & \end{array}$$

(f)

An answer:

$$\begin{array}{llll}\hfill & -90^{\circ }\le \angle H_1(\mathit{j\omega })\le 0^{\circ }& \hfill & \\ \hfill & -90^{\circ }\le \angle H_2(\mathit{j\omega })\le 0^{\circ }& \hfill & \\ \hfill & \angle H_3(\mathit{j\omega })=90^{\circ }& \hfill & \end{array}$$

(g)

An answer:

The loop gain is $A_{\mathit{loop}}(\mathit{j\omega })=-A_{\mathit{buf}}\cdot \frac{1+2\mathit{j\omega L}∕R}{\mathit{j\omega L}∕R}\cdot \frac{1+\mathit{j\omega RC}}{\mathit{j\omega RC}}\cdot \frac{1}{\mathit{j\omega RC}}$

The denominator is imaginary. If the numerator can be set to an imaginary value for a non-zero finite $\omega$ then this circuit can be dimensioned to oscillate harmonically. Working things out a little leads to:

$$\begin{array}{llll}\hfill & A_{\mathit{loop}}(\mathit{j\omega })=-A_{\mathit{buf}}\cdot \frac{1+\mathit{j\omega }(2L∕R+\mathit{RC})+2{(\mathit{j\omega })}^2\mathit{LC}}{{(\mathit{j\omega })}^{3}L{C}^{2}R}& \hfill & \\ \hfill & {\omega }_{\mathit{osc}}=\frac{1}{\sqrt{2\mathit{LC}}}& \hfill & \\ \hfill & A_{\mathit{buf}}=\frac{\mathit{RC}}{2\mathit{RC}+4L∕R}& \hfill & \end{array}$$