Exercise 9.11 An harmonic oscillator?

(a)

(b)

An answer:
$$\begin{array}{llll}\hfill & H_1(\mathit{j\omega })=A_v\cdot \frac{\mathit{j\omega L}∕R}{1+\mathit{j\omega L}∕R}& \hfill & \\ \hfill & H_2(\mathit{j\omega })=\frac{\mathit{j\omega L}∕R}{1+\mathit{j\omega L}∕R}& \hfill & \\ \hfill & H_3(\mathit{j\omega })=\frac{1}{1+\mathit{j\omega RC}}& \hfill & \\ \hfill & H_4(\mathit{j\omega })=\frac{1}{1+\mathit{j\omega RC}}& \hfill & \end{array}$$

(c)

An answer:
$$\begin{array}{llll}\hfill H_{\mathit{loop}}(\mathit{j\omega })& =A_v\cdot \frac{\mathit{j\omega L}∕R}{1+\mathit{j\omega L}∕R}\cdot \frac{\mathit{j\omega L}∕R}{1+\mathit{j\omega L}∕R}\cdot \frac{1}{1+\mathit{j\omega RC}}\cdot \frac{1}{1+\mathit{j\omega RC}}& \hfill & \\ \hfill & =A_v\cdot \frac{(\mathit{j\omega L}∕R)\cdot (\mathit{j\omega L}∕R)}{(1+\mathit{j\omega L}∕R)\cdot (1+\mathit{j\omega L}∕R)\cdot (1+\mathit{j\omega RC})\cdot (1+\mathit{j\omega RC})}& \hfill & \end{array}$$

This has a real numerator and a complex denominator. If the denominator is real for a specific non-zero and finite $\omega$ then the circuit can oscillate harmonically.

$$\begin{array}{llll}\hfill H_{\mathit{loop}}(\mathit{j\omega })& =\frac{A_v\cdot (\mathit{j\omega L}∕R)\cdot (\mathit{j\omega L}∕R)}{1+2\mathit{j\omega }(\frac{L}{R}+\mathit{RC})+j^2{\omega }^2(\frac{L^2}{R^2}+4\mathit{LC}+R^2{C}^2)+2j^3{\omega }^3(\frac{L^{2}C}{R}+L{C}^{2}R)+j^4{\omega }^4{L}^2{C}^2}& \hfill & \end{array}$$

This is obviously possible for the derived loopgain.

(d)

An answer:
For this, the denominator of the loop gain needs to be set to a real value for a finite, non-zero $\omega$. This follows from solving the equation $$\begin{array}{lll}\hfill 2(\frac{L}{R}+\mathit{RC})-j{\omega }^2\cdot 2(\frac{L^{2}C}{R}+L{C}^{2}R)=0& & \hfill \end{array}$$

This is obviously possible for the derived loopgain.

(e)

An answer:
Not applicable.