Exercise 7.3 Phase margins

See the schematic below. For the opamp, $r_{\mathit{in}}\to \infty$, $r_{\mathit{out}}=0\mathrm{\Omega }$) and the voltage gain is $A$.

a)

Under assumption of $A\to \infty$, derive the transfer function from $v_{\mathit{in}}$ to $v_{\mathit{out}}$.

b)

Now assume that the opamp gain has a finite frequency-independent value A. Calculate the transfer function $H(\mathit{j\omega })=v_{\mathit{out}}(\mathit{j\omega })∕v_{\mathit{in}}(\mathit{j\omega })$ for this case and draw a magnitude part of the bode plot, indicating the values of poles, zeros (x-axis) and flat parts (y-axis).

c)

Now assume that the opamp gain has a frequency-dependent gain $A(\mathit{j\omega })=A_0∕(1+\mathit{j\omega }∕{\omega }_1)$. Determine whether the circuit has a phase margin $>45°$ for $A_0=100$dB, ${\omega }_1=2\pi \cdot 10$, $L_1=100\mathit{\mu H}$, $R_2=1k\mathrm{\Omega }$.