Frequency Response – How to Visually Determine Damping Ratio from Peak Amplitude

Let me label the intermediate nodes in the circuit using the letters A, B and C as shown below.

The nodal equations at the nodes A,B and C can be re-arranged to get the three equations given below.

$$\begin{align}U_AS_A &= CsU_1 + CsU_B\tag1\\ U_BS_B &= \frac{G}{2}U_1 + GU_2+CsU_A\tag2\\ U_CS_C &= GU_1 + G'U_2\tag3\end{align}$$

Where, \$G=\frac{1}{R}\$, \$G'=\frac{1}{K}\$. \$U_A, U_B\$ and \$U_C\$ are the potential at the nodes A, B and C respectively. And \$S_A, S_B\$ and \$S_C\$ are defined as follows:

$$\begin{align}S_A &= G+2Cs\\ S_B &= \frac{3}{2}G + Cs\\ S_C &=G+G'\end{align}$$

Let the gain of the operational amplifier be \$A_{op}\$ and \$A_{op}\rightarrow \infty\$. Now the output voltage of op-amp can be written as:

$$U_2 = A_{op}(U_B-U_C)|_{A_{op}\rightarrow\infty}\tag4$$

From the equations (1) to (3) potential at nodes can be written as: $$\begin{align}U_A &= \frac{Cs}{S_A}U_1+\frac{Cs}{S_A}U_B\tag5\\ U_B &= \frac{G}{2S_B}U_1+\frac{G}{S_B}U_2+\frac{Cs}{S_B}U_A\tag6\\ U_C &= \frac{G}{S_C}U_1+\frac{G'}{S_C}U_2\tag7\end{align}$$

The signal flow graph can be drawn using the equations from (4) to (7) as given below:

You can apply the limit \$A_{op}\rightarrow \infty\$ while simplifying the calculations.

Now, Aβ would typically be a value ranging 1000000 to 10000 (in opamp amplifier systems, where open-loop gain is usually around 120dB).

That's at DC where nobody really worries about phase margin because it's never going to be an issue. Look at a typical open-loop response of an op-amp: -

Picture source and other relevant information that could be useful.

At 1 kHz the open loop response might have dropped to 60 dB (G = 1000). At 1 MHz, the o/l gain is only 10 and this is an area where quite a few op-amp circuits have problems. Arctan of 10 (assuming a unity gain situation) is 84 degrees and consistent with the graph above. At 10 MHz the gain is unity and arctan of 1 is 45 degrees i.e. not a million miles off.

If you know the T.F. of the forward gain device and you know the feedback T.F. then certainly you can calculate phase margin by considering the loop broken with a signal being injected at the input and the output being taken from where the break is. But you have to respect impedances and loading when you open the loop and sometimes it can be difficult to realize.