I was trying to understand the following journal article about how to analyze loop-gain using design-oriented analysis.
First, I can now derive the loop-gain expression neglecting load impedance and damping network (\$T_\text{NLND}\$) with the following nodal equations at node \$V_F, V_y, V_z\$ and circuit below. I used breaking loop method by injecting \$V_t\$ to find this loop-gain.
$$\begin{align} \displaystyle V_o &= -A_\text{OL}V_t\\ \frac{V_F-V_z}{Z_G} + \frac{V_F-V_o}{Z_F} &= 0\\ \frac{V_y}{Z_{F1}} + \frac{V_y-V_z}{Z_{F2}} + \frac{V_y-V_o}{Z_S+Z_{C\text{INJ}}} &= 0\\ \frac{V_z}{Z_{F3}} + \frac{V_z-V_y}{Z_{F2}} + \frac{V_z-V_F}{Z_G} &= 0 \end{align}$$
By solving the above equations, I get the same result of \$T_\text{NLND} = \dfrac{V_F}{V_t} = -A_\text{OL}\dfrac{\mathscr{N}}{\mathscr{D}}\$ compared to one in the article.$$\begin{align} \mathscr{N} ={} & Z_{F_1}\left[Z_{F_2}(Z_{F_3}+Z_G) + Z_{F_3}(Z_F+Z_G+Z_S+Z_{C\text{INJ}}) + Z_G(Z_S+Z_{C\text{INJ}})\right]\\& +\left[Z_{F_3}Z_G+Z_{F_2}(Z_{F_3}+Z_G)\right]\left(Z_S+Z_{C\text{INJ}}\right)\label{eq:4-5}\\ \mathscr{D} ={} & Z_{F_1}\left[Z_{F_2}(Z_{F_3}+Z_G+Z_F) + Z_{F_3}(Z_F+Z_G+Z_S+Z_{C\text{INJ}}) + (Z_G+Z_F)(Z_S+Z_{C\text{INJ}})\right]\\& +\left[Z_{F_3}(Z_G+Z_F)+Z_{F_2}(Z_{F_3}+Z_G)\right]\left(Z_S+Z_{C\text{INJ}}\right) \end{align}$$
My 1st question: according to the article, he suggested breaking the loop by injecting signal like a circuit shown below. How can I find the loop-gain from this? i.e. what are \$V_F\$ and \$V_t\$ of it?
My 2nd question: when I applied nodal analysis in my first circuit at node \$V_o\$ instead of previous work (node \$V_F\$) and I got the loop-gain \$T_\text{NLND}\$ below which are not the same result. What is wrong with it? Both numerators are the same, but not for the denominators.
$$\begin{align} \mathscr{N}_\text{false} ={} & Z_{F_1}\left[Z_{F_2}(Z_{F_3}+Z_G) + Z_{F_3}(Z_F+Z_G+Z_S+Z_{C\text{INJ}}) + Z_G(Z_S+Z_{C\text{INJ}})\right]\\& +\left[Z_{F_3}Z_G+Z_{F_2}(Z_{F_3}+Z_G)\right]\left(Z_S+Z_{C\text{INJ}}\right)\\ \mathscr{D}_\text{false} ={} & Z_{F1}\left[(Z_{F3}Z_G+Z_{F_2}(Z_{F_3}+Z_G)) + (Z_{F3}+Z_G)(Z_F+Z_S+Z_{C\text{INJ}})\right]\\& +\left[Z_{F3}Z_G+Z_{F2}(Z_{F3}+Z_G)\right]\left(Z_F+Z_S+Z_{C\text{INJ}}\right) \end{align}$$
After that, in the next section, I begin to use open-circuit version of the Extra Element Theorem by realizing load impedance \$Z_\text{LOAD,th}\$ as an open-circuit extra element. Since if my work on \$T_\text{NLND}\$ is corresponded, this shows that the input and output of loop-gain transfer function should follow my first circuit (output is \$V_F\$ and input is \$V_t\$)
For the driving-point impedance, using my first circuit, it is quite straightforward since \$V_t\$ was shorted and by inspection, it is really
$$Z_D = Z_{C\text{INJ}} \parallel (Z_S + (Z_{F1} \parallel (Z_{F2}+(Z_{F3} \parallel (Z_G+Z_F)))))$$
My 3rd question: However, I have no idea for finding nulling impedance. I do find clarification about null condition. In this case, double-null injection of input \$V_t\$ and \$V_{test}\$ (at node where load impedance would be connected) must null the output voltage \$V_F \rightarrow 0\$ (which is not short-circuit but just null). So, I just say that \$V_F = 0\$ and find \$Z_N = \dfrac{V_{test}}{I_{test}}\$ at node where load impedance would be connected (the top node), but I got \$Z_N\$ like this below which is not the same as in article. How should I do?
$$Z_N = (Z_{C\text{INJ}}+Z_F) \parallel (Z_S + (Z_{F1} \parallel (Z_{F2}+(Z_{F3} \parallel Z_G))))$$
Below is the nulling impedance from the article:
$$Z_N = Z_{C\text{INJ}} \parallel (Z_S + (Z_{F1} \parallel (Z_{F2}+(Z_{F3} \parallel Z_G)))) \parallel \left(\frac{Z_G}{Z_{F1}}+\left(1+\frac{Z_{F2}}{Z_{F1}}\right)\left(1+\frac{Z_G}{Z_{F3}}\right)\right)(Z_{F2}+(Z_{F3} \parallel Z_G))))\frac{Z_{C\text{INJ}}}{Z_F}$$
Also, in the next section is to consider damping network as short-circuit Extra Element. I guess I would confuse about that nulling impedance. If you could guide me for that nulling impedance too, I would appreciate it!
P.S. Apologize if my post is somehow weird, this is my first post in stackexchange.