I would like to know how to solve for Zin of the non ideal non-inverting amplifier shown below:
I wonder how to get to this expression: $$ \beta = \frac{R_1}{R_1 + R_2} $$ $$ Z_{in} \approx 2R_{icm} || (R_{id} + \beta A(s) R_{id}) $$
Where A(s) is the gain of the op-amp in the s-domain (again, it's non-ideal). Obviously there are some approximations I do not see. I know \$R_{icm} > R_{id}\$, and probably \$R_0 >> R_1 \: or \: R_2\$, but otherwise I'm in the dark, totally in the dark in fact.
Thanks!
EDIT: Maybe I should have been more precise regarding my question. I'm dealing with a textbook not really showing anything. Of course I can use shortcuts regarding negative feedback, I can also use google (altough I haven't found any links properly explaining what I'm seeing here), and I understand the basics of negative feedback.
I want to solve it simply using standard analysis method, \$Z_{in} = V_{i}/I_{i}\$. The author does the following steps:
- \$V_x = V_o \frac{2R_{icm} || R_1}{2R_{icm} || R_1 + R_2} \approx \beta V_o = \beta A(s) V_{id}\$
\$I_x = V_{id}/R_{id} \$
\$ Z_x = \frac{V_x}{I_{x}} = \frac{\beta A(s) V_{id}}{V_{id}/R_{id}} = \beta A(s) R_{id} \$
\$ \implies Z_{in} = 2R_{icm} || (R_{id} + Z_x) = 2R_{icm} || (R_{id} + \beta A(s) R_{id}) \$
What I don't understand is:
I don't even know where is \$V_x\$ because no schematics show it. I guess it's the voltage at \$ V_{IN-} \$, and I also guess they approximated the current entering \$V_{IN-}\$ to be zero in order to get that relation (but again, why is it a valid approx considering we assume finite resistances, because \$R_1 \: or \: R_2 << R_{icm} \: or \: R_{id}\$?). For the rest I understand, including the simplification to \$ \beta \$.
What is \$ I_x \$? The current flowing in V_{IN-} seems to be approximated to \$ V_{id}/R_{id} \$, correct? But why? Is it because \$R_icm >> R_{id}\$ (so that the current in the icm resistors are negligible)?
I don't have any idea what is \$Z_{x}\$ at this point? What is this equivalent impedance, or what is the equivalent circuit with \$Z_x\$?
I have no idea why \$Z_x\$ is in series with \$R_{id}\$. And why \$2R_{icm}\$ is in parallel with those 2.
If someone could explain these steps that would be really useful to me so I can get on with my reading. I'm more interested into the analytical solution than why I get the final result.
Thanks!