Question about obtaining the input network transfer function of an inverting OPA

[opa627bm]

Hi all,
I am trying to review some of the fundamentals that has been long forgotten since school.
by reading this
https://peer.asee.org/a-method-for-obtaining-the-transfer-function-of-inverting-and-non-inverting-op-amp-circuits-based-on-classical-feedback-theory.pdf

I can follow the non-inverting section pretty well, but when comes to the inverting section:
page 8,
The input network transfer function is defined as - RF/(RF+RA), and feedback transfer function still RA/(RF+RA)

Since now Vin is a term in the equation,

Vi-Vout/(RA+RF) = Vi-V_ /RA = V_ - Vout)/RF

I know the gain is simply  -RF/RA , but i am trying to make sense of it.

any input will be appreciated, thank you !


 
My question is now did the author come out with - RF/(RF+RA) as for the input network transfer function..
thank you !

Regards,
Li

[ledtester]

If we naively translate the inverting op-amp circuit to a transfer network we might begin with something like:



The key to figuring out what goes in the ? ? ? area is to realize that the formula for the inverting input is

$$
\text{Inverting input} = \frac{v_i R_f + v_o R_a}{R_a + R_f}
$$

and so we can fill in the ? ? ? as follows:



Note how the minus sign in the \$v_i\$ transfer function is canceled by sending it the minus input of the new summing node.

Now we can short circuit the top summing node because we are just adding 0 -- i.e. the output of the lower summing node can go directly to \$G(s)\$ -- and that leads to the diagram in the paper.

Correction: We can short circuit the lower summing node to \$G(s)\$ with a multiplier of -1 since the lower summing node goes the negative input of the upper summing node.

[opa627bm]

Hi Ledtester,
thank you so much for the reply.
I can get the V_ transfer function using superposition.
but how to add the correct sign (input network (some people call it alpha ) is negative) The feedback network (some people call it beta ) ,
how to separate the combined term correctly into alpha and beta and with correct sign ?

Thank you for the help!

Regards,
Li

PS: sorry the pic I attached is rotated for some reason, from my viewer it is in the correct orientation.

[ledtester]

I'm not sure exactly what you're asking about, but perhaps this will help...

The equation \$V_A = V_B B + V_C C\$ is the same as this network:



Of course, you can decide to put the minus sign on C rather than B if you also switch the inputs to the summing node.

You have:

$$
\begin{align}
V &= \frac{ V_{\text{IN}} R_F + V_{\text{OUT}} R_A } { R_F + R_A } \\

   &= V_{\text{IN}} \frac{R_F}{R_F+R_A} + V_{\text{OUT}} \frac{R_A}{R_F+R_A} \\
\end{align}
$$
So set:
$$
\begin{alignat*}{2}
V_B &= V_{\text{IN}},   &B = \frac{R_F}{R_F+R_A} \\
V_C &= V_{\text{OUT}}, &C = \frac{R_A}{R_F+R_A} \\
\end{alignat*}
$$
and apply the diagram.

[opa627bm]

Hi Ledtester,
Sorry for not making my question clear.
I have used PDF this time, that should me the orientation correct.

My question is, how to map the terms in the V_ transfer function with correct sign to the input network transfer function and feedback network transfer function. and how to determine the correct sign on the summing block (+） and (-)


Regards,
Li

[ledtester]

Both of these are equivalent - you can decide to use either one. Note carefully where the + and - signs are on both the transfer functions and the summing node inputs.