I am developing a circuit to convert single-ended video to differential, based around the TI article "Converting single-ended video to differential video in single-supply systems" by Jim Karki. The article describes how to bias a FDA using a Thevenin equivalent to eliminate offset errors in the output.
Conceptually it all makes sense, and I can make sense of everything except for Equation 4 on the 3rd page (see the included image). The equation states that the gain of the circuit is G = 2 x (Rf/Rth)x(Rt/(Rs+Rt)). What I don't understand is why the second term, Rt/(Rs+Rt) is included here, when it already is included in the calculation of Rth (the Thevenin resistance) along with the video input's source resistance. It isn't as simple as just saying that the effects of Rt & Rs cancel out the 2x at the beginning of the equation (as would be the case with most video systems), because Rt and Rs are not the same resistance in the final solution.
Part of me thinks I can see the logic, but I can't formulate a consistent explanation. Because the circuit I am designing is slightly different, I really need to understand all of the steps rather than lazily copying what is in the paper!
EDIT: I corrected the erroneous Rs to Rt in one of the equations as pointed out by one of the commenters
Here is a link to the full article: https://www.ti.com/lit/an/slyt427/slyt427.pdf