# Any transfer function has the same denominator 1+T

Sometime ago, I came across some theory saying that in a control loop any transfer functions such as input/output impedance, voltage gain can be derived from loop gain. Maybe something like any transfer function related to loop gain will have the denominator in the form (1+T) with T is loop gain. However, I couldn't find it now. Hope anyone here can help with some material or name of the theory.

Actually, in a system featuring a unity return as shown below, the perturbation $u_1$ is rejected by an amount depending on $S=\frac{1}{1+T(s)}$. $S$ is the sensitivity function and it shows how robust the system is to incoming perturbations. You can see that if you have a large open-loop gain in dc, for $s=0$, then $S$ is close to 0 and the rejection is perfect. As the magnitude of $T$ reduces when frequency increases and you approach crossover (the 0-dB point), the system is less and less at ease to reject these disturbances. When you pass the 0-dB point, the system runs in open-loop ac-wise and does what it can to react: no gain, no control system.

Classical perturbations are the input voltage $V_{in}$ and the output current $I_{out}$. The rejection of $V_{in}$ is called the audio-susceptibility of the considered equipment, also called input line rejection. It is the capability of a converter, for instance, to maintain a perfectly-regulated $V_{out}$ despite input voltage variations. If you consider $A_{S,OL}$ as the open-loop audio susceptibility of a given control system operated in open-loop, then once you close the loop, the new audio susceptibility becomes: $A_{S,CL}=\frac{A_{S,OL}}{1+T(s)}$.

For the output current, the small-signal open-loop output voltage variations are linked to the output impedance: $\Delta V_{out}=Z_{out,OL}\times \Delta I_{out}$. When you close the loop, the new output impedance becomes: $Z_{out,CL}=\frac{Z_{out,OL}}{1+T(s)}$. You may find more information in this PPT.