Blackman's theorem

${\displaystyle Z=Z_D\frac{1+T_{SC}}{1+T_{OC}},}$

where Z_D = impedance with the feedback disabled, T_SC = loop transmission with a small-signal short across the selected terminal pair, and T_OC = loop transmission with an open circuit across the terminal pair. The loop transmission also is referred to as the return ratio. Blackman's formula can be compared with Middlebrook's result for the input impedance Z_in of a circuit based upon the extra-element theorem:

${\displaystyle Z_{in}=Z_{in}^{\mathrm{\infty }}\left[\frac{1+Z_e^0/Z}{1+Z_e^{\mathrm{\infty }}/Z}\right]}$

where:

${\displaystyle Z}$ is the impedance of the extra element; ${\displaystyle Z_{in}^{\mathrm{\infty }}}$ is the input impedance with ${\displaystyle Z}$ removed (or made infinite); ${\displaystyle Z_e^0}$ is the impedance seen by the extra element ${\displaystyle Z}$ with the input shorted (or made zero); ${\displaystyle Z_e^{\mathrm{\infty }}}$ is the impedance seen by the extra element ${\displaystyle Z}$ with the input open (or made infinite).

Blackman's formula also can be compared with Choma's signal-flow result:

${\displaystyle Z_{SS}=Z_{S0}\left[\frac{1+T_I}{1+T_Z}\right],}$

where ${\displaystyle Z_{S0}}$ is the value of ${\displaystyle Z_{SS}}$ under the condition that a selected parameter P is set to zero, return ratio ${\displaystyle T_Z}$ is evaluated with zero excitation and ${\displaystyle T_I}$ is ${\displaystyle T_Z}$ for the case of short-circuited source resistance. As with the extra-element result, differences are in the perspective leading to the formula.