Noise-modeling in two-ports

From the four port variables (see Chapter Two-ports), two can be selected as independent variables. This results in six equivalent noise models for two-ports. They are shown in Fig. 588.

Fig. 588 Six ways to model a noisy two-port. The positive directions for the signs of the port voltages \((V_{i} ,V_{0})\) and of the port currents \((I_{i},I_{o} )\) have been indicated by means of plus and minus signs and arrows at the port terminals, respectively.

In the example below, we will demonstrate the transformation of the noise representation according to Fig. 588A into the representation from Fig. 588C.

Example

The two-port equations for the model according to Fig. 588A are

\[\begin{split}\left( \begin{array} [c]{c} V_{i}+V_{ni_{A}}\\ I_{i}+I_{ni_{A}} \end{array} \right) =\left( \begin{array} [c]{cc} A & B\\ C & D \end{array} \right) \left( \begin{array} [c]{c} V_{o}\\ I_{o} \end{array} \right) .\end{split}\]

We eliminate \(I_{ni_{A}}\) from the input current vector by subtracting it from the current equation (row 2):

(247)\[I_{i}=CV_{o}-I_{ni_{A}}+DI_{o}. \label{eq-Itransformed}\]

We then define the new output voltage of the noisy two-port as \(V_{o} -I_{ni_{A}}\frac{1}{C}\), and write expression (247) as

(248)\[I_{i}=C\left( V_{o}-I_{ni_{A}}\frac{1}{C}\right) +DI_{o}. \label{eq-Ifinal}\]

We then substitute this new output voltage into the voltage equation (row 1). This changes the voltage equation to

(249)\[V_{i}+V_{ni_{A}}=A\left( V_{o}-I_{ni_{A}}\frac{1}{C}\right) +\frac{A}{C}I_{ni_{A}}+BI_{o}. \label{eq-Vtransformed}\]

After bringing the voltage \(\frac{A}{C}I_{ni}\) from the right side of ((249)) to the left side of this equation, we obtain the corrected input voltage:

(250)\[V_{i}+V_{ni_{A}}-\frac{A}{C}I_{ni_{A}}=A\left( V_{o}-I_{ni_{A}}\frac{1}{C}\right) +BI_{o}. \label{eq-Vfinal}\]

Equations ((248)) and ((250)) are the new two-port equations:

\[\begin{split}\left( \begin{array} [c]{c} V_{i}+V_{ni_{A}}-\frac{A}{C}I_{ni_{A}}\\ I_{i} \end{array} \right) =\left( \begin{array} [c]{cc} A & B\\ C & D \end{array} \right) \left( \begin{array} [c]{c} V_{o}-I_{ni_{A}}\frac{1}{C}\\ I_{o} \end{array} \right) ,\end{split}\]

from which we find the equivalent noise voltage sources according to the representation in Fig. 588C:

\[\begin{split}V_{ni_{C}} & =V_{ni_{A}}-\frac{A}{C}I_{ni_{A}},\\ V_{no_{C}} & =-I_{ni_{A}}\frac{1}{C}.\end{split}\]

Note: if \(V_{ni_{A}}\) and \(I_{ni_{A}}\) are uncorrelated, then \(V_{ni_{C}}\) and \(V_{no_{C}}\) are partially correlated.