Classical Noise Analysis of a Two-Port Network


A noisy two port network can be represented by a noiseless two port network along with noise sources referred to input. Figure 1 illustrates the noiseless two port network with noise sources referred to input.

  • The input is represented by it’s Norton equivalent – current source(\overline{i_s}) in parallel with source admittance (Y_s =G_s+j B_s)
  • Input referred noise sources \overline{e_n} and \overline{i_n} have correlated components
  • \overline{i_s} is uncorrelated with \overline{e_n} and \overline{i_n}

The noise factor for this two port network can be expressed as

(1)   \begin{equation*} F = 1+ {\overline{|i_n + Y_s e_n|^2} \over \overline{i_s^2}} \end{equation*}

By splitting the input referred noise \overline{i_n} into uncorrelated and correlated components, i_n = i_u + i_c. Now we can related the correlated component (i_c) to \overline{e_n}  with correlation admittance Y_c (=G_c+j B_c) as i_c = Y_c e_n.

Substituting in Eq-(1), we get

(2)   \begin{equation*} F = 1+ {\overline{i_u^2} + |Y_c + Y_s|^2 \overline{e_n^2} \over \overline{i_s^2}} \end{equation*}

These noise sources are now uncorrelated, can be treated as thermal noise sources with equivalent thermal noise resistance given by

(3)   \begin{eqnarray*} R_n &=& { \overline{e_n^2} \over 4kT \Delta f}\\ G_u &=& { \overline{i_u^2} \over 4kT \Delta f}\\ G_s &=& { \overline{i_s^2} \over 4kT \Delta f} \end{eqnarray*}

Substituting Eq-(3) in Eq-(2),

(4)   \begin{eqnarray*} F &=& 1+ {G_u + |Y_c + Y_s|^2 R_n \over R_s} \\ &=& 1+ {G_u + \left[ (G_c + G_s)^2+(B_c + B_s)^2 \right] R_n \over G_s} \\ \end{eqnarray*}

The noise factor of the two-port network (from Eq-(4) ) characterized by G_c, B_c, G_u and R_n is also a function of source admittance(G_s and B_s). So the noise factor of network is minimized by right selection of source admittance Y_s.

The noise factor as a function of B_s is minimum when B_s = B_c^*.

The condition for minimizing noise factor w.r.t G_s is given by {\partial F \over \partial G_s} = 0

(5)   \begin{eqnarray*} {\partial F \over \partial G_s} &=& {\partial \over \partial G_s}\left( 1 + {G_u + (G_c + G_s)^2 R_n \over G_s}\right) \\ &=& - {G_u \over G_s^2} + {2(G_c+G_s)\over G_s}R_n - {(G_c+G_s)^2  \over G_s^2}R_n =0\\ \end{eqnarray*}

The solution for minimum noise figure is

(6)   \begin{eqnarray*} G_{s,opt} &=& \sqrt{G_c^2 + {G_u \over R_n}} \\ B_{s,opt} &=& -B_c \end{eqnarray*}

The minimum noise figure is given by

(7)   \begin{equation*} F_{min} = 1+2R_n(G_c + G_{s,opt}) = 1 + 2 R_n \left(G_c + \sqrt{G_c^2 + {G_u \over R_n}}\right) \end{equation*}

If source impedanceY_s \neq Y_{s,opt}, then noise factor is given by

(8)   \begin{equation*} F = F_{min} + {R_n \over G_s} |Y_s - Y_{s,opt}|^2 \end{equation*}

For maximum power transfer Y_s = Y_{in}^*, but for minimum or optimum noise figure Y_{s,opt} \neq Y_{in}^* . Therefore the noise match does not correspond to the power match, and thus a compromise is necessary to find the best performance.

 

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