Mason's Invariant - Derivation of U

Derivation of U

Mason first defined the device being studied with the three constraints listed below.

  1. The device has only two ports (at which power can be transferred between it and outside devices).
  2. The device is linear (in its relationships of currents and voltages at the two ports).
  3. The device is used in a specified manner (connected as an amplifier between a linear one-port source and a linear one-port load).

Then, according to Madhu Gupta in Power Gain in Feedback Amplifiers, a Classic Revisited, Mason defined the problem as "being the search for device properties that are invariant with respect to transformations as represented by an embedding network" that satisfy the four constraints listed below.

  1. The embedding network is a four-port.
  2. The embedding network is linear.
  3. The embedding network is lossless.
  4. The embedding network is reciprocal.

He next showed that all transformations that satisfy the above constraints can be accomplished with just three simple transformations performed sequentially. Similarly, this is the same as representing an embedding network by a set of three embedding networks nested within one another. The three mathematical expressions can be seen below.

1. Reactance padding: \begin{bmatrix} Z'_{11} & Z'_{12} \\ Z'_{21} & Z'_{22} \end{bmatrix} = \begin{bmatrix} Z_{11}+jx_{11} & Z_{12}+jx_{12} \\ Z_{21}+jx_{21} & Z_{22}+jx_{22} \end{bmatrix}

2. Real Transformations: \begin{bmatrix} Z'_{11} & Z'_{12} \\ Z'_{21} & Z'_{22} \end{bmatrix} = \begin{bmatrix} n_{11} & n_{12} \\ n_{21} & n_{22} \end{bmatrix} \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} n_{11} & n_{12} \\ n_{21} & n_{22} \end{bmatrix}

3. Inversion: \begin{bmatrix} Z'_{11} & Z'_{12} \\ Z'_{21} & Z'_{22} \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix}^{-1}

Mason then considered which quantities remained invariant under each of these three transformations. His conclusions, listed respectively to the transformations above, are shown below. Each transformation left the values below unchanged.

1. Reactance padding: \left and \left

2. Real transformations: \left \left and \dfrac{\det{\left }}{\det{\left }}

3. Inversion: The magnitudes of the two determinants and the sign of the denominator in the above fraction remain unchanged in the inversion transformation. Consequently, the quantity invariant under all three conditions is:

\begin{align} U & =\dfrac{|\det{\left }|}{\det{\left }} \\ & = \dfrac{|Z_{12}-Z_{21}|^{2}}{4 (\operatorname{Re} Re-\operatorname{Re} \operatorname{Re})} \\ & = \dfrac{|Y_{21}-Y_{12}|^{2}}{4 (\operatorname{Re} \operatorname{Re}-\operatorname{Re} \operatorname{Re})} \end{align}

Read more about this topic:  Mason's Invariant