Editing Two-port network (section)

==Collapsing a two-port to a one port==

A two-port network has four variables with two of them being independent.  If one of the ports is terminated by a load with no independent sources, then the load enforces a relationship between the voltage and current of that port.  A degree of freedom is lost.  The circuit now has only one independent parameter.  The two-port becomes a [[one-port]] impedance to the remaining independent variable.

For example, consider impedance parameters

:<math> \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} </math>

Connecting a load, {{math|''Z''<sub>L</sub>}} onto port 2 effectively adds the constraint 
:<math> V_2 = -Z_\mathrm{L} I_2 \, </math>

The negative sign is because the positive direction for {{math|''I''{{sub|2}}}} is directed into the two-port instead of into the load.  The augmented equations become

:<math>\begin{align}
       V_1 &= Z_{11} I_1  +  Z_{12} I_2 \\
  -Z_\mathrm{L} I_2 &= Z_{21} I_1  +  Z_{22} I_2
\end{align}</math>

The second equation can be easily solved for {{math|''I''{{sub|2}}}} as a function of {{math|''I''{{sub|1}}}} and that expression can replace {{math|''I''{{sub|2}}}}  in the first equation  leaving {{math|''V''{{sub|1}}}} ( and {{math|''V''{{sub|2}}}} and {{math|''I''{{sub|2}}}} ) as functions of {{math|''I''{{sub|1}}}}

:<math>\begin{align}
  I_2 &= -\frac{Z_{21}}{Z_\mathrm{L} + Z_{22}} I_1 \\[3pt]
  V_1 &= Z_{11} I_1 - \frac{Z_{12} Z_{21}}{Z_\mathrm{L} + Z_{22}} I_1 \\[2pt]
      &= \left(Z_{11} - \frac{Z_{12} Z_{21}}{Z_\mathrm{L} + Z_{22}}\right) I_1 = Z_\text{in} I_1
\end{align}</math>

So, in effect, {{math|''I''{{sub|1}}}} sees an input impedance {{math|''Z''{{sub|in}}}} and the two-port's effect on the input circuit has been effectively collapsed down to a one-port; i.e., a simple two terminal impedance.