Editing Two-port network (section)

===Series-series connection===
[[File:two-port series-series.svg|thumb|'''Fig. 10.''' Two two-port networks with input ports connected in series and output ports connected in series.]]
When two-ports are connected in a series-series configuration as shown in figure 10, the best choice of two-port parameter is the {{mvar|z}}-parameters.  The {{mvar|z}}-parameters of the combined network are found by matrix addition of the two individual {{mvar|z}}-parameter matrices.<ref>Ghosh, p. 371.</ref><ref>Farago, p. 128.</ref>

:<math>[\mathbf z] = [\mathbf z]_1 + [\mathbf z]_2</math>

[[File:two-port series-series improper.svg|thumb|left|'''Fig. 11.''' Example of an improper connection of two-ports. {{math|''R''<sub>1</sub>}} of the lower two-port has been by-passed by a short circuit.]]
[[File:two-port series-series proper.svg|thumb|'''Fig. 12.''' Use of ideal transformers to restore the port condition to interconnected networks.]]
As mentioned above, there are some networks which will not yield directly to this analysis.<ref name=Farago1227/>  A simple example is a two-port consisting of a {{mvar|L}}-network of resistors {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}}.  The {{mvar|z}}-parameters for this network are;

:<math>[\mathbf z]_1 = \begin{bmatrix} R_1 + R_2 & R_2 \\ R_2 & R_2 \end{bmatrix}</math>

Figure 11 shows two identical such networks connected in series-series.  The total {{mvar|z}}-parameters predicted by matrix addition are;

:<math>[\mathbf z] = [\mathbf z]_1 + [\mathbf z]_2 = 2[\mathbf z]_1 = \begin{bmatrix} 2R_1 + 2R_2 & 2R_2 \\ 2R_2 & 2R_2 \end{bmatrix}</math>

However, direct analysis of the combined circuit shows that,

:<math>[\mathbf z] = \begin{bmatrix} R_1 + 2R_2 & 2R_2 \\ 2R_2 & 2R_2 \end{bmatrix}</math>

The discrepancy is explained by observing that {{math|''R''{{sub|1}}}} of the lower two-port has been by-passed by the short-circuit between two terminals of the output ports.  This results in no current flowing through one terminal in each of the input ports of the two individual networks.  Consequently, the port condition is broken for both the input ports of the original networks since current is still able to flow into the other terminal.  This problem can be resolved by inserting an ideal transformer in the output port of at least one of the two-port networks.  While this is a common text-book approach to presenting the theory of two-ports, the practicality of using transformers is a matter to be decided for each individual design.