Again two reference planes are chosen; one for the input and one for the output. The E and H fields entering and leaving the ports are replaced by the incident and reflected voltage and current amplitudes.

As a convention, the current directions are taken as entering into the network.

The following figure shows an N-port microwave network

If each mechanical port supports a single mode of propagating field, it can be represented as an electrical port. If in a given mechanical port, more than one mode of propagating fields are supported, than that mechanical port will be represented by that many electrical ports. In many practical cases, only one mode propagates through a given port. Therefore, the above complication is removed.

The dependent variables ( total voltages) at each port can be related to the independent variables ( total currents) at each port through the [Z] matrix defined by

in short hand notation

The elements Z_ij are can be found by

i.e.,

• by open-circuiting all the other ports, except i'th port,
• by applying a voltage source at port j,
• by measuring current at j'th port (I_j), the total current at port j, and V_i, the voltage at port i.
• The ratio of V_i/I_j gives Z_ij.

Similarly, an admittance matrix is defined by relating Currents to Voltages through

in short

The elements Y_ij are found by

i.e.,

• by short circuiting all the other ports, except i'th port,
• by applying a voltage source V_j at port j, and
• by measuring I_i, the total current at port i. The ratio of I_i/V_j gives Y_ij.

Properties of lossless Networks.

1. The rate of change of reactance (or susceptance) as a function of w is positive. This is Foster's Reactance Theorem. The result implies that the poles and zeros of a reactance (or susceptance) function alternate in position along the w axis.
2. Even and odd properties of Zin =R(w) + JX(w) for a single port microwave network
   R (-w) = R(w) (even function) and X(-w) = -X(w) (odd function)
3. For networks with linear e and m, the impedance (admittance) matrix is symmetric. i.e.,
   Z_ij =Z_ji or Y_ij =Y_ji

4. Impedance and admittance matrices are reciprocal of each other. i.e.,
   
5. For a lossless network, Re (Z_nm) =0. i.e., the network elements are purely imaginary.

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   Examples of Two port networks

   .
   • Coax-Waveguide adapter.
     
   • Coax-Microstrip fixture.
     
   • Waveguide transition
     
   Typical representation of the two port network will be
   

   And as a T-network

   

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   Copyright; Altan M. Ferendeci, University of Cincinnati