Lumped Port Parameters
In transmission line theory voltages and currents are dealt with rather than electric and magnetic fields, so the lumped port provides an interface between them. The requirement on a lumped port is that the feed point must be similar to a transmission line feed, so its gap must be much less than the wavelength. It is then possible to define the electric field from the voltage as
where h is a line between the terminals at the beginning of the transmission line, and the integration is going from positive (phase) V to ground. The current is positive going into the terminal at positive V.
The transmission line current can be represented with a surface current at the lumped port boundary directed opposite to the electric field.
The impedance of a transmission line is defined as
and in analogy to this an equivalent surface impedance is defined at the lumped port boundary
To calculate the surface current density from the current, integrate along the width, w, of the transmission line
where the integration is taken in the direction of ah × n. This gives the following relation between the transmission line impedance and the surface impedance
where the last approximation assumed that the electric field is constant over the integrations. A similar relationship can be derived for coaxial cables
The transfer equations above are used in an impedance type boundary condition, relating surface current density to tangential electric field via the surface impedance.
where E is the total field and E0 the incident field, corresponding to the total voltage, V, and incident voltage, V0, at the port.
When using the lumped port as a circuit port, the port voltage is fed as input to the circuit and the current computed by the circuit is applied as a uniform current density, that is as a surface current condition. Thus, an open (unconnected) circuit port is just a continuity condition.