MD-passivity

We begin by defining a domain in the vector space defined by the new coordinates under a transformation of the type (3.21) (which may be an embedding). Consider an -port defined over the domain , with port voltages and currents , for . The instantaneous absorbed power density, at any point in the interior of is defined by

and the

In addition, we can define the source and dissipated power densities within to be and . The energy balance of the -port can then be generalized directly from (2.3):

where is the -element row vector outward unit normal to the surface of , is a surface element of , and is a volume element internal to . See Figure 3.3

and MD-lossless if (3.28) holds with equality. The total stored energy lost through the boundary of must be less than the energy supplied through the ports in ; this is equivalent, from (3.27) to saying that the energy dissipated in must be greater than the energy coming from the source. The previous definition of MD-passivity has been more precisely called

An -port which is differentially MD-passive everywhere throughout a domain will also be integrally MD-passive with respect to . The converse is not necessarily true.

It is also useful to define, for an -port, a scalar total energy [85] by

Here a spatial region defined as the cross-section of at time , and is a column unit vector in the time direction; note that this definition is framed in terms of the untransformed coordinates , and has been projected onto these coordinates under (3.21a). It can also be used as a measure of the total energy at time in a given circuit, as we will see in §3.7.4.

Fettweis [44] looks at an extension of the idea of positive realness (see §2.2.2) to two dimensions, for the case of a real linear and shift-invariant -port. This idea generalizes easily to higher dimensions, as per some very early work in MD system theory [135]. Consider a real linear and shift-invariant (LSI) -dimensional -port, where the port quantities are in an exponential state of frequency , where

are the frequency variables conjugate to . Thus we have the real instantaneous voltages and currents

where and are complex amplitudes. If there is an impedance relation between the voltages and currents, then we can write

where and . The total complex MD power density at frequency can be defined as

and the average or active power density as

The positive realness condition on for MD-passivity follows immediately, and is similar to (2.5), except that we now must have

Thus the impedance must be positive real in

(3.31) |

where is the vector of frequencies in the untransformed coordinates . Thus, due to the positivity condition on the elements of the last row of and the last column of , we will have that

for |

so that for an MD-passive -port,

It is thus seen that MD-passivity can be interpreted as passivity, but spread over a new system of coordinates (regardless of whether the new coordinates number more than the old).