According to this paper, Section V-C, a rational impedance matrix Z(s) is the impedance of a passive lumped system if and only if the following four conditions are satisfied:

1) each element of Z(s) is dedined and analytic in \$Re(s)>0\$;

2) \$Z(j\omega)+Z^H(j\omega)\$ is nonnegative-definite for all \$\omega\in R\$;

3) \$Z(-j\omega)=Z^*(j\omega)\$;

4) asymptotically, \$Z(s)\rightarrow As\$ in \$Re(s)>0\$, where \$A\$ is real, constant, symmetric, and nonnegative-definite.

Condition 2 ensures positive dissipated energy. Condition 3 ensures real impulse response in the time domain. My question is about the exact meaning of the fourth condition. For example, suppose I have a one-port which consists of nothing but a series of a positive resistor \$R\$ and a capacitance \$C\$. The input impedance, which is equivalent to the 1x1 impedance "matrix", is simply \$Z(s)=R+1/sC\$. So asymptotically \$Z(s) \rightarrow R\$, which is not in the form of \$As\$, for whatever \$A\$. But this simple one-port is doubtlessly passive. In this case, should we consider the 4th condition as violated? Or, does it actually mean that the asymptotic behavior of \$Z(s)\$ cannot grow faster than \$As\$?

  • \$\begingroup\$ The paper, in later sections, shows a simple Low Pass Filter causes modeling problems. Does that apply to this High Pass Filter ? \$\endgroup\$ – analogsystemsrf Apr 4 '19 at 8:32
  • \$\begingroup\$ @analogsystemsrf That example begins with a noncausal "scattering parameter", which might not be realizable by circuit elements exactly. My example above, on the other hand, starts from a "circuit model", and then its impedance is calculated trivially but found to be not consistent with the theorem's requirement, seemingly. \$\endgroup\$ – George C Apr 4 '19 at 11:58

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