# Passivity of impedance matrix on the imarinary axis

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\$$?

• The paper, in later sections, shows a simple Low Pass Filter causes modeling problems. Does that apply to this High Pass Filter ? – analogsystemsrf Apr 4 '19 at 8:32
• @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. – George C Apr 4 '19 at 11:58