If I have a transfer function H(s) = C*(sI-A)^-1*B + D.
Does the matrix D has to be positive definite in order for the transfer function be positive definite thus passive?
my assumption is that D has to be +ve definite because at s=inf , C*(sI-A)^-1*B = 0 and therefore D has to be +ve in order for H(s) to be +ve at s=inf.
is my assumption correct?
thank you
Mina
Hi Mina;
Happy new year; Let me value this opportunity for communicating with an old, dear and knowledgeable friend!
What I think is:
We need to differentiate between the passivity conditions for the immittance-transfer Functions (e.g. Z and Y parameters) and the s-parameter functions.
For the immittances, where the positive realness is necessary and sufficient condition for the passivity, we need to have D+D^t>=0 (+ve real).
For the S-parameters where the passivity requires bounded-realness then we need norm(D,2)=max(D) \infty. (please also note the initial Laplace theorem to see how it is related to the system behaviour at t=0).
When dealing with a matrix-valued transfer function, we have to adapt these conditions to the matrix notation, e.g. while every driving point transfer functions (on the diagonal) must be +ve real (/bounded), this condition is not directly applied to the off-diagonal transfer functions; instead, the entire transfer matrix should be +ve real (/bounded).
More over we have to consider that the two following necessary conditions needs to be satisfied before even checking +ve realness (/boundedness)
1) Poles should be stable, real(P)
Happy new year my Friend.
Tha you so much for your detailed reply. As usual you are my resource to get back to :)
One more question. Regarding the S parameter, do we still need the transfers function to be positive definite as well as norm of transfers function< 1 for all frequencies? Or is it enough to have norm of transfer function < 1?
let me try and answer ur question about transfer functions having +sE terms. In case of S-parameters then the transfers function will never be passive because at s -> infinity ||S|| > 1. Regarding Y and Z parameters, them E must be positive definite in order to have positive definite transfer function and this passive transfer function.
my answer correct?!
thanks
Mina
Hi Mina;
Nice to hear from you in 2019.
1) About S-para: You are right real-bounded-ness is the necessary and sufficient condition for the passivity of a network characterized by its s-parameters.
Why: Let intuitively see the driving point s-parameter as a ratio of the reflected energy over the incident energy causing that refelection. When the network (structure) is passive and unable to generate the energy the magnitude of the reflected energy wave will always remain less the incident one (for lossy case) etc.
2)You are right (as always). Intuitively it can be said that,
- for the case of z-parameter having a linear term can be seen as a passive network (realized by the proper part of the transfer function) in series with an inductor (whose impedance is represented by hs) . Then, we need the value of the inductor to be positive h>0.
- For the Y-parameter the linear term can be interpreted as a parallel capacitor at input and hence, a simillar explanation remains valid.
-For s-parameter as you mentioned we need the transfer function remains bounded real (passive) at w=\inft. This furthermore requires that the norm of the matrix of direct coupling constants \mathbf{D}=[d_{ij}] to be less than one.
3) For passivity check you may want to revisit the concept of Hamiltonian matrix (or half-Hamiltonian for as proposed by Bjorn Gustavsen)
4) Let us act selfish at least once :) by mentioning our firend's valuable book: I really like the
"Passive Macromodeling: Theory and Applications" (by Professor Stefano Grivet-Talocia and Dr. Gustavsen)
You may also find other references intresting, such as:
- Linear Multiport Synthesis, by Robert W. Newcomb (McGraw-Hill 1966)
- Master and PhD thesis by "Dr. Saraswat Dharmendra" graduated from our group (here in Carleton Univ.) and the references he is mentioning therein
All the best!
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