HFSS15: Overview of the Technical Approach for Derivatives in in HFSS

HFSS includes the ability to compute derivatives of S-parameters and related matrix quantities. The variables of differentiation are project variables or design properties. By far the most useful variable type is geometric, though these also require the most computational effort.

The general approach by which derivatives are computed in HFSS is not really new and the general subject of linear systems has long contained the high-level plan in use. However, the application to full-wave field solvers is relatively recent, and of course the details of the HFSS implementation is where original work in this regard was done.

It is important to realize that in HFSS, it is the discrete (approximate) solution which is differentiated. In a nutshell, the differentiation process can be described as follows. The global matrix equation that HFSS sets up and solves can be written as Ax = b where A is a very large matrix, x is the unknown vector containing fields and S-parameters, and b represents the excitations. Given a model parameter “g” (e.g. the diameter of an iris in a filter) each of the quantities in the matrix equation is in turn a function of g.

Formal differentiation of the global equation with respect to g and simple rearrangement yields Ax'= b' - A'x, where the primes indicate derivatives. The quantity sought is x' and the matrix multiplying this quantity is A, just as in the original global matrix equation. The right-hand side contains b', A', and x. The latter quantity is the solution to the global system and is assumed already known. Thus if b' and A' can be computed, x' may be obtained by solving a matrix equation of the same form as the original Ax=b.

Now, consider the quantities A' and b'. Restriction is currently made to the important case where the excitation is from ports only, in which case b'= 0, leaving the equation Ax'= -A'x. To generate the right-hand side of this matrix equation, the key remaining quantity to be supplied by the field solver is therefore A'. In HFSS this is done analytically for the most part. Thus a matrix assembly algorithm has been crafted that fills A' in a manner similar to how A is filled. So ”analytical”, rather than “numerical” differentiation is a principal aspect of the HFSS method.

Also, it can be shown that in most cases the computation of x' can avoid even an additional matrix solution (the system is “self adjoint”) and S' is obtained by a purely algebraic post-process operation once A' is assembled and x is known.

Finally, it should be noted that if there are multiple variables of differentiation, each one is simply considered individually. Differentiation of response functions is not currently supported.

In regards to fundamental limitations, mostly the issue is computation of A' when the variable of differentiation affects ports. Considerable original work was successfully performed to include such variables, but not all cases are doable yet, in particular certain ports containing degenerate modes.

Some technical details on the implementation of derivatives in HFSS may be found in: “Sensitivity Analysis of S-parameters Including Port Variations Using the Transfinite Element Method”, L Vardapetyan, J. Manges, Z. Cendes, 2008 IEEE International Microwave Symposium, Atlanta Georgia, June 2008.