Calculating the W-Elements

W-elements are a distributed model for transmission lines used in certain HSPICE-compatible circuit simulators. The W-element models in HFSS are computed from the 2D port solutions.

	Z = R + jwL = g * Z₀	(1)
	Y = G + jwC = g * inverse(Z₀)	(2)

Extensions to the multi conductor case are possible but are beyond the scope of this discussion.

There are two flavors of W-element export in HFSS: Tabular and RLGC. (You see them in the Equivalent Circuit Export Options panel.) The more basic "RLGC" format fits the computed RLGC data to the model equations repeated below. For RLGC format RG parameters are frequency dependent such that HSPICE computes them as

	R(f) = Ro + Rs * sqrt(f)	(3)
	G(f) = Go + Gd * f	(4)

where Ro is the low frequency resistance and Go is the low frequency conductance. Rs and Gd represent the high-frequency asymptotes due to skin effect and loss tangent, respectively. L and C are assumed to be fixed over the frequency band.

Thus the "RLGC" format uses four matrices (Ro, Rs, Go, Gd) to model frequency dependence for R(f) and G(f), and it neglects the frequency dependence of L and C.

In contrast the "Tabular" format is not fitting the data to this simplified model: it is printing out the actual RLGC values at each frequency as computed by HFSS from the port solution.

Under some conditions, during W-Element export, HFSS may issue a warning that "One or more of the diagonal terms in the W-element <designated> matrix for some frequency are negative for a <specified port>." There may also be warnings if there are insufficient low-frequency points for the W- element data.

This signifies a non-physical result. A negative diagonal entry in the RLGC matrices indicates that the model is non-passive at that frequency (i.e., can produce energy, which is probably not realistic for a metal interconnect).

This can be caused by a number of things:

Interpolating sweeps.
Non-causal dielectric material models, such as the traditional constant loss tangent, constant permittivity dielectric model. The Djordjevic or Debye models in HFSS would be the preferred ones to use here.
Allowing radiation boundaries on the ports.
Solving to a very low frequency (one so low that the field solution from the full-wave solver contains significant numerical errors.) For frequencies under 10 MHz, you must have at least 10 sampling points per decade.