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68 DESIGN007 MAGAZINE I SEPTEMBER 2018 Examples of non-localized structures are coupled traces, striplines with non-connected reference planes, traces crossing gaps in refer- ence planes, vias with far, no or insufficient stitching vias (vias connecting reference planes of the connected traces). Analysis of non-local- ized structures is usually possible only at the post-layout stage with substantial model sim- plifications that degrade accuracy at higher fre- quencies. To design predictable interconnects, only localized structures must be used—this is one of the most important elements for design success. The localization is always bandwidth limited for striplines (two reference conduc- tors) and for vias (two or more reference con- ductors). How do we estimate the localization property of a transition? One way is to run an electromagnetic analysis of the structure with different boundary conditions or simply change simulation area size without changing phase reference planes and evaluate the dif- ferences in the computed S-parameters. If the difference is small, the structure may be con- sidered localized and suitable for final design [2]. Alternatively, compute and plot the power- flow density and literally see the localization of the signal in space as illustrated in Figure 2. First, let's get familiar with the power-flow density concept using a simple example and analogy with the circuit theory for a stripline structure in Figure 2. Voltage in the circuit theory corresponds to the modal electric field intensity E, current cor- responds to the modal magnetic field intensity H. Cross-product of the electric field and mag- netic field intensities is the vector of power- flow density (or Poynting vector), measured in Watt/m^2. It is energy through unit area in space transferred in one second. When we look at the power-flow density vectors, we basically see where the energy of the signal is in space around a trace or via hole. Total power through a cross-section of the stripline corresponds to the power flow in corresponding transmission line model, equal to product of the voltage and current. To understand the localization concept, it is very important to know that the signal energy is actually distributed in space around each element of interconnect struc- ture. For instance, the power-flow density of the dominant quasi-TEM mode in stripline is shown in Figure 3 at four frequencies. The strip is 1.2 mil thick, 7 mil wide trace, in homogeneous dielectric with Dk=3.76, LT = 0.006 @ 1 GHz, planes 0.77 mil thick and 17.2 mil apart, 1 V excitation and 50-ohm termina- tors. The power-flow density is depicted by vec- tors with the direction along the t-line (into the picture) outside of the conductors. The value of the vectors is expressed with color scale in dB from zero (red color) to -60 dB (blue color). Figure 2: Simple example of the power-flow density concept.