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46 DESIGN007 MAGAZINE I JUNE 2019 measure, it's no surprise that there are a vari- ety of measurement methods and some more appropriate to some applications than others. Split post resonator methods, for example, are ideal for bulk measurement of loss tangent when manufacturing base materials. When choosing a value of loss tangent for use in signal integrity applications, you will get best results if you use a value derived by using transmission line techniques. The loss tangent in a data sheet may have been measured in a variety of ways depending on application and frequency of measurement (most data sheets note this), but if in doubt, ask. Above 3 GHz (and maybe less if the lines are long or the traces very thin and narrow), more factors come into play and modelling becomes more important and nuanced. Insertion loss needs to be predicted, and the designer and fabricator involved in material selection or design rule selection to ensure the design oper- ates as expected at a reasonable cost. You will need to model the effects of loss tangent on the loss as well as copper losses, which are further split into skin effect derived losses and losses owing to surface roughness. Modelling with loss tangent is relatively straightforward, and the loss is proportional to the loss tangent value and the frequency of operation. Loss tan- gent itself does vary with frequency, but solv- ers can model this for you (Figures 2 and 3). Using the above modelling to predict Er and Tan D from a known spot frequency—in whichever is your field solver of choice and measured with an appropriate model for use in a transmission line application—is the opti- mum solution when faced with a variety of Tan D values, methods, and frequen- cies. The effects of Tan D can be seen in Figure 4 for the same geometrical struc- ture built with low-, medium-, and high- loss materials (Tan D values of 0.01, 0.02, and 0.03). Skin depth losses are also fairly straightforward to model. The skin depth reduces predictably with frequency. Copper roughness is a far more devi- ous animal. The mechanical metrology is somewhat open to interpretation, and then a whole gamut of models is avail- able, including Hammerstad, Groisse, Huray, and Cannonball Huray. They all behave in slightly different ways and obtaining input roughness data for the models is of varying complexity. Hammerstad is the oldest method dat- ing from WWII and the early days of radar where it was used to model the increased losses owing to machining marks on antenna waveguides. It needs Rz as input but is still a reasonable tool up to around 4 GHz. You can see in Fig- ure 5 that Hammerstad modelling satu- rates both for higher roughness and fre- quency. Groisse is similar to Hammer- stad but holds its own for a few more GHz. Figure 2: Tan D versus frequency. Figure 3: Er versus frequency.