Coplanar waveguides (CPWs) are frequently used in all different kinds of planar circuits like (printed circuit boards) PCBs and on-wafer applications. Especially the latter – ubiquitous nano- and microelectronics – utilize this transmission line type particularly. The exact knowledge of its high-frequency properties thus is essential for modern electronics with increasing data-rates and consequently increasing application frequencies.

Nowadays numerical full-wave electro-magnetic simulators are available that are capable of accounting for parasitic effects like dispersion, radiation and conductor loss effects . Nevertheless analytical transmission line models are required for developing reliable uncertainty budgets for calibration purposes and moreover have fundamental importance to microwave design.

Such an analytical model was presented in the early nineties . This quasi-TEM approach describes the electrical behavior of a CPW from DC to a certain frequency – depending on CPW dimensions – from where on radiation and additional dispersion, due to coupling to higher order modes, take place. These effects were added to the CPW model in , so that modeling of CPW properties at high frequencies is possible now. But at high frequencies the skin depth decreases to the order of surface roughness, so that the assumption of ideal smooth conductor walls does not hold anymore and roughness strongly impacts on transmission properties of any kind of waveguide.

This paper shows the application of a roughness model that is capable of accounting for impact on loss and delay of transmission lines to the contemplated CPW model. This new extension finally allows for precisely describing properties of CPWs including all known parasitic effects. The model predictions are compared to measurements of CPWs on Al2O3 with both metal and ceramic chuck and show a very good agreement in both scenarios.

In the following the geometrical parameters wg, w, t, s and h are utilized as shown in the sketched cross-section of a CPW in Fig. . Additionally the material parameters relative permittivity εr of the substrate, dielectric loss tangent tan⁡δ and conductivity σ of the traces are necessary.

The regarded transmission lines are purely coplanar, i.e. there is no additional ground plane beneath the dielectric. However, to rest the DUT during measurement, a so called chuck is used. Depending on its material, metal or ceramic, there is an impact on the measurement and consequently has to be considered during simulation, which is done by assuming an infinite substrate in the case of the ceramic chuck. This assumption is justified, since the height of the chuck is very large compared to the substrate, so that the electromagnetic field does not penetrate through it. Additionally, the relative permittivity of the chuck and substrate are assumed to be equal. The metal chuck results in a boundary condition respectively after the specified substrate height. A more detailed insight of the impact from different chuck permittivities is given in .

The CPW was fabricated on an Al2O3 substrate, which was characterized separately with a split cylinder resonator . The relative permittivity in the frequency range from 8 to 22 GHz was measured as 9.7, the loss tangent about 1.25×10-4 to 1.5×10-4. The nominal dimensions of the lines are w=50µm, s=25µm, wg=270µm. Due to the electroplating process the conductive traces are subject to a variation of their thickness across the wafer. A figure of 6.5 µm therefore is used in the following investigations. Likewise the conductivity σDC is affected by the electroplating process, since it is always lower compared to bulk metal. Because of that we use a conductivity value of 36 MS m-1.

Thus, measurements were performed whereby the complex propagation coefficient γ of the CPW is obtained by a multiline calibration method . The measured responses are presented as attenuation α=Re(γ) and effective permittivity εr,eff=c0/vp2, in which vp in turn is the phase velocity vp=ω/Im(γ).

In the next sections, the CPW model with and without radiation and dispersion effects is compared to the measured propagation properties. Then the application of the surface roughness model is explained. For this purpose the relationship between independently measured surface profile data, magnetic field and resulting effective, frequency dependent material parameters and necessary adaptions in the CPW model are shown.

As a result we obtain a model including most known parasitics, that can precisely predict propagation properties of CPWs. The additional consideration of surface roughness impact leads to an improvement in the frequency range from 50 to 100 GHz regarding the attenuation in Fig.  utilizing a metal chuck. The relative error stays below 5 % up to 100 GHz. Above 100 GHz measurement uncertainties become larger due to influences from parasitic modes such as e.g. substrate modes.

The surface roughness impact on phase delay and effective rel. permittivity respectively is typically nearly constant over the regarded frequency range (Fig. ), so that the simulated and measured responses only lead to an effective error below 0.5 %.

Measurements of the same wafer on a ceramic chuck lead to slightly different results. Especially, the attenuation does not show a distinctive bend above 60 GHz. Figure  shows, that it is much closer to a f- and f-dependency than the responses obtained from measurements utilizing a metal chuck.

In the simulation the ceramic chuck was considered by assuming an infinite substrate height, since it consists of a similar material.

Like in the case with metal chuck, the relative error only raises to noteworthy figures above 100 GHz.

Also the effective relative permittivity in case of a ceramic chuck is predicted very well, as can be seen in Fig.  with a similar relative error as for the measurements on a metal chuck.

An existing CPW model was expanded utilizing frequency dependent, effective material parameters. The necessary modifications were explained in detail. Those effective parameters are calculated with a surface roughness model that considers roughness impact on both phase and delay. Its input parameter, the RMS-roughness Rq, was estimated from measured bearing area curves. The obtained model now includes parasitics like dispersion, radiation and conductor surface roughness.

The DUT was fabricated on an Al2O3 substrate and was measured utilizing a ceramic and a metal chuck. The simulations with the extended model precisely predict the measured responses, i.e. attenuation and phase delay in both cases up to 100 GHz. The impact from dispersion and surface roughness on attenuation and effective relative permittivity was shown by successively including those parasitics in simulations.

Precise analytical transmission line models including parasitics are fundamental for developing reliable uncertainty budgets for calibration purposes and also have their importance in design.