An existing analytical transmission line model to describe propagation
properties of coplanar waveguides including dispersion and radiation effects
was extended to take into account surface roughness of conductor traces. The
influence of parasitics is successively included in the simulation and
compared to measurements. The device under test (DUT) was fabricated on an

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

Such an analytical model was presented in the early nineties

This paper shows the application of a roughness model

In the following the geometrical parameters

CPW cross-section with geometrical and material parameters.

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

The CPW was fabricated on an

CPW Parameter.

Thus, measurements were performed whereby the complex propagation coefficient

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.

In analytical transmission line models, the electromagnetic properties are
described with per unit length (pul) parameters

It's worth to note, that both outer and inner inductance contribute to the
total pul inductance

The quasi-TEM model approach

Measured and simulated attenuation on metal chuck. The simulation does neither include radiation and dispersion nor surface roughness.

The simulation is in good agreement with the measured responses at lower
frequencies. The deviation of simulated and measured attenuation gets
significant above 40

Measured and simulated effective rel. permittivity on metal chuck. The simulation does neither include radiation and dispersion nor surface roughness.

The CPW model

Particularly, with regard to the bend towards higher attenuation that is
observed above 60

Measured and simulated attenuation on metal chuck. The simulation includes radiation and dispersion.

Measured and simulated effective rel. permittivity on metal chuck. The simulation includes radiation and dispersion.

Generally a model should have an underlying physical notion, a minimum number
of parameters and of course it must predict measurement results with
reasonable accuracy. Especially a model for the interaction of
electromagnetic fields with rough surfaces should utilize input parameters
for surface properties, that are well known and specified in surface
metrology. For example the RMS

root mean square.

-roughnessThe Gradient Model presented by

Many surface profile measurement systems like tactile or optical ones not
only deliver statistical parameters, but also the bearing area curve (BAC) or
Abott-Firestone curve

The mean response from four BAC measurements at the top of the

Measured BAC, CDF from Eq. (

From the magnetic field all other relevant quantities, e.g. loss power
density, magnetic field energy, etc. can be calculated. Since most analytical
transmission line models assume ideally smooth surfaces, the application of
the Gradient Model utilizing effective, frequency dependent material
parameter was shown in

Decreasing effective conductivity

The advantage of using effective material parameters to depict roughness impact is, that they can be utilized in any model assuming ideally smooth surfaces. This is done by simply replacing material with effective material parameters, one only has to take care of their frequency dependence.

The CPW model distinguishes three frequency sections: The quasi-static case,
a transition region and the skin effect region. Since the Gradient Model
assumes the skin effect, only skin effect and transition region has to be
adapted. In the latter, the conductivity

In the skin effect region the modifications of the pul resistances

The pul inductance

From the original document

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

The surface roughness impact on phase delay and effective rel. permittivity
respectively is typically nearly constant over the regarded frequency range
(Fig.

Measured and simulated attenuation on metal chuck. The simulation includes radiation, dispersion and roughness.

Measured and simulated effective rel. permittivity on metal chuck. The simulation includes radiation, dispersion and roughness.

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

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

Also the effective relative permittivity in case of a ceramic chuck is
predicted very well, as can be seen in Fig.

Measured and simulated attenuation on ceramic chuck. The simulation includes radiation, dispersion and roughness.

Measured and simulated effective rel. permittivity on ceramic chuck. The simulation includes radiation, dispersion and roughness.

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

The DUT was fabricated on an

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

The data are available from the authors upon request.

The authors declare that they have no conflict of interest.

The authors are grateful to Rohde & Schwarz for manufacturing the calibration substrate, to Thorsten Probst from PTB for performing the on-wafer measurements, and to Dylan Williams from NIST for providing the initial layout of the substrate.

Furthermore, the authors would like to thank Franz-Josef Schmückle from Ferdinand-Braun-Institute for supplying program code on radiation and dispersion effects of a CPW.

The authors acknowledge support by the European Metrology Programme for Innovation and Research (EMPIR) Project 14IND02 “Microwave measurements for planar circuits and components”. The EMPIR program is co-financed by the participating countries and from the European Union's Horizon 2020 research and innovation program.

This paper was edited by Thorsten Schrader and reviewed by two anonymous referees.