1. An Accurate Methodology
for Modeling High Frequency Electrical Impact of
Package Conductor Roughness
Dr. Kemal Aygun, Dr. Zhichao Zhang
Mentors
Intel Corporation,
Chandler, AZ, USA
Raj Sarvan, Anuj Shah, Chris Lieb
School of Electrical, Computer and Energy Engineering
Arizona State University
Tempe, AZ, USA
raj.sarvan@asu.edu, anuj.shah@asu.edu, chris.lieb@asu.edu
Abstract— This paper presents the modeling of the surface
roughness of the trace using different empirical models as well as
the accurate smooth model using computer aided design (CAD)
tool High Frequency Structural Simulator (HFSS). The smooth
trace model is prepared in HFSS. It is compared with the two
empirical model designs by Hammerstad & Bekkadal [1] and
Huray [2]. Geometrical tuning of the model parameters is done to
get the best match for the insertion loss between the two models.
Index Terms—Surface roughness, HFSS, Hammerstad &
Bekkadal model, Huray model, smooth trace, insertion loss
I. INTRODUCTION
This paper mainly focuses on the insertion loss obtained
from the smooth trace and is compared with the rough trace at
operating frequency of 20 GHz. Insertion loss is the loss in
transmitting the power from one port to the another port. It is
generally expressed in terms of decibels.
𝐼𝐿 = −10 log
𝑃2
𝑃1
(1)
where P2 is power transmitted to port 2 and P1 is input power.
First, the 3D model for the trace on a substrate is created
using HFSS. Here, the trace is a cuboid with no roughness at
all. A trace is placed on the substrate, which is connected to the
ground plane of same conductivity as the trace. Finally, the
whole structure is kept in an air box with some radiation
boundary condition to mimic an infinitely large propagation
space.
Next, the Hammerstad & Bekkadal model was studied and
based on the equation given in paper [1], the roughness
parameter w.r.t. smooth surface was calculated. This roughness
parameter depends on the root-mean-square (RMS) value of
roughness and skin depth of the material. Skin depth in turn
depends on frequency. Hence, the roughness parameter has a
direct variation with the frequency. Now, the conductivity due
to roughness depends on the roughness parameter and hence
conductivity of the surface varies inversely with frequency.
Then, the trace is assigned the frequency dependent
conductivity material and insertion loss is plotted using HFSS
simulation.
For the third model based on paper [2] by Huray, snowballs
were used to model the roughness of the surface. Different
radii of snowballs were chosen and their corresponding number
of snowballs was calculated. The roughness parameter was
then calculated based upon the frequency. Finally, conductivity
based on the frequency was assigned to the model and
simulated.
The last part was to calculate the RMS error to match
Hammerstad & Bekkadal model with different radii of
snowballs from the Huray model. One radius was selected that
would give the least RMS error between Hammerstad &
Bekkadal model and Huray model.
Figure 1 shows the surface roughness on both trace and
ground plane [3]. This roughness will cause the increase of loss
as compared to the smooth copper case. This particular effect is
modeled by scaling down the conductivity of the copper trace
for both Hammerstad & Bekkadal model and Huray model as
discussed earlier.
Fig.1 Microstrip line with rough surface on both the trace and
ground plane [3].
II. MODEL WITH SMOOTH TRACE IN HFSS
As shown in the Figure 2, the HFSS smooth trace model is
created. At the bottom is ground plane with the same
conductivity and material as the trace, then there is the
substrate (also called build up material) and finally the solder
resist material that contains the trace. The whole system is
placed in the radiation boundary box. The trace and ground
plane are connected with two lumped ports i.e. input and output

2. ports (used to give excitation to trace from port 1 and received
at port 2).
Fig. 2. HFSS model for smooth surface.
Thus, this model was simulated for the frequency range of
50 MHz to 40 GHz with the solving frequency of 20 GHz and
the plot of insertion loss vs frequency was created. This plot is
shown in Figure 3.
The insertion loss at the operating frequency of 20 GHz
was found to be -0.1318 dB and at 40 GHz, it was -0.2341 dB.
Fig. 3. Insertion loss Plot for smooth surface.
III. HAMMERSTAD AND BEKKADAL MODEL FOR SURFACE
ROUGHNESS
Using finite difference modelling method, Hammerstad and
Bekkadal [1] came up with the following equation to calculate
the roughness parameter that depends on RMS value of surface
roughness and skin depth. The RMS value of roughness used
here is 0.846814 um and skin depth was calculated as
𝛿 =
1
√𝜋𝑓𝜎𝜇
(2)
where μ is permeability = 4π * 10-7
H/m, f=1 Hz to 40 GHz
and σ=3.5 * 107
S/m.
The roughness parameter given by Hammerstad &
Bekkadal [1] is as follows
𝑓𝑟𝑜𝑢𝑔ℎ =
𝑃𝑟𝑜𝑢𝑔ℎ
𝑃𝑠𝑚𝑜𝑜𝑡ℎ
(3)
𝑓𝑟𝑜𝑢𝑔ℎ = 1 +
2
𝜋
tan−1
(1.4 ∗ (
𝑅𝑞
𝛿
)
2
) (4)
where Rq = RMS value of surface roughness=0.846814 * 10-6
m.
Finally, the effective conductivity was calculated using the
result published in [3] as
𝜎𝑟𝑜𝑢𝑔ℎ =
𝜎
𝑓𝑟𝑜𝑢𝑔ℎ2 . (5)
The roughness parameter and effective conductivity were
plotted vs. frequency using MATLAB and the results can be
seen in Figure 4 and Figure 5, respectively.
Fig. 4. Roughness parameter plot for Hammerstad and Bekkadal
model.
The roughness parameter at the operating frequency of 20
GHz is found to be 1.78. See Appendix for the MATLAB
code.
Fig. 5. Effective conductivity plot for Hammerstad and Bekkadal
model.
The conductivity data points with respect to frequency were
imported to HFSS and a new material was created that had the
above effective conductivity variation with frequency. This
new material was assigned to the ground plane as well as the
trace, then the insertion loss vs frequency plot was created.
Figure 6 shows the comparison of insertion loss with the
smooth surface.
0 5 10 15 20 25 30 35 40
-0.25
-0.2
-0.15
-0.1
-0.05
0
Frequency (GHz)
InsertionLoss(dB)
Smooth Surface

3. Fig. 6. Insertion loss comparison with smooth surface.
At the operating frequency of 20 GHz, the insertion loss for
the rough surface is -0.1788 dB while that the smooth surface
was -0.1312 dB. The values at 40 GHz for the rough case is -
0.3185 dB and the smooth case is -0.2341 dB.
IV. HURAY MODEL FOR SURFACE ROUGHNESS
This model [2] takes into account the surface roughness as
snowballs of some particular radius and using those values the
roughness parameter is calculated. The roughness parameter is
given as
𝑓𝑟𝑜𝑢𝑔ℎ = 1 +
3
2
(
𝑁𝑒𝑓𝑓∗4𝜋𝑎2
𝐴
)
1
(1+
𝛿
𝑎
+
1
2
(
𝛿
𝑎
)
2
)
(6)
where N is number of snowballs, a is radius of each snowball,
δ is skin depth and A is area of surface profile.
To calculate the number of snowballs, we calculated the
volume under the rough surface using trapz function in
MATLAB and the volume came out be 1.0052e-15 m3
. Then,
volume of each sphere is found out to calculate number of
snowballs.
𝑁 =
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑠𝑛𝑜𝑤𝑏𝑎𝑙𝑙(𝑆𝑝ℎ𝑒𝑟𝑒)
𝑁 =
1.0052∗10−15 𝑚3
4
3
𝜋𝑎3
(7)
As the volume of surface due to roughness is irregular, it is the
case of irregular packing of spheres so a density factor needs
to be multiplied with the number of sphere. For irregular
packing of spheres, the density factor is 0.634 [4], which is
multiplied with N to get effective N.
𝑁𝑒𝑓𝑓 = 0.634 ∗ 𝑁 = 0.634 ∗
1.0052∗10−15
4
3
𝜋𝑎3
(8)
Finally, the conductivity is tweaked according to the value
of roughness parameter as mentioned in Hammerstad &
Bekkadal model.
Simulations were run for the radii ranging from 0.1 um to
5um with the step size of 0.5um. The upper limit to radius is
set as 5 because above 5 um, the absolute value of the number
of snowball goes to zero. Different materials were created
based on the frequency dependent conductivity and assigned to
the trace and ground plane to plot insertion loss. Figure 7
shows the insertion loss curves for all radii and is compared
with the smooth copper case.
Fig. 7 Insertion loss for different radii compared with smooth surface.
Table 1. Insertion loss for different radii of snowballs.
The last column of the Table 1 shows insertion loss at 20
GHz operating frequency. The radius is in um and the
insertion loss is in dB.
We think that a good model can be made only if there are
more number of snow balls with smaller radius. So snowballs
can be selected for radius of 0.1 um with roughness parameter
of 2.827 and an insertion loss of -0.2324 at 20 GHz frequency.
Figure 8 shows plot of roughness parameter vs. frequency.
Figure 9 shows plot of effective conductivity vs. frequency.
Fig. 8 Roughness parameter vs. frequency for a=0.1um.
0 5 10 15 20 25 30 35 40
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
X: 20
Y: -0.1788
Frequency (GHz)
InsertionLoss(dB)
Smooth Surface
Hammerstad & Bekkadal Model
0 5 10 15 20 25 30 35 40
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Frequency (GHz)
InsertionLoss(dB)
Smooth Surface
Huray Model a=0.1 um
Huray Model a=3 um
Huray Model a=3.5 um
Huray Model a=4 um
Huray Model a=4.5 um
Huray Model a=4.7 um
0 5 10 15 20 25 30 35 40
1
1.5
2
2.5
3
3.5
4
4.5
X: 20
Y: 2.827
Frequency (GHz)
frough

4. Fig. 9 Effective conductivity vs. frequency for a=0.1 um.
Fig. 10 Insertion loss vs. frequency for smooth surface,
Hammerstad & Bekkadal model and Huray model for a=0.1 um.
As seen in Figure 10, there is a notable difference between the
Hammerstad & Bekkadal model and Huray model. It can be observed
that the losses are more in Huray model as compared to Hammerstad
& Bekkadal model.
V. RMS ERROR BETWEEN HAMMERSTAD & BEKKADAL
MODEL AND HURAY MODEL
To find the RMS error, we started with collecting the
insertion loss values at all frequencies from 1 GHz to 40 GHz.
Then, the following RMS error [5] formula was used to
calculate the error percentage for all frequency ranges.
𝑅𝑀𝑆 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = √
1
𝑁
∑
(𝑥 𝑖−𝑦 𝑖)2
𝑥 𝑖
2 ∗ 100 (9)
where, xi is the value of insertion loss (dimensionless) at that
particular frequency for Hammerstad & Bekkadal Model,
yi is the value of insertion loss (dimensionless) at that
particular frequency for Huray Model, and N is the total
number of data points which is 3900 from 1 GHz to 40 GHz
with step of 0.1 GHz.
This procedure was repeated for all radii until the minimum
RMS error percentage was found. Table 2 shows different
radii, effective number of snowballs and the RMS error
percentage, found using the equation above.
Table 2. RMS error between Hammerstad & Bekkadal Model and
Huray Model.
It can be observed from Table 2 that the best match between
the two models can be achieved if the radius of snowball is
taken as 4.5 um and number of snowballs is 2. Here, we got an
error of 0.0878% in the frequency range of 1 GHz to 40 GHz.
Fig. 11 Insertion loss vs. frequency for smooth surface,
Hammerstad & Bekkadal model and Huray model for a=4.5 um.
Figure 11 shows a good match between the Hammerstad &
Bekkadal model and Huray model for a=4.5 um.
VI. FUTURE WORK
More accurate model can be prepared using the brute force
approach, i.e. a trace can be created using cuboids whose Z
height depends on the surface profile roughness e.g. if the
surface profile is 30 um x 30 um and 256 x 256 data points are
given which varies in height then 256 x 256= 65,536 cuboids
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
X: 20
Y: 4.378e+06
Frequency (GHz)
sigmarough
0 5 10 15 20 25 30 35 40
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Frequency (GHz)
InsertionLoss(dB)
Smooth Surface
Hammerstad & Bekkadal model
Huray Model a=0.1 um
0 5 10 15 20 25 30 35 40
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Frequency (GHz)
InsertionLoss(dB)
Smooth Surface
Hammerstad & Bekkadal model
Huray Model a=4.5 um

5. can be created having variation in height. The X and Y length
can be calculated as 30 um / 256 =117.18 nm. Also, the
position of each cuboid can be determined from above data.
This can be done in a Visual Basic script as it is the case of
repetition of cubes. The Visual Basic Script can be imported
and run in HFSS. After that all the cuboids can be united to
form a rough surface of 30 um x 30 um. This piece can be used
to make the trace and then can be simulated to plot insertion
loss. We wrote the Visual Basic Script for creating 9 cuboids
i.e. 3x3 of different heights, which can be found in Appendix.
VII. CONCLUSION
Hammerstad & Bekkadal and Huray models can be used to
characterize the loss increase from copper surface roughness.
Comparing Hammerstad & Bekkadal model with Huray
model, some interesting results came out. The RMS error
between the two models is the least at 0.0878 % so correlation
can be achieved when the radius of snowball is 4.5 um and
number of snowballs is 2.
As discussed in future work about the brute force approach,
it would be really interesting to see what radius of snowballs
matches that to the accurate result and how close Hammerstad
& Bekkadal model is to the brute force approach.
VIII. APPENDIX
Matlab code to get roughness parameter and effective
conductivity for Hammerstad & Bekkadal Model
f=1:1e7:40e9; %Frequency Range
Rq=0.846814e-6; %Surface Roughness RMS value
sigma=3.5e7; %Bulk Conductivity
mu=4*pi*1e-7; %Permeability
delta=1./sqrt(pi.*mu.*sigma.*f);%Skin Depth
frough=1+(2./pi).*(atan(1.4.*(Rq./delta).^2)); %Equation to
find frough
figure(1);
plot(f/1e9,frough); %plotting frequency vs f rough
grid ON;
xlabel('Frequency (GHz)');
ylabel('f rough');
figure(2);
sigmarough=sigma./((frough).^2); %Calculating roughness
conductivity
plot(f/1e9,sigmarough); %Plotting frequency vs sigma rough
grid ON;
xlabel('Frequency (GHz)');
ylabel('sigma rough');
Matlab code to get roughness parameter and effective
conductivity for Huray Model
f=1:1e7:40e9;
r1=input('Enter radius of sphere in um:');
r=r1*1e-6;
N=0.634*1.0052e-15./((4./3.*pi).*(r.^3))
Ahex=25e-6*25e-6;
Rq=0.846814e-6;
sigma=3.5e7;
mu=4*pi*1e-7;
delta=1./sqrt(pi.*mu.*sigma.*f);
frough=1+(1.5).*(N.*4.*pi.*(r.^2))./Ahex./(1+(delta./r)+((delt
a.^2)./(2.*(r.^2))));
figure(1);
plot(f/1e9,frough);
grid ON;
xlabel('Frequency (GHz)');
ylabel('f rough');
figure(2);
sigmarough=sigma./((frough).^2);
plot(f/1e9,sigmarough);
grid ON;
xlabel('Frequency (GHz)');
ylabel('sigma rough');
Visual Basic script for 9 cuboids i.e. 3x3
Set oAnsoftApp =
CreateObject("AnsoftHfss.HfssScriptInterface")
Set oDesktop = oAnsoftApp.GetAppDesktop()
Set oProject = oDesktop.GetActiveProject
Set oDesign = oProject.SetActiveDesign("HFSSDesign1")
Set oEditor = oDesign.SetActiveEditor("3D Modeler")
Dim ArrayPos
Dim number1
Dim number2
Dim YPos
Dim XPos
Dim Points
Points = Array (10e-07, 2e-06, -2e-06, -4e-06, 5e-06,
2e-06, 7e-06, -10e-06, 5e-06)
ArrayPos = 0
YPos = 0
For number1 = 0 To 2
Xpos = 0
For number2 = 0 To 2
oEditor.CreateBox Array("NAME:BoxParameters", _
"XPosition:=", XPos, _
"YPosition:=", YPos, _
"ZPosition:=", 0, _
"XSize:=", "117.18nm", _
"YSize:=", "117.18nm", _
"ZSize:=", Points(ArrayPos)), _
Array("NAME:Attributes", _
"NAME:=", "SurfaceBox", _
"FLAGS:=", "", _
"Color:=", "(132 132 193)", _
"Transparency:=", "0", _
"PartCoordinateSystem:=", "Global", _
"MaterialName:=", "vacuum", _
"SolveInside:=", true)
ArrayPos = ArrayPos + 1
XPos = XPos + 117.18e-09
Next
YPos = YPos + 117.18e-09
Next

6. ACKNOWLEDGMENT
We are really grateful to our mentors at Intel Corporation,
Dr. Aygun and Dr. Zhang for their constant motivation,
suggestions, advice, help and monitoring this project very
closely. This would not have been possible without our
mentors. Next, we would like to thank our instructor Dr.
Tasooji at Arizona State University for her support during the
course and believing in us for this project. Last but not the
least, we would like to mention all lab managers here at High
performance Lab who allowed us to work on HFSS and helped
to resolve all the technical issues regarding the CAD tools.
REFERENCES
[1] E.O Hammerstad and F. Bekkadal, Microstrip Handbook,
Trondheim, Norway: University of Trondheim, 1975, pp.
4-8.
[2] P.G. Huray, O. Oluwafemi, J. Loyer, E. Bogatin, X. Ye,
Impact of Copper Surface texture on Loss: A Model that
works, DesignCon 2010.
[3] H. Braunisch, X. Gu, A. Camacho-Bragado, and L.
Tsang, “Off-chip rough-metal-surface propagation loss
modeling and correlation with measurements,” in Proc.
IEEE Electron. Compon. Technol. Conf., Reno, NV, May
29–Jun. 1, 2007, pp. 785–791.
[4] http://en.wikipedia.org/wiki/Sphere_packing
[5] http://en.wikipedia.org/wiki/Root_mean_square