The role of surface roughness on the electrical behavior of flexible and stretchable screen-printed silver ink on Kapton substrate

Changes in the morphology and profile of the printed electronic traces affect the radio frequency (RF) performance, especially when compared to the simulated designs, which are primarily rectangular and smooth. The shape and morphology of the printed traces depend on the printing system, a combination of ink and substrate properties, and the printed technology used for fabrication. This paper investigates the significance of printing parameters like roughness, height, width, and profile shape on the RF performance of screen-printed traces statistically. This paper also derives a model based on actual profile shape and roughness measurements from incorporating these effects into the simulations. From the statistical study, we found that the role of surface roughness becomes more significant as the frequency increases. Thus, modeling surface roughness in flexible hybrid electronics simulations is fundamental for accurate results. Different simulated profile shapes also showed their effect on the RF performance, when combined with roughness, showed better correlation with experimental data. Our proposed model combines the modified Morgan and Hammerstad equation with the compound conductor model. Our proposed model uniquely integrates the modified Morgan and Hammerstad equation with the compound conductor model, leveraging a comprehensive approach focused on the quality and uniformity of printed electronics trace. This integration is aimed at constructing a model that is firmly grounded in actual roughness data, resulting in a solution that is both straightforward to implement and exceptionally accurate in its outcomes. The model agreed well with experimental measurement data over 7 GHz–40 GHz with a root-mean-square error average of 5.7%.


Introduction
Over the last decade, form factors for the hardware design of electronics started moving from rigid and rectangular towards flexible and stretchable [1][2][3].Flexible hybrid electronics (FHE) technology integrates conventional subtractive silicon electronics and additive printable electronics, providing highperformance, low-power capabilities in large-size form factors. Essentially, FHE is a powerful and versatile electronic platform that is flexible, efficient, inexpensive, and compatible with large areas.This technology is emerging with huge market potential; according to IDTechEX, the total market for FHE was estimated at 30 billion USD in 2017 and is expected to reach 73 billion USD by 2027 [4].
Large-scale, high-throughput, and roll-to-roll (R2R) production are printed electronics' main strengths alongside printing devices' capability on soft, flexible, and non-traditional substrates [29,30].The quality and productivity of the printed electronic components remain critical since they affect the electrical RF performance.The RF performance of the printed components depends on mechanical factors (such as cell stylus, dot size of a cell, and nip force) and chemical characteristics of the materials (such as viscosity and surface tension of the ink and adhesiveness of the substrate) [31][32][33][34][35][36].Small changes in the printing technique, ink, and substrate characteristics can lead to significant fluctuations in the print quality (such as thickness and surface roughness) [31,32].
The trace profile is an essential property of a conductor that affects its RF performance.For example, the width of a trace affects its impedance [37].In additive manufacturing, the printed trace widths usually come out slightly more than the specified design widths, owing to the phenomenon of certain inks spreading after printing and during the sintering processes [38].This gives rise to impedance mismatches and reflections along the lines.The heights of the printed lines are also non-uniform along the width of the traces.For these reasons, it is important to accurately model and study the various profiles of the printed traces.
Electromagnetic waves propagating within finite conductivity media suffer from attenuation because of the ohmic losses [39].This attenuation is quantified in terms of the skin depth, δ, which is defined as the distance over which the amplitude of the power drops by 1/e.For conductors, δ = √ 1/fσπ µ, where f is the frequency, σ is the conductivity, and µ is the permeability of the conductor [39].In this case, the density of alternating current (AC) concentrates at the surface and decreases deeper toward the conductor.Most of the AC travels within a thin layer at the conductor surface, referred to as skin depth.When the skin depth is less than the cross-section of the conductor, the skin depth effect reduces the effective cross-section and increases the effective resistance.
The conductor surface roughness effect refers to the impact of surface roughness on the AC flow within the conductor surface and how it affects the electromagnetic properties of the conductor [39].Surface roughness interrupts the current flow resulting in higher than usual ohmic loss than that predicted by the skin effect model.
As the speed of electronic devices increases, the operating frequency and bandwidths increase.The conductor surface roughness effect becomes more crucial [39].Thus, surface roughness effect modeling has become critical.At the multi-GHz range, the skin depth becomes comparable to the root-means-square (RMS) height of the rough conductor surface [39].Failure to model the increased attenuation caused by conductor surface roughness can result in losses past the design margin and cause severe signal integrity problems [39].
This paper focuses on theoretically modeling screen-printed silver ink (ACI FE3124) microstrip lines on Kapton substrates [40].The relationship between profile shape and RF performance was analyzed based on theoretical studies, and compared the results to experimental measurements.The effects of the significant parameters (thickness, width, roughness, and waviness) on the RF insertion loss were investigated using regression analysis and analysis of variance (ANOVA).Finally, we delved into theoretical modeling of trace roughness, focusing on the quality and uniformity of printed electronics traces.Our aim was to create a model that's straightforward to use while still giving accuracy.The distinct feature of our model is its exclusive reliance on actual surface roughness measurements, unlike models from the literature that require an understanding of particle shapes and metrics.We then compared these theoretical results with our experimental measurements.
The structure of this paper is as follows.In the next section, the past efforts on the relation of power absorption in conductors with trace profile and surface roughness are covered.In section 3, we present our study of different trace profiles.In section 4, we statistically investigate the significance of surface roughness on the power absorption of printed traces.In section 5, we discuss how to model the surface roughness of screen-printed microstrip lines.Finally, a summary and concluding notes are conveyed.

Trace profile
Lim et al in [41], studied the effects of trace edges for a 2 mm long transmission line from DC to 30 GHz.It was observed that modeling the transmission line with sharp trapezoidal edges overestimated the insertion loss by about 14%.However, using curved edges for the trapezoidal corners gave a good match between simulation and measurement.This work studies the effect of the trace profiles on the insertion losses for a longer transmission line (5.5 mm) up to 40 GHz and performs correlations between simulations and measurement.

Power absorption due to surface roughness effect
Much research has been conducted to understand the effect of conductor surface roughness on electromagnetic performance.The concept of the power absorption enhancement factor was introduced to characterize the power absorption property of a rough conductor [42].Authors of [42] described the power absorption enhancement factor, K sr , as 'the ratio of power dissipated by eddy currents in a conductor per unit area when the conductor surface is rough, P a,rough , to the power dissipated when the surface is smooth, P a,smooth .' K sr is used when calculating the attenuation as a correction factor to account for the conductor surface roughness effect.Many analytical models were derived from calculating K sr based on the material and geometry of the conductor traces.K sr is usually a function of skin depth and RMS height of the surface roughness profile [39].These models are only valid for a specific range of frequencies [39].This section will provide an overview of the most common models.
Over the years, researchers analyzed the effect of conductor surface roughness on electromagnetic waves.Two main approaches were used for these analyses: (1) periodic structure assumption and (2) random surface assumption.

Models based on periodic structure assumption
Morgan and Hammerstad Model is the first to analyze the power absorption of conductor surface roughness [42].The model is based on modeling the conductor's rough surface using a periodic groove profile, as shown in figure 1(a).Then, a finite difference method was used to evaluate Maxwell's equation.This model assumes that current travels a longer path due to periodicity surface roughness, causing additional power absorption [39,42].Later, Hammerstad and Jensen model was proposed leading to an empirical formula for calculating K sr , correlating K sr to the ratio of the RMS height of the surface roughness profile, ∆, and the skin depth, δ. . ( This model was modified later to account for increased surface area by multiplying the basic Morgan and Hammerstad equation by a scaling factor (SF) [43].The SF represents the ratio of the length of the rough surface to the spatial length.This model is limited by surface roughness of less than two µm RMS [44].
(3) Hall et al modeled the rough conductor surface based on hemispheres protruding from the surface of the conductor plane, as shown in figure 1(b) [45].This model is accurate up to 30 GHz; this model requires additional geometrical factors like the rootmean-square radius of hemispheres and the spacing leading to a more accurate result [45].
Huray et al utilized a pyramidal stack-up of spherical conductor snow-balls on a conductor surface to model the roughness, as shown in figure 1(c) [46,47].The results of this model agree with the measured insertion loss up to 50 GHz achieving the best broadband accuracy [46,47].However, this model requires many parameters to define the surface roughness profile.

Models based on the random surface assumption
In recent years, researchers used stochastic analyses to study the effect of conductor surface roughness on power absorption.Random processes with specific statistical parameters are used to model conductor surface roughness.The results of stochastic analyses were more flexible than those based on periodic structure assumption [39].
Chen and Wong developed a stochastic integral method that uses a surface boundary condition for a conductor with a random surface roughness profile [49].This method calculates the power loss based on a quasi-periodic assumption and solves it using the method of moments.Later, they improved this model using order reduction techniques to improve the accuracy and lower the computational cost [50].However, this model is restricted by the assumption that the conductor surface is smooth in one direction.
Tsang et al built a two-dimensional random rough surface model based on deriving a closedform expression for K sr using a second-order small perturbation method (SPM2) [51].Also, Gu et al and Braunisch et al extracted a two-dimensional power spectral density function of an isotropic rough surface from height measurements.Then, used SPM2 method to calculate K sr [52,53].The results based on this method are accurate up to 20 GHz.[54].Reproduced from [48].CC BY 4.0.

Compound conductor model
Simulation tools to describe the conductor surface roughness are limited.Currently, available simulation tools only use the RMS height of the roughness profile to adjust the conductivity of the conductor; the original conductor with a rough surface is shown in figure 2(a).Research has shown that this method needs to be revised for accurate results [39].Also, the computational cost to analyze and model a moderate structure is high.The only option is to use equivalent models with deterministic geometry and known material properties to obtain approximate results [39].
Layered impedance boundary conditions can be specified at the conductor surface to account for the conductor surface roughness effect.This method provides more flexibility in defining the conductor surface roughness.However, obtaining the related parameters still needs to be clarified.Ma et al proposed an equivalent model called the compound conductor model [39].
The main principle of the compound conductor model is to replace the interface of the conductor's rough surface with a smooth and effective conductive layer that adapts to the contour of the conductor while the inside cross-section of the conductor remains unchanged as shown in figure 2(b) [39].Replacing the rough geometry with a smooth later simplifies electromagnetic simulation and improves efficiency.The aim of the compound conductor model is that the composite conductor can replace the conductor with a rough surface in the electromagnetic simulation to achieve more accurate results [39].In other words, the conductor cross-section is divided into two sections: a bulk conductor region that keeps the original conductor's conductivity and a conformal outer layer with different conductivity [39].
Since K sr is widely accepted to evaluate power absorption enhancement of a rough-surface conductor, the compound conductor model is designed to result in the same K sr ; this results in accurately explaining the effect of conductor surface roughness [39].The smooth outer layer should mimic the behavior of the conductor's rough surface.The thickness and the conductivity of this layer are frequencydependent and can be obtained by matching the power absorption enhancement factor of the roughsurface conductor [39].Once the outer layer's thickness and conductivity are known, the compound conductor model can be implemented in electromagnetic simulations.
As proposed in [39], the power absorption enhancement factor can be obtained using the compound conductor model as: ( In equation ( 4), P a,rough,c is the time-average power absorbed in the compound conductor model, h e is the thickness of the effective conducting layer, σ e is the conductivity of the effective conducting layer, δ e = 1/πµσ e f is the skin depth of it, and k 1z ≈ 1 − j/δ e is the propagation constant inside it in the direction perpendicular to the air-conductor interface [39].The combinations of he and σ e that result in the desired K sr , defined in equation ( 4), are not unique.Thus, additional constrain must be defined to determine them [39].Since the effective conductor layer occupies a space that is initially the bulk conductor, this layer should be confined to be as thin as possible [39].Thus, the smallest possible he should be determined, then the corresponding σ e .Ma et al explained the procedure to determine the optimal he and σ e in detail, resulting in frequencydependent values for effective layer thickness and conductivity that can be used later for electromagnetic simulations [39].

Study of different trace profiles
The analyzed trace is depicted in figure 3. It is divided into different sections as shown in [54].The trace height is taken to be around 10 times the skin depth to minimize losses that will arise due to the skin effect and profile height is any additional thickness above the height of the trace.The trace width is broken down into rectangular and slope widths respectively.Various combinations of the sections produce the analyzed profiles as depicted in figure 4. Each of the sections is simulated and compared to the experimentally measured data.The simulation is performed using the high-frequency structure simulator finite element modeler (HFSS-FEM) simulator from ANSYS with frequencies between 10 MHz and 40 GHz [55].Results for attenuation constant, phase constant, and impedance were discussed in [54].The insertion loss results are shown in figure 5.The losses are lowest for the wide and wide rectangular dome profiles.The thickness of these profiles is at least 10 times that of the skin depth which minimizes losses due to the skin effect.Furthermore, the trace width (275 µm) of these profiles results in a 50 ohm impedance which reduces reflections that arise due to impedance mismatches between the microstrip lines and ports used for the simulations.
Compared to the wide and wide rectangular traces discussed above, the trapezoid and wide dome profiles have more insertion loss.This is due to the heights of the traces at the edges of these profiles being much less than the skin depth.
The dome profile shape has the worst radio frequency (RF) performance.Firstly, the width of the profile is 184 µm which results in an impedance greater than 50 ohm leading to reflections.This gives rise to the oscillations as seen in figure 5. Similar to the trapezoid and wide dome, the trace thickness at the edges of the trace is much less than the skin depth which results in higher insertion losses.
The correlation studies above only consider smooth traces and do not account for the surface roughness of the traces.For this reason, we see a mismatch between the simulation and experimental measurements.Section 5 of the paper will discuss the roughness and porosity of the traces.

Statistical study of significant parameters affecting the RF insertion loss
We did a statistical study using design of experiments techniques to investigate the significance of surface roughness of screen-printed silver ink traces on the electromagnetic performance.We started by printing eight sheets of screen-printed microstrip lines on Kapton using silver ink; each sheet has 12 microstrip lines, totaling 96 microstrip lines.All the samples were cured for 10 mins at 150 • C in a conveyer reflow thermal oven.Then we used VK-X1050 laser confocal microscope from Keyence to optically characterize all the printed samples; we measured the average cross-sectional area, average width, average roughness, and average waviness for all the samples.The average thickness was calculated from each sample's cross-sectional area and average width.Figures 6(a    the microscopic cross-section of one of the samples.Figures 6(d) and (e) show the roughness and waviness of the same sample, respectively.Also, we used ZNB 40 vector network analyzer from Rohde & Schwarz to measure the S-parameters for all the samples; then, we extracted Insertion-loss at 400 MHz, 10 GHz, 24.5 GHz, and 39 GHz.To stabilize our data, we used the average of 5 points around the insertion loss at each frequency (the point itself, two points to the right, and two points to the lift).The optical and RF data collected were used to perform some simple statistical analysis to remove outliers and check independence and normality assumptions.Minitab ® was used for all the statistical analysis performed [56].We consider the average thickness, average width, average roughness, and average waviness to be our independent variables.We plotted the histograms for the independent variables.We realized that some of the data were outliers.By investigating, we found that all these outlier points belong to samples of sheet 1; we excluded them from our study.Now, we have 84 microstrip line samples.Table 1 summarizes simple statistics for the independent variables after excluding the outliers.
We plotted matrix and probability plots to check independence and normality assumptions.Figures 7  and 8 show matrix plots and probability plots, respectively.From these plots, the independence and normality assumptions are not violated.Also, we checked the p-values of the probability plots to ensure that the normality assumption was not violated.The data can be used to build a regression model and ANOVA.
We performed regression analysis and Best Subsets Regression four times; each time, we considered each insertion-loss (400 MHz, 10 GHz, 24.5 GHz, and 39 GHz) as the dependent variable.These analyses aim to investigate which independent variables (the average thickness, average width, average roughness, and average waviness) are significant  to the insertion-loss at different frequencies.We based our conclusions on three criteria: Pareto chart, pvalue of ANOVA analysis, and best subsets regression.The three techniques lead to the same conclusion.More details about identifying outliers, checking assumptions, and regression analysis are discussed in previous work [57].
At 400 MHz, the best regression model mainly utilizes two independent variables: average width and average thickness.From the ANOVA study, waviness is a slightly less significant factor.The Pareto chart shown in figure 9(a) draws the same conclusion.At 10 GHz, the best regression model utilizes mainly three independent variables, width, thickness, and roughness.The Pareto chart shown in figure 9(b) shows the same conclusion; the width is the most significant factor, then thickness and roughness, respectively.At 24.5 GHz, the best regression model utilizes mainly three independent variables, width, thickness, and roughness.The effect of the waviness diminishes.The Pareto chart in figure 9(c) shows the same conclusion.At 39 GHz, the best regression model utilizes mainly three independent variables, width, thickness, and roughness.At this high frequency, the effect of roughness becomes the most significant factor; this is apparent from the Pareto chart shown in figure 9(d).
From the statistical analysis, we were able to conclude that surface roughness becomes more significant as the frequency increases for screen-printed FHEs.Therefore, modeling surface roughness for FHE simulations is crucial for accurate analysis.

Theoretical study of the roughness and porosity
The main objective for building our model is to have a model based on experimental roughness measurements that can be used to simulate flexible electronic conductors.This way, it can be used for different inks printed using different additive manufacturing techniques on different substrate materials.
We used the modified Morgan and Hammerstad equation and the compound model described in [39] to determine frequency-dependent values for effective layer thickness and conductivity for a screenprinted silver ink trace.The concept of our model is illustrated in figure 10.Our model is not entirely theoretical but based on real roughness measurements as input.We chose this parameter as it offers a representative measure of structural uniformity's influence on RF performance.Additionally, the compound model we integrated with the modified Morgan and Hammerstad equation relies on mean roughness.This selection aligns with our aim for a versatile model applicable across various inks and printing processes.Initial results based on this model were presented in [58].
The first step is determining the K sr using the modified Morgan and Hammerstad equation for screen-printed silver ink trace.We used σ = 7.81 × 10 6 S m −1 (based on measurement) as the conductivity of the silver ink, and ∆ = 1.8488 µm as the average roughness of all the samples, and SF from measurements using a Keyence microscope.Figure 11 shows the results for SF; in this case, we got SF = 1.02.
Then, we used the procedure described in [39] to determine the optimal values for h e and σ e .We generate plots for the power absorption enhancement factor obtained using the compound model, K sr,c , vs. σ e for a different h e .K sr,c can be calculated using equation ( 4), as shown in figure 12 [39].
Then, we trace the maximum value of K sr , K sr,max , for all h e values.We cannot achieve K sr values more than the maximum value, K sr,max , if we only vary the σ e [39].Then, we plot K sr,max vs. h e , and interpolate to get the optimal (minimum) h e , h e,op , that corresponds to the desired K sr , as shown in figure 13(a).We can obtain σ e,op easily by doing one more interpolation from K sr,c vs. σ e curve at h e,op , as shown in figure 13(b).shown at a fixed frequency of 10 GHz.We developed a MATLAB ® code to obtain h e,op and σe,op for a frequency sweep up to 40 GHz; this way, we can build our frequencydependent model for the effective layer thickness and conductivity for a screen-printed silver ink trace [59].The results for this frequency sweep are shown in figure 14.These results can be used for electromagnetic simulations.
We used this compound model results to define our frequency-dependent model for the effective layer thickness and conductivity in ANSYS ® HFSS.From the results, we can see that there is no significant change for h e over our frequency range.We used the average of these values to be the (∼0.36 µm) to be h e over our range of frequencies to build our model in ANSYS ® HFSS.Also, we imported the values of σe vs.  frequency to ANSYS ® HFSS to define our frequencydependent model for the effective layer conductivity.We used this model to obtain insertion loss (S 21 ) for the microstrip line shown in figure 15.We have a thin layer around the trace of the microstrip line and the ground representing the thin effective conducting layer.We used our model to simulate both rectangle and wide rectangle shape profiles.Figure 16 shows the S 21 results of the microstrip line structure using our roughness model for rectangle and wide rectangle shape profiles versus the S 21 results without roughness modeling and experimental measurements of a screen-printed silver ink microstrip  line structure with the same roughness parameters used to feed our model.Figure 16 also shows the simulation without roughness modeling as well as simulation using ANSYS ® HFSS built-in roughness model (Groisse model) [60,61].
A very good agreement is observed between S 21 using our model and the experimental measurements, especially above 7.5 GHz.This conclusion agrees with our conclusion from the previous section that roughness affects the electromagnetic results at higher frequencies.Also, our model can capture more details compared with ANSYS ® HFSS built-in roughness model (Groisse model).We calculated the RMS error (RMSE) between each of the simulation results and the experimental measurement.We got an RMSE of 33.3% between the experimental measurement and the simulated data using a rectangle shape profile without modeling the roughness.We got RMSE of  13.1% and 7.7%, respectively, between experimental measurement and the simulated data using rectangle and wide rectangle shape profiles with modeling the roughness using our model.By focusing only on the results between 7 GHz and 40 GHz, we can achieve an RMSE of 5.7%.We tried to apply our model to other shape profiles discussed in section 3. Using our model with other shape profiles made the design more complicated and did not yield better results.
We picked two other samples and followed the same procedure; the simulated samples details are summarized in table 2. We achieved similar results; we got RMSE of 8.5% and 7.5% between experimental measurement and the simulated data using wide rectangle shape profile with modeling the roughness using our model for the two samples.Figure 17 shows the S 21 results of the microstrip line structure using our model for the two samples.

Conclusions
This paper provides an overview of the background related to printed electronics trace profiles and conductor surface roughness.We explore the impact of various trace profile models on the RF performance of simulated microstrip transmission lines.Notably, wider profiles such as wide rectangles and wide rectangular domes exhibit the closest alignment with experimental measurement data.
We also conducted a statistical study using regression analysis and ANOVA to assess the significance of conductor surface roughness across different frequency ranges.The results indicate an increasingly pronounced influence of conductor surface roughness as frequency rises.
In this context, we introduce and validate our model for accurately depicting the surface roughness of screen-printed silver ink microstrip lines on Kapton substrates.It is important to note that a limitation of this model involves including surface roughness in simulations, especially at lower frequencies.However, the model relies on real roughness measurements and exhibits strong agreement (RMSE of 5.7% between 7 GHz and 40 GHz) with experimental data.
Our objective is to establish a straightforward technique for modeling additively manufactured printed microstrip lines.This technique ensures accurate alignment with the actual measurements of fabricated (screen-printed) microstrip lines.Furthermore, this adaptable model holds potential for various materials and additive manufacturing methods.

Figure 3 .
Figure 3. Actual cross-sectional screen-printed profile of one of the samples.

Figure 4 .
Figure 4. Proposed different trace profiles for modeling.

Figure 5 .
Figure 5. Insertion loss for different modeled trace profiles versus the experimentally measured data.
) and (b) show the setup of taking parallel lines along and across the microstrip line trace to measure the average cross-sectional area, average width, average roughness, and average waviness.

Figure 6 .
Figure 6.(a) and (b) Keyence setup of taking parallel lines along and across the microstrip line trace for measurements.(c) the microscopic cross-section of one of the samples.(d) and (e) the roughness and waviness of the same sample, respectively.

Figure 7 .
Figure 7. Matrix plots to check the independence assumption.

Figure 8 .
Figure 8. Probability plots to check the normality assumption.

Figure 10 .
Figure 10.General concept diagram of our model.

Figure 11 .
Figure 11.(a) Keyence setup for measuring the scaling factor, and (b) measuring the actual length and the horizontal length to determine the scaling factor.

Figures 12 and 13
Figures 12 and 13 are shown at a fixed frequency of 10 GHz.We developed a MATLAB ® code to obtain h e,op and σe,op for a frequency sweep up to 40 GHz; this way, we can build our frequencydependent model for the effective layer thickness and conductivity for a screen-printed silver ink trace[59].The results for this frequency sweep are shown in figure14.These results can be used for electromagnetic simulations.We used this compound model results to define our frequency-dependent model for the effective layer thickness and conductivity in ANSYS ® HFSS.From the results, we can see that there is no significant change for h e over our frequency range.We used the average of these values to be the (∼0.36 µm) to be h e over our range of frequencies to build our model in ANSYS ® HFSS.Also, we imported the values of σe vs.
Figures 12 and 13 are shown at a fixed frequency of 10 GHz.We developed a MATLAB ® code to obtain h e,op and σe,op for a frequency sweep up to 40 GHz; this way, we can build our frequencydependent model for the effective layer thickness and conductivity for a screen-printed silver ink trace[59].The results for this frequency sweep are shown in figure14.These results can be used for electromagnetic simulations.We used this compound model results to define our frequency-dependent model for the effective layer thickness and conductivity in ANSYS ® HFSS.From the results, we can see that there is no significant change for h e over our frequency range.We used the average of these values to be the (∼0.36 µm) to be h e over our range of frequencies to build our model in ANSYS ® HFSS.Also, we imported the values of σe vs.

Figure 14 .
Figure 14.(a) he and (b) σe for a frequency sweep up to 40 GHz.

Figure 15 .
Figure 15.Cross-section of the microstrip transmission line model in ANSYS ® HFSS.

Figure 16 .
Figure 16.The S21 results of the microstrip line structure using our roughness model for rectangle and wide rectangle versus the S21 results of without using our model and experimental measurements of a screen-printed silver ink microstrip line structure with the same roughness parameters used to feed our model.Simulation without roughness modeling and simulation using ANSYS ® HFSS built-in roughness model (Groisse model) are included for comparison.

Figure 17 .
Figure 17.The S21 results of the microstrip line using our model for wide rectangle versus the experimental measurements for (a) sample 2 and (b) sample 3.

Table 1 .
Simple statistics summary of the independent variables.