Multi-Physics Modeling and Characterization of Components

1730 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 9, NO. 9, SEPTEMBER 2019

Multi-Physics Modeling and Characterization ofComponents on Flexible Substrates

Sridhar Sivapurapu , Rui Chen, Chirag Mehta, Yi Zhou, Mohamed L. F. Bellaredj , Xiaotong Jia,

Paul A. Kohl , Tsung-Ching Huang, Suresh K. Sitaraman, and Madhavan Swaminathan

Abstract— Due to the increased popularity of wearable devices,designing flexible hybrid electronics (FHE) is becoming increas-ingly critical. Some emerging systems that require FHE includewearables, energy-harvesting devices, sensory networks, andelectrocardiogram monitors. These systems require designers toaccount for different flexible actions, such as stretching, bending,twisting, etc. Accounting for different flexible actions requiresmodels for different components that capture the behaviorunder these actions. This work focuses on characterizing theeffects of bending two components, namely, printed microstriptransmission lines and power inductors. In this work, both themechanical and electrical aspects are captured specifically foraerosol jet-printed (AJP) transmission lines, screen-printed (SP)transmission lines, and SP power inductors on flexible substratesusing 3-D finite element modeling (FEM), which are then cor-related with the measurements. The challenges in such multi-physics modeling and measurements are discussed, including theeffect of bending.

Index Terms— Aerosol jet printing, finite element modeling(FEM), flexible hybrid electronics (FHE), power inductors, screenprinting, transmission lines.

I. INTRODUCTION

THE increasing demand for wearable devices has resultedin further developments in flexible technology. For this

technology, the ultimate goal is maximizing mechanical flexi-bility while improving electrical performance at high frequen-cies, increasing system-level integration, and miniaturizingcomponents. In order to design these systems, a process designkit (PDK) [1] would be useful for designers to use during thedesign process. Including these effects early in the design cyclecan improve performance [2].

Manuscript received July 10, 2019; accepted July 15, 2019. Date of publi-cation July 26, 2019; date of current version September 26, 2019. This workwas supported in part by Semiconductor Research Corporation (SRC) andthe Texas Analog Center of Excellence (TxACE) under Grant 2712.024. Thismaterial is based, in part, on research sponsored by the Air Force ResearchLaboratory under Agreement No. FA8650-15-2-5401, as conducted throughthe flexible hybrid electronics manufacturing innovation institute, NextFlex.The U.S. Government is authorized to reproduce and distribute reprints forGovernmental purposes notwithstanding any copyright notation thereon. Theviews and conclusions contained herein are those of the authors and shouldnot be interpreted as necessarily representing the official policies or endorse-ments, either expressed or implied, of the Air Force Research Laboratoryor the U.S. Government. Recommended for publication by Associate EditorX. Fan upon evaluation of reviewers’ comments. (Corresponding author:Sridhar Sivapurapu.)

S. Sivapurapu, R. Chen, C. Mehta, Y. Zhou, M. L. F. Bellaredj,X. Jia, P. A. Kohl, S. K. Sitaraman, and M. Swaminathan are with theGeorgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]).

T.-C. Huang is with HP Enterprises, Palo Alto, CA 94304 USA (email:[email protected]).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCPMT.2019.2931452

Ideally, at the component level, devices and interconnectsshould maintain stable performance under different flexibleactions, such as stretching, bending, and twisting. However,as shown in [3] and [4], in order to maintain relatively similarperformance for transmission lines while stretching, a longer,serpentine-like structure must be fabricated. The fundamentaldisadvantage of this design stems from the extra length,which degrades insertion loss (S21). Even with these modifiedstructures, there is still a noticeable change in dc resistancewith elongation caused by structural changes, such as plasticdeformation and microcracks. Therefore, a multi-physics mod-eling approach that includes the effects of these deformationsis necessary to capture the behavior of flexible devices fullywith respect to both its mechanical and electrical performance.Most multi-physics approaches for modeling package-levelcomponents focus on electrical-thermal or mechanical-thermalmodeling, missing the key interaction for flexible electron-ics between mechanical and electrical effects. Additionally,the change in electrical performance differs for different typesof components. For example, [5] shows that there is minimalchange in the S-parameters (where S-parameters, or scatteringparameters, represent the relationship between the incident andreflected power waves for a given component) in coplanarwaveguides, for small bending compared to the length of thelines. However, for other components, such as antennas, [6]has shown that there is a noticeable change in the resonantfrequency during various degrees of bending. In both cases,the mechanical effects of bending are neither included norconsidered, which are necessary to properly model the impactof bending (if sufficient strain occurs). Therefore, propermodels that account for both electrical and mechanical changesdue to bending must be developed for each component toadequately characterize these changes. Most electrical modelscan account for the changing shape of the structure, but do notinclude the impact of mechanical deformation on the device,which is necessary if any deformations occur, especially whilecharacterizing the impact of bending on different components.Similarly, bending models account for the change in strain,but do not include the impact on material parameters, suchas conductivity. Prior published work in [3]–[6] presents theeffects of bending on electrical performance without providingdetails on the root cause for these effects.

This article discusses the electrical and mechanical effectsdue to bending for three components on flexible substrates,namely: 1) aerosol jet-printed (AJP) microstrip transmissionlines; 2) screen-printed (SP) microstrip transmission lines; and3) SP power inductors. We have picked these two basic typesof components since the transmission line is distributed along

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SIVAPURAPU et al.: MULTI-PHYSICS MODELING AND CHARACTERIZATION OF COMPONENTS ON FLEXIBLE SUBSTRATES 1731

an extended length while the inductor is concentrated in asmall area. The purpose is to determine if there is a differ-ence in the change in electrical performance (S-parameters,inductance, and resistance) for these components when theyare bent, and if there is a change, the root cause for thischange. Using finite element modeling (FEM) in both themechanical and electrical domains, the change in the responsefor both types of transmission lines and power inductorwhile bending is discussed and correlated with high-frequencymeasurement results. These measurements are conducted insitu, where the high-frequency response is measured duringbending along with deembedding structures. In this article,we provide a comprehensive set of calibrated proceduresfor modeling and measurements that enables the electricalcharacterization of integrated components on flexible sub-strates, such as transmission lines and power inductors undermechanical bending. The importance of process variations forsuch components is also discussed along with gaps identifiedfor the modeling of such structures using current commercialtools.

In [7], AJP microstrip lines on flexible substrates werediscussed. We significantly extend this discussion, in thisarticle, to include SP microstrip lines and power inductors.We derive a relationship between mechanical strain, resistancechange, inductance change, and S-parameters during bendingto explain the physics behind the phenomena. To the best ofour knowledge, the connectivity between mechanical strainand high-frequency electrical performance due to bending hasnot been previously reported in the open literature. Otherflexible operations, such as stretching, twisting, folding, andcombined operations are not discussed in this article, but arebeing studied in parallel at and will be presented in futurepublications.

II. FABRICATION PROCESS

For all three components, the following procedure was usedto prepare the substrate for printing. First, a flexible double-sided, Cu-clad laminate of DuPont Pyralux AP 9151R wasused as the starting substrate. One side of the copper wasremoved before printing the components onto the substrate.A copper etching solution consisting of 100 mL of hydrochlo-ric acid (38%) and 100 mL of hydrogen peroxide was used toremove one side of copper to print the components. Kaptontape was used to protect one side of copper (to be usedas the ground plane). After completing the copper etching,the Kapton tape was removed, and the substrates were cleanedthoroughly in acetone, isopropyl alcohol, and deionized waterbefore printing the structures.

For the AJP process, UTD nano silver ink was depositedonto the substrate using the Optomec Aersol Jet 300 Sys-tem. For the SP process, the MPM SPM 7279 Semiauto-matic Printer [pulse-position modulation (PPM) Inc] printedboth the SP transmission line and the power inductors. TheSP transmission line used a 50-μm-thick stencil, and thepower inductors used a 50-μm-thick, 325 mesh stainless steelscreen. An optical microscope measured dimensions, suchas the width, length, and separation for all the components.A Veeco Dektak 150 surface profilometer measured the cross-section dimensions, such as the thickness and roughness.

Fig. 1. Generalized cross-section view of the substrate and ground planewith the component fabricated on top.

Fig. 2. Top view of the microstrip with important dimensions labeled.

TABLE I

MICROSTRIP TRANSMISSION LINE DIMENSIONS

The dimensions of all of the components are included inSections III-A and III-B.

III. COMPONENT STRUCTURE

Both types of transmission lines (AJP and SP) and the powerinductor were fabricated on a Kapton polyimide substrate witha copper ground plane. The Kapton polyimide and copperground plane are 127 and 38 μm thick, respectively. Kaptonpolyimide has an elastic modulus of 2.5 GPa, a dielectricconstant of 3.5, and loss tangent of 0.0026 [8]. A generalizedcross section for all three components is shown in Fig. 1.

A. Microstrip Transmission Line Structure

Both types of microstrip transmission lines use the samebasic microstrip structure, as shown in Fig. 2, with thedimensions defined in Table I. The major differences, betweenboth transmission lines, are the printing process (AJP vs.SP), the line thickness (650 nm vs. 10 μm), the length ofeach transmission line (107.74 mm vs. 152.28 mm), and theconductor used to print the structures (UTD nano silver ink vs.Dupont 5025 silver conductor). A comparison of the measuredS21 (insertion loss, a measure of loss between the two portsof the transmission line), as well as S21 per unit length ofeach line under flat condition, is shown in Fig. 3(a) and (b).A per-unit-length comparison is used to compensate for the44.54-mm difference in length between the AJP and SP lines.As can be seen in Fig. 3, the SP lines are less lossy than theAJP lines primarily due to the larger conductor thickness.

B. Power Inductor

The top view of the power inductor is shown in Fig. 4. Morespecifically, Fig. 4(a) shows the power inductor on the entire


Fig. 3. Measured (a) S21 magnitude and (b) S21 per unit length for both theAJP and SP transmission lines.

Fig. 4. Top view of the inductor with dimensions labeled (a) full substrateand (b) zoomed-in view of an inductor.

TABLE II

POWER INDUCTOR DIMENSIONS

substrate, and Fig. 4(b) shows a zoom of the power inductor.The extra feed length is required for the bending structure.The dimensions of the power inductor are shown in Table II.Similar to the SP transmission line, the power inductorsuse Dupont 5025 silver conductor. The key parameters toconsider for power inductor’s performance are the side length,linewidth, line spacing, and the number of turns. Increasing theside length and the number of turns increases the inductancewhile increasing the linewidth decreases inductance. However,wider lines decrease the resistive losses of the inductor.The dimensions were chosen to represent a good tradeoffbetween maximizing inductance and minimizing the resistivelosses.

Fig. 5. Examples of (a) cylindrical bending and (b) adaptive curvaturebending. For cylindrical bending, “r” denotes the radius of curvature of thecylinder used for the bending test.

Fig. 6. Examples of (a) tensile and (b) compressive bending. Tensile bendinghas the microstrip facing outward and compressive bending has it facinginward.

IV. BENDING PROCEDURE

For most bending tests, cylindrical bending is used,as shown in Fig. 5(a). This type of bend test is straightforwardas the component under test is wrapped around a cylinder ofa given radius of curvature. The disadvantage of this type ofbend test is that for every radius of curvature used, a new teststructure must be fabricated. Therefore, in this article, adaptivecurvature bending is used, which can be seen in Fig. 5(b). Forthis procedure, the component under test is placed in betweentwo panels, one of which is fixed while the other moves freely.Therefore, the distance between the two panels can be changedby the user while using the same test structure. Anotheradvantage provided by this test procedure is its ease forcompleting cyclical tests. A cyclical test starts with a flat com-ponent, bending the component under test to a minimum panelseparation and then unbending the component to its flat state,and repeating the process multiple times. While the componentis undergoing a change in the panel separation, measurementsare taken at intermediate points in situ using a digital multi-meter (DMM) for dc measurements or a vector networkanalyzer (VNA) for high-frequency measurements. With thecylindrical bending test setup, measurements on a continuouslychanging radius of curvature cannot be performed as many teststructures are required. One other advantage of the adaptivecurvature bending test is the relative ease of switching betweentensile (conductor facing outward) and compressive (conductorfacing inward) bending while keeping the component in theair (as opposed to backed by a dielectric). Examples of bothtypes of bending for a microstrip line are shown in Fig. 6. Forthe inductor, bending occurs along Direction 1 (Fig. 4) basedon the setup described.

The fully assembled test setup can be seen in Fig. 7 for bothcompressive and tensile bending for the AJP transmissionlines. For all of the high-frequency measurements, an AgilentE8363B VNA was used and calibrated with an HP 85052DCalibration Kit, using the short-open-load-thru (SOLT)


Fig. 7. High-frequency measurement setup for the AJP microstrip for both(a) compressive and (b) tensile bending.

standards [10]. The VNA was connected to each componentusing a VLF40-002 SMA, which is a vertical launch SMAthat uses pressure contacts. In addition, each of the panelsused for the adaptive bending test structure has cutouts toconnect the VNA to the vertical launch SMAs. For bothtypes of components (transmission lines and inductors),deembedding structures were used to remove the effectsof the SMAs and the panels on the measurements. Moredetails for deembedding each component is described in theirrespective measurement sections. Due to the length of theinductor’s substrate, only tensile bending could be completedas the minimum achievable panel separation for compressivebending for this particular setup is 60 mm, which is only5 mm smaller than the total length of the inductor substrate.The dc resistance and the conductivity of all the componentswere measured using a DMM. The resulting conductivity wascalculated using σ = (L/Rdc A) where σ is the conductivity,L is the length of the component, Rdc is the dc resistancemeasured by the DMM, and A is the cross-sectional area ofthe component. For the conductivity measurement, the averagecross-sectional area, defined as the product of the averagewidth and the average thickness of the component, was used.

V. MECHANICAL FEM MODELING

The electrical properties of both the transmission line andinductor will change due to geometric deformation. The impactof geometric deformation on the electrical performance can beseen through a change in the dc resistance. From mechanics,there are two types of geometric deformation, elastic andinelastic (plastic), and if there is sufficiently large deformation,cracking along the body will occur. The difference betweenelastic and inelastic deformation is the behavior of the com-ponent after the mechanical load occurs. If the componentreturns to its original state (properties are similar before andafter the mechanical load), elastic deformation occurs. If thecomponent does not return to its original properties before themechanical load, plastic deformation has occurred. When plas-tic deformation occurs, property changes, such as dc resistancebecome permanent, meaning that complete recovery will notoccur after the load is removed. To determine the relationshipbetween the geometric deformation and the dc resistance, both

Fig. 8. Model used for mechanical finite-element modeling where (a) showsthe model and (b) shows the simulated result at 70-mm panel separation whileundergoing tensile bending.

FEM and scanning electron microscope (SEM) imaging wereperformed. The FEM focused on the strain distribution becauseit directly relates to geometric deformation and can be usedto estimate the change in relative resistance. To relate thestrain distribution to a change in resistance, the followingsteps were used: 1) the relative-resistivity, or the ratio of theresistivity at a specific strain to the nominal resistivity, wasfound using ρ = ρ(�) where ρ is the resistivity and � is thestrain; 2) next, we determined the strain distribution along thelongitudinal direction of the conductor using � = �(x) where xis the distance from a specific position on the conductor to theleft edge of the conductor; and 3) we integrated the relativeresistance along the longitudinal direction of the conductor,taking the geometric deformation into consideration using thefollowing equation:

R

R0=

∫ L

0

(1 + �(s))2ρ(�(s))

ρ0 Lds. (1)

In (1), (R/R0) is the relative resistance, L is the length ofthe conductor and ρ(�) is the relationship between resistivityand tensile strain, and �(s) is the tensile strain at position s.From experimental measurements, the function ρ(�) for SPDupont 5025 silver conductor was determined to be ρ =ρo(0.9654 + 20.685�) and ρ = ρo(1 + 2.21�) [11] for theUTD nano silver ink used on the AJP transmission line. Therelationships between conductivity and strain for both SP andAJP lines were found by measuring the relative change in dcresistance due to strain.

A. FEM Modeling—Transmission Lines

For all the mechanical simulations, ANSYS 18.1 [12] isused. Fig. 8(a) shows the 3-D SP microstrip model undergoingtensile bending. In this model, the bottom plate is fixed whilethe top plate moves toward the bottom plate until the gapbetween the two plates is 20 mm. The substrate is bondedto the two plates at both ends of the substrate by the edge.Table III shows the mechanical properties of the materialsused in the finite-element simulations. As seen in Table III,the materials are modeled as linear elastic. The elastic modulusof the silver conductor is based on the averaged data measuredon the polyimide substrate in [15]. Although [15] focused oninkjet printed silver conductor, there is no existing data of


TABLE III

MECHANICAL MATERIAL PROPERTIES

TABLE IV

MAXIMUM PRINCIPAL STRAIN VERSUS MESH LAYERS

screen-printed silver conductor, which is used in this article.The mechanical properties of parallel plate are set as those ofstructural steel in ANSYS 18.1. Since the deformation in theparallel plates is not interested, and in experiments the sampleis too soft to make any significant deformation to the parallelplates, the mechanical properties of the parallel plates will notaffect the results as long as they are large enough.

HEX20 solid elements are applied to the substrate andparallel plates, while QUAD8 shell elements are applied tothe transmission line and copper layer. HEX20 is a 20-nodedquadratic solid element, and QUAD8 is an 8-noded shellelement. As the parallel plates are rigid and do not undergodeformation, only one layer of HEX20 is used for modelingthe parallel plates. For the flexible substrate, two layers ofHEX20 are used; for the transmission line and the copperground plane, one layer of QUAD8 is used, as these layersare sufficiently thin, and the shell elements are capable ofaccounting for bending. Since the number of mesh layersthrough thickness is sensitive [16], to ensure convergence,simulations are also completed with both three and fourlayers of HEX20 for the substrate, and it is seen that thestrain values in the transmission line with the higher meshdensity differ from the strain values with the lower meshdensity by less than 0.01%. The maximum principal strainin the printed conductor results calculated by different meshlayers is shown in Table IV. Therefore, all of the followingsimulations are completed with two layers of HEX20 for thesubstrate and one layer of QUAD8 for the transmission lineand the copper plane to be able to save computational time andstorage. Thus, in total, the model consists of approximately18 000 elements and 90 000 nodes. At the beginning of thesimulation, the sample is almost flat but has a large panelseparation so that the sample can be bent. Both tensile andcompressive bending simulations are performed.

Fig. 8(b) shows the maximum principal strain distributionfor the model under tensile strain at 70-mm panel separation.

Based on Fig. 8(b), it is apparent that there is a negligibledifference in maximum principal strain for the conductor andsubstrate. This is due to the thickness and width of the con-ductor are small in relation to the same dimensions of thesubstrate. Here, the substrate and the copper layer construct themain body of the transmission line, so that they also dominatethe deformation. As seen in Fig. 8(b), the maximum principalstrain is uniform across most of the width of the substrate.However, near the edges where edge effects are dominant,the strains show minor variation. This edge effect is presentthrough a distance of approximately 250 μm on both sides ofthe substrate, which is approximately double the thickness ofthe substrate. This is consistent with what is seen in moststructures that have edge effects [17]. Also, it should benoted that adaptive curvature bending, the radius of curvatureis not uniform, and this radius continues to decrease fromthe fixed edge toward the center of the substrate as onemoves longitudinally along the substrate. Thus, pure bendingis not expected. Furthermore, near the center of the substrate,plane-strain conditions prevail, while the free edges of thesubstrate are traction-free and plane-strain conditions do notprevail.

From the simulation, when the panel separation is 70 mm,the maximum strain on the conductor for tensile bending is0.419% (tensile strain) and for compressive bending is 0.423%(compressive strain), which is small. Therefore, significantdamage to the conductor is not expected, as discussed later.Additionally, the maximum principal strain on the copperground plane is approximately 0.049% and 0.098% for tensileand compressive bending respectively, which are much smallerthan the general 0.2% yield strain, for both tensile and com-pressive bending cases. When the panel separation decreasesto 20 mm, the maximum strain for tensile bending is 1.50%,and for compressive bending is 1.53%, which are again small.The maximum principal strain in the copper is between 0.18%and 0.35%, implying that the copper ground plane may yield.However, since the copper ground plane is copper foil insteadof bulk metal, and for copper foil, the yield strain is greaterthan 0.5% [18], thus the yielding of copper is not consideredin this work.

In the longitudinal direction, the maximum principal strainreaches its peak value at the center of the substrate anddecreases moving toward the two ends, as shown in Fig. 9.As shown in Fig. 9, as the panel separation decreases, the partof the conductor under strain decreases though the maximumstrain increases. As a consequence, when the two plates cometogether, the resistance at the center of the line increases, butcloser to the edges. However, the total resistance may notincrease as much due to the dimensions.

After using (1) and based on the strain distribution for ten-sile bending at 20-mm panel separation for the SP transmissionlines with the length specified in Table I, the change in totalresistance was found to be 2.77%. At 70-mm panel separationfor these transmission lines, the resistance change was 2.17%.In other words, there is a noticeable, but a small change inresistance due to bending (compared to the flat version of thistransmission line), but the difference between different panelseparations is not large.


Fig. 9. Typical strain distribution along the conductor length for panelseparation of 70 and 20 mm for the SP transmission line.

Fig. 10. Mechanical FEM result for strain modeling of the power inductor.

B. FEM Modeling—Power Inductor

The FEM model for power inductor is similar totransmission-line FEM model. TET10 and HEX20 solid ele-ments are used for the substrate and parallel plates, respec-tively. There are 3–4 elements through the thickness of thesubstrate. QUAD8 shell elements are used for the printedconductor and copper ground plane (similar to the transmissionline). In total, the simulation includes 12 269 elements and22 403 nodes. Fig. 10 shows the maximum principal straindistribution of the power inductor when the panel separationis 15 mm. Here, the peak strain is approximately 2.57%, whichis considerably larger than the peak strain for the transmissionline at a 20-mm panel separation. The peak strain increases fastwith the decrease of panel separation when panel separationis small.

The resistance change for the power inductor is dependenton the tangential strain distribution through the component.The tangential strain is the strain along the tangential directionof the trace. Since the strain is along the longitudinal directionof the power inductor, as shown in Fig. 10, there will bea resistance change along Direction 1 in Fig. 4. Using (1)and the simulated strain distribution for the power inductor,the relative resistance (normalized to flat) at panel separationsof 35, 25, and 15 mm are shown in Table V. As a comparison,the relative resistance for a transmission line that is 65 mmlong (the length of the substrate for the inductor) is alsoshown in Table V. At larger panel separations, such as 35 mm,both the transmission line and inductor are comparable inaverage change in resistance, but at smaller panel separations,the inductor has a significantly larger change in resistance.

TABLE V

ESTIMATED CHANGE IN AVERAGE RELATIVE RESISTANCEDUE TO BENDING

This is due to the location of the power inductor as itis centered on the substrate, which is where the maximumstrain occurs for maximum bending. Therefore, the inductoris expected to experience a larger change in resistance (andelectrical performance) due to bending as compared to thetransmission line.

VI. ELECTRICAL FEM MODELING AND MEASUREMENT

For electrical FEM modeling, an adaptive curvature modelwas developed in ANSYS HFSS v19.1 [19] by creating threesections, two flat and one curved. The two flat sections are forthe parts of the component that are flat during the test, and thecurved section represents the portion of the component that isbeing bent. As the panel separation decreases, a larger portionof the transmission line becomes flattened, which is consistentwith the nature of adaptive curvature bending and similarto the mechanical simulation geometry. In these simulations,the mechanical effects of bending are included as a changein conductivity across the entire structure (from Section V,Table IV). The conductivity change is determined by taking theinverse of the change in resistance. For example, an increasein relative resistance of 8% (relative resistance of 1.08)corresponds to a relative conductivity decrease of 7.41%(relative conductivity of 0.9259). For reference, the changein conductivity in relation to the change in resistance for theinductor is shown in Table IV as well. Additionally, for all ofthe simulated components, an adaptive mesh with tetrahedral,curvilinear elements is used.

A. FEM Modeling—Transmission Lines

For both types of bending (compressive and tensile),the panel separation was changed from 30 to 100 mm inincrements of 10 mm for the SP microstrip to determinewhether the differing panel separation would impact theelectrical response. The results of these parametric sweepsare shown in Fig. 11 for the SP lines. The largest devi-ation in S21 for both types of microstrips is less than0.5 dB for all panel separations. Additionally, because S11(a measure of how close the component matches the sys-tem impedance of 50�) in all cases for both types oftransmission lines is below −10 dB, a good match wasmaintained throughout the frequency of operation. Therefore,although the deviation in S11 across different panel separationsis larger than 1 dB, this does not present an importantdifference.


Fig. 11. Magnitude of S11 (right) and S21 (left) for tensile (top) andcompressive (bottom) bending for simulated adaptive curvature bending forthe SP microstrip, where PS represents the panel separation.

Fig. 12. Simulated inductance versus frequency with varying panel separa-tions accounting for the conductivity changes in Table III.

B. FEM Modeling—Power Inductor

Since the length of the power inductor substrate is smallerthan the transmission line, the panel separations were variedfrom 15 to 35 mm in increments of 10 mm, similar to themechanical FEM modeling. Additionally, the inductor in theflat state was used as a baseline comparison for the differentpanel separations. These results are shown in Fig. 12 as afrequency sweep from 10 to 100 MHz. The inductance wasextracted from the single port simulation using the followingequation:

L = Imag(Z11)

2π f. (2)

In (2), Z11 corresponds to the Z-parameters (opencircuit impedance parameters) obtained from the simu-lated S-parameters, and f corresponds to the frequency.From the simulation, the inductance decreases with decreas-ing panel separation when the panel separation is small(25 mm or smaller). Otherwise, when the panel separationis larger than 25 mm, the inductance change is minor up to80 MHz.

Fig. 13. Surface current of a flat inductor (left) and an inductor with 15-mmpanel separation (right). Both inductors use the same scale (left).

The inductor modeled in Fig. 12 has a skin depth thatis much larger than the thickness of the conductor over thefrequency range of 100 MHz. Therefore, the dc resistance andtotal inductance are of interest here, as all of the current willbe carried through the entire cross section of the printed ink.This is based on the skin depth given by

δ =√

1

σπ f μ0(3)

where σ is the conductivity of the silver conductor, f is thefrequency of operation, and μ0 is the magnetic permeabilityof free space. Using (3), the skin depths at 50 and 100 MHzfor the power inductor are 38 and 26.6 μm, respectively,both of which are significantly larger than the thickness ofthe conductor meaning that skin effect is minimal. Hence,the inductance is determined by both the internal and externalinductance of the inductor. For other situations, as seen in [20],when the conductivity and thickness of the conductor are muchlarger, the skin depth will have a noticeable impact on thehigh-frequency performance of the component even at a loweroperating frequency. In [20], the skin depth at 20 MHz, forexample, is 43.9 μm, which is smaller than the dimensions ofthe solder joint (112 μm diameter and 105 μm thickness). Forthe other simulated components, the AJP transmission line’sskin depth at 8 GHz (the highest simulated and measuredfrequency) is 2.5 μm, which is significantly larger than thethickness of the ink, meaning that skin effect should benegligible. For the SP transmission line, the skin depth isnoticeable at higher frequencies (the skin depth at 1 GHz is8.42 μm), but the skin effect is already considered in ANSYSHFSS v19.1 and is included in these simulations.

Since Table III shows the change in dc resistance, we focushere on the change in inductance, which comprises thesum total of the internal and external inductance. Based onFig. 12, there is some change in inductance for panel separa-tions between 25 and 65 mm. However, for a panel separationof 15 mm, the change in inductance is substantial and has asignificant frequency dependent effect. Since the inductor issimulated in ANSYS HFSS v19.1, the source of this changeis purely due to electromagnetic effects and modification inconductivity from Table III based on mechanical deformation,as the strain behavior in Fig. 9 is not directly captured. Theimpact of geometry change due to panel separation can be seenas a change in the surface current density in Fig. 13 for 15-mmpanel separation and flat condition at 50 MHz, respectively.


Fig. 14. Measurements for AJP microstrip transmission line S21 (left) andS11 (right) for tensile (top) and compressive (bottom) bending for differentpanel separations.

Fig. 15. Measurements for SP microstrip transmission line S21 (left) andS11 (right) for tensile (top) and compressive (bottom) bending for differentpanel separations.

Here, the surface current density appears to decrease withbending, and both are using the same scale.

C. Measurement Results—Transmission Line—Single CycleThe panel separations were measured from 40 to 100 mm

for tensile bending and from 60 to 100 mm for compressivebending for both the AJP and SP microstrip lines. The portionof the transmission line and the surrounding substrate near thelaunch were taped to each panel to connect the transmissionlines to the panels. The effect of taping this part of the trans-mission line (the SMA and the small length taped) was takeninto account through thru-reflect-load (TRL) deembedding,as described in [21]. The results of both types of bending forboth printing processes across the different panel separationsare shown in Figs. 14 and 15. From the measurements,there is no noticeable change in either S11 or S21 when thepanel separation was changed, consistent with the results in

Fig. 16. Measured S21 after four cycles for compressive (right) andtensile (left) bending at 4 GHz. A noticeable change in S21 is seen whenmultiple cycles do not occur for the transmission line. Here, a 152.28-mmpanel separation corresponds to a flat transmission line.

Fig. 17. Correlation between measured and simulated AJP microstriptransmission line for the flat condition.

Section V-A where minimal resistance change was seen dueto strain (the measured dc resistance change due to tensilebending for SP transmission lines was 3.043% and 5.217% at70- and 20-mm panel separations, respectively).

D. Measurement Results—TransmissionLine—Multiple Cycles

Cyclical bending measurements were conducted on the SPtransmission lines to determine whether a more pronouncedchange in S21 would occur. This would be seen through alack of full recovery, or full elastic relaxation, resulting in achange in the magnitude of S21, making the structure inelastic.The results after four cycles are shown in Fig. 16, revealinga minimal change in the normalized S21 magnitude causedby cyclical compressive and tensile bending at 4 GHz (exceptfor a single outlier at 100 mm). Here, the magnitudes arenormalized to −4.548 and −4.8125 dB for compressive andtensile bending, respectively.

E. Simulation to Measurement Correlation—Transmission Line

For both types of transmission lines, a reasonably goodcorrelation was obtained between FEM simulation and mea-surements after accounting for the appropriate values forconductivity. For the AJP lines, the conductivity of the linewas found to be half the expected value, or 5 × 106 S/m.This was determined by measuring the total dc resistance ofthe line using a DMM and using the dc resistance and thecross section of the conductor to determine conductivity. Thecorrelation between model and measurement with the adjustedconductivity is shown in Fig. 17, where the deviation in S21is ∼1 dB, and S11 is better than −10 dB indicating a goodmatch up to 3 GHz.


Fig. 18. SP microstrip aerial profile at six separate sections of thetransmission line showing the width variation of the line. The widths alongthe line from left to right are: 334.79, 338.72, 336.07, 350.7, 361.39, 353.4,369.38, 344.04, 332.05, 325.37, 329.36, 312.07, 324.05, 321.36, 324.03,329.39, and 329.43 μm.

Fig. 19. Correlation between modeled and measured (red) SP microstriptransmission line for the flat case with 80 combinations of widths to accountfor the line variability during simulation where the red curve represents themeasurement.

For the SP lines, the discrepancies found were related tolinewidth variation, as seen in Fig. 18. The profile in Fig. 18shows six separate sections of the transmission line. Thechange in linewidths between sections was approximately57 μm or 17% of the average width. Without accountingfor these variations, the correlation between modeling andmeasurements can be poor. Therefore, using Monte-Carlosimulation, 80 cases with 18 separate sections of equal-lengthmicrostrip lines with widths varying by ±20% along thelength of the SP line were modeled to improve the correlation,as shown in Fig. 19. For S11, the correlation is good, whileS21 provided reasonably good agreement up to 6 GHz.

F. Measurement Results—Power Inductor—Single Cycle

Similar to the power inductor simulations, the panel sep-aration was changed from 15 to 35 mm for tensile bending.A similar procedure used to connect to the transmission lineswas used for the power inductor as well. Additionally, priorto the measurements, the VNA was calibrated using the pro-cedure described in Section IV. For deembedding the feedlinefrom the power inductor, a separate structure consisting ofonly the feedline was fabricated, and the S-parameters weremeasured. The S-parameters were converted to Z-parameters,and the impedance of the feedline was subtracted to obtain theimpedance of the standalone inductor. Using (2) the standaloneinductor’s inductance was obtained.

The results for inductance versus frequency at differentpanel separations for tensile bending at 50 MHz is shownin Fig. 20. The simulated inductance was normalized to56.8 nH, and the measured inductance was normalized to55.7 nH. For panel separations of 25 mm and larger, a similartrend emerges for both the simulation and measurement. In this

Fig. 20. Simulated and measured normalized inductance at 50 MHz fordifferent panel separations for tensile bending.

Fig. 21. Measured normalized resistance after four cycles of bending at50 MHz. The resistance is normalized to the resistance of the first flatmeasurement. The straight lines correspond to decreasing panel separation,and the dashed lines correspond to increasing panel separation.

range of panel separations, the change in inductance causedby bending is relatively small. However, below 25-mm panelseparation, the impact of bending is far more noticeable inthe simulation compared to the measurement. This differenceis due to the nonlinear impact of strain on the inductor. Inthe simulation, the impact is assumed to be constant acrossthe inductor as opposed to localizing the maximum strainat the center of the substrate and decreasing the strain thefarther away from the center (as seen in Fig. 9). Since thestrain is more concentrated at the center, a smaller portion ofthe inductor is being impacted, as seen in the measurement.As a result, since ANSYS HFSS v19.1 only simulates theelectromagnetic effects of bending while using a constantconductivity, a decrease in the panel separation causes a largerdifference to emerge.

G. Measurement Results—Power Inductor—Multiple Cycles

A cyclical test was performed to determine the relation-ship between panel separation and normalized resistance todetermine if the geometric deformation caused by bendingis elastic or inelastic. The results are shown in Fig. 21 withthe resistance normalized to 28 � (65-mm panel separationat the beginning of the test). After four cycles, the change


Fig. 22. Measured inductance versus panel separation at 50 MHz after fourcomplete bending cycles. Straight lines: decreasing panel separation. Dashedlines: increasing panel separation.

in resistance was measured to be ∼15% when the inductorreturned to flat. Between the initial flat and 15-mm panelseparation in the fourth cycle, the change in resistance was∼30%. Also, in Fig. 21, a hysteresis effect associated withinelastic strain is present as the resistance does not fullyrelax back to its initial value and is indicative of the damageoccurring along the length of the inductor.

Cyclical bending was used to determine if the inductanceexperienced similar inelastic behavior. Similar to resistancemeasurements, four cycles were completed, with the resultsshown in Fig. 22 for a normalized inductance of 55.7 nH at50 MHz. As shown in Fig. 22, the inductance does not fullyrecover at the conclusion of every cycle, similar to resistance.The change in inductance was ∼10% after four cycles (whenthe inductor returned to flat) and ∼25% between initial flatand 15-mm panel separation in the fourth cycle. Therefore,the inelastic deformation and corresponding hysteresis fromcyclical bending impacts both the resistance and inductance.This can be expected due to the causal relationship betweenresistance and inductance. We initially believed that the causeof the fatiguing effect was due to cracking along the lengthof the inductor. However, as can be seen from the SEMimage in Fig. 23, there is no cracking under flat and 15-mmseparation conditions. However, the silver particles appearto be separated more for 15-mm panel separation comparedto the initial flat condition. It is important to note that theinductor resistance and inductance regained close to theiroriginal values (within 2%) 24–48 h after completing the four-cycle test.

H. Simulation to Measurement Correlation—Power Inductor

In order to obtain a better model to measurement corre-lation, we measured the resistance of the inductors similarto the transmission lines and found the conductivity of thesilver ink to be 2.193 × 106 S/m, or 40% smaller thanexpected. By using the new conductivity, the simulated induc-tance is shown in Fig. 24 along with the measurements.Here, three separate panel separations were simulated, namely15, 30, and 65 mm (flat). From Fig. 24, it is clear that themodel under predicts inductance with the simulations showing

Fig. 23. SEM images showing (a) flat inductor and (b) inductor at 15-mmpanel separation after one cycle.

Fig. 24. Inductance comparison between modeling and measurements(single cycle).

a much stronger frequency dependence as compared to mea-surements. The discrepancy between model and measure-ments is larger for smaller panel separation. This discrepancy,we believe, is caused by the increased strain at the center ofthe component at smaller panel separations. For an SP processusing Dupont 5025, electrical continuity occurs through par-ticle connectivity, and these particles move farther apart withincreased strain, as can be seen in Fig. 23. This behavior causesthe electrical characteristics to exhibit a nonlinear relationshipwith strain. Furthermore, the impact becomes increasinglypronounced with increasing frequency because of the non-linear relationship between frequency, inductance, and strain.Currently, since ANSYS HFSS v19.1 focuses solely on theelectromagnetic effects, this phenomenon is not appropriatelycaptured, causing the discrepancy in Fig. 24.

VII. DIFFERENCE IN ELECTRICAL PERFORMANCE DUE TO

BENDING BETWEEN TRANSMISSION LINES

AND INDUCTORS

Due to the structural differences of the microstrip trans-mission line and power inductor, the impact of bending eachcomponent on their respective electrical performance is differ-ent. As shown, for the transmission line, there was minimalchange in the S-parameters regardless of the panel separation.However, when the panel separation for the inductor decreases,the resistance and inductance for the power inductor dochange. The cause for these differences is due to the structuraldifferences between the two components. Since the inductoris centered on the substrate with multiple conductive paths in


the form of a spiral, and since the strain is concentrated atthe center as the panel separation decreases, the net effect oninductance is noticeable. This is in contrast to the transmissionline where a single path of the conductor is distributed acrossthe entire length of the substrate. Since the concentrationof strain with reduced panel separation only affects a smallportion of the transmission line, resulting in minimal changein the electrical response. The strain distribution and its effecton conductive particle separations could not be captured inthe electrical simulations, showing a discrepancy between themodel and measurements for the power inductor. The impacton the response of the power inductor due to bending isincreasingly observable during cyclical tests because both theresistance and inductance do not fully recover to their originalvalues. This is because the inductor has moved out of theelastic deformation region, indicative of significant strain. Thiseffect could not be captured through electrical modeling aswell. Finally, it is important to note that the printed substratesare prone to significant process variations due to the spreadingof the ink. In addition, the printed conductivity of the inksis significantly different from their bulk conductivity. Unlessthese effects are included, it is difficult to obtain a good modelto measurement correlation.

VIII. CONCLUSION

Both mechanical and electrical models were created tosimulate the effect of bending on microstrip transmission linesand power inductors. Although the transmission lines showedsome change with bending, the change was insignificant.However, inductors showed a much larger change. We havetried to explain this change through measurements by relatingthe mechanical strain distribution, resistance, conductivity, andhigh-frequency electrical response.

Measurements were performed on the components, bothfor single and multiple cycles. Based on the measurements,the transmission line did not experience much change in bothS11 and S21 due to bending for single and multiple cycles.However, for inductors, single-cycle bending resulted in a7% increase in resistance and a 5% decrease in inductance,respectively. The inductance further decreased by ∼25% andresistance increased by ∼30% after four cycles between theinitial flat and 15-mm panel separation. At the completionof four cycles, resistance increased by ∼12% and inductancedecreased by ∼11%.

The modeling-to-measurement correlation was reasonablefor the transmission lines after accounting for process vari-ations. For the AJP lines, a good correlation was foundafter accounting for a decrease in the conductivity of theink. However, for the SP lines, an improved correlation wasrealized after discretizing the line into multiple sections ofdifferent widths to account for process variations. The inductormodels, on the other hand, correlated less favorably with

measurements, especially at higher frequencies and withsmaller panel separation. We believe this is because the toolsdo not accurately capture the nonlinear relationship betweenstrain and electromagnetic effects. Moreover, multiple cycleeffects containing semi-elastic behavior cannot currently becaptured in the tools.

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Multi-Physics Modeling and Characterization of Components