Electrical lengths and phase constants of stretchable coplanar transmission lines at GHz frequencies

Elastic, bendable and stretchable electronics establish a new and promising area of multi-physics engineering for a variety of applications, e.g. on wearables or in complex-shaped machine parts. While the area of metamorphic electronics has been investigated comprehensively, the behavior at radio frequencies (RFs), especially in the GHz range, is much less well studied. The mechanical deformation of the soft substrates, for instance, due to stretching, changes the geometrical dimensions and the electrical properties of RF transmission lines. This effect could be desirable in some cases, e.g. for smart devices with shape-dependent transmission or radiation characteristics, or undesirable in other cases, e.g. in feed and distribution networks due to the variable electrical lengths and thus phase variations. This contribution describes the results of a systematic study of the broadband RF properties of coplanar transmission lines on Ecoflex® substrates, based on numerical simulations and experimental data. Two types of stretchable transmission line structures were studied: Meander- and circular ring-segmented lines. Modeling and simulation were performed combining a 2D circuit simulation software with electromagnetic full-wave simulations. The experimental part of the work included the fabrication of metamorphic substrates metallized with thin copper layers and systematic measurements of the electrical lengths and phase constants of coplanar waveguides in the frequency range from 1 to 5 GHz based on vector network analysis for different stretching levels. With the given substrate technology, we succeeded in demonstrating stretchability up to a level of 21%, while the theoretical limit is expected at 57%. The meander- and circular-shaped line structures revealed markedly different sensitivities to the stretching level, which was lower for circular structures compared to the meander structures by approximately a factor of three.


Introduction and background
Stretchable electronics mark a rapidly evolving technology which paves the way to new classes of electronic devices, where the printed circuit board is no longer restricted to rigid materials but can adapt its shape to the design of a given mount of complex or variable geometries.This enables a new field of applications which additionally benefit from the low weight and a fairly high circuit density of the stretchable devices.Devices that can change their shape, bend or assume the form of the object into which they are integrated, are becoming more and more attractive.This is especially important for wearable devices which have become quite popular recently [1][2][3].A stretchable circuit board can be exposed to different types of mechanical deformation such as stretching, twisting or bending.The capability of taking different shapes is described by the term 'metamorphic' .One intuitive example of metamorphic devices is a membrane equipped with several microphones, which can adapt to different acoustic environments by changing its shape [4] and [5].
While low-frequency flexible and stretchable electronics have been well developed and found many applications [5][6][7][8], the situation is different for studies of their high-frequency properties in the GHz frequency range [1,9,10].In addition, radio frequency (RF) and microwave circuit technologies bear their own specific design challenges.Beside the common mechanical requirements like good adhesion of the metal structures to the substrate and sufficiently high structural precision on the scale of the guided wavelengths, additional effects like the frequency-dependent properties of complex-valued dielectric permittivity and electrical conductivity may give rise to increasing dispersion, losses due to dissipation [11], mode conversion, or (parasitic) radiation [12].Variations of the electrical lengths may cause impedance mismatch or detuning of frequencyselective devices.In addition to substrates with singlesided metallization, at high frequencies, double-sided metallizations or even multi-layer circuits are frequently employed.As a result, the development of passive stretchable RF electronics necessitates systematic studies of the properties of their constituting elements like transmission lines and devices composed thereof, like planar filters, signal distribution networks, or antennas.Active RF components or devices connected to DC bias like amplifiers, detectors or modulators necessitate additionally suitable assembly and packaging technologies, but are out of scope of this paper.
Despite developments in the field of stretchable microwave electronics, multiple challenges towards circuits entirely composed of stretchable components remain to be solved at present.Numerous research groups have addressed RF applications [13][14][15], but there is still little systematic research.The issue of interconnection and stretching effects on microwave propagation was studied in [16].The results reveal, not surprisingly, an inherent trade-off between stretchability and microwave performance.
The focus of this work is on the study of coplanar waveguide (CPW) structures as key RF transmission element accessible, given that double-sided versions are lacking at present, with single-sided metallized stretchable substrates.The most important geometrical design parameters of a CPW are the length and width of the signal and ground conductors and their separation [17].While the conductor lines are laid out mostly by contiguous metal regions if used on rigid substrates, their layout must be optimized to achieve stretchability without compromising the transmission properties.The electrodynamic performance of a transmission line is mainly described, for a given propagating mode, geometrical layout and electromagnetic material parameters, by the complex-valued and frequency dependent propagation constant γ(f ) = α(f ) + j β(f ), where α and β denote the attenuation and phase constants, respectively [18].The characteristic line impedance Z c (f ) is another key parameter of RF transmission lines; the adjustment of specific Z c -values for a given type of transmission line, e.g. for the design of matching networks or filters, is well understood and not further elucidated here [19].Rather, the focus of the present study is on the electrical lengths and phase constants of stretchable CPW, including the impact of the layout of the conductor lines and the effect of stretching.

Fabrication of stretchable RF transmission lines
For the layout of stretchable RF transmission lines, two elements must be considered in combination: The conductive structures and the stretchable substrate.The basic requirements for the conductor material are usually high electrical conductivity and good mechanical stretchability.In practice, it is necessary to follow one of two basic approaches, namely the use of stretchable materials or stretchable structures.
The former approach is to compose novel types of materials that are inherently stretchable (materialbased stretchability).Such materials usually include conductive polymers, transition-metal dichalcogenides, fluids, nanowires or nanoribbons, carbonbased nanomaterials or other.Despite significant research in this direction [20,21], the application of such materials is still challenging on industrial scales.Basically, this results from toxicity, insufficient thermal and electrical performance, and the complexity and cost of the manufacturing process [20].
The second approach consists of employing rigid materials such as metals and transforming them into structures that can be stretched to a certain extent (geometry-based stretchability).While it is well known that metals are not inherently stretchable, it was argued in [22] that any metal conductor of sufficiently small cross section becomes flexible due to bending deformations that decrease linearly with thickness.Typical examples of stretchable geometrical shapes are circular arcs or twisted lines, similar to the design of bellows [22][23][24][25][26].The challenge of this approach is to achieve reliable stretchability of the structures over a large number of cycles [27].Cracks caused by elongation may lead to changes of the transmission parameters or even to a complete interruption of the conductor paths [28].
The fabrication of stretchable electronic circuits does not generally follow standard routes or fixed procedures.Rather, the technology selection is tailored to the required properties or intended applications, the production volume, and the related unit costs.Notwithstanding, different technologies can be distinguished in terms of the order of process steps involved.In case of the geometry-based stretchable electronics, two main fabrication routes have been established [4]: • Soft substrate first: Direct fabrication of the circuit on the stretchable organic material • Soft substrate last: Fabrication of the circuit on a regular silicon substrate and subsequent transfer to a stretchable polymer substrate.
The soft substrate first approach refers to a manufacturing process in which a soft polymer substrate is manufactured at the beginning of the fabrication.
All subsequent processing steps are then carried out directly on this substrate.This type of processing is widely used to create stretchable electronics as it allows for through-holes and multilayer structures.
On the other side, processing on a soft substrate is difficult and often complicated by various limitations.For example, high temperature processing is not supported due to the thermal instability of the soft materials; further, accurate alignment and adjustment is always a challenge on soft substrates.The soft substrate last approach is more compatible with standard substrate technologies based on rigid substrates.The fabrication begins with a rigid carrier substrate, e.g. a glass plate or a silicon wafer, which enables high temperature processing, precise alignment and also the use of standard assembly and packaging techniques including surface-mount devices.The entire chain of manufacturing, assembly, and test processes is performed on the hard carrier.Only at the end of the processing sequence is the circuit eventually transferred to a stretchable substrate.
Our soft substrate last process followed three steps: 1. Preparation of the hard carrier for applying the conductive structure 2. Photolithographic structuring of the transmission lines 3. Transfer of the conductive structure to the elastic substrate Steps two and three of the fabrication process are illustrated in details in figure 1. Steps A.1 to A.4 were performed on the hard carrier, steps B.1 to B.3 were used for the transfer to the elastomer substrate.The process begins with hard carrier cleaning.While conventional silicon wafers (dummy 4 ′′ Cz-Si Wafer, 500 ± 25 µm thickness, Microchemicals GmbH) were used in this study, any hard carrier would be suitable for this process.The wafers were subject to an ultrasonic bath with a universal cleaner (Tickopur R33, Dr H. Stamm) diluted 20:1 in de-ionized water, for 5 min.The substrate was rinsed, dried, and then kept in an oven at 105 • C. A PMMA layer (AR-P 672.11, ≈4 µm thickness, Allresist) was spin-coated (500 rpm for 2 s, 1000 rpm for 2 s, 1500 rpm for 60 s at a constant acceleration of 500 rpm s −1 ), and then placed on a hot plate at 150 • C for 3 min, to evaporate the solvents and enable adhesion to the hard carrier.Polyamide (PI 2610, ≈4 µm thickness, HD Microsystems) was spin-coated (500 rpm for 2 s, 1500 rpm for 3 s, 2500 rpm for 70 s at a constant acceleration of 500 rpm s −1 ) and placed on a hot plate for 5 min at 180 • C and then cured in an oven at 200 • C for 5 h under nitrogen atmosphere.This PMMA/PI interface is necessary for the later transfer to the stretchable substrate, since there is low adhesion between PMMA and PI.
Metallized films of 20 nm-thick titanium and 200 nm thick copper were sputter-coated on top of the PI using the Ardenne CS400S system at a pressure of 5 × 10 −3 hPa and a power of 200 W.The Ti deposition rate was 0.217 nm s −1 , the Cu deposition rate amounted to 0.754 nm s −1 .A negative resist (AZ 15nXT, ≈10 µm, Microchemicals) was spin-coated and patterned by photolithography (see step A.1 in figure 1).This negative resist is suitable for electroplating.The wafer was immersed in an electrolyte solution (NB semiplate Cu 100, Mirochemicals) and a constant current of 80 mA was applied between a Cu target and the surface of the substrate for 1 h to achieve the desired copper thickness of 10 µm for the meanders (A.2).The pads reached a thickness of 8.5 ± 1 µm.We noticed that the deposition rate was not homogeneous across all areas.The resist was then stripped off (Technistrip NI 555, Microchemicals) in two immersion steps, first 5 min at a temperature of 80 • C, and then 5 min at room temperature (A.3).A copper thickness of 200 nm was wet-etched by immersion in Ammonium persulfate diluted in water (15 g per 100 ml) during 30 s, and then immersed in BOE 6% for 20 s, to etch away the Ti adhesion layer (A.4).Only the desired electrical copper line structures remained on the wafer.Ecoflex ® 00-30 was poured on top of the structure (B.1).Based on systematic studies, we experienced that the elastic material was ready to be peeled off after 14 h of curing at room temperature [29,30], and all layers embedded up to and including the PI could be transferred, while the PMMA remained on the hard carrier (B.2).Ecloflex ® 00-30 was chosen instead of PDMS because of its significant better mechanical performance as soft elastomer substrate: To highlight two key parameters, the Young's modulus of Ecoflex ® 00-30 is 0.1 MPa, while for the PDMS Sylgard 184 (10:1) it amounts to 2.4 MPa; second, the elongation at break for Ecoflex ® 00-30 is 835% compared to 135% for the PDMA [31].The final step consisted of a dry etch of the PI layer using a combination of CF 4 /O 2 plasma as described in detail in [4].The electroplating limited the structural resolution to 5 µm, according to the manufacturer.
The manufacturing process is challenged by avoiding short circuits between the signal and ground conductors or interruptions of the folded lines.The most serious problem, however, is that the copper structures may detach from the substrate when stretched.This issue was addressed by the layout and handling of the RF test packages as described in further detail below.

Geometrical considerations
For the purpose of our research outlined above, the ideal type of transmission line is a CPW, which consists of a center conductor of width w 1 , separated by a gap of width g from two adjacent ground conductors, each having a width w 2 , as illustrated in figure 2. This line topology fits best to the soft substrate last technology as all metal structures are located on a single side of the substrate.
The conventional CPW layout employs conductors as rigid metal strips and thus is not inherently stretchable.Following the geometry-based stretchability approach, the flat metal conductors were replaced by three bent conductor lines, in order to form a stretchable coplanar transmission line as sketched in the right-hand part of figure 2. A ring-shaped unitcell element of the stretchable transmission line was devised, as depicted in figure 3(a).The geometrical construction of the semi-circle was based on the outer and inner arcs AB and СD, respectively.The radius R corresponds to the average radii of both arcs, as indicated by the dotted line.In the initial position, when the element is not subject to stretching, the length of the arc is given by l = πR and the chord width is given by w = 2R.The stretching model assumes the length l to remain constant upon stretching, and the chord length to increase until it reaches the maximum value of w max = l = πR.Consequently, the initial and the maximum lengths of the basic element are related by the equation: The maximum stretching level expected from this model hence amounts to approximately 57%.In this paper, the level of stretching refers to the ratio d of the stretched length l s to the unstretched length l 0 , expressed as a percentage value: Figure 3 shows in the panels (b) to (i) the transmission line structures eventually used.Two different shapes of stretchable conductor elements were investigated in this work: Meander lines (M) as shown in figure 3(b) and circular ring-segmented lines (C) in figure 3(c), both consisting of the same arc-shaped basic elements.The ring-type version results from the meander line by shifting the semi-arcs on one side of the symmetry line.Table 1 summarizes the main geometric parameters of the investigated structures.Each structure has been given a name consisting of a letter and a number, with the letter 'M' for meander 'M' and circle 'C' , while the number identifies the version defined by a specific set of geometric parameters.
Apart from the different basic types (M versus C), varying the radius R allowed us to study the effect of conductor shape on the electrical length and the stretchability of the

RF modeling of stretchable transmission lines
Our working hypothesis is that the RF transmission properties of the stretchable versions of the transmission lines can be described analogously to their rigid counterparts, namely the Telegrapher's equations [18]: where v(z,t) and i(z,t) denote the voltage and current waves along the transmission line.The parameters R, L, G, and C represent the line resistance, inductance, conductance and capacitance per unit length ∆z as depicted in figure 4(a) [18].The propagation constant γ results from the line parameters: with its real and imaginary parts referred to as attenuation constant α and phase constant β = 2π/λ•√ε reff of the line, respectively, λ the free-space wavelength, ε reff the effective permittivity, and ω = 2πf the angular frequency.The line impedance Z c follows from Both, propagation constant and line impedance can be derived from the scattering (S) parameters [18].The calculation starts with the ABCDparameters of a transmission line: The matrix coefficients are related to the Sparameters by:  (11) where Z ref is the reference impedance of the system.The propagation constant is related to the matrix parameter A by: leading to the electrical length of the transmission line θ = Im{γ•l}.The phase constant β can be derived eventually from the electrical length by division by the effective line length l.
The line impedance results from √B/C [19]: At this point, it is important to note that the transmission line parameter accessible directly through simulations or measurements is the electrical length θ but not the phase constant β.In contrast to rigid CPW structures, the line length l of the stretchable meandered versions cannot be determined easily, reflecting the impact of the line shapes and their parasitic interaction on the propagating modes and their dispersive properties; these effects were studied rigorously in [23,32].Consequently, the following analysis focuses on the electrical length θ.In line with our working hypothesis, the electrical length of the stretchable transmission lines is expected to scale linearly with frequency.

Numerical simulations-TEM and quasi-TEM modes
For the numerical simulations, case #1 according to table 2, the ideal case of TEM mode propagation was adopted initially, with the stretchable transmission line fully embedded inside the homogeneous dielectric substrate, as illustrated in figure 4(c).Except for potential dispersion from the meandered structure [23] which would, however be correctly reproduced through electromagnetic full-wave simulations, additional dispersive effects due to the effective permittivity of the structure are excluded.The thickness of the substrate was kept constant for all stretching levels.The dielectric properties of the substrate were assumed independent of frequency, which was confirmed by measurements 3 .The dielectric permittivity derived from measurement varied between 3.15 and 3.0 at frequencies ranging from 1 GHz to 10 GHz.We used a value of 3 in the simulation, in good agreement with other studies and measurements [19,[33][34][35].Towards a more realistic case, and to study the effect of inhomogeneous interfaces on the transmission properties of realistic stretchable transmission lines, a second set of simulations (#2 in table 2) was performed in which the metal layer was embedded between the flexible substrate and air, as depicted in figure 5(a).
The third set of simulations was the same as #2 but included the probe ports, thus being closest to the real conditions.
The numerical simulations were carried out for different levels of stretching from d = 0% up to 3 Keysight, '85070E Dielectric Probe Kit 200 MHz to 50 GHz,' Technical Overview, 2017.d = 57% in steps of 10%, using the software tool MOMENTUM 4 ; a mesh density of 20 cells/wavelength was applied.The simulation allows for the computation of the field distribution in a given structure when excited at defined ports.From this, Sparameter data were obtained and used to extract the transmission line parameters as described above.All normalized curves in figure 6(a), for both line types, merge onto a single line for a given line geometry, indicating the absence of any frequency dispersion caused by the meandered shape of the transmission line.This behavior is expected for TEMmodes but indicates at the same time the insensitivity of coplanar transmission lines to inhomogeneous permittivity values underneath and above the structures.

Simulated RF properties
For the cases #1 and #2, the electrical length θ increases in proportion to the stretching level d (panels a and b), though markedly differently for both line types.The sample C1 revealed a three times steeper slope of about 5.5 × 10 −3 rad/% for the ideal case (panel a), respectively 4.4 × 10 −3 rad/% for the quasireal case (panel b), compared to the sample M1 with slopes of 1.8 × 10 −3 rad/% and 1.4 × 10 −3 rad/%, respectively.The slope values for the quasi-real simulation conditions (case #2) are about 25% smaller than for the idealized case #1.This value reflects the different effective permittivities: For case #1, we expect ε reff = 3 due to the symmetric embedding of the transmission line into the dielectric soft material.For case #2, we expect ε reff = 2, assuming equal weights of electrical field density in the substrate (ε r = 3) and in air (ε r = 1).This change in permittivity from the ideal to the quasi-real boundary conditions leads to an expected change by (3/2) 1/2 = 22%, which comes close to the simulated variation by 25%.
The simulation under real conditions (case #3 and figure 6(c)) leads to a pronounced frequency dependence and variation of the slopes across the ranges (1.2…6.5)× 10 −3 rad/% and (0.2…4.7) × 10 −3 rad/% for the C-and M-structures, respectively.We conclude that this variability results from the pad structures, giving rise to markedly frequency dependent effects.
The different susceptibilities to strain for the Mand C-structures remains evident for all three simulation conditions.The markedly different stretching susceptibilities for the two line types were found for all other geometrical variations.This systematic difference gives an important clue on the design of stretchable transmission lines optimized for intended applications where the sensitivity on stretching should be weak (M-type) or strong (C-type).
The electric length θ follows directly from the S-parameters, while the determination of the phase constant β requires knowledge of the effective line length l.For a classical CPW, this parameter is well defined [17]; for meandered structures, on the contrary, the line length is not well defined [23].Therefore, in addition to the results for the electrical length, figure 6(d) shows the calculated line

Measurement setup and samples
In order to assess the RF transmission line properties through waferprober measurements, groundsignal-ground contact pads were integrated into the transmission lines as depicted in figures 5(b) and (c).The pads with dimensions 6 × 11.75 mm (Mtype) and 6.5 × 11.75 mm (C-type) were laid out such as to maintain the structural stability especially under stretching.The S-parameter measurements were performed with a vector network analyzer (Agilent E8361A) connected to the probe tips (Cascade FPC-GSG-650) of a waferprober station as shown in figure 7(d, e).
A special test fixture (figure 7(a)) was devised to enable RF measurements of the transmission lines at different well-defined stretching levels.The fixture allowed to fix both ends of the test structures by thin clamps attached to plexiglass blocks with screws.The clamps were designed with holes for the pads making it possible to reach the pads with the probe tips.Stretching was achieved by changing the position of one of the blocks using a threaded screw, with the probe tips released.(figure 7(b, c)) C-and M-type CPW structures were manufactured according to the method described in section 1.The Ecoflex ® substrates contained all six transmission line structures shown in figures 3(d-i).The substrates were 2 mm thick, the copper metallization was 10 µm thick.Initially, multiple geometrical modifications of the CPW samples were investigated, including different radii and linewidths of the stretchable conductor lines, and different numbers of ground conductors.As an example, figure 8(a) shows M-and C-type structures with a linewidth of w s = 50 µm and n = 4 ground conductors at each side of the center conductor.Figure 8(b) displays the same structures for a stretching level d = 32%.It can be seen that the meander structure was stretchable while the circular structure detached from the substrate without stretching.It eventually turned out that a smaller width w s = 15 µm of the conductors, a reduction of their number n = 1, and a larger radius R = 0.6 mm of the segments improved the stretchability (table 1), as indicated for selected M-and Csamples in figures 8(c)-(f).

Measurement results and outlook
Figure 9 compares the measured results (solid lines) for the electrical length θ versus stretching level d at the two selected frequencies of 1 and 5 GHz with the simulated data (dotted lines, case #3) for two M-type samples (blue color tones) and three C-type  samples (red color tones).The maximum stretching level reachable in the experiment without damaging the sample amounted to 21%.Higher stretching levels could presumably be reached with optimized structures.Even across this limited range of stretching levels, the measured data confirm the systematic differences between meander and circular line structures expected from the numerical simulations, including the real conditions with probe pads.While the slope values agree on average, the measured data reveal significant scatter and a major influence of the contact pads.In addition, the impact of the structural parameters such as the width and the number of the conductors and the radii R of the segments on the stretchability were confirmed by the measurements.

Conclusions
We presented a systematic study of the transmission properties of stretchable transmission line structures, fabricated with the organic substrate material Ecoflex ® and single-sided copper metallization, in the RF range; the research focus was on the spatial phase variation as a function of the stretching level.Measurements of the dielectric permittivity ε r ≈ 3 across the frequency range 1…10 GHz revealed negligible variation below 5%, in accordance with literature data.The stretchability of the transmission lines was achieved by circular line segments.Conductor lines of reduced width, enhanced radius of the circular unit-cell elements, and a reduced number of ground conductors improved the stretchability compared to their respective counterparts.CPW transmission lines with a maximum stretchability of 40% in a singular case could be successfully fabricated, packaged, and measured.The maximum stretching level for the investigated type of structures was predicted to be limited at π/2 − 1 ≈ 57%.
The RF properties were studied systematically for two different types (meander-shaped and circular segments) with respect to their frequency-and stretching level-dependent electrical lengths with circuit and electromagnetic fullwave simulations.The transmission parameters were additionally measured in the GHz frequency range.
A key finding was that the C-type structures displayed a sensitivity of the electrical length to the stretching level about three times higher than the Mtype structures.The systematic analysis of the geometrical line lengths and the phase constant indicates that the observed stretchability does not only result from the geometrical variation alone but additionally from the nature of the propagating modes.These features remained unaffected by the boundary conditions used for the numerical simulations.Compared to the strong impact of the contact pad structures on the sensitivity on the stretching level, dispersive effects played a minor role.Hence, for given intended applications, the layout of the contact pads should be in the focus of the design.The experimental data confirmed the numerical results, even though the small number of functional samples leaves corresponding statistical uncertainties.Further studies would need to reveal the impact of different substrate thicknesses and of dielectric losses on the findings.
In order to overcome the difficulties in fabricating and contacting fully stretchable transmission line circuits, one step could be to create hybrid structures [36][37][38], combining rigid and flexible sections.In addition, our findings help to design capacitive RF elements as constituting parts of tunable devices.The project results provide a valuable basis for follow-up research, in which metamorphic structures may purposedly display either weak or strong sensitivity to stretching.In this context, the electrical length represents a key parameter for the design of RF devices like line-coupled filters, antenna arrays, or active circuits like oscillators.

Figure 1 .
Figure 1.Fabrication of the stretchable RF transmission lines according the soft substrate last process flow.

Figure 2 .
Figure 2. Typical layout of a coplanar waveguide (left) and its representation in a geometry-based stretchable layout (right), where w1 is width of the center conductor, w2 the width of the ground conductors, and g the separation between them.

Figure 3 .
Figure 3. Basic circular ring-shaped line element (a) of the stretchable RF transmission lines with the following geometrical parameters: arcs AB and СD, radius R, arc length l, chord width w, center point o.Sketch of the meander (b) and circular type (c) structures.The symmetry axis is marked by the grey line and the basic element highlighted in red.The various CPW layouts are divided into meander M1 (d), M2 (f), M3 (h) and circular type C1 (e), C2 (g), C3 (i).Table 1 provides an overview of the geometrical dimensions.

Figure 4 .
Figure 4. (a) Lumped-element representation of an infinitesimal transmission line element of length ∆z with line parameters and voltages and currents indicated.(c) Schematic representation of the substrate.The orange strip indicates the metallic transmission line (thickness 10 µm), the grey regions denote Ecoflex ® substrates below and above (εr = 3, tan δ = 0.015, thickness t = 2 mm), and the blue regions represent air.(b) Sketch of selected geometrical shapes of M-type transmission lines at different stretching levels 0% ⩽ d ⩽ 57%.(d) Same as (b) for C-type transmission lines.

Figure 5 .
Figure 5. (a) Schematic representation of the substrate.The orange strip indicates the transmission line (thickness 10 µm), the grey region denotes the Ecoflex ® substrate underneath (εr = 3, tan δ = 0.015, thickness t = 2 mm) and the air region (in blue) above.Schematic integration of the probe ports into the stretchable transmission line structures, exemplified for the two types M1 (b) and C1 (c), cf figures 2(d) and (e).

Figure 6
Figure 6 displays the simulated dependence of the electrical length θ on the stretching level d for a set of frequencies ranging from 1 GHz to 10 GHz (color-coded) for the two representative samples M1 (circles) and C1 (squares).Panel (a) refers to the idealized boundary conditions #1 (table 2 and figure 4(c)), panel (b) to the inhomogeneous case #2 (figure 5(a)), and panel (c) to the real conditions #3.The θ-values were normalized to each respective frequency.All normalized curves in figure6(a), for both line types, merge onto a single line for a given line geometry, indicating the absence of any frequency dispersion caused by the meandered shape of the transmission line.This behavior is expected for TEMmodes but indicates at the same time the insensitivity of coplanar transmission lines to inhomogeneous permittivity values underneath and above the structures.For the cases #1 and #2, the electrical length θ increases in proportion to the stretching level d (panels a and b), though markedly differently for both line

Figure 6 .
Figure 6.Simulated dependence of the electrical length θ (a,b,c), normalized to the respective frequencies, on the stretching level d at frequencies from 1 to 10 GHz (different colors) for the transmission line samples M1 and C1 for the ideal (a), the quasi-real (b) and the real simulation conditions (c).Effective normalized transmission line lengths derived for both line types from the electrical length, assuming a constant β-value, as a function of stretching in comparison with measured line lengths (d).

Figure 6 (
d) also contains measured l-values, determined from the geometrical distance between the input and output probe ports.The calculated effective lengths l(d) differ for both line types (M-or C-structures) despite their nominally identical dimensions.This discrepancy between geometric and effective line lengths implies that the phase constants differ for both line types and are strain dependent.Apparently, the transmission modes are influenced by the respective line geometry and substrate thickness, especially for the less stretching-sensitive meandershaped structures.

Figure 9 .
Figure 9. Measured (solid lines) and simulated (dotted lines) dependences of the electrical length θ on the stretching level d at f = 1 GHz (a) and f = 5 GHz (b) for M-type (bluish colors) and C-type structures (reddish colors).

Table 1
provides an overview of the geometrical dimensions.

Table 1 .
Types of investigated meander (M) and ring-shaped circular (C) structures and their basic geometrical parameters.R denotes the arc radius, n the number of ground conductors, and ws the width of the base element.
structure.Figures3(d)-(i) provide an overview of the transmission line designs used in this study.

Table 2 .
Overview of the types of simulations performed with different complexity including dispersive effects.