# Reconfigurable single-layer air coil inductor with bistable compliant windings

Norbert Seliger, Nico Leirich

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Norbert Seliger, Nico Leirich. Reconfigurable single-layer air coil inductor with bistable compliant windings. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0012. hal-05069635

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# Reconfigurable single-layer air coil inductor with bistable
compliant windings

Norbert Seliger

Nico Leirich

TECHNICAL UNIVERSITY OF APPLIED SCIENCES ROSENHEIM

Power Electronics and EMC Laboratory

Hochschulstrasse 1

83024 Rosenheim, Germany

Tel.: +49 / (08031) – 8052624.

E-Mail: norbert seliger@th-rosenheim.de

URL: https://www.th-rosenheim.de

## ACKNOWLEDGMENT

The authors are very grateful to Mrs. Cordula Helmbrecht for assistance in material supply, and to Wolfgang Alversammer and Dr. Christoph Jäger from Research&Innovation Department for their continuous support.

*Index Terms*—Reconfigurable resonant network, Passive component, Magnetic device, Winding topology, S-Parameters.

*Abstract*—We present an air coil with bistable compliant single layer windings. Due to stable elastic snapping deformation of wire segments, the self-inductance of the air coil can be precisely varied in multiple steps. A numerical and analytical design methodology is proposed and successfully validated by magneto-quasi-static field simulations and experiments on prototypes.

## I. INTRODUCTION

Key requirements for modern power electronic systems are high power density, low costs, modularity [1], flexibility and sustainability [2], [3]. This can be achieved by high efficient resonant power conversion topologies with a typical small range of optimal performance.

In many applications, therefore, resonant circuits have to be tuned to compensate for changing loads or different supply conditions. This can be accomplished economically by variable reactance elements, such as inductors and capacitors. In RF electronics, tunable capacitors based on diodes or switched capacitors are preferred, whereas latter components have been recently developed

for power applications [4]. Variable inductors, on the other hand, have been proposed in numerous studies based on different technological approaches [5]. Among them, voltage control [6] and switch control [7], [8] are readily available. Shape forming approaches ([9]–[11]), however, are still under development.

In this paper, we propose compliant mechanism [12] to create different loop geometries of a single layer air coil with multiple windings in order to achieve a reconfigurable inductor. We follow the approach of Seliger in [13], [14] to design and analyze wire loops with elastically snapped wire segments. In this research, however, we further develop this methodology to be applicable to multiple winding single layer structures. Two different air coil geometries are investigated: in Section II a coil with two deformable segments and in Section III loops with edge points forming a regular polygon. The reconfigurable inductance is analyzed numerically, calculated from field simulation and measured on prototype structures in Section IV. We demonstrate a high range of tunability (see Section V) which makes this technique attractive for impedance matching and resonance tuning, where these tasks find numerous applications in wireless power transfer [15], radio-frequency (RF) power [16] and class E amplifiers [17].

## II. BISTABLE COMPLIANT AIR COIL INDUCTOR WITH TWO ELASTIC SEGMENTS

Figure 1 shows our first proposed structure for a variable inductor. A single layer air coil is formed between terminals A and B, consisting of N windings of ribbon wires of thickness h and width w, adding up to an axial length τ. Two longer segments of length l are
connected by short wires of length $d$. Applying an axial force in $x$-direction on the longer wires shortens the end-to-end points by $\Delta l$ and causes elastic snapping of these longer windings in either positive or negative $y$-direction (clamped-clamped bistable compliant mechanism).

<figure><img src="image_2.png" /><figcaption>Fig. 1. Air coil with axially compressed winding segments (bistable deflection): $N = 4$ windings.</figcaption></figure>

Referring to [18] the form of the deflected (buckled) winding segment, $w(x)$, is normalized in order to get the maximum deflection $w_0$ at the midpoint at $x = 0$. Thus between $x = \pm \frac{1}{2}$, corresponding to $x = \pm \frac{l}{2}(1 - \frac{\Delta l}{l})$, the midpoint deflection named $w'_0$ is then $\frac{w_0}{l}$. A stable position pointing outwards, as indicated by the gray colored uppermost winding segment connected to terminal A in Figure 1, is expressed in accordance with [18]:

$$w(x) = -\frac{w'_0}{2} \left(1 + \cos\left(2\pi x\right)\right), \quad (1)$$

where $w'_0$ is given by:

$$w'_{0} = \frac{2}{\pi} \sqrt{\frac{\Delta l}{l} - 4\pi^2 S}, \quad (2)$$

with the stretchability $S = \frac{h^2}{12l^2}$. Applying a small lateral force on this winding in positive $y$-direction causes the segment to snap inwards into the second stable position (illustrated in gray color for the bottom air coil in Figure 1). As a consequence, the loop area of the very top winding and the coupling distance to the other windings below are simultaneously altered, resulting in a change of the air coil inductance. Assuming the lateral separation $d < 2w_0$, only one segment per winding loop can be deflected inside. Hence the number of snapping variations, $m$, is simply:

$$m = 2^N \quad (3)$$

The air coil of Figure 1 with $N = 4$ offers $m = 2^4 = 16$ different snapping geometries, which are depicted in Figure 2. Due to symmetries between snapped windings along the coil axis in $z$-direction, however, the final altered inductance values are always smaller than $m$. As an example, configuration No.3 is equivalent to No.12 in Figure 2.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Bistable snapped winding conditions for the coil in Figure 1 with $N = 4$ windings. The numbers indicate the index of variation.</figcaption></figure>

The largest inductance is obtained for all windings being snapped outside (configuration No. 0), while No.15 represents the arrangement for smallest inductance. Both above mentioned special conditions (No.0, No.15) are characterized by equal snapping directions for all of the windings of a segment. Such a single layer winding structure can be approximated by the shell of a right prism with a rhombic base (Section II-A) or by the shell of a cuboid (Section II-B) in the thin current sheet approximation.

## A. Highest Inductance

The shape of a single-layer air coil with two elastic wire segments and lateral separation $d$ can be approximated with a rhombic geometry for the case of all wire segments being deflected outside for achieving a
maximum inductance value. Figure 3(a) illustrates this approach.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Resemblances for the snapped winding geometry defining the base shapes of the right prism representing thin current sheet air coils: (a) Rhombic basis, (b) Rectangular basis, (c) Regular triangle with all wire elements deflected inside, (d) Regular triangle with all wire elements deflected outside, equivalent circle.</figcaption></figure>

For the bistable compliant mechanism geometry the distance a is one-half of the end-to-end distance, namely $a = \frac{1}{2}l(1 - \frac{\Delta l}{l})$, and $b = \frac{d}{2} + w_0$, i.e. the sum of mid-point deflection and one-half of the lateral distance d. By application of the Fourier transform method presented in [19] we calculate the loop inductance $L_{loop}$ for a right prism of length $\tau$, with rhombic base and replacing the current in the single-layer windings by a sheet current:

$$L_{loop} = \frac{2\mu_0 ab N^2}{\tau} G \qquad (4) \\ G = 1 - \frac{8ab}{\pi^2 \tau} \int_0^\tau \int_0^\infty \frac{\{\cos(a\xi) - \cos(b\eta)\}^2}{a^2\xi^2 - b^2\eta^2} \times \frac{1}{\sqrt{\xi^2 + \eta^2}} (1 - e^{-\tau\sqrt{\xi^2 + \eta^2}}) d\xi d\eta$$

## B. Lowest Inductance

In this case a rectangle as basis shape for a cuboid can be assumed, as demonstrated for a single winding in [13] (cf. Figure 3(b)). With side-lengths $a = l/2$ and $b = d/2$, we apply the Fourier transform method again

and have [20]:

$$L_{loop} = \frac{4\mu_0 N^2 ab}{\tau} G \qquad (5) \\ G = 1 - \frac{1}{\pi^2 \tau ab} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\sin(a\xi)^2 \sin(b\eta)^2}{\xi^2 \eta^2} \times \frac{1}{\sqrt{\xi^2 + \eta^2}} (1 - e^{-\tau\sqrt{\xi^2 + \eta^2}}) d\xi d\eta$$

Thus the design for a variable inductor with specified inductance range $L_{min}...L_{max}$ is analytically feasible by calculating $L_{loop}$ from Equation (5) and Equation (4), respectively, choosing iteratively appropriate lengths $l$ and $d$, a shortening distance $\Delta l$ and a number of windings $N$. The axial length $\tau$ is close to $\tau = Nw$ provided a small vertical space between adjacent windings.

## III. AIR COIL WITH POLYGONAL WINDING ARRANGEMENT

A wider range of tunability is achieved as follows. As demonstrated in [14], the connections between two elastic wire segments can be arranged as sides of a regular polygon forming a winding loop with $n$ bistable snapping elements. For an $n$-sided polygonal loop shape building up an air coil with $N$ windings, we find $m = (2^n)^N$ variations in snapping elements. Due to symmetry in the winding geometry, however, the total number of deflections resulting in different coil inductance values would be smaller.

Enabling further individual snapping of each of $N$ windings within one side segment only, but keeping the other remaining sides in the same snapping state (i.e. each wire of such a side segment deflected either inside or outside), we find:

$$m = \frac{n(n-1)}{2} 2^N \qquad (6)$$

different snapping scenarios. As in the general case, however, the resulting reconfigurable inductance is expecting to be smaller due to symmetries along the coil axis. For the exemplary geometry chosen in this paper featuring a regular triangle ($n = 3$) with $N = 4$ windings, the maximum number of possible variations is $m = (2^3)^4 = 4096$. For modeling and verification purposes, however, we study snapping of a single wall only while keeping the remaining winding side elements being either deflected inside or outside. Thus, the variational number in snapping arrangements, according to Equation (6), reduces then to $m = \frac{3 \cdot 2}{2} \cdot 2^4 = 48$. Figures 4 to 6 depict the corresponding realized snapping winding arrangements.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Snapped variations for triangular shaped bistable windings. Two side segments are each kept deflected outside, the windings of the third side segment are configured individually.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Snapped variations for triangular shaped bistable windings. One side segment is each kept deflected inside and outside, respectively. The windings of the third remaining side segment are deformed individually.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Snapped variations for triangular shaped bistable windings. Two side segments are kept deflected inside, the windings of the third side segment are bended individually.</figcaption></figure>

A. Thin current sheet approximation

According to Seliger in [14], the loop area $s_b$ of a single winding air coil with $n$ bistable deflected segments of length $a = l(1 - \frac{\Delta l}{l})$ can be expressed by a triangular approximation. This simplified shape is exemplary illustrated in Figure 3(c),(d) for the regular polygon with $n = 3$ sides and with all segments snapped inside and outside, respectively. Note that the end points of the winding segments are outlined on a regular polygon, whereas the triangular approximation of the snapped windings adds the deflected midpoints, forming the shape of an irregular polygon. The loop area $s_b$ is calculated according to [14] as:

$$s_b = n \frac{a^2}{4} \cot\left(\frac{\pi}{n}\right) + w_0 a \left(\frac{n}{2} - k\right), \quad (7)$$

where $k$ is the number of wire segments deflected in and $w_0$ is the midpoint deflection distance. This result can be adopted to an air coil with multiple windings, requesting that all windings of a segmented side to be kept deflected in the same direction. Hence $k$ in Equation (7) represents then the number of side segments with inside deflection for each of the $N$ windings. Note that $k$ ranges from $0 \le k \le n$.

Since the air coil geometry with bistable snapping side segments is similar to a right prism with irregular polygonal base, the area $s_b$ and the coil length $\tau$ are the governing factors for the loop inductance rather than the exact shape of the loop windings. Notably the thin sheet approximation for a cylindrical air coil is found applicable, if the area of the circular base is equivalent to the area $s_b$ calculated from Equation (7) for the irregular polygon.

As stated by Ishida et al. [20], the cylindrical air coil inductance in the thin sheet approach corresponds to:

$$L_{loop} = \frac{\mu_0 N^2 \pi r^2}{\tau} G \quad (8)$$

$$G = 1 - \frac{2}{\tau} \int_{0}^{\infty} \frac{J_{1}^{2}(r\zeta)}{\zeta^{2}} \times (1 - e^{-\tau\sqrt{\zeta}}) d\zeta,$$

with the base circle radius $r$ and $J_1()$ the Bessel function of the first type. For an air coil ($N$ windings) with an $n$-sided regular polygon with length $a = l(1 - \frac{\Delta l}{l})$ as base, with $k$ side segments snapped in and thus a loop area $s_b$ as given by Equation (7), the loop inductance can be guessed from Equation (8) by taking into account an equivalent loop area for that circle, i.e. $r = \sqrt{\frac{s_b}{\pi}}$.
IV. PROTOTYPES, EXPERIMENTAL
CHARACTERIZATION, SIMULATION

For the purpose of validation of the theoretical find-
ings prototypes of reconfigurable inductors have been
designed and built. We use ribbon wire segments of
phosphorous bronze with a thickness of h = 0.2 mm and
width of w = 5 mm. The ends of the segments of initial
length l = 85 mm are compressed by a Δl = 5 mm,
then mounted on insulated rods of wood, and finally
connected by lead-free solder at the ends in order to
form the intended loop structures.

The inductor terminations are either coaxial SMA con-
nectors for impedance measurements (S-parameter char-
acterization) or cable connections for tests in power
electronics or RF resonant circuits. The mock-up of the
tunable air coils introduced in Section II and Section III,
respectively, are shown in Figures 7 and 8.

Note that the inductor measurements and simulations
have the SMA connectors directly attached to the top
and bottom Cu windings, respectively. The additional
inductance of the cable connections of Figures 7 and 8
are therefore not included.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Experimental coil with two elastic wire segments per loop and N = 4 windings, configuration index according to Figure 2: (a) No. 0, (b) No. 5, (c) No. 15.</figcaption></figure>

The loop inductance $L_{loop}$ is extracted from the input
power reflection coefficient ($S_{11}$-parameter), measured
with a vector network analyzer (R&S ZVRE):

$$
z_{in} = \frac{1 + S_{11}}{1 - S_{11}}, \qquad (9)
$$

with $z_{in}$ the normalized input impedance. Denoting the
wave impedance by $Z_w = 50 \Omega$ and by taking the
imaginary part $\mathrm{Im}(z_{in})$, we obtain $L_{loop}$ for a specific
angular frequency $\omega = 2\pi f$:

$$
L_{loop} = \frac{1}{\omega} Z_w \mathrm{Im}(z_{in}). \qquad (10)
$$

<figure><img src="image_9.png" /><figcaption>Fig. 8. Experimental coil with three elastic wire segments per loop and N = 4 windings (regular triangular base): (a) Index no. 0 of Figure 4, (b) Index no. 1 of Figure 4, (c) Index no. 15 of Figure 6, (d) Various wire segments snapped inside at different loops in each side segment.</figcaption></figure>

This extraction is valid as long as the total conductor
length of the loop $l_{tot}$ is much smaller than the free-
space wavelength $\lambda_0$ at this frequency, i.e. satisfying the
quasi-static field limit $l_{tot} \ll \lambda_0$, and:

$$
\lambda_0 = \frac{c_0}{f}, \qquad (11)
$$

with $c_0$ the speed of light in air. For the remainder of
this paper we extract the loop inductance for $f = 5$ MHz,
i.e. $\lambda_0 = 60$ m. For the coil geometry in Section II, the
total length is $l_{tot} = 2N(l+d) = 0.728$ m, where for the
triangular air coil from Section III: $l_{tot} = nNl = 1.02$ m.
We further apply the method-of-moments based numer-
ical magneto-quasi-static field solver FastHenry [21] for
the computation of the loop inductance. Our simula-
tion approach and the lumped element extraction from
measurements would be valid up to 20 MHz in both
structures.

V. RESULTS AND DISCUSSION

This section summarizes the modeling, experimental
and analytical results for a reconfigurable coil achieved
for the two geometries of Section II and Section III,
respectively.
### A. Air coil with two elastic segments

The variation in loop inductance for the air coil with four windings and two elastic wire elements is summarized in Figure 9.

<figure><img src="image_10.png" /><figcaption>Fig. 9. Variable loop inductance for the geometry of Figure 7: Thin sheet approximation, FastHenry simulation and measurement result.</figcaption></figure>

Figure 10 repeats the inductance variation of Figure 9, but highlights the $2^4 = 16$ possible variations in snapping arrangements according to Figure 2 with index numbers 0-15.

<figure><img src="image_11.png" /><figcaption>Fig. 10. Variable air coil inductor with four windings and two elastic wire segments.</figcaption></figure>

If all windings are bended outside (zero windings snapped inside in Figure 9, index No.0 in Figure 2), the highest loop inductance is achieved. As a next step, snapping a single winding from outside to inside, reduces apparently the inductance. Due to symmetry in mutual coupling to other windings, two different values are obtained in such a configuration: one winding inside (see Figure 9 and index No.1/No.8 respective index No.2/No.4 in Figure 2). With two windings snapped in (Figure 9), four different configurations are observed:

index No.3/No.12, index No.5/No.10, index No.6 and index No.9 (Figure 2). Three snapped-in windings result in two values: index No.7/No.14 and index No.11/No.13 with almost identical value in Figure 9. Finally, the smallest inductor is realized with configuration named index No.15 in Figure 2 (four windings snapped inside, Figure 9).

Summing up, we find a very good matching between the simulation, analytical modeling result and measurements for this geometry. We would like to emphasize that the thin layer approximations based on Equations (4) and (5) reveal accurate results for the highest inductance of 650 nH and the lowest inductance of 350 nH, respectively.

### B. Air coil with triangular wire arrangement

A first variational study covering the snapping configurations as listed in Figure 4 reveals inductance values as summarized in Figure 11. All windings of two side segments remain deflected outside while bistable snapping is carried out in the third winding stack. The

<figure><img src="image_12.png" /><figcaption>Fig. 11. Simulated and measured loop inductance for the case of two wall segments deflected out (Figure 4). Thin sheet approximations for a circular coil with an equivalent area are also shown.</figcaption></figure>

thin sheet approximation with a circular base (cf. Equation (8)) is applicable to conditions No.0 and No.15, respectively, where all windings of each side segment are deflected in the same direction. Note that measurements for configurations exhibiting symmetry (i.e. index No.1 and No.8) are carried out only once.

Figure 12 depicts the reconfigurable inductance obtained for the snapping geometries given by Figure 5. Note that the element with index No.0 is equivalent to element No.15 of Figure 11. All windings of a first side segment are kept in an outside position, all windings of a second side segment remain in a stable inside position. The third winding stack is configured individually.
<figure><img src="image_13.png" /><figcaption>Fig. 12. Simulated and measured loop inductance for the case of one wall segment kept deflected in and on kept bended out (Figure 5). Thin sheet approximations for a circular coil with an equivalent area are also shown.</figcaption></figure>

The arrangement for two side segments having inside snapping conditions provides tuning in the lowest possible inductance range. This is confirmed by the results shown in Figure 13 for the winding variations related to Figure 6. We obtain a minimum inductance value of 1050 nH.

<figure><img src="image_14.png" /><figcaption>Fig. 13. Simulated and measured loop inductance for the case of two wall segments kept deflected in (Figure 6). Thin sheet approximations for a circular coil with an equivalent area are also shown.</figcaption></figure>

The loop inductance measurements on our prototype match accurately with the FastHenry simulations over the full tuning range. As has been already discussed, the thin sheet cylindrical air coil estimation (cf. Equation (8)) is applicable for specific winding configurations only, since each side segment has to be aligned in a common snapping direction in order to represent an irregular polygonal basis for all windings along the coil axis. The maximum error between the

analytically computed and the simulated inductance is 6%. The demonstrated tuning range for the proposed triangular coil is 1.05µH...1.6µH. For an exemplary untuned loop inductance of 1300 nH (configuration index No.3 in Figure 12), a reconfigurability by ±23% has been demonstrated.

A main advantage of the bistable compliant winding configuration is an easy design and a high flexibility regarding the inductance values. The ribbon wire geometry supports snapping, because a small height *h* to length *l* ratio keeps the stretchability, and consequently the force needed for snap-through, small. Furthermore a single layer ribbon winding is beneficial in terms of reduced parasitic winding capacitance. The winding capacitance is obviously changing with the snapping arrangement, which could be an interesting topic for future studies. In this research we focus on a good snapping behavior rather than on minimized resistive losses. Thus we have preferred the ribbon wire against other wire geometries [22] with known superior performance regarding eddy current losses (skin and proximity effect).

Since bistability can be employed for different wire materials like Copper and Aluminum, and since it is scalable in geometry, we see potential applications in both radio frequency (RF low and high power) and in power electronics in general. As an example, the coil of Figure 1 has been already successfully operated at a current of 10 A. From a systems perspective, among many others reactance compensation in wireless power transfer (capacitive and inductive transfer), resonant tuning in class-E power converters or antenna input impedance matching have been identified.

We believe that the bistable tuning method is interesting from the economic and sustainable point of view, as various target inductance values are realizable within a single mechanical construction. This obviously reduces the amount of parts, decreases manufacturing complexity and finally cuts costs down.

## VI. CONCLUSION

A novel method for tuning the self inductance of a single-layer air coil based on snapping of bistable compliant winding segments has been proposed. We demonstrate reconfigurable inductors with two and three elastic wire deflection segments per loop and present analytical approximations for the design based on a thin current sheet approach. The inductance measurements
on prototypes prove the theoretical findings and match
very well with quasi-static magnetic field simulation of
the loop inductance. We achieve tuning from 1 to 1.6 µH
with a coil of four windings and a regular triangular base
of 85 mm side length.

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