*2.2. Calculation of Sensor Inductance*

The behavior of the proposed sensor was first predicted by simulations based on calculation. Due to the practical limitation that finite element method (FEM) simulation of electromagnetic fields is time consuming, we simplified the serpentine pattern as a straight line, so the shape change of the serpentine pattern could be simplified to the change in aspect ratio of a rectangle as illustrated in Figure 4. The inductance of the simplified model was calculated as the sum of self-inductances and mutual inductances of all line segments, based on the Greenhouse method [24]. In the Greenhouse method, the inductance is expressed as the sum of self-inductances and mutual inductances of each segments. The inductance *L* is given by

$$L = \Sigma L\_{\\$} + \Sigma M\_{+} - \Sigma M\_{-}$$

where *Ls* is the self-inductance of each segment, and *M+* and *M*<sup>−</sup> are the positive and negative mutual inductance between segments, respectively. Since there is no segment combination in which current flows in the same direction, the sum of the positive mutual inductance is zero. The four segments used in the calculation are illustrated in Figure 4: left, right, top, and bottom segment.

**Figure 4.** Simplification of the serpentine sensor coil as a rectangle to calculate the inductance.

Based on Equation (1), the inductance of the 1-turn rectangular coil has only two components of self-inductance *Ls* and negative mutual inductance *M*−, and it can be calculated with the following equation:

$$L = L\_{\rm s, Rottom} + L\_{\rm s, Lrft} + L\_{\rm s, Top} + L\_{\rm s, Right} + M\_{-, Top, Rottom} + M\_{-, Lrft, Right} \tag{2}$$

where *Ls*, *Bottom*, *Ls*,*Lef t*, *Ls*, *Top*, and *Ls*,*Right* are the self-inductances of the bottom, left, top and right segment, respectively, and *M*−, *Top*,*Bottom* is the negative mutual inductance between the top and bottom segments, and *M*−,*Lef t*,*Right* is the negative mutual inductance between the left and right segments. *Ls* and *M*<sup>−</sup> can be calculated by the following equations:

$$L\_s = 0.002l \times \ln\left(\frac{2l}{l\_1 + l\_2} + 0.50049 + \frac{l\_1 + l\_2}{3l}\right) \tag{3}$$

$$M\_{-}=2l\left[\ln\left\{\frac{l}{d}+\left(1+\frac{l^2}{d}\right)^{\frac{1}{2}}\right\}-\left(1+\frac{d^2}{l^2}\right)^{\frac{1}{2}}+\left(\frac{d}{l}\right)\right] \tag{4}$$

where *l* is the length of the selected segment, *l*<sup>1</sup> is the horizontal length of the coil, *l*<sup>2</sup> is the vertical length of the coil, and *d* is the distance between the segment centers. The relationship between the length of the coil segments and the strain applied to the sensor is based on the measurement that when the sensor was stretched 100%, the length of the top and bottom segments were increased by 63% and the length of the left and right segments were decreased by 30% as illustrated in Figure 3. Based on this, *l*<sup>1</sup> and *l*<sup>2</sup> of the coil when a strain *ε* is applied to the whole sensor are expressed as below:

$$l\_1 = 1.63 \varepsilon l\_{1, 0\%} \tag{5}$$

$$l\_2 = 0.7 \varepsilon l\_{2,0\%} \tag{6}$$

where *l*1,0% is the horizontal length at 0% strain and *l*2,0% is the vertical length at 0% strain. Thus, the inductance of the sensor when a strain is applied can be calculated using Equation (2), substituted by Equations (3) to (6). This calculation was implemented in MATLAB (MATLAB 2016b, Mathworks Inc., Natick, MA, USA). Based on this method, the relationship between the inductance and strain could be obtained by linear fitting based on the calculated inductance values, expressed by

$$L \approx L\_0 + \mathfrak{a}\_0 \varepsilon \tag{7}$$

where *L* is the inductance of the sensor, *α*<sup>0</sup> is the stretching coefficient, *L*<sup>0</sup> is the non-stretched inductance, and *ε* is the applied strain. As the sensor consisted of an LC circuit to resonate at a specific frequency, the resonant frequency *f* of which is calculated as:

$$f \approx \frac{1}{2\pi\sqrt{(L\_0 + \alpha\_0\varepsilon)C}}\tag{8}$$

where *C* is the additional tuning capacitance.
