With parallel split-gap orientation of SRR

Resorting to single-mode approximation, a lumped-element equivalent circuit (as in Ref. [Reference Bojanic, Milosevic, Jokanovic, Medina-Mena and Mesa13]) in Fig. 4 is considered to calculate the resonant mode frequency of this coupled system.

Fig. 4.

Equivalent circuit model of coupled SRR and microstrip line is shown.

The Lagrangian formalism for this case runs as follows:

In view of the high Q-value (≥10) of the device, and so, a small rate of energy dissipation and radiation in the microwave range, the kinetic energy term including magnetic interaction of the oscillators can be written as (using Refs [Reference Li, Liu, Li, Wang, Cao, Zhu, Dong, Zhu and Zhang15–Reference Jing, Guang-Yong and Wei-Ren17])

(5)$$T_M = \displaystyle{1 \over 2}L\dot{q}_2^2 + \displaystyle{1 \over 2}L_S\dot{q}_4^2 + M_H\dot{q}_2\dot{q}_4, \;$$

where the first two terms are energy stored in the inductors and the last term, having M _H, the mutual inductance, represents the energy associated with magnetic interaction. This quantity, M _H, can be positive or negative.

Again, the potential energy of this coupled system is

(6)$$V_E = \displaystyle{1 \over 2}\displaystyle{{q_1^2 } \over {{{C_d} \over 2}}} + \displaystyle{1 \over 2}\displaystyle{{q_3^2 } \over {{{C_d} \over 2}}} + \displaystyle{1 \over 2}\displaystyle{{q_4^2 } \over {C_c}}-V_1( q_1 + q_2) + V_2( q_2-q_3) , \;$$

where C _c and C _d are, respectively, the effective capacitance of the coupled SRR and microstrip line. The capacitance of a charged conductor increases if a metal sheet or strip is brought near it. This happens on account of the lowering of the potential of the charged conductor due to the induced charges of opposite sign on the strip. We may write C _c = C _S + C _SRR−MS; the change (C _SRR−MS) is roughly proportional to the inverse cube of the distance between SRR and MTL, and may also involve higher inverse-powers. A similar expression can be written for C _d. Basically, the distance between the two conductors is the distance between the split-gap and the microstrip line. The mutual inductance between the SRR and the MTL, on the other hand, varies inversely as (and also, the next few higher inverse-powers of) the distance between them since the magnetic field over the ring, due to the current in the MTL, varies roughly in that manner. Variation of resonance frequency can be explained on the basis of the natures of variation of C _SRR−MS and M _H.

Now, the Lagrangian for the system can be written as

(7)$$\eqalign{\Im _E & = \displaystyle{1 \over 2}L\dot{q}_2^2 + \displaystyle{1 \over 2}L_S\dot{q}_4^2 + M_H\dot{q}_2\dot{q}_4-\displaystyle{1 \over 2}\displaystyle{{q_1^2 } \over {{{C_d} \over 2}}}-\displaystyle{1 \over 2}\displaystyle{{q_3^2 } \over {{{C_d} \over 2}}}-\displaystyle{1 \over 2}\displaystyle{{q_4^2 } \over {C_c}} + V_1q_1 \cr & \quad + V_2q_3 + q_2( V_1-V_2)}.$$

The Euler–Lagrange equations of this coupled system are

(8)$$\displaystyle{d \over {dt}}\left({\displaystyle{{\partial \Im_E} \over {\partial {\dot{q}}_i}}} \right)-\displaystyle{{\partial \Im _E} \over {\partial q_i}} = 0\,\;( {{\rm where}\,i{\rm} = {\rm 1, \;\ 2, \;\ 3, \;\ 4}} ).$$

From (7) and (8), we obtain

(9)$$q_1 = \displaystyle{{C_dV_1} \over 2}, \;$$

(10)$$L\ddot{q}_2 + M_H\ddot{q}_4-( V_1-V_2) = 0, \;$$

(11)$$q_3 = \displaystyle{{C_dV_2} \over 2}, \;$$

and

(12)$$L_S\ddot{q}_4 + M_H\ddot{q}_2 + \displaystyle{{q_4} \over {C_c}} = 0.$$

From (10) and (12), we get

(13)$$L_S\ddot{q}_4 + M_H\left({\displaystyle{{-M_H{\ddot{q}}_4} \over L}} \right) + \displaystyle{{q_4} \over {C_c}} + \displaystyle{{M_H} \over L}( V_1-V_2) = 0.$$

From (13), it is observed that when V ₁ = V ₂ (suggesting band pass filtering action) the SRR loaded microstrip line can be described by an oscillatory equation for the charge q ₄:

(14)$$\ddot{q}_4 + \displaystyle{{q_4} \over {C_c\left({L_S-\displaystyle{{M_H^2 } \over L}} \right)}} = 0.$$

So, the resonance mode frequency is

(15)$$f_\parallel{ = } \displaystyle{1 \over {2\pi \sqrt {L_SC_c( {1-k_m^2 } ) } }}.$$

Here, $k_m\left({ = M_H/\sqrt {LL_S} } \right)$ is the magnetic coupling co-efficient between microstrip line and SRR.

Note that when the distance between the split-gap and the microstrip is small, the variation of C _SRR−MS with distance (rapid rise with decreasing distance) plays the dominating role in deciding how the resonance frequency changes. However, when the distance is large, the variation of M _H determines how resonance frequency changes.

With perpendicular split-gap orientation of SRR

An equivalent circuit (as in Ref. [Reference Bojanic, Milosevic, Jokanovic, Medina-Mena and Mesa13]), for the configuration having the split-gap of the SRR in perpendicular orientation with respect to the MTL, is shown in Fig. 5. The perpendicular component of the electric field of the MTL is parallel to the split-gap which causes a strong electric coupling. In the equivalent circuit model, C_m (called coupling capacitance) comes from this additional electric coupling between the microstrip line and the SRR.

Fig. 5.

Equivalent circuit model, for a loaded microstrip line with vertical split-gap SRR, is shown.

Now, to calculate the resonance frequency, we again consider the Lagrangian formalism. Assuming no dissipation of energy as for the parallel split-gap orientation of SRR, the kinetic energy (using Refs [Reference Li, Liu, Li, Wang, Cao, Zhu, Dong, Zhu and Zhang15–Reference Jing, Guang-Yong and Wei-Ren17]) is

(16)$$T_M = \displaystyle{1 \over 2}L\dot{q}_1^2 + \displaystyle{1 \over 2}L_S\dot{q}^2_2 + M_H\dot{q}_1\dot{q}_2.$$

Also, the electrostatic energy is

(17)$$ \!\!\!\!\!\!V_E = \displaystyle{1 \over 2}\displaystyle{{{( q-q_1) }^2} \over {C_m}} + \displaystyle{1 \over 2}\displaystyle{{{( q-q_1-q_2) }^2} \over {C_c-C_m}}-( V_1-V_2) q_1-V_1( q-q_1).$$

Combining (16) and (17), and using the Euler–Lagrange formalism, the equations of motion are found to be

(18)$$L\ddot{q}_1 + M_H\ddot{q}_2-\displaystyle{{q-q_1} \over {C_m}}-\displaystyle{{q-q_1-q_2} \over {C_c-C_m}} + V_2 = 0, \;$$

(19)$$L_S\ddot{q}_2 + M_H\ddot{q}_1-\displaystyle{{( q-q_1-q_2) } \over {( C_c-C_m) }} = 0, \;$$

and

(20)$$\displaystyle{{( q-q_1) } \over {C_m}} + \displaystyle{{( q-q_1-q_2) } \over {C_c-C_m}}-V_1 = 0.$$

The condition for the resonance can be expressed as having nontrivial solutions for q and q ₂ when the output voltage at port B, V ₂ = 0 (i.e. q ₁ = 0 at all times; hence, $\dot{q}_1 = \ddot{q}_1 = 0$.) So, our equations reduce to

(21)$$M_H\ddot{q}_2-\displaystyle{q \over {C_m}}-\displaystyle{{q-q_2} \over {C_c-C_m}} = 0, \;$$

(22)$$L_S\ddot{q}_2-\displaystyle{{( q-q_2) } \over {( C_c-C_m) }} = 0, \;$$

and

(23)$$\displaystyle{q \over {C_m}} + \displaystyle{{( q-q_2) } \over {C_c-C_m}}-V_1 = 0.$$

From (23), we have

(24)$$\displaystyle{{( q-q_2) } \over {C_c-C_m}} = \displaystyle{{C_m} \over {C_c}}.\left({V_1-\displaystyle{{q_2} \over {C_m}}} \right).$$

Moreover, using (23) in (21) we get

(25)$$M_H\ddot{q}_2 = V_1.$$

Again, using (25), (23), and (24), we get

(26)$$\left({L_S-\displaystyle{{C_m} \over {C_c}}.M_H} \right)\ddot{q}_2 + \displaystyle{{q_2} \over {C_c}} = 0.$$

This leads to the resonance mode frequency of the coupled system

(27)$$\eqalign{\,f_\bot = & \displaystyle{1 \over {2\pi \sqrt {L_SC_c-C_mM_H} }} \cr {\rm} = & \displaystyle{{\,f_0} \over {\sqrt {1 + \displaystyle{{( {L_SC_{SRR-MS}-C_mM_H} ) } \over {L_SC_S}}} }}.} $$

The magnitude of ${{( {L_SC_{SRR-MS}-C_mM_H} ) } \over {L_SC_S}}$ is the square of the effective coupling co-efficient of this coupled system, and (27) can be re-written as

(28)$$f_\bot = \displaystyle{{\,f_0} \over {\sqrt {( {1 \pm k_E^2 } ) } }}.$$

In (28), the positive sign applies when L _SC _SRR−MS > C _mM _H and the negative sign applies when L _SC _SRR−MS < C _mM _H. The quantity k _E is the effective coupling co-efficient between the microstrip line and the SRR for perpendicular split-gap orientation of SRR. So, the microstrip line may show a lower frequency resonance mode depending on the sign of C_m when coupled with an SRR with perpendicular split-gap. As V ₂ = 0, this is the band stop or notch configuration. Note that there will be no effect on the resonance frequency if, for the perpendicular orientation of the gap, the split-gap appears in the opposite arm of the ring.

Variation with substrate height for a particular side-length of a square loop (we have taken l = 3 mm)

At first, we show in Fig. 6, the resonance mode frequencies as obtained for different coupling distances, which is the separation (denoted by s) between the nearest edges of an SRR and MTL, for the substrate height (h) of 0.65 and 1.27 mm. Also, the fundamental simulated resonance frequency of uncoupled SRR is shown for both cases in the mentioned figures.

Fig. 6.

Variations of resonance mode frequency with edge-to-edge separation between MTL and SRR, at a substrate height of (a) 0.65 mm, and (b) 1.27 mm, are shown for near, far, and vertical split-gap orientation of SRR.

It is observed that when the split-gap of the SRR is on the distant parallel arm, the coupled system shows a higher frequency resonance mode which decreases with the increase of separation. But, the resonance mode frequency increases with the increase in separation distance when the split-gap of the SRR is on the nearer parallel arm and also, with the perpendicularly oriented split-gap of SRR. Again, for the last two split-gap orientations, a lower frequency resonance mode occurs at shorter distances and the higher frequency resonance mode is obtained at larger separations. Due to a decrease of mutual inductance, beyond a certain distance, the mode frequencies settle down to the fundamental resonance frequency of uncoupled SRR at the substrate height of 1.27 mm (see Fig. 6(b)). But, for the decreased substrate height of 0.65 mm, the resonance mode frequency for SRR, with parallel split-gap near to the line and also for SRR with perpendicularly-oriented split-gap, increases above the fundamental resonance frequency of uncoupled SRR (i.e. 6.288 GHz) with an increase of separation from 0.3 to 0.7 mm as shown in Fig. 6(a). This happens due to the fact that the coupling capacitance (i.e. C_m) becomes negative (the current through it has reversed direction) and its magnitude increases after a certain separation gap between the MTL and the SRR. The variations of coupled mode frequency, as noted above, are consistent with theoretical expectations as outlined at the end of sub-section “Variation with substrate height for a particular side-length of a square loop (we have taken l = 3 mm)”.

Now to calculate the value of mutual inductance (M _H) between MTL and SRR for various distances, we may use (15) for parallel split-gap far from the line. At first, the value of C _SRR−MS is assumed to be negligible with respect to C _S. Then, the resonance frequency of this coupled system can be written as

(29)$$\eqalign{& f_{{\parallel} \_f}\approx \displaystyle{1 \over {2\pi \sqrt {L_SC_S( {1-k_m^2 } ) } }} \cr & \approx \displaystyle{{\,f_0} \over {\sqrt {\left({1-\displaystyle{{M_H^2 } \over {LL_S}}} \right)} }}, \;} $$

where L is the inductance of the uncoupled MTL. The inductance of an MTL can be written as [Reference Paul21]

(30)$$L = \displaystyle{{60l_1} \over v}\ln \left[{\displaystyle{{8h} \over W} + \displaystyle{W \over {4h}}} \right]{\rm for\ }\displaystyle{W \over h} \le 1$$

or,

(31)$$L = \displaystyle{{120\pi l_1} \over v}\left[{\displaystyle{1 \over {\displaystyle{W \over h} + 1.393 + 0.667\ln \left({\displaystyle{W \over h} + 1.444} \right)}}} \right]{\rm for\ \displaystyle{W \over h} > 1, \;}$$

where l ₁ and W are the length and width of the MTL, respectively. Also, v is the free space light velocity. Again, the self-inductance of a single gap square-shaped SRR can be written as [Reference Samanta, Pradhan and Syam22]

(32)$$ L_S = \displaystyle{{2\mu _0} \over \pi }\left[{\ln ( l ) -\ln ( t ) + \left\{{\sqrt 2 + 2\ln 2-\ln \left({\sqrt 2 + 1} \right)-2} \right\}} \right]l, \;$$

where l and t are the side-length and width of the SRR, respectively. Further the capacitance of the single gap square-shaped SRR may be written as [Reference Samanta, Pradhan and Syam22]

(33)$$C_S = \displaystyle{{\varepsilon _0\varepsilon _gd.t} \over b} + \varepsilon _0\varepsilon _g( {d + b + t} ) + 2( d + t) \varepsilon _sC_{surf}^{\,pul} , \;$$

where d and b are the thickness and split-gap of the SRR. Also, ɛ_g is the effective dielectric constant of the substrate (on which the SRR is fabricated) for the capacitance associated with the split-gap and ɛ_s is the corresponding quantity for the capacitance associated with the side-surface of the ring.

To determine the analytical expression of C _S, the height of the dielectric substrate was considered to be infinite; in reality, this height is finite. Also, the effect of the metal sheet (on which the substrate is deposited) has not been considered for finding L _S and C _S. So, to rectify the error (i.e. in order to find the actual uncoupled resonance frequency from the analytical value), it is necessary to calibrate an SRR against simulated resonance frequency, for different substrate heights. For three different sides length of SRR (keeping fixed the other geometrical parameters), the calibration factor (=ratio of simulated SRR resonance frequency to analytical SRR resonance frequency) is plotted in Fig. 7 for various substrate heights. It is observed that for the side-length of 3 mm and the substrate height of 1.27 mm, the simulated SRR resonance frequency is close to analytical SRR resonance frequency.

Fig. 7.

Variation of calibration factor with substrate height is shown for three different values of side-length keeping fixed the other geometrical parameters of the coupled system.

Now, using simulation data for $f_{{\parallel} \_f}$ and f ₀, and putting the analytical values of L and L _S, we can evaluate the apparent value of M _H for different intermediate distances (i.e. the distance of mid-point of the loop from the line which is assumed to be d ₂ = (s + l/2), where l is the side-length of SRR loop). Plotting M _H versus (s + l/2) graph and fitting the nature of M _H by $b_1/d_2 + b_2/d_2^2 + b_3/d_2^3 + b_4/d_2^4$, the coefficients b ₁, b ₂, b ₃, and b ₄ could be determined. After that, putting the value of M _H, the values of C _SRR−MS at different separation gaps (i.e. where d ₁ = (s + t/2), t is the width of the SRR), could be calculated from resonance frequency data for SRR with near parallel split-gap, given by

(34)$$f_{{\parallel} \_n} = \displaystyle{{\,f_0} \over {\sqrt {\left({1 + \displaystyle{{C_{SRR-MS}} \over {C_S}}} \right)\left({1-\displaystyle{{M_H^2 } \over {LL_S}}} \right)} }}.$$

Next, plotting C _SRR−MS versus (s + t/2) for near parallel split-gap orientation, and fitting the nature of C _SRR−MS by $a_1/d_1^3 + a_2/d_1^4$, we can work out approximate values of the two constants a ₁ and a ₂. Now, the contribution of C _SRR−MS may be considered in (15) for parallel split-gap far from the line, when the separation gap is to be considered as d ₃ = (s + l − t/2). After that, a revised value of M _H could be determined from (34) and thereafter the constants b ₁, b ₂, b ₃, and b ₄ could be found properly from the plot of M _H versus (s + l/2). Also, plotting k _m versus (s + l/2) graph and fitting the nature of k _m by $c_1/d_2 + c_2/d_2^2 + c_3/d_2^3 + c_4/d_2^4$, the coefficients c ₁, c ₂, c ₃, and c ₄ could be determined. Once more, a plot of C _SRR−MS versus (s + t/2) would have to be made and fitting the data for C _SRR−MS, we can determine improved values of the constants a ₁ and a ₂.

At first, for the substrate height of 0.65 mm, we have calculated the values of M _H, k _m, and C _SRR−MS at various distances from MTL and plotted these in Figs 8(a)–8(c), respectively. Moreover, considering the analytical values of C _S and C _SRR−MS at vertical split-gap with different separation gaps, and using (27), the C _m may be calculated for different distances. Also, the value of electric coupling co-efficient (k _e) between C and C _S can be calculated for different separation gaps.

Fig. 8.

Variations with distance of (a) mutual inductance, (b) magnetic coupling coefficient, and (c) the change in SRR capacitance are plotted for SRR at a side-length of 3 mm and a substrate height (h) of 0.65 mm.

The MTL capacitance per unit length (C ₁) may be written as [Reference Paul23]

(35)$$C_1 = \displaystyle{{0.67( {1.41 + \varepsilon_r} ) } \over {\ln \left\{{\displaystyle{{5.98h} \over {0.8W + d}}} \right\}}}, \;$$

where d is the thickness of the MTL which is identical to the thickness of the SRR. Also, ɛ_r is the dielectric constant of the substrate.

A list of analytical and simulated parameters is given in Table 1 for the substrate heights of 0.65 and 1.27 mm.

Table 1.

List of parameters to draw Figs 8 and 9

Fig. 9.

For MTL ,with SRR having a side-length of 3 mm and a dielectric substrate height of 1.27 mm, the (a) mutual inductance, (b) magnetic coupling co-efficient, and (c) the change in SRR capacitance.

Similar calculations have also been made for the substrate height of 1.27 mm and the corresponding plots are shown in Figs 9(a)–9(c). With perpendicular orientation of SRR split-gap, the mode frequency of the coupled MTL is above the uncoupled SRR resonance frequency for edge-to-edge separation between MTL and SRR of 0.05–0.7 mm. The list of calculated values of constants from Figs 9(a)–9(c) is given in Table 2 for two different substrate heights.

Table 2.

List of fitted constants for a split-ring with a side-length of 3 mm and the substrate heights of 0.65 and 1.27 mm

Variation with side-length of a square loop for a particular substrate height (h = 1.27 mm)

In this sub-section, Figs 10(a) and 10(b) are drawn to display the effect of coupling for different separations (i.e. s) between the nearest edges of the SRR and MTL for a particular substrate height of 1.27 mm, and SRR side-lengths of 3.5 and 4.0 mm. It is observed that the uncoupled resonance frequency of SRR decreases with an increase of side-length and also, the resonance mode frequency of MTL decreases at different separation distances.

Fig. 10.

Variations of resonance mode frequency of MTL, with separation from SRR, are shown for near, far, and vertical split-gap configurations, for SRRs with a side-length of 3.5 mm (Fig. 10(a)) and 4 mm (Fig. 10(b)) when substrate height is 1.27 mm.

For the SRR with a side-length of 3.5 mm, numerical calculations have been carried out to study the variations of different parameters such as M _H, k _m, and C _SRR−MS as shown in Figs 11(a)–11(c), as functions of separation distances. The results of similar calculations for the SRR of side-length of 4 mm are shown in Figs 12(a)–12(c).

Fig. 11.

Variations, with separation distance, of (a) mutual inductance, (b) magnetic coupling co-efficient, and (c) C _SRR−MS, are plotted for MTL with SRR having a side-length of 3.5 mm and a substrate height (h) of 1.27 mm.

Fig. 12.

Plots of (a) mutual inductance, (b) magnetic coupling co-efficient, and (c) C _SRR−MS are shown for different separation distances for MTL with SRR having a side-length of 4 mm and a substrate height (h) of 1.27 mm.

In Table 3 are shown the fitted constants for the SRR having a side-length of 3.5 and 4 mm, respectively. Other analytical and simulation parameters are presented in Table 4 for this coupled system for a substrate height of 1.27 mm. Again, the calculated value of C _m has been shown for different distances in Fig. 13 (substrate height = 1.27 mm) for a side-length of 3.0 and 3.5 mm. It is observed that the coupling capacitance decreases with distance and it shows larger value for SRR of a side-length of 3.5 mm. The C _m is not considered at higher edge-to-edge separation between MTL and SRR due to un-physical (negative) result in C _m value. For a side-length of 4 mm, C _m is positive over a very small range of SRR to MTL distance. This is the reason why we have no data corresponding to a side-length of 4 mm in Fig. 13.

Fig. 13.

The coupling capacitance is plotted for different separation distances between MTL and SRR, with SRR having a side-length of 3 and 3.5 mm.

Table 3.

List of fitted constants for a side-length of 3.5 and 4 mm for a substrate height of 1.27 mm

Table 4.

List of parameters to draw Figs 11 and 12

We can also calculate the electric coupling coefficient (k _e) between C and C _S at perpendicular split-gap orientation of SRR with respect to the MTL.

For our calculations, the analytical values of L _S and C _S are considered. But, the simulated resonance frequency of uncoupled SRR is used to calculate M _H, k _m, C _SRR−MS, and C _m. The analytical resonance frequencies are 6.557, 5.748, and 5.145 GHz for SRR side-length of 3, 3.5, and 4 mm, respectively, which differ from the simulated values, i.e. 6.404, 5.284, and 4.410 GHz, respectively.

The list of other geometrical parameters which were used in simulations appears in Table 5. In our FW simulation, the Rogers (RO3010) has been assumed to be the substrate with a dielectric constant ɛ_r = 10.2.

Table 5.

List of geometrical parameters of the coupled system

To compare with analytical results in [Reference Bojanic, Milosevic, Jokanovic, Medina-Mena and Mesa13], our analytical formula has been used to determine magnetic coupling co-efficient (i.e. k _m) and electric coupling co-efficient (i.e. k _e) for the given geometrical parameters in that reference. It is observed that our analytical value for k _m and k _e match well with the analytical results in that reference. The variation may be attributed to the difference in simulated resonance frequencies due to the use of different software. These are shown with the geometrical parameters in Table 6. The magnetic coupling coefficient between MTL and SRR, and the coupling capacitance between MTL capacitance and SRR capacitance for perpendicular split-gap orientation are shown in Table 6. In that table, the geometrical parameter values shown are the same as in Ref. [Reference Bojanic, Milosevic, Jokanovic, Medina-Mena and Mesa13]. In [Reference Bojanic, Milosevic, Jokanovic, Medina-Mena and Mesa13], the MTL inductance and capacitance, and also, the SRR inductance and capacitance have been found under coupled condition for a particular separation distance. The line parameters (i.e. L, C) have been obtained using LINPAR software under coupled condition. Then, the magnetic coupling co-efficient has been calculated using closed-form expressions for the resonance frequency and the minimum reflection frequency. To calculate the coupling capacitance, and also to find both sets of coupling coefficients at different coupling distances, it was necessary to repeat the analysis process. But, in our method, we do not need to repeat the analysis. Just putting the separation distance in the fitting formula and using appropriate coefficient values, we can find the coupling capacitance and the coupling coefficients.

Table 6.

List of geometrical parameters of the coupled system in [Reference Bojanic, Milosevic, Jokanovic, Medina-Mena and Mesa13] and comparison between analytical results

Using our parameters, we have calculated the Q-factor of this coupled system taking 1.27 mm as the substrate height and the edge-to-edge separation between SRR and MTL of 0.1 mm which is given in Table 7. The Q-factor is defined as f ₀/Δf, where Δf is the bandwidth at −3 dB and also, f ₀ is the center frequency. It is seen that the Q-factors are almost similar (about 22) for parallel split-gap of SRR far from the line and perpendicular split-gap SRR. But, for parallel split-gap of SRR near to the line, it is very high (about 100).

Table 7.

List of Q-factors with variation of side-length for different split-gap orientation.