By connecting terminal B₁ to A₂, two coupled MCs can be structured as a serial circuit with identical polarities as shown in Fig. 2. By using Kirchhoff’s Voltage Law (KVL), the voltage and current across A₁ and B₂ can be written as 5a 5b By integrating both sides of Eqs. (5a) and (5b), we have 6a 6b Based on Eqs. (1), (4), (5), and (6), the following set of differential equations for describing this serial circuit can be obtained: 7a 7b Since equation (7) is too complex to be resolved by using analytical methods, numerical methods are generally used to calculate φ₁ and φ₂, as well as the MCA of each MC.

Fig. 2.

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	Fig. 2. Coupled MCs serially connected with identical polarities.

For the special case of α₁ = α₂ = κ₁ = κ₂ = α, which can be precisely achieved by physical fabrication, equation (7) can be simplified into 8a 8b Assuming that the initial values of φ₁ and φ₂ are both zero, the analytical expressions for φ₁ and φ₂ can be obtained as follows: 9a 9b Hence, the MCA of individual MC can be obtained by substituting Eq. (9) into Eq. (4). For further simplification we assume that the initial MCA values β₁ and β₂ are both equal to β, equation (9) can be further simplified into 10 According to Eqs. (4) and (10), the MCA of each MC can be expressed as 11 Based on the preceding analysis, it can be observed that two coupled MCs serially connected with identical polarities can be operated also as the MC but with new MCA value 12

For the special case of α₁ = κ₂ = 0, these two MCs are in fact operated in serial connection without coupling effect. Besides, as α₁ = α₂ = α and β₁ = β₂ = β, the values of C_m1 and C_m2 can be expressed as 13

By connecting B₁ to B₂, the diagram of two coupled MCs in serial connection with opposite polarities is shown in Fig. 3, where the excitation voltage v₁₂ is applied to terminals A₁ and A₂, while terminal T is floating. The voltage, flux, current and charge of this circuit can be expressed by 14a 14b Therefore, the following set of differential equations can be obtained: 15a 15b Like Eq. (7), neither of the analytical solutions for φ₁ and φ₂ can be obtained from equation (15). By taking the special case of α₁ = α₂ = κ₁ = κ₂ = α into consideration, equation (15) can be simplified into 16a 16b Subsequently, C_m1 and C_m2 can be obtained by substituting Eq. (16) into Eq. (4), and the equivalent MCA can be expressed as C_m1C_m2/(C_m1 + C_m2). For the case of κ₁ = κ₂ = 0, these two coupled MCs in serial connection with opposite polarities are in fact operated without coupling effects. Furthermore, as α₁ = α₂ = α and β₁ = β₂ = β, C_m1 and C_m2 can be calculated from 17a 17b

Fig. 3.

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	Fig. 3. Coupled MCs serially connected with opposite polarities.

For further analyzing the dynamic behavior of this coupled MC circuit, a normal capacitor is serially connected with this coupled MC circuit as shown in Fig. 3, where v₁₂ represents the voltage across terminals T and A₂ instead of A₁ and A₂. The voltage v₁₂, flux φ₁₂, current i as well as charge q can be represented by 18a 18b From Eqs. (1), (4), and (18), we can derive the following equations: 19a 19b where 20 From Eq. (19), it can be clearly disclosed that even a linear capacitor connected with this coupled MC circuit can result in high complexities in resolving flux and MCA. For the special case of α₁ = α₂ = α, κ₁ = κ₂ = κ, and β₁ = β₂ = β, we have 21a 21b 21c where C_m is the MCA of two serially connected MCs. In this simplified case, the coupled MC circuit connected in series with opposite polarities evidently behaves as a regular capacitor with a capacitance value of β/2.

The memcapacitive circuit of two coupled MCs in parallel connection with identical polarities is shown in Fig. 4, of which the current going through terminals A₁ and B₂, the charge, voltage and flux can be expressed by 22a 22b Hence, we have 23 By combining Eqs. (1) and (23), the total MCA of the dual coupled MCs in parallel connection can be calculated from 24 Hence, it can be deduced from Eq. (24) that the flux coupled MCs in parallel connection can behave as a new flux-controlled MC, and the equivalent MCA value is equal to the sum of two individual MCAs.

Fig. 4.

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	Fig. 4. Coupled MCs connected in parallel with the same polarities.

The diagram of two coupled MCs connected in parallel with opposite polarities is shown in Fig. 5, the current, charge, voltage and flux can be shown by 25a 25b Based on Eq. (1) and (25a), it can be derived that 26 By combining Eq. (1) with Eq. (26), the total MCA of the coupled MCs in parallel connection can be calculated by 27 Equation (27) reveals that the variation of MCA in terms of flux φ₁₂ is dependent on the difference between α₁ + κ₁ and α₂ + κ₂. For the case of α₁ + κ₁ > α₂+κ₂, the MCA is in direct proportion to the excitation flux φ₁₂, and higher flux will result in greater MCA. Likewise, for the case of α₁ + κ₁ < α₂ + κ₂, the MCA is inversely proportional to flux φ₁₂, and the MCA decreases with the increase of flux. For the special case of α₁ + κ₁ = α₂ + κ₂, this coupled MC is operated as a linear capacitor with the capacitance of β₁ + β₂ independent of flux φ₁₂.

Fig. 5.

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	Fig. 5. Coupled MCs connected in parallel with opposite polarities.

Based on Eqs. (24) and (27), it can be observed that for the uncoupled case of κ₁ = κ₂ = 0, the parallel connected MCs are also operated as a new MC but the MCA variation differs from the parallel MCs with coupling effect.

In order to further evaluate the coupled-MC-based parallel circuit, a common capacitor is serially connected with this dual coupled MC as shown in Fig. 5. The excitation voltage v₁₂ is now imposed on terminals T and A₂ instead of A₁ and A₂. According to KVL and KCL, the following set of equations can be deduced: 28a 28b 28c where v_C is the voltage across capacitor C_s. From Eq. (27) it can be deduced that the MCA of the two coupled MCs is 29 By combining Eqs. (26) and (28a), we have 30 Based on Eqs. (28a) and (30), the following equation holds: 31 By referring to Eq. (4) and integrating both sides of Eq. (31), the following equations can be obtained: 32a 32b Therefore, the equivalent MCA across terminals T and A₂ can be analytically expressed as 33 It can be concluded from Eq. (33) that when the coupled MCs are serially connected with a capacitor, the MCA value of the circuit is only dependent on the flux φ₁₂ (or voltage v₁₂) across the terminals T and A₂.

It is worth noting that the coupling coefficients κ₁ and κ₂ can be positive or negative. Positive coefficient indicates coupling reinforcement while the negative one manifests counteraction of coupling effects.

The coupled MCs connected in series with opposite polarities are used to structure an RO as shown in Fig. 6. According to input and output performance of the operational amplifier U1, we have 34 Based on Eqs. (4) and (34), the set of system equations for describing the RO can be expressed by 35 By setting α₁ = α₂ = κ₁ = κ₂ = α, the equivalent MCA can be expressed by 36 where 37

Fig. 6.

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	Fig. 6. Coupled MCs connected in series with opposite polarities and the RO.

Equation (35) displays the variation relationship between v_o and v_A1A2. By adjusting the coupling coefficients κ₁ and κ₂, the output voltage v_o can be purposely controlled. However, equation (35) is very complex and can hardly be solved analytically. Normally, the numerical method is suggestively employed to inspect the inherent dynamic behaviors and output performance of the RO. For the special case of α₁ = α₂ = α, β₁ = β₂ = β, and κ₁ = κ₂ = 0, the calculations of C_m1 and C_m2 can be simplified and expressed by Eq. (17), then the set of system equations for describing the RO can be simplified into 38 In this case, with κ₁ = κ₂ = 0, the output voltage v_o is only controlled by v_A1A2.

The coupled MCs in parallel connection with opposite polarities are then used to build the RO, and the schematic diagram is shown in Fig. 7. Based on Eqs. (35), (36), and (27), the system functions of the coupled-MCs-based RO can be shown by 39 where v_A1B1 is equivalent to −v_A2B2 and v_o is controlled by v_A1B1. Meanwhile, by changing the values of κ₁ and κ₂ the output voltage of the RO can be controlled and operated inside the target range. For the uncoupled case of κ₁ = κ₂ = 0, the parallel MCs can also be operated as another MC with the MCA different from the parallel MCs with coupling effects. Also, the output voltage v_o is only determined by v_A1B1 and we have 40 Equation (40) possesses high nonlinearities and can hardly be analytically solved either. However, the voltage v_A1B1 across MC1 can be calculated numerically.

Fig. 7.

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	Fig. 7. Coupled MCs connected in parallel with opposite polarities and the RO.

The MCA of the serially connected MC circuit can be expressed as 41

With regard to the case of serial connection with identical polarities, the simulation parameters are configured as α₁ = α₂ = 4.914 μF/Wb, β₁ = β₂ = 0.5528 μF, κ₁ = 0.2κ₂, and κ₂ = κ₁. A sinusoidal voltage of v_A1B2(t) = 3sin(20πt) V is adopted to excite this coupled-MC-based circuit. Based on Eq. (9), for consistence with theoretical analysis, the initial condition of φ₁(0) = φ₂(0) = 0 Wb is considered in the remaining simulation studies. Three curves of voltage versus charge corresponding to two individual MCs and the coupled MC in serial connection are shown in Fig. 8(a). It can be seen that the serially connected coupled MC circuit also possesses typical memcapacitive characteristics with pinched hysteresis loops (PHLs) behaving as an inclined “8”. The voltage across MC1 with the amplitude of 1.028 V is clearly less than the voltage across MC2, which indicates that the MCA of C_m1 is time variant and greater than C_m2.

Fig. 8.

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	Fig. 8. (color online) Simulation results of serial connection. (a) PHLs; (b) Curves of MCA varying with current.

Three curves of MCA varying with current are shown in Fig. 8(b), which demonstrates that the MCA of the dual coupled MCs in serial connection is less than the individual MCAs C_m1 or C_m2. The MCAs increases with the increase of the current passing through them. The variations of intervals C_m1, C_m2, and C_m12 are measured to be [0.634,1.013], [0.329,0.529], and [0.217,0.347] μF, respectively.

For observing the influence of coupling coefficients on MCA C_m12, different values of coupling coefficient κ₂ are chosen. Curves of MCA varying with flux under the condition of different values of coupling coefficient κ₂ are shown in Fig. 9, from which we can see that the MCA C_m12 can increase from the initial value 0.093 μF with the increase of flux under the condition of coefficients κ₂ varying from 0 to 2.5α₁. The curve slope of C_m12 versus φ₁₂ can also increases with the increase of coefficient κ₂.

Fig. 9.

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	Fig. 9. (color online) Curves of MCA varying with flux under the conditions of different values of coupling coefficient κ2.

For the connection condition with opposite polarities, the simulation parameters are configured as α₁ = α₂ = 0.1659 μF/Wb, β₁ = β₂ = 1.269 μF, κ₁ = 0.5α₂, and κ₂ = 0.5α₁. A sinusoidal voltage of v_A1A2(t) = 3sin(20πt) V is used for exciting the coupled MC circuit. Three curves of voltage versus charge corresponding to two individual MCs and the coupled MC circuit are shown in Fig. 10(a). Like the case of identical polarities, the two serial MCs also behave together as a new MC with PHL passing through the origin. The MCAs of C_m1, C_m2, and C_m12 in terms of current are shown in Fig. 10(b). Unlike the case of identical polarities, the C_m1 can be nonlinearly increased by increasing the current, and the C_m2 can nonlinearly decrease, while the MCA C_m12 first increases and then decreases. For the maximal charge quantities, the values of C_m1, C_m2, and C_m12 are 1.475, 1.075, and 0.622 μF, respectively. Figure 11 shows the variation characteristics of MCA against flux, with κ₁ changing from 0.039α₁ to α₁. Obviously, the curve of C_m12 versus φ₁₂ first increases and then decreases when κ₁ is equivalent to 0.039α₂, 0.39α₂, and 0.5α₂, respectively, and then values of C_m12 reach maximal values of 0.6224, 0.6209, and 0.6196 μF respectively when φ₁₂ = −0.0193 Wb.

Fig. 10.

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	Fig. 10. (color online) Simulation results of serial connection. (a) PHLs; (b) Curves of MCA varying with current.

Fig. 11.

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	Fig. 11. (color online) Curves of MCA varying with flux under the condition of different values of coupling coefficient κ1.

According to the preceding analysis, the voltage across each MC also periodically evolves but no longer sinusoidally due to nonlinear variation of MCA. The curves of MCAs C_m1, C_m2, and C_m12 varying with terminal voltage v₁, v₂, and v₁₂ are displayed in Fig. 12 for comparatively analyzing the influences from different coupling coefficients. The curve of C_m1 versus v₁ evolves anticlockwise, which means that positive v₁ results in the increase of C_m1 and negative v₁ leads to the decrease of C_m1. The curves of C_m2 versus v₂ and C_m12 versus v₁₂ evolve clockwise, corresponding to the alternating voltages v₂ and v₁₂, respectively. In consideration of the symmetry, only the positive interval is needed for demonstration. When two MCs are positively coupled with κ₁ = 0.5α₂ and κ₂ = 0.5α₁, C_m12 first increases and then decreases with the positive voltage increasing as shown in Fig. 12(a). However, C_m12 is monotonically decreased and the variation range of C_m12 can be comparatively widened by κ₁ = 0.5α₂ and κ₂ = α₁ as shown in Fig. 12(b).

Fig. 12.

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	Fig. 12. (color online) Curves of MCA varying with terminal voltage. (a) κ1 = 0.5α2 and κ2 = 0.5α1; (b) κ1 = 0.5α2 and κ2 = α1.

A regular capacitor of 330 nF is serially connected with the coupled MC circuit for further discovering the dynamic behaviors and connectivity of this serial memcapacitive circuit. Curves of MCA C_m varying with φ_m for different values of coupling coefficient κ₁ but constant coupling coefficient κ₂ (κ₂ = 0.5α₁) are shown in Fig. 13(a). It reveals that κ₁ ≥ 0.66α₂C_m can nonlinearly increase with the increase of flux φ_m, while 0.39α₂ ≤ κ₁ < 0.66α₂, C_m can first increase and then decrease with the increase of flux φ_m. Evidently, for κ₁ ≤ 0.39α₂, C_m can nonlinearly decrease with the increment of flux φ_m. The enlarged curve of κ₁ = 0.5α₂ is shown in Fig. 13(b) for demonstration.

Fig. 13.

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	Fig. 13. (color online) Curves of MCA varying with flux. (a) κ1 equivalent to 0.039α2, 0.39α2, 0.5α2, 0.66α2, 0.78α2, and α2; (b) Enlarged curve of κ1 = 0.5α2.

The MCA of the parallel connected coupled MC circuit can be expressed as 42

For the case of coupled MC in parallel connection with identical polarities, simulation parameters are employed as follows: α₁ = α₂ = 16.59 μF/Wb, β₁ = β₂ = 1.269 μF, κ₁ = α₂, and κ₂ = 0.5α₁. The curves of charge versus voltage corresponding to C_m1, C_m2, and C_m12 behave as typical PHLs as shown in Fig. 14(a), which reveals that the two MCs in parallel connection can also be operated together as a new MC. The MCA of this memcapacitive circuit is shown in Fig. 14(b). The values of MCA C_m12 vary with flux φ₁₂ for different values of coupling strength κ₁. The magnitude of the MCA can be linearly enlarged by increasing the coupling strength κ₁ and κ₂, which is in good agreement with Eq. (24).

Fig. 14.

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	Fig. 14. (color online) Simulated curves of parallel connections. (a) PHLs; (b) MCA varying with flux for different values of κ1.

The parameters configured for the two coupled MCs in parallel connection with opposite polarities are the same as those in the case of identical polarities. The variations of MCA C_m12 with flux φ₁₂ for different values of coupling coefficient κ₁, when κ₂ is equal to 0.5α₁, is depicted in Fig. 15. It can be observed that the MCA constantly remains at 2.558 μF for the special case of κ₁ = κ₂ = 0.5α₁. For the case of κ₁ < κ₂, the MCA linearly decreases with increasing the flux while the MCA can linearly increase with the flux increasing as κ₁ > κ₂, which is in good agreement with Eq. (29).

Fig. 15.

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	Fig. 15. (color online) Curves of MCA varying with flux for different values of coupling coefficient κ2.

The simulated curves of MCAs varying with charge for the case of κ₁ = 0.39α₂ and κ₂ = 0.5α₁ are shown in Fig. 16(a). Due to curve symmetry, only the interval of positive charge is needed for demonstration. Unlike C_m2 and C_m12, C_m1 can increase with the increase of positive charge. The maximal values of charge q₁ and C_m1 are much less than q₁₂ and C_m12, respectively. Curves of MCA for another coupling case of κ₁ = 0.75α₂ and κ₂ = 0.5α₁ are displayed in Fig. 17(a). In this case, both C_m1 and C_m2 can first increase and then decrease with increasing positive charge. Hence, C_m12 can first increase and then decrease with the increase of positive charge. The waveforms of MCA and charge in time domain are correspondingly given in Figs. 16(b) and 17(b) for comparative illustration, respectively.

Fig. 16.

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	Fig. 16. (color online) Simulated curves of MCA and charge. (a) MCA varies with current for κ1 = 0.39α2 and κ2 = 0.5α1; (b) time-domain waveforms of MCA and charge.

Fig. 17.

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	Fig. 17. (color online) Simulated curves MCA and charge. (a) MCA curves varying with current for κ1 = 0.75α2 and κ2 = 0.5α1; (b) time-domain waveform of MCA and charge.

A common capacitor of 330nF is serially connected to the coupled MC circuit with κ₂ = 0.5α₁ for exploring the MCA variation with flux as shown in Fig. 18. These results reveal that for the case of coupling strength κ₁ lower than κ₂, the MCA can decrease linearly with the increase of flux. While κ₁ is greater than κ₂, the MCA can linearly increase with the increase of flux. Note that for the case of κ₁ = κ₂ = 0.5α₁, the equivalent MCA keeps unchanged and equal to the sum of β₁ and β₂.

Fig. 18.

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	Fig. 18. (color online) Curves of MCA varying with flux for different values of coupling coefficient κ1.

The simulation results of coupled-MCs-based RO are demonstrated in this subsection for further discovering the possible practical application of the coupled MCs.

The RO circuit structured by the coupled MCs in serial connection with opposite polarities is first analyzed. The simulation parameters of the coupled MCs are configured as α₁ = α₂ = 1.918 μF/Wb, β₁ = β₂ = 0.4315 μF, κ₁ = 0.087α₂, and κ₂ = 0.071α₁. Three curves of voltage versus charge are displayed in Fig. 19(a), which indicates that the coupled MC circuit in serial connection also possesses the memcapacitive characteristics with pinched hysteresis loops (PHLs) behaving as an inclined “8”. The maximal voltage across MC1 is 2.727 V, which is less than that of MC2. Besides, the minimal voltage across MC1 is −3.367 V, which is also less than that of MC2. The time-domain waveforms of voltage and charge are shown in Fig. 19(b) for comparative demonstration.

Fig. 19.

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	Fig. 19. (color online) Curves of voltage and charge. (a) PHLs; (b) time-domain waveforms of voltage and charge.

By configuring R_s1 = 200 kΩ, R_s2 = 100 kΩ, and R_s3 = 200 kΩ, simulation figures are presented to display the output characteristics of the MC-based ROs as demonstratively shown in Fig. 20. The output oscillating period and duty cycle of the RO for κ₁ = 3.9α₂ and κ₂ = 0.071α₁ (Fig. 20(a)) are 235.55 ms and 62.4%, respectively. For the case of κ₁ = 0.39α₂ and κ₂ = 0.071α₁ as shown in Figs. 20(a) and 20(b), the output oscillating period and duty cycle of RO are 143.63 ms and 51.36%, respectively. Meanwhile, for the case of κ₁ = 0.087κ₂ and κ₂ = 0.071κ₁ (Fig. 20(b)), the output oscillating period and duty cycle of RO are measured to be 140.51 ms and 50.44%. Likewise, by changing the value of coupling coefficient κ₂, the oscillation period and duty cycle of the RO can also be regulated, which is in good agreement with the scenario in Fig. 20.

Fig. 20.

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	Fig. 20. (color online) Simulation results of serially connected coupled-MCs-based RO. (a) κ1 = 3.9α2 and κ1 = 0.39α2; (b) κ1 = 0.39α2 and κ1 = 0.087α2.

Then, the RO circuit established by the coupled MCs in parallel connection with opposite polarities is analyzed. The simulation parameters of coupled-MCs-based RO are set to be as follows: κ₁ = 3.9α₂, κ₂ = 0.49α₁, α₁ = α₂ = 1.918 μF/Wb, β₁ = β₂ = 0.4315 μF, R_s1 = 200 kΩ, R_s2 = 100 kΩ, and R_s3 = 23 kΩ. Three PHLs corresponding to C_m1, C_m2, and C_m12 are shown in Fig. 21(a), which reveals that the coupled MCs in parallel connection have typical memcapacitive behaviors even when operated in the RO circuit. The measured maximal charge passing through MC1 in positive direction is 1.2686 μC, which is greater than that of MC2. The charge passing through MC1 in negative direction is −1.0658 μC, which is greater than that of MC2 as well. Hence, the intersection occurs between trajectories of C_m1 and C_m2 as shown in Fig. 21(b). These three curves shown in Fig. 21(b) are in good agreement with the theoretical calculations from Eqs. (25) and (27).

Fig. 21.

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	Fig. 21. (color online) Voltage, charge and MCA. (a) Charge varying with voltage; (b) MCA varying with charge.

The waveforms of output voltage and MCA in time-domain are displayed in Fig. 22, which reveals the dynamic behaviors of the output oscillation voltage in terms of MCA. According to the simulation results exhibited by Figs. 20 and 22, we can see that high coupling coefficient κ₁ can actually lead to high periods and duty cycles. Also, as shown in Fig. 22, the output oscillation period and duty cycle of the RO for κ₁ = 3.9α₂ and κ₂ = 0.49α₁ (Fig. 22(a)) are measured to be 259.7 ms and 52.33%, respectively. For the operation case of κ₁ = 0.078α₂ and κ₂ = 0.49α₁ as shown in Figs. 22(a) and 22(b), the output oscillating period and duty cycle of the RO are 212.6 ms and 49.95%, respectively. Under the condition of κ₁ = 0.039α₂ and κ₂ = 0.49α₁ (Fig. 22(b)), the output oscillation period and duty cycle of the RO are 211.6 ms and 49.9%, respectively.

Fig. 22.

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	Fig. 22. (color online) Simulation results of coupled parallel connected MCs-based RO. (a) κ1 = 3.9α2 and κ1 = 0.078α2; (b) κ1 = 0.078α2 and κ1 = 0.039α2.

The proposed schematic of two coupled MC emulators is shown in Fig. 23. One op amp (TL084), one multiplier (AD633), and four current feedback op amps (CFOA) are necessarily required for building each of the two emulators, i.e., MC1 (inside the upper red dotted frame) and MC2 (inside the lower blue dotted frame). Without considering the coupling effects, by referring to the operational function of CFOA AD844, we have 43a 43b where φ_A1B1 is the time integral of terminal voltage v_A1B1, v_u11, and v_u31 are the output voltages of CFOAs U11 and U31, respectively. The coupling action is achieved by two adder circuits made by op amps U51 and U52.The output voltage of op amp U51 can be calculated from 44 Based on the datasheet of AD633, v_w1 can be derived as 45 Due to the actions of U21 and U41, the current i_MC1 going through terminals A₁ and B₁ is in fact determined by voltage v_w1. Therefore, we have 46a 46b where q_MC1 is the time integral of i_MC1. By combining Eqs. (43a), (45) and (46b), we can deduce the following equation: 47 Hence, the MCA of MC1 can be derived by considering the coupling action, 48 where α₁ is corresponding to the variation rate of MCA, and parameters α₁ and β₁ can be expressed as 49 Therefore, it can be deduced from Eq. (49) that the initial value of MCA can be adjusted by resistor R₄₁ and DC voltage supply v_s1.

Fig. 23.

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	Fig. 23. (color online) Schematic diagram of two flux-controlled coupled MC emulators.

By taking the coupling strength into consideration, the coupling connection between MC1 and MC2 is achieved by two inverting adders (U51 and U52). According to Eqs. (44), (47), (48), and (49), the MCA of each MC can be expressed by 50a 50b where the parameters α₂ and β₂ can be deduced by referring to Eqs. (48) and (49), and the coupling coefficients κ₁ and κ₂ are 51 Hence, the coupling strengths can be conveniently controlled by adjusting the values of coupling resistors R_c1 and R_c2 in the emulator as shown in Fig. 23.

In the experimental implementation, it has been confirmed that the possible variation range of the MCA is mainly dependent on the saturation output of the active chips. As shown in Fig. 23, by taking MC1 for demonstration, U31 possesses the highest chance to be operated with saturation output, while the amplitude and period of the input voltage v_A1B1 are both large. Assuming that the power supply voltages of the active chips are V_OH and V_OL, when positive saturation happens to U31, namely, v_u31 = V_OH. According to Eq. (44), we have 52 53 54 Hence, the upper boundary of MCA variation range C_M1max can be calculated from 55 Likewise, when negative saturation to U31, v_u31 = V_OL, the lower boundary of MCA variation range C_M1min can be obtained from 56 Similarly, saturation could happen to the adder circuit U51. For the case of v_u51 = V_OH, according to Eq. (45), we have 57 58 In this case, based on Eq. (58), the maximal value of MCA C_M1max can be expressed by 59 Likewise, the minimal value of MCA C_M1min can be deduced by 60 The saturation could also happen to the remaining active chips shown in Fig. 23. The value of MCA remains constant as the saturation takes place. Therefore, in the MC-based RO circuit, neither the output oscillation period nor duty cycle can be controlled by the coupling strength as the MCA reaches its maximal or minimal boundary. However, output saturations of the active chips could be avoided by properly adjusting the system parameters of the circuit, such as the values of the resistors.

The case of two coupled MC circuits serially combined with opposite polarities (B₁ directly connected to B₂) is taken for demonstration to verify the theoretical analysis. The experimental parameters are given as follows: R₁₁ = R₁₂ = 51 kΩ, R₂₁ = R₂₂ = 51 kΩ, R₃₁ = R₃₂ = 39 kΩ, R₄₁ = R₄₂ = 39 kΩ, R₅₁ = R₅₂ = 10 kΩ, R₆₁ = R₆₂ = 10 kΩ, R₇₁ = R₇₂ = 90 kΩ, C₁ = C₂ = 100 nF, C_1m = C_2m = 330 nF, v_s1 = v_s2 = −15 V. Based on the parameter configuration, the variation rates α₁ for MC1 and α₂ for MC2 are both equal to 16.59 μF/Wb, and the initial MCA values of β₁ for MC1 and β₂ for MC2 are equal to 1.269 μF. The excitation voltage v₁₂ can be characterized by a sinusoidal voltage 1.9sin(20πt) V. Two potentiometers R_c1 and R_c2 with 1-MΩ maximal value are used for smoothly adjusting the coupled coefficients. For the sake of verifying the consistency between experimental and simulation results, the experimental data are sampled by oscilloscope TDS20114 and then transferred into OriginPro8.0 software to draw the curves.

According to Eqs. (49) and (51), we can obtain that the coupling coefficient κ₁ is in proportion to the variation rate κ₂ with the coefficient of R₁₂C₂R₃₂/(R₁₁R_c2C₁). Based on the above experimental parameter configuration, the coefficient can be calculated and equal to R₃₂/R_c2. The charge q passing through the MC can be measured by sampling the output voltage of AD633 according to Eq. (46b). Under the condition of R_c1 = 79 kΩ and R_c2 = 78 kΩ, the experimental data of voltage and charge relating to terminals A₁B₁, A₂B₂, and A₁A₂ are measured to draw the PHLs at κ₁ = 0.5α₂ and κ₂ = 0.49α₁ as shown in Fig. 24(a). It can be seen that all the three PHLs behaving as an inclined “8” are together passing through the origin, and the curves of v₁–v_w1 and v₂–v_w1 are closely coincided with each other. Obviously, this experimental PHL is in good agreement with the simulation result shown in Fig. 10(a), which also reflects that each of the two MCs can behave as individual MC and the coupled MC circuit can also be operated as individual MC even in serial connection coupled MC with opposite polarities. The MCAs C_m1, C_m2, and C_m12 varying with terminal voltages at κ₁ = 0.5α₂ and κ₂ = 0.49α₁ are shown in Fig. 24(b), which demonstrates that C_m2 and C_m12 decrease while C_m1 increases due to positive terminal voltage. Evidently, the curves of C_m1–v₁ and C_m2–v₂ are almost identical to each other, which is in good consistence with Fig. 24(a).

Fig. 24.

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	Fig. 24. (color online) Measured curves of two coupled MC circuits in serial connection. (a) PHLs; (b) MCA variation with terminal voltages at κ1 = 0.5α2 and κ2 = 0.49α1.

By setting R_s1 = 200 kΩ, R_s2 = 100 kΩ, and R_s3 = 200 kΩ, two coupled MCs in serial connection with opposite polarities are used to structure the RO circuit. R_c1 = 500 kΩ and R_c2 = 78 kΩ are adopted to verify the MCA characteristics of MC1 and MC2 as operated in the RO circuit, as well as the output performance of the RO. Two identical Zener diodes of 1N4739 are utilized to restrict the output voltage within the interval of [−9.1, 9.1] V. For the special case of κ₁ = 0.5α₂ and κ₂ = 0.078α₁, the measured experimental results are displayed in Fig. 25, from which we can see that all the three PHL curves behave as an inclined “8” as shown in Fig. 25(a). Note that these three PHLs stably evolve but with non-smooth ends, which can be attributed to the square waveform of the output oscillation voltage of RO. The square output voltage leads to approximate triangle terminal voltage of the MCs as shown in Fig. 25(b).

Fig. 25.

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	Fig. 25. (color online) Experimental results at κ1 = 0.5α2, κ2 = 0.078α1. (a) PHLs of MC1, MC2, and serial connection coupled MC; (b) Three terminal voltages and the output voltage of U61 in time domain.

The MCAs varying with terminal voltage in the special case of κ₁ = 0.5α₂, κ₂ = 0.078α₁ are shown in Fig. 26(a), in which the MCAs C_m2 and C_m12 decrease while C_m1 increases due to positive terminal voltage. The measured curves shown in Fig. 26(b) indicate that the coupled-MC-based RO can be self-excited and output square voltage waveforms with stable period and duty cycle. Meanwhile, by maintaining R_c2 = 79 kΩ leading to κ₁ = 0.5α₂, four different cases with coupling coefficients of κ₂ = 0.78α₁, κ₂ = 0.5α₁, κ₂ = 0.13α₁, and κ₂ = 0.078α₁ are used to demonstrate the control effects of different coupling strengths on the output voltage of RO as shown in Fig. 26(b). It can be derived that with the increase of coupling coefficient κ₂, the duty cycle and the period of output voltage can both increase.

Fig. 26.

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	Fig. 26. (color online) Experimental results. (a) MCA varying with terminal voltages at κ1 = 0.5α2, κ2 = 0.078α1; (b) The output voltages of the RO under different values of coupling coefficient κ2.

This dual coupled MC emulator can be connected in parallel with opposite polarities by linking A₁ with B₂, and B₁ with A₂. The parameters for experimentally testing this parallel MC circuit are configured as R₁₁ = R₁₂ = 51 kΩ, R₂₁ = R₂₂ = 51 kΩ, R₃₁ = R₃₂ = 39 kΩ, R₄₁ = R₄₁ = 39 kΩ, R₅₁ = R₅₂ = 10 kΩ, R₆₁ = R₆₂ = 10 kΩ, R₇₁ = R₇₂ = 90 kΩ, C₁ = C₂ = 100 nF, C_1m = C_2m = 220 nF, v_s1 = v_s2 = −15 V. The variation rates α₁ and α₂ can be calculated and equal to 11.06 μF/Wb. The initial MCA value β₁ is the same as β₂, and both are equal to 0.8462 μF. The excitation voltage v₁₂ = 2sin(20πt) V is employed for testing these MCs.

By maintaining the potentiometers R_c1 = 78 kΩ and R_c2 = 800 kΩ, we have κ₁ = 0.78α₂ and κ₂ = 0.5α₁. Based on Eq. (46b), v_w1 is measured for representing the charge q₁ passing through MC1, while v_w2 is measured for representing the charge q₂. The three PHLs of this dual coupled MC circuit in parallel connection with opposite polarities are shown in Fig. 27(a), where the PHLs with red, blue and black colors are the PHLs corresponding to MC1, MC2, and the MC circuit in parallel. The curves of MCAs varying with charges at κ₁ = 0.049α₂ and κ₂ = 0.5α₁ are given in Fig. 27(b). The measured MCA of the coupled MC circuit is equal to the sum of the MCAs of MC1 and MC2, hence it is possible that C_m1 and C_m12 increase while C_m2 decreases for positive charge, as the curves are shown in Fig. 27(b).

Fig. 27.

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	Fig. 27. (color online) Measured curves of two coupled MC circuits in parallel connection with opposite polarities. (a) PHLs; (b) MCA variation with charge at κ1 = 0.049α2 and κ2 = 0.5α1.

The curves of C_m12 varying with flux φ₁₂ are displayed in Fig. 28, in which the coupling coefficient κ₂ = 0.5α₁ and the coupling coefficient κ₁ changes from 0.078α₂ to 2.5α₂. It can be observed that the MCA can be linearly reduced by the increase of the flux as κ₁ is greater than 0.5α₂, while the MCA can linearly increase with increasing flux as κ₁ is less than 0.5α₂. For the special case of κ₁ = 0.5α₂, the MCA remains constantly equal to its initial value of 1.74 μF no matter whether the flux is varying.

Fig. 28.

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	Fig. 28. (color online) Measured curves of MCA varying with flux under different values of coupling coefficient κ1.

The parameters for implementing the RO with coupled MCs in the parallel connection are configured as R_s1 = 200 kΩ, R_s2 = 100 kΩ, and R_s3 = 23 kΩ. By setting R_c1 = 78 kΩ and R_c2 = 10 kΩ, it can be calculated that coupling strengths satisfy the condition of κ₁ > κ₂. The experimental curves are displayed in Fig. 29(a), from which we can see that the PHLs of MC1, MC2 and the coupled MC circuit are together passing through the origin but with very different shapes and variation rates. Hence, it can be speculated that the variation of MCA with terminal voltage across each MC is also different from the other MC due to the influence from coupling strength. Three terminal voltages and the output voltage of U61 in time domain are shown in Fig. 29(b) for comparative exhibition.

Fig. 29.

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	Fig. 29. (color online) Experimental results at κ1 = 3.9α2 and κ2 = 0.5α1. (a) PHLs of MC1, MC2, and the coupled MC circuit; (b) three terminal voltages and the output voltage of U61 in time domain.

By keeping R_c1 = 78 kΩ and R_c2 = 10 kΩ, we can calculate that the coupling coefficients κ₁ = 3.9α₂ and κ₂ = 0.5α₁. Three curves of MCA versus charge are displayed in Fig. 30(a), in which the MCA of the coupled MC circuit is equal to the sum of the MCAs of MC1 and MC2. Obviously, the MCAs of C_m1 and C_m12 can increase while C_m2 decreases by positive terminal voltage. Shown in Fig. 30(b) are the three output voltages under different values of coupling coefficient κ₁ and a fixed value of coupling coefficient κ₂. It can be concluded that with the increase of coupling coefficient κ₁, the duty cycle and the period of output voltage both increase.

Fig. 30.

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	Fig. 30. (color online) Experimental results. (a) MCA varying with charge at κ1 = 3.9α2, κ2 = 0.5α1; (b) output voltages of the RO under different values of coupling coefficient κ1.

In order to verify the controllability of different coupling strengths, κ₁ = 0.5α₂ and κ₂ varying from 0 to α₁ are adopted to carry out the experimental validation for the case of the serial connection as shown in Fig. 31(a), while the parameter configuration of κ₂ = 0.5α₁ and κ₁ varying from 0 to 2.6α₂ are employed for the experimental implementation for the case of parallel connection as shown in Fig. 31(b). Clearly, the high coupling coefficients κ₁ and κ₂ can lead to high oscillation periods and duty cycles. As shown in Fig. 31(a), the variation ranges of oscillation duty cycle and period are [49.2%, 52.5%] and [120.9, 170] ms, respectively. Meanwhile, the variation ranges of oscillation duty cycle and period are measured to be [48%, 59%] and [160.2, 270.3] ms, respectively, as depicted in Fig. 31(b).

Fig. 31.

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	Fig. 31. (color online) Experimental results of controllability. (a) Serial connection; (b) parallel connection.

These experimental results show that the proposed emulator circuit is capable of mimicking the dynamic behaviors of coupled MCs. At the same time, the coupled MC with multiple circuitry connections can be used to structure the RO with adjustable period and duty cycles. These experimental results exhibit good agreement with the theoretical and simulation results.