Analysis of Sliding-Mode Controlled Impedance Matching Circuits for Inductive Harvesting Devices

Abstract

Most systems converting wave energy into longitudinal or angular mechanical energy use the upward and downward movement of the waves at a fixed sea point. [...]in [4,5,6] electricity is produced by an angular movement using a pulley and a rod to convert a longitudinal movement into an angular one. [...]the Laplace transform can be directly applied in the ideal dynamics (20), and a small signal model is not required. [...]the proposed sliding surface (23) regulates the output current iL1(t) to a given value that, considering the converter voltage gain, imply the desired input current, that is iLFR(t). [...]the matching efficiency is always over 95% whatever be the input voltage, as can be seen in Figure 22c,d. To reduce the input impedance variation in terms of the input voltage, the hysteresis width used to implement the sliding surfaces by means of hysteretic comparators must be reduced.

1. Introduction

Modern technology consumes large amounts of electrical energy. This energy is usually generated in power plants where different energy sources are converted into electrical energy. Each power plant type and energy source have their own advantages, drawbacks, and conversion efficiencies, but in addition, many of them raise environmental concerns due to the excess of pollutants produced in the conversion procedure.

Although the energy conversion efficiency of power plants can be improved, the pollution and residues generated cannot always be reduced. In this context, renewable energy sources, as hydro power, wind farms, PV (photovoltaic) plants, and the oceans energy, must be profitable to favor a more sustainable development that is respectful with the natural environment.

Depending on the physical characteristic considered, the oceans and sea waters offer different ways to collect the energy of water in movement, namely: Tides, sea-waves, and marine currents. Each water displacement type requires different technologies and transducers for collecting its energy [1].

Wave energy can be a promising energy source. Indeed, the areas of the world having larger wave energy resources are those subjected to regular wind fluxes. There are different harvesting methods. Linear-oscillating magnetic transducers can be used for direct wave energy conversion to electricity. These transducers are usually made with a coil and a permanent magnet [2].

Sea waves are generated when wind passes over the water surface. As the sea waves propagate slower than the wind speed causing them, the energy is transferred from the wind to the waves. Wind friction on the water surface, and the difference of air pressure between the two wave sides, makes stress on the water, causing the waves growth [3]. Then, generated waves propagate on the sea surface, and transport their energy to the shore with the group velocity c_wg [4].

The oscillatory motion is higher at the sea surface, and decreases exponentially with depth, making the waves more independent of sea floor contour conditions as the sea depth increases. For this reason, floating harvesters are more competitive than bottom standing ones. Besides, the available energy is higher in near-shore and off-shore locations than in on-shore placements. Figure 1 depicts possible placements for floating harvesting devices.

The wave height H is given by the wind speed, the sea depth, the fetch angle, and the seafloor topography. The average energy density d_s[E_w] per unit area (1) is the sum of the kinetic and potential energy of the wave [3], where ρ is the density of sea water, H is the wave height, and g is the acceleration of gravitational force. According to the equipartition theorem, both energy types contribute equally to the wave energy. Parameter k_w accounts for the wave periodicity level. Thus, k_w = 1 for random waves, and k_w = 2 for periodic ones.

_EW=_EK+_EP→_dS[_EW]=_kw16ρg^H2 [Joule/^m2]

The energy flux, per unit-width, through a vertical plane that is perpendicular to the wave propagation direction is the average power density (2) per unit-width d_l[P_w]. Then, the average power available P_w(L_e) in a wave-train with T period and L_e width, can be accounted for with (3). As an example, in a wave train of H = 1 m, with a period of T = 10 s, the power density is around 5 kW/m.

_dl[_Pw]=_cwg⋅_ds[_Ew]=_{kw cwg}16ρg^H2 [W/m]

_Pw(_Le)=_kwρ^g264πT_Le ^H2 [W],where_cwg≈g4πT

Notice that although wave parameters {T, L_e, H} determine the available wave energy, the real electric power extracted from the sea will depend additionally on the transducer kind, rating, and conversion efficiency, and finally on the efficiency of the power processing circuits used.

Most systems converting wave energy into longitudinal or angular mechanical energy use the upward and downward movement of the waves at a fixed sea point. Thus, in [4,5,6] electricity is produced by an angular movement using a pulley and a rod to convert a longitudinal movement into an angular one. In this sense, different floating buoys with its respective efficiencies are given in [7].

A difficulty with extracting energy from inductive magnetic energy harvesters is that they normally produce a low AC voltage magnitude at a very low frequency (<10 Hz), and boost transformers cannot be directly used because of their large size. To step-up the voltage many applications propose using a step-up converter after the rectifying stage. Thus, to maximize the harvested energy, the rectifier losses must be reduced. Some literature works propose using voltage multipliers [8], others propose reducing the rectifier voltage drop using mosfet active rectifiers [9,10] as depicted in Figure 2a. The use of two equal generators in counter-phase connected to a three-wire, two-diode full-wave rectifier, has also been proposed in several works, for example in [11,12], and depicted in Figure 2b.

The harvesting device presented here, shown in Figure 3, is made with three inductive transducers with a high output voltage peak. As this device delivers a high voltage output, in clear contrast with previous works, no special rectifier circuits are required. Nevertheless, to reduce the rectifier losses, Schottky diodes have been considered for the full bridge rectifier input stage. Indeed, instead of maximizing the device output voltage combining the three transducer coils in series, a different interconnection has been preferred to reduce the device output impedance, as explained in Section 3.

Impedance matching has been proposed in different works to maximize the energy transfer between the harvesting generator and the load. References [13,14,15,16,17,18] propose the use of a buck-boost power converter operating in discontinuous conduction mode (DCM) as an impedance adaptor, some of them without voltage sensors [13,14,15] because input and output adaptor voltages are quite constant; or including voltage feedback to compensate the generator voltage variations [15,16,17]. Nevertheless, none of them use a harvesting generator with a similar degree of generator voltage variation as proposed in this work. We propose the use of a sliding-mode control to force a switching converter to behave like a loss free resistor (LFR) [19] for the impedance adaptor. The inherent robustness to parametric variations of sliding control, can assure a good impedance matching for a wide range of input voltage variation voltage as here.

After this introduction, this work continues as follows. In Section 2, the operation of the harvesting device presented in this work is analyzed, modelled, and verified experimentally. The principle of using an impedance matching circuit to maximize the collected energy is described in Section 3. Next, two sliding mode control laws are proposed to force two different switching converters to behave like a LFR-based impedance adaptors. The first converter, the hybrid buck-boost (HBB) is analyzed in Section 4, and next, the Sepic converter is studied in Section 5. The realization of both converters is described in Section 6. Section 7 is dedicated to the experimental results, and finally in Section 8, some conclusions and future research lines, are given.

2. Transducer and Harvesting Device Description

In this work we propose to use this wave energy for small-power, stand-alone applications. The extraction of small amounts of energy from each wave, conveniently stored in a battery, is a solution to supply intermittent operation small equipment, where the peak power consumption is occasional and brief: Weather stations, telecom relays, and similar equipment.

Figure 3c shows the wooden hand-made, articulated harvesting device, developed in our laboratory. This device includes three magnet-coil transducers (mA, mB, and mC). Figure 3a is depicts one of these transducers. To understand the operation of the real harvesting device appearing in Figure 3c, Figure 3b depicts in a simplified way, the front and profile views of the whole generator, and Figure 3d shows schematically the interaction between the sea waves and the developed generator. To track appropriately the train waves, the balloon floats must have a given amount of water inside, as the floats must only compensate the device weight.

Most harvesting systems that convert wave energy into a longitudinal displacement to drive a coil-magnet based linear generator, use the upward and downward movement of the wave at a fixed point [11,19]. In contrast, the system presented in Figure 3, uses the differential movement between two points of the water surface.

By means of levers, the differential movement caused by a sea wave propagating over the sea is converted into a synchronized horizontal displacement of each magnet through the corresponding coil, inducing a given voltage pulse according to Faraday’s law. As seen in Figure 3b,d, the transducer mB is moving in counterphase compared to mA and mC, that are moving in phase. This can be compensated electrically by modifying the coils connections, or placing two magnets facing a given magnetic pole, and the remaining magnet, to the opposite one.”

2.1. Harvesting Transducer Operation Principle

The transducer shown in photograph Figure 3a, a permanent magnet linear generator [20], is depicted schematically in Figure 4. According to its operation principle, a permanent magnet is moving inside and outside of a transducer coil winding. This causes a magnetic flux variation through it, that considering the Faraday’s Law, creates a voltage difference across its terminals.

The N-turns coil in Figure 4a, has been idealized in the equivalent diagram of Figure 4b, where all the turns are collapsed in a single turn, of zero thickness and radius R. To produce electricity a magnet of length L, crosses the coil at a variable speed v(t). The distance from the magnet center to the coil center is the variable z. The magnet is modeled as two fictitious magnetic charges of equal value and opposite sign, realizing a magnetic dipole, with both charges separated by the distance L, the magnet length. This approach shown in Figure 5, is sufficient for our purposes.

Considering the magnet length L, and its dipolar moment m, the equivalent magnetic charges are q=±m/L , and the magnetic flux density B_q(z) in the vicinity of a magnetic charge/pole is

_B→q=_μoq4π^r2r^⇒_Bq(z)=_μom4πL^r2

The total magnetic field B(z) created by both model charges at a given point of the permanent magnet axis placed at a distance z from its center (5), is the sum of contributions of both charges.

B(z)=_B+q+_B−q=_μom4π L[^(z−L2)−2−^(z+L2)−2]

The electromagnetic force (EMF) generated by the transducer can be calculated from the total magnetic flux φ created by the dipole charges ±q through the equivalent collapsed loop.

Φ(z)=∫SB⋅dS→=−∫SB⋅dScos(θ)=_Φ+q(z)+_Φ−q(z)

_Φ+q(z)=−sgn(z−L2)_μoqN2{1−|z−L2|⋅^{[(z−L2)2+R2]−1/2}}

_Φ−q(z)=sgn(z+L2)_μoqN2{1−|z+L2|⋅^{[(z+L2)2+R2]−1/2}}

This flux φ is variable because the magnet position z in respect to the coil is changing because the magnet is moving with a certain speed v(t), along the axis spire. Applying Faraday's law to the whole flux produced (6), the EMF induced ε(t) in the spire is given by (9), where the charges values ±q have been changed by their equivalent value in terms of the magnetic dipolar moment (±m/L). When the transducer is placed in a vertical position, the velocity v(t) and the position z(t) are the result of gravitational acceleration, and the potential ε(t) becomes (10)

ε(t)=−dΦdt=v(t)N_μom^R22Lv(t)⋅(^{[(z(t)−L2)2+R2]−32}−^{[(z(t)+L2)2+R2]−32})

ε(t)=gtN_μom^R22L(^{[(gt22−L2)2+R2]−32}−^{[(gt22+L2)2+R2]−32})

The theoretical voltage waveform ε (t) given by expression (10) is shown in Figure 6a, and the real transducer voltage at no-load conditions is given in Figure 6b. The magnet parameters are R = 2.5 cm, L = 6 cm, m = 5.15·10⁻⁴ Am², and the coil has N = 7000 turns of a copper wire with a diameter of φ = 0.3 mm. The measured inductance and resistance are respectively L_T = 3 H and R_T = 424 Ω.

The experimental pulse waveform shown in Figure 6b corresponds to a single transducer with all the partial coil windings are connected in series, and therefore N = 7000 turns. These 7000 turns are organized in five partial windings, three of 1000 turns, and two of 2000 turns. These windings are organized as described in Table 1. Realize that the resistance and inductance of each partial winding (at equal number of turns) is smaller in the inner windings than in the outer ones.

2.2. Final Harvesting Device

The harvesting device has three interconnected single transducers. The connections realized between the different windings and coils are given in Figure 7. The final goal is to reduce the device output impedance, increasing also the energy yield.

According to the circuit of Figure 7, the specific details of the coil-interconnection parameters are given in Table 2.

At no-load conditions, the voltage pulse produced by the proposed device is shown in Figure 8. The final parameters of this generator are: R_int = 26 Ω, L_int = 52.8 mH.

3. Impedance Matching with a Loss Free Resistor

The main purpose of the work presented here is to charge the pay-load system battery taking as much as energy as possible from the harvesting generator. If the battery is connected directly to the generator, using only a bridge rectifier, the transferred energy cannot be maximized, in part due to impedance mismatching between the battery and the generator, and in addition, because only the parts of the input voltage pulse waveform exceeding the battery voltage will be profited. Figure 9 depicts the proposed solution, a matching circuit between the generator and the battery.

Therefore, the impedance adaptation is the only solution to maximize the energy transferred at any working condition. However, even in this case, only 50% of the energy produced in the transducer will be transferred to the load. Indeed, if the conjugate input impedance of the power adaptor were equal to the generator output impedance (11), a perfect adaptation would occur.

For its simplicity, a DC-DC converter behaving as a loss free resistor [21] is proposed as impedance matching circuit. The LFR is a two-port circuit with a resistive input impedance R_LFR that can be adjusted. The output port, with a power source characteristic, delivers to the load, in this case the system battery, all the power absorbed by the LFR input resistance.

_Zo=_Rint+jω_Lint=26+j0.33[Ω]_Zin*=^{(_RLFR∥−jω _Cf)*}=j_RLFR/ω _CfRLFR+j/ω _Cf=25.96+j0.045[Ω]}then_Zo=_Zin*⇔_Rint≈_RLFR

Expression (11) shows the generator output impedance Z_o, and Z*_in, that is the conjugate of the matching circuit input impedance. The numerical values for Z_o and Z*_in appearing in (11) have been calculated for C_f = 10 μF, and considering the worst case shown in Table 3. This means a time lapse between consecutive waves of one second (ΔT = 1 s), although the real one is longer, between 10 and 30 seconds.”

The design of filter capacitor C_f implies three different concerns: (a) the low-pass filter effect must be small because its voltage v_cf(t) must track appropriately the generator pulses v_p(t), (b) the capacitor must compensate the inductive impedance of the generator, and (c) the capacitor must filter the high frequency switching noise. Finally, the selected value for the capacitor is C_f = 10 μF.

According to circuit of Figure 8, the instantaneous capacitor voltage v_cf(t) is given in (12). Next, neglecting the rectifier bridge voltage drop 2V_d, and assuming impedance matching (R_LFR = 26 Ω), the capacitor voltage v_cf(t) should be (13) the half of the input pulse absolute value.

_vCf(t)≈|_vp(t)−_{Rint Iint}(t)−2_Vd|

_vCf(t)≈|_vint(t)|=|_vp(t)|2

The LFR behavior is required for impedance matching, but the appropriate converter to implement the LFR depends on its input and output voltage ranges.

According to the pulse waveform in Figure 7, the capacitor voltage v_Cf(t) experiences a large variation following the voltage pulse v_p evolution. During 250 ms, the generator voltage varies from −100 V < v_p(t) < 160 V. Once rectified, if impedance matching occurs, the capacitor voltage changes from 0 < v_Cf(t) < V_max = 80 V. Thus, using a common 12 V battery, the matching circuit input voltage v_Cf(t) can be greater (v_Cf(t) > V_bat) or smaller (v_Cf(t) < V_bat) than the battery voltage. Consequently, any converter proposed as matching circuit must exhibit a buck-boost characteristic.

3.1. DCM Operated Buck-Boost Converter

Since the buck-boost operated at DCM [19] has a natural LFR behavior, that converter could be proposed as an impedance matching circuit. From the buck-boost input power in DCM, the expression of the converter input impedance R_LFR can be easily calculated (14)

_Pin=_vcf2 ^D2 _TS2L=_vcf2RLFR⇒_RLFR=2L^D2(t)_TS(t)≈_Rint=26Ω

where D is the converter duty ratio, and T_S is the switching period. Theoretically, any pair of constant D and T_S can be used to regulate the input impedance R_LFR = 26 Ω. However, to guarantee that the converter is working in the discontinuous mode, the converter duty ratio D(t) must change continuously (15) to follow the v_Cf(t). Therefore, to keep constant, at 26 Ω, the switching frequency f_S(t) = 1/T_S(t) must also be adapted continuously.

DCM<_vL>=0}⇒D(t)<_VbatVbat+_vCf(t)

It can be easily concluded that designing a low consumption control circuit able to adjust continuously the values of duty cycle D(t) and T_S (t) to fulfill simultaneously nonlinear Equation (14) and in Equation (15) is very complicated. Besides, the buck-boost converter has output voltage sign inversion, input and output pulsating currents, and, therefore, other solutions must be explored.

3.2. Buck/Boost Hybrid Converter (HBB)

The buck/boost hybrid converter is shown in Figure 10. While v_Cf(t) be smaller than the battery voltage V_bat, the circuit will operate in boost mode, but when v_Cf(t) be greater than the battery one, it will work in buck mode. When the converter is in step-up mode, S_Buck is permanently at ON-state, whereas in the buck working mode S_Boost is permanently at OFF-state.

This converter has a pulsating output current in boost mode and a pulsating input current in buck mode, so there is an electromagnetic interference (EMI) reduction.

To impose a LFR behavior to both working modes, assuring impedance matching, two sliding control laws must be used, one per each mode. Both control surfaces are analyzed in Section 4.

The mode selection is done with a comparator, but there is a small dead-zone around V_bat where the converter is not switching to reduce switching losses. This is an advantage compared to a classic step-up/down converter like the buck-boost, Ćuk, or Sepic, that are switching continuously regardless the value of v_Cf(t) and V_bat. Conversely, in the dead zone, there is no control, and the impedance matching is lost.

3.3. Sepic Converter

Figure 11 depicts the Sepic converter. As in the HBB converter case, there is no output voltage sign change. As happens with the HBB converter in the boost mode, the input current has a triangular waveform whereas the output current is pulsating. In this case, as there is no dead-zone around V_bat, the impedance matching occurs for any v_cf(t) value. Besides, as additional advantage, only one sliding-mode control law is required to impose the LFR behavior for all v_cf(t) voltage range.

4. HBB Converter Matching Circuit

The hybrid converter is a combination of two converters. Therefore, it has two operation modes, buck and boost. Two different sliding mode surfaces are required to force this circuit to behave like a loss free resistor. The controllable inductor is L₁. In the boost mode, the inductor current i_L1(t) is the converter input current, so i_L1(t) = i_LFR(t). Conversely, in the buck mode, i_L1(t) is the converter output current, that is, the current charging the battery, and therefore i_LFR(t) = i_L1(t)·[V_bat(t)/v_cf(t)].

4.1. Boost Mode Operation (HBB)

When v_Cf(t) < V_bat, the switch S_Buck is always at ON state, and i_L1(t) = i_int(t). The resulting circuit is an input filtered boost converter. The output capacitor C_o is not included because it is in parallel with the battery and has no dynamics. For simplicity, we use v_p(t) in the analysis, instead of |v_p(t)|. Figure 12 depicts the two circuit topologies, ON and OFF.

Using the switching variable u(t) = {0, 1} the equations of ON and OFF topologies can be compacted, and the converter dynamics at any instant is described with a sole set of Equations, (16)

x(t)=[_iintiL1vcf]⇒{_Lintd_iintdt=_vP−_{Rint iint}−_vcfL1d_{iL 1}dt=_vcf−_VBat(1−u) _Cfd_vcfdt=_iint−_iL1

The sliding surface (17) imposed to the inductor current i_L1(t), guarantees the impedance matching. From the Equation set (16), taking into account the control surface S(x) and its existence conditions, the equivalent control (18) is obtained.

S(x)=_iL1−_vcfRint=0⇒_iLFR=_vcfRintS(x)S˙(x)<0

_ueq=1−_vcfVBat+_{L1Rint VBat}⋅d_vcfdt

Forcing to zero the dynamics of Equation set (16), and considering the equivalent control (18), the system equilibrium point (19) and the related ideal sliding dynamics (20) can be obtained. The resulting dynamics is linear. At the equilibrium point, the capacitor voltage results to be the half part of the input voltage pulse (V_cf^* = V_p^*/2), as expected from an impedance matching system.

^X*=^{[_Iint*,_Vcf*,_IL1*]T}=^{[_VP *2_Rint,_VP *2,_VP *2_Rint]T}

{_g1(x)=d_iintdt=1_LintvP−_RintLintiint−1_Lintvcfg2(x)=d_vcfdt=1_Cfiint−_{vcfRint Cf}

The ideal sliding dynamics is linear, as can be seen in (20). Consequently, the Laplace transform can be directly applied in the ideal dynamics (20), and a small signal model is not required. After some manipulations the transfer function _Vcf(s)/_Vp(s) (21) is obtained. This transfer function evidences the impedances matching at DC, because V_Cf/V_p = 1/2. The tracking of the input pulse v_p(t) is controlled by a second order low-pass filter, with a characteristic polynomial P(s) is unconditionally stable, because all its coefficients exist and are positive. Considering that C_f = 10 μF, the system natural frequency (21) is ω_n ≈ 1900 rad/s, and the bandwidth is around BW_-3dB–310 Hz.

_Vcf(s)_VP(s)=_V^cf(s)_V^P(s)=KP(s)=1_{Lin Cf}^s2+(_RinLin+1_{Rin Cf})s+2_{Lin Cf}

4.2. Buck Mode Operation (HBB)

When v_Cf (t) < V_bat, the switch S_Boost will be permanently at OFF state. The resulting circuit is a common buck converter with an input filter. The output capacitor C_o is not depicted because is in parallel with the battery. The topologies ON and OFF, are shown respectively in Figure 13a,b.

Repeating the process described for the boost mode, a single set of differential Equation (22) describes, at any instant, the dynamics of the converter.

x(t)=[_iintiL1vcf]⇒{_Lintd_iintdt=_vP−_{Rint iint}−_vcfL1d_{iL 1}dt=_vcfu−_VBatCfd_vcfdt=_iint−_iL1u

To assure impedance matching, the converter input current should be forced to be proportional to the input voltage v_Cf(t). However, the only state variable leading to a stable control surface is the output current i_L1(t). As a result, the proposed sliding surface (23) regulates the output current i_L1(t) to a given value that, considering the converter voltage gain, imply the desired input current, that is i_LFR(t). From the Equation set (22), considering the surface S(x) and its existence conditions, the equivalent control (24) is obtained.

S(x)=_iL1−_vcfRint⋅_vcfVBat=0⇒_iLFR=_vcfRintS(x)S˙(x)<0

_ueq=1−_vcfVBat+_{L1Rint VBat}⋅d_vcfdt

Forcing to zero the system dynamics in (22), and considering the equivalent control (24), we obtain the equilibrium point (25) and the corresponding ideal sliding dynamics (26), that is non-linear. As happened in the boost mode, the equilibrium point capacitor voltage is the half part of the pulse voltage is (V*_cf = V*_p/2), as expected from impedance matching.

^X*=^{[_Iin*,_Vcf*,_IL1*]T}=^{[_VP *2_Rint,_VP *2,_Vcf*24_{Rint VBat}]T}

{_g1(x)=d_iintdt=1_LintvP−_RintLintiint−1_Lintvcfg2(x)=d_iL1dt=2_{vcfRint VBat}⋅d_vcfdt_g3(x)=d_vcfdt=1_Cfiint−2_{CfL1Rint VBat}d_vcfdt_iL1−_VBatvcfiL1Cf

To evaluate the local stability, the ideal dynamics (26) is linearized around the equilibrium point X^* using the typical small signal perturbation model (27), (28), whose coefficients are given in (29)

x(t)=^X*+x^(t)

{_g1(x)≈a⋅_v^P(t)+b⋅_i^int(t)+c⋅_v^cf(t)_g2(x)≈d⋅_v^cf(t)+e⋅w^(t)wherew=d_vcfdt_g3(x)≈f⋅_i^int(t)+g⋅_i^L1(t)+h⋅_v^cf(t)+k⋅w^(t),^W*=0

a=_{∂g1∂vP|^x*}=1_Linb=_{∂g1∂iint|^x*}=−_RintLinc=_{∂g1∂vcf|^x*}=−1_Lind=_{∂g2∂vcf|^x*}=0e=_{∂g2∂w|^x*}=_{VP*Rint VBat}f=_{∂g3∂iint|^x*}=1_Cfg=_{∂g3∂iL1|^x*}=−2_{VBatVP* Cf}h=_{∂g3∂vcf|^x*}=1_{Rint Cf}k=_{∂g3∂w|^x*}=−_L12_Cf^{(_{VP*Rint VBat})2}

Applying the Laplace transform to the linearized model (28) with the coefficients of (29), the _V^cf(s)/_V^p(s) transfer function (30) is obtained. This function evidences the impedance matching at DC because V_Cf/V_p = 1/2. As in the previous case, the tracking of v_p(t) is given by a low-pass filter of second order. The system stability is verified because all coefficients of P(s) exist and are positive.

_V^cf(s)_V^P(s)=1_{Lint Cf}^s2[1+_L12_Cf^{(_{VP*Rint VBat})2}]+s{1_{Rint Cf}[1+_L12_Lint^(_VP*VBat)2]+_RintLint}+2_{Lint Cf}

As expected from the nonlinear sliding dynamics (26), in the buck mode, the bandwidth B_-3d, and the natural frequency (31) are not constant, depending on the generator pulse V_P and the converter gain. Realize that in the boost mode (18) the natural frequency was constant.

_ωn2=2_{Lint Cf}⋅^{[1+_L12_Cf(_{VP*Rint VBat})2]−1}

5. Sepic Converter Matching Circuit

In the Sepic converter the controllable inductor is L₁. As in the HBB boost mode, the current through this inductor i_L1(t) corresponds to the converter input current i_LFR(t). This circumstance allows using the same control surface S(x) to force the LFR behavior that was proposed for the HBB boost mode in Section 4. Figure 14 depicts the two converter topologies.

Again, in the previous cases by means of the switching variable u(t) = {0, 1}, the dynamics of the converter at any instant can be described with a single set of differential Equation (32).

x(t)=[_{iintiL1iL2vc1vcf}]⇒{_Lintd_iintdt=_vP−_{Rint iint}−_vcfL1d_{iL 1}dt=_vCf−(1−u)(_vC1+_VBat)_L2d_{iL 2}dt=−_VBat+u(_vC1+_VBat)_Cfd_vCfdt=_iint−_iL1C1d_vC1dt=_iL1−u(_iL1+_iL2)

The sliding surface (33) imposed to the inductor current i_L1(t) assures the impedance matching. The equivalent control (34) is deduced from the Equation set (32), using the surface S(x) and its related existence conditions.

S(x)=_iL1−_vCfRint=0⇒_iLFR=_vcfRintS(x)S˙(x)<0

_ueq(t)=1−_vCf(_vC1+_VBat)+_L1Rint(_vC1+_VBat)⋅d_vCfdt

Forcing to zero the dynamics shown in (32), and considering the equivalent control (34), the equilibrium point (35) and the corresponding ideal dynamics (36) are deduced, which result to be non-linear. As in boost mode, the capacitor voltage at the equilibrium point is again, the half part of the pulse voltage is (V_cf^* = V_p^*/2), corresponding to a situation of impedance matching.

^X*=^{[_Iint*,_IL1*,_IL2*,_VCf*,_VC1*]T}=^{[_VP*2⋅_Rint,_VP*2⋅_Rint,_VP* 24⋅_{Rint VBat},_VP*2,_VP*2]T}

{_g1(x)=d_iintdt=1_Lint[_vP−_{Rint iint}−_vcf]_g2(x)=d_{iL 1}dt=1_Rint d_vCfdt_g3(x)=d_{iL 2}dt=1_L2[_vC1−_vCf+_L1Rint⋅d_vCfdt]_g4(x)=d_vCfdt=1_Cf[_iint−_iL1]_g5(x)=d_vC1dt=1_C1{_iL1+_iL2vC1+_VBat⋅[_vCf−_L1Rint·d_vCfdt]−_iL2}

To evaluate the local stability of the proposed surface, the ideal dynamics (32) should be linearized around the equilibrium point X* using the well-known small signal perturbation model (37), (38). The coefficients of the small signal model are given in (39).

x(t)=^X*+x^(t)

{_g1(x)=d_iintdt=a⋅_v^P(t)+b⋅_i^int(t)+c⋅_v^cf(t)_g2(x)=d_{iL 1}dt=d⋅w^(t)where,w(t)=d_vCfdt_g3(x)=d_{iL 2}dt=e⋅_v^C1(t)+f⋅w^(t) +h⋅_v^Cf(t)_g4(x)=d_vCfdt=j⋅_i^int(t)+k⋅_i^L1(t)_g5(x)=d_vC1dt≈m⋅_i^L2(t)+n⋅_i^L1(t)+p⋅w^(t)+q⋅_v^C1(t)+r⋅_v^Cf(t)

{a=_{∂g1∂vP|^X*}=1_Linte=_{∂g3∂vC1|^X*}=1_L2m=_{∂g5∂iL 2|^X*}=−2_VBatC1(_VP*+2_VBat)b=_{∂g1∂iint|^X*}=−_RintLintf=_{∂g3∂w|^X*}=_{L1L2 Rint}n=_{∂g5∂iL1|^X*}=_VP*C1(_VP*+2_VBat)k=_{∂g4∂iL 1|^X*}=−1_Cfc=_{∂g1∂vCf|^X*}=−1_Linth=_{∂g3∂vCf|^X*}=−1_L2q=_{∂g5∂vC1|^X*}=−_VP* ²2_{Rint C1 VBat}(2_VBat+_VP*)p=_{∂g5∂w|^X*}=−_VP*2_{C1 Rint2 VBat}d=_∂g2δw|^X*=1_Rintj=_{∂g4∂iint|^X*}=1_Cfr=_{∂g5∂vCf|^X*}= _VP*2_{Rint C1 VBat}

After linearizing (38) the ideal dynamics, the small signal _V^cf(s)/_V^p(s) transfer function (40) is obtained, that evidences the perfect impedance matching is at DC, and good at low frequency, like the sea waves-trains. Realize in (40) that at DC V_Cf = 1/2V_p. When frequency increases the module of _V^cf(s)/_V^p(s) decreases from its maximum value at DC (1/2) and the phase-shift between V_Cf and V_p increases. As in the HBB boost mode, the natural frequency is given by L_int and the capacitor C_f.

_{HCf P}(s)=_V^Cf(s)_V^P(s)=1/_{Lint Cf}^s2+s(_Lint+_{Cf Rint} ²_{Cf Rint Lint})+2_{Lint Cf}=12⋅_ωna2^s2+2_{ζa ωna}s+_ωna2

The Sepic converter is a fifth order system, and the ideal system dynamics should be of fourth order, but expression (40) denotes a second order system. The fourth order dynamics consists of two decoupled pairs of complex conjugated poles that can be factorized in two second order functions as seen in (41). The converter is stable, since all its coefficients of D_a(s) and D_b(s) exist and are positive. In a clear contrast to H_CfP(s), where a pair of complex conjugated poles is hidden, in other system transfer functions the whole dynamics are visible. One of these cases is the H_C1P(s) transfer function, where according to expressions (42) and (43), the full dynamics is shown. Realize that s expected, at DC conditions V_Cf = V_C1.

P(s)=(^s2+2_{ζa ωna}s+_ωna2)⋅(^s2+2_{ζb ωnb}s+_ωnb2)=_Da(s)⋅_Db(s)

_HC1P(s)=_V^C1(s)_V^P(s)=_V^C1(s)_V^Cf(s)⋅_V^Cf(s)_V^P(s)=_HC1Cf(s)⋅_HCfP(s)

_HC1Cf(s)=_V^C1(s)_V^Cf(s) =−^s2 _L2(_VP*+2_VBat)+[4_{VBat2 L1}+2_{VP* L2 VBat}+_{VP* L2}(_VP*+2_VBat)]_Rint⋅s+4_{Rint2 VBat2}2^s2[_{Rint2 L2 C1 VBat}(_VP*+2_VBat)]+s⋅_{Rint L2 VP*2}+4_{Rint2 VBat2}

6. Experimental Circuits of the Impedance Adaptors

Figure 15 and Figure 16 depict the Sepic converter and its control board, respectively.

Figure 17 and Figure 18 depict the HBB power stage and its control board. The boost mode HBB and the Sepic control circuits mode are equal, because the control surface is the same, see (17) and (33).

7. Experimental Verification

The performance of both converters, the hybrid and the Sepic, have been simulated and verified experimentally with two types of input signals. The first signal v_p(t) is the pulse waveform provided by the harvesting device of Figure 8. The simulated results are shown in Figure 19a–f. The second signal v_p(t) is a sinusoidal waveform, and the simulated and experimental results are given in Figure 20a–f. Experimental results with a third pulse waveform are in Figure 21a–d.

In the simulation captions of Figure 19, the signals shown are: i_L(t) in blue, v_cf(t) in black, i_int(t) in light green, and finally v_int(t) in red. The currents are shown amplified by a factor 10. Current i_int(t) is the adaptor input current, this is, the absolute value of the harvesting device current. Signal v_int(t) is the absolute value of the harvesting device internal voltage drop, this is v_int(t) = R_int · i_int(t). If impedance matching occurs, then v_int(t) must coincide with v_cf(t), the adaptor input voltage.

The simulated cases are the following: Figure 19a HBB with only the boost mode enabled; Figure 19b HBB only enabling the buck mode; in cases Figure 19c,d the HBB adaptor can work with both modes, but the capacitor value is C_f = 10 μF in case Figure 19c, and C_f = 200 μF in case Figure 19d; and finally cases Figure 19e,f depict the Sepic converter, using C_f = 10 μF in Figure 19e and with C_f = 200 μF in Figure 19f.

As expected, and verified in Figure 19, in the Sepic converter, the adaptor input current i_int(t) and the inductor current i_L1(t) are equal, but in the HBB converter both currents only coincide in the boost mode, in the buck mode both currents are different. The effect of the C_f filter capacitor can also be observed. When its value is 200 μF, v_cf(t) and v_int(t) are different. There is a phase shift between them, and therefore the impedance matching is not perfect. Finally, in Figure 19a,b it can be appreciated that impedance matching occurs only when a converter is switching. In the HBB boost mode, the impedance matching only happens when v_cf(t) < Vbat, see Figure 19a, whereas in the buck mode, impedance matching takes place only when v_cf(t) > Vbat, as can be seen in Figure 19b.

Figure 20 depicts the simulated and experimental results obtained with a sinusoidal waveform for v_p(t). Both converters have been tested at different frequencies from 50 Hz to 200 Hz. Realize that in a sea wave-train, waves are separated 10–20 seconds, implying a wave-train frequency of 0.05 Hz–0.1 Hz. In this way, a wave pulse can stand for 0.25–2.5 seconds, that corresponds to a range of fundamental frequencies of 0.4 Hz–4 Hz. To assure a good behavior with sea waves, the tests have been made at unrealistically high frequencies for the harvesting device. If the tests are satisfactory, the v_Cf(t) tracking behavior of a real sea-wave pulse voltage v_p(t) will deliver even better results.

The signals shown at Figure 20 are the following: The input adaptor voltage v_Cf(t) in shown in dark blue, the inductor current i_L1(t) in cyan color, the HBB buck mode switch gate signal S_buck(t) is shown light green, and finally the Sepic switch, and the HBB boost mode switch gate signals S_sepic(t) and S_boost(t) are shown in pink.

The oscilloscope captions at Figure 20c,d corresponding to the real experimental test at 50 Hz show a good agreement with the corresponding simulated results, also at 50 Hz. The experimental results at 200 Hz, show that at his frequency the filtering effect of C_f = 10 μF does not allow the adaptor input voltage v_Cf(t) to drop at zero volts. Figure 20a shows clearly the dead-zone effect around V_bat = 12 V, where the HBB operating in boost mode changes to buck operation mode and vice-versa. Comparing the HBB and the Sepic converter experimental results, the adaptor input voltage v_Cf(t) exhibit a small ripple in the Sepic case and in the HBB operation mode because the adaptor input current i_int(t) is the inductor one i_L1(t). Conversely, in the HBB buck mode, the adaptor input current i_L1(t) is a pulsating one, and the ripple is higher.

Figure 21 depicts the experimental results using a laboratory programmable power supply that delivers a train of voltage pulses imitating a sea wave-train. Each voltage pulse reproduces the harvesting-device pulse given in Figure 8. This power supply supplies the impedance matching circuit through a series resistor of 26 Ω reproducing the harvesting device output impedance.

The captions in Figure 21a–d show two waveforms. The voltage-drop in the 26 Ω resistance v_int(t) is shown in pink color, whereas the impedance matching circuit input voltage v_Cf(t) is shown in dark blue color. As can be seen in these figures, there is a good agreement between both curves proving the correct behavior of the impedance matching circuit, for both converters: The HBB and the Sepic converter. The remaining captions, Figure 21e,f, show various signals from the HBB and the Sepic converters, respectively. The signals shown in these captions are: v_int(t) in cyan color, v_Cf(t) in dark blue, S_boost(t) and S_sepic(t) in pink color, and finally S_buck(t) in light grey.

The input impedance Z_in, and the η_M matching efficiency (44) for both converters are depicted in Figure 22a–d. The data shown in these graphs have been obtained along with the sinusoidal experiments of Figure 20. Three different frequencies are used: 50 Hz (red), 100 Hz (blue), and 200 Hz (green). The HBB and the Sepic converter input impedances Z_in are plotted, respectively, in Figure 22a,b.

The impedance matching efficiency η_M (44) of both converters is shown Figure 22c,d. That matching efficiency has been defined as the ratio between the power absorbed by the converter, and the power absorbed by the converter in case of an ideal matching, when Z_in = R_int = 26 Ω.

_ηM=_Zin⋅_VP2^(26+_Zin)2⋅^(26+26)226⋅_VP2=104⋅_Zin^(26+_Zin)2

This experimental section ends with two photographs given in Figure 23. Thus, Figure 23a shows the experimental workbench testing the HBB converter, whereas Figure 23b show the same environment with the Sepic converter.

8. Conclusions

Although extracting electric power from sea-waves at great scale is worldwide developing, the extraction of small amounts of energy from the waves, conveniently stored in a battery, can be solution to supply small stand-alone equipment, with occasional and short-lived power consumption peaks: Weather stations, telecom relays, and similar equipment.

The design, simulation, realization, and verification of an inductive harvesting generator that profits the differential position of two sea-surface points has been presented. This device supposes a novelty compared to sea-wave energy buoys and harvesting devices based on the vertical oscillation of a single sea point.

The proposed harvesting generator is made with three transducers, that are based on a coil-magnet arrangement, where a permanent magnet moves linearly inside the coil. The generator operation has also been simulated and verified experimentally.

Next, two switching converters, a hybrid buck-boost cell and a Sepic converter have been proposed, analyzed, and verified experimentally as matching circuits to transfer the energy from the harvesting device to the storage 12 V battery. Two different sliding mode control surfaces are required to guarantee the impedance matching using the HBB converter. Thus, during the boost mode the corresponding surface controls the input current, whereas in the buck mode, the surface controls the output current. Conversely, a clear advantage of the Sepic converter is that only a single surface is needed for the same purpose, because in this case the adaptor circuit input current is always the controllable inductor current. Indeed, the Sepic converter control law is the same that has been used for the boost mode operation in the HBB converter.

The main difference between both adaptor circuits is their behavior when the adaptor input voltage v_Cf(t) coincides with the battery voltage. While the hybrid converter must change its operation mode, nothing happens with the Sepic converter. In the hybrid converter case, there is an input voltage dead-zone where the converter is not switching. Although the switching losses disappear, as the converter becomes uncontrolled, the impedance matching is lost during those instants.

Both converters behave reasonably well, and show good results matching the generator output impedance and the converter input impedance, as shown in the extensive experimental and simulation results. In both cases, as can be seen in Figure 23a,b there is a slight variation of the adaptor input resistance, due the current sensor lack of linearity working in a wide range of current values, the variable delay caused by the mosfet drivers, and finally the converter efficiency, which is not obviously constant along the input voltage range. Anyway, the matching efficiency is always over 95% whatever be the input voltage, as can be seen in Figure 22c,d.

To reduce the input impedance variation in terms of the input voltage, the hysteresis width used to implement the sliding surfaces by means of hysteretic comparators must be reduced. Nevertheless, although this would increase the tracking precision, leading to a slight increment of the matching efficiency, maybe the switching losses would be greatly increased leading to a global reduction of the energy delivered to the storage battery.

In this paper two switching converters operating as impedance matching circuits have been studied. To achieve this, they must be forced to behave like a loss free resistor, and here sliding mode control is introduced as an easy technique to assure this behavior in a wide range of input and output voltage variations. Buck-boost and similar converters in DCM operation are used in the literature for this proposal, in harvesting applications, but the input voltage variation range is usually small. The solution proposed here allows impedance matching and loss free resistor behavior with harvesting generators with a wide input voltage variation range. The simplicity of the sliding mode control technique, its inherent robustness against parametric variation, and input and output disturbances, are the main advantages of the technique proposed here.

These issues, as other related to a different configuration of the harvesting device coils will be the subject of a future research.

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Winding	Winding Name	Winding Terminals	Turns	Resistance (Ω)	Inductance (mH)
1st	L1n	1n–2n	1000	43	50
2nd	L2n	3n–4n	2000	104	250
3rd	L3n	5n–6n	1000	61	85
4th	L4n	7n–8n	2000	138	390
5th	L5n	9n–10n	1000	78	120
full coil	LT	1n–10n	7000	434	3000

From Coil	Windings Association	Resistance (Ω)	Inductance (mH)
A	WA = (L1a + L4a) || (L2a + L5a)	91.15	200
B	WB = (L1b + L4b) || (L2b + L5b)	91.15	200
C	WC = (L1c + L4c) || (L2c + L5c)	91.15	200
A, B, C	WD = L3a + L3b + L3c	183.3	255
device	WA || WB || WC || WD	26.06	52.86

[Ω]	∞	30 s	10 s	3 s	1 s
Re [Zo]	26	26	26	26	26
Re [Z*in]	RLFR	RLFR	RLFR	RLFR	RLFR
Im [Zo]	0	10−2	3.3·10−2	10−1	0.33
Im [Z*in]	0	1.5·10−3	4.5·10−3	1.5·10−2	4.5·10−2

Author Contributions

Conceptualization, J.A.G.-C., and H.V.-B.; methodology, H.V.-B., J.A.G.-C., J.A.B.-R. and À.C.-P.; validation, J.A.G.-C.; formal analysis, H.V.-B. and J.A.G.-C.; investigation, H.V.-B., J.A.G.-C. and J.A.B.-R.; resources, H.V.-B.; writing-original draft preparation, J.A.G.-C., and H.V.-B.; writing-review and editing, H.V.-B., J.A.G.-C., J.A.B.-R, and À.C.-P.; visualization, J.A.G.-C. and À.C.-P.; supervision, H.V.-B.; project administration, H.V.-B.; funding acquisition, H.V.-B.

Funding

This research was funded by the Spanish Agencia Estatal de Investigación (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) under research projects DPI2015-67292-R (AEI/FEDER, UE).

Conflicts of Interest

The authors declare no conflict of interest.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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