Fuel cell system

Abstract

The present invention provides a fuel cell system capable of appropriately measuring the AC impedance of a fuel cell. The fuel cell system is characterized in that a control unit performs on-off control of switches of n phases so that the switches of the n phases are respectively operated at different phases and the switches of the n phases are respectively operated at the same duty ratio, and when it is determined that a specific condition is satisfied, the switches of the n phases are respectively operated at the same phase, and the AC impedance of the fuel cell is measured based on the current waveform and voltage waveform of the fuel cell.

Description

Technical Field

[0001] The present disclosure relates to fuel cell systems. Background Art

[0002] Various studies have been conducted on converters used in systems mounted on vehicles such as fuel cell electric vehicles (FCEVs). A circuit composed of a reactor, a switch, a diode, and a capacitor is often used in a DC / DC converter used in various electronic devices. The DC / DC converter controls the increase or decrease of the current flowing to the reactor according to the on / off signal of the switch.

[0003] For example, Patent Document 1 discloses a fuel cell system that measures the AC impedance of a fuel cell stack with high accuracy.

[0004] Furthermore, Patent Document 2 discloses a fuel cell system mounted on a vehicle that can measure the impedance of a fuel cell with high accuracy without being affected by load fluctuations of auxiliary machines.

[0005] In addition, Patent Document 3 discloses an impedance measurement method for calculating impedance with high accuracy in a short time using a simple device and structure.

[0006] Patent Document 4 discloses a fuel cell system that uses a bidirectional buck-boost converter provided with a dead time to accurately apply an alternating current to a fuel cell for measuring the impedance of the fuel cell.

[0007] Patent Document 1: Japanese Patent Application Publication No. 2008-098134

[0008] Patent Document 2: Japanese Patent Application Publication No. 2014-232681

[0009] Patent Document 3: Japanese Patent Application Publication No. 2013-145692

[0010] Patent Document 4: Japanese Patent Application Publication No. 2014-235781

[0011] As one of the indicators for controlling the operating state of the fuel cell to be optimal, the AC impedance of the fuel cell is used. The converter performs step-up and step-down control on the output voltage of the fuel cell according to the switching action performed by the switching element (switch). It is known that there is a region of reduced response performance relative to the duty cycle change in the converter. If the AC impedance of the fuel cell is measured in such a region of reduced response performance, the accuracy of the superposition of the high-frequency signal of the fuel cell by the converter is reduced, thereby causing the inconvenience of significantly reduced AC impedance measurement accuracy.

[0012] As a goal of miniaturization of the converter, as an option, a magnetic coupling converter with a magnetic coupling reactor that magnetically couples multiple coils on the same core can be considered. In a magnetic coupling converter, in order to alleviate the magnetic saturation of the core of the reactor and reduce the output current fluctuation, the magnetically coupled coils are generally switched on and off alternately with the same duty ratio and at equal intervals with a phase difference. When the output of the fuel cell is low, the magnetic coupling reactor operates in a discontinuous mode including the time when the current flowing in the coil becomes zero.

[0013] The researchers have newly discovered the existence of a "dead zone" in which the negative current from other coils caused by mutual inductance, which is a phenomenon unique to magnetic coupling reactors, becomes a pseudo switch-on state when operating in the discontinuous mode of current discontinuity, and the switch control from off to on cannot be smoothly switched, and the output current value of the fuel cell does not change even if the duty cycle is changed. In the dead zone, even if the duty cycle is changed to apply an AC signal for AC impedance measurement, the output current value of the fuel cell hardly changes as described above, so the AC impedance of the fuel cell cannot be properly measured. This is equivalent to the response performance reduction area of ​​the above-mentioned patent document 1, but the frequent occurrence of time periods in which the AC impedance is not measured or the time period in which the AC impedance cannot be measured may result in the inability to optimally control the operating state of the fuel cell. Summary of the invention

[0014] The present disclosure has been made in view of the above-mentioned actual situation, and a main object thereof is to provide a fuel cell system capable of appropriately measuring the AC impedance of a fuel cell.

[0015] The fuel cell system disclosed in the present invention is characterized in that the above-mentioned fuel cell system includes a fuel cell and a converter that performs at least one selected from the group consisting of stepping up and stepping down the output voltage of the fuel cell, the above-mentioned converter includes an inductor that magnetically couples coils of n phases to each other, switches of n phases that are respectively connected to the above coils, and a control unit, wherein n is an integer greater than or equal to 2, the above-mentioned control unit performs on-off control of the above-mentioned switches of n phases, the above-mentioned control unit monitors the current value of the above coils, the above-mentioned control unit measures the AC impedance of the above-mentioned fuel cell based on the current waveform and voltage waveform of the above-mentioned fuel cell, the above-mentioned control unit operates the above-mentioned switches of n phases at different phases, and operates the above-mentioned switches of n phases at the same duty ratio, and the above-mentioned control unit operates the above-mentioned switches of n phases at the same phase when it is determined that the following condition 1 is satisfied, wherein the condition 1 is an operating condition that the above-mentioned switches connected to the coil of at least one other phase are switched from on to off when the current flowing in the above coil is in a discontinuous mode after the above-mentioned switches of n phases are respectively operated at different phases and the current value flowing to the above coil of one phase is maintained at zero.

[0016] In the fuel cell system of the present disclosure, the control unit may operate the switches of the n phases at a phase difference of (360 / n)° when operating the switches of the n phases at different phases.

[0017] It can also be configured as follows: based on the fuel cell system disclosed in the present invention, the above-mentioned control unit pre-stores the operating conditions of the above-mentioned converter equivalent to the above-mentioned condition 1 as a data group, and the above-mentioned control unit compares the current operating conditions of the above-mentioned converter with the data group to determine whether the above-mentioned condition 1 is satisfied.

[0018] It can also be constructed as follows: on the basis of the fuel cell system disclosed in the present invention, the above-mentioned control unit confirms whether there is a requirement for measuring the AC impedance of the above-mentioned fuel cell when it is determined that the above-mentioned condition 1 is met. When it is confirmed that there is a requirement for measuring the AC impedance of the above-mentioned fuel cell, the above-mentioned control unit causes the above-mentioned switches of the n phases to operate in the same phase respectively, and measures the AC impedance of the above-mentioned fuel cell.

[0019] It can also be configured as follows: based on the fuel cell system disclosed in the present invention, the control unit causes the switches of the n phases to operate in the same phase and measures the AC impedance of the fuel cell, and then determines whether the average current value flowing in the coil of the reactor is greater than a predetermined threshold value, and when it is determined that the average current value flowing in the coil of the reactor is greater than the predetermined threshold value, causes the switches of the n phases to operate in different phases.

[0020] It can also be configured as follows: based on the fuel cell system disclosed in the present invention, when n=2 and the switches of the two phases magnetically coupled to each other are operated with a phase difference of 180 degrees, when the current flowing in the coil is in a discontinuous mode and the duty ratio of the switches of the two phases is less than 50% (D<0.5), and when either the following formula (A) or the formula (B) is satisfied, the control unit determines that the above condition 1 is satisfied,

[0021] Formula (A): D<{(1 / 2)(L-M)(V H -V L )} / (LV L +MV L -MV H )

[0022] Formula (B): D<(1 / 2){1-(V L / V H )}

[0023] [In equations (A) and (B), L is the self-inductance of the reactor, M is the mutual inductance of the reactor, V H is the output voltage of the above converter, V Lis the input voltage of the above converter, and D is the duty cycle (-). ]

[0024] According to the fuel cell system of the present disclosure, the AC impedance of the fuel cell can be appropriately measured. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 FIG. 1 is a diagram showing an example of a circuit configuration of a fuel cell system including a boost converter and peripheral components.

[0026] Figure 2 It means that the input voltage Vfc is 200V and the output voltage V H The figure shows the current change of the average current value (single-phase average current value) flowing to each coil of the reactor when the voltage is 350 V, the step-up ratio is constant, and the two-phase coils that are magnetically coupled to each other are driven to gradually increase the duty ratio.

[0027] Figure 3 1 is a diagram showing current waveforms of respective two-phase coils that are magnetically coupled in a region where a dead zone occurs.

[0028] Figure 4 The figure shows the current waveform when the coils of two phases magnetically coupled to each other are driven to superimpose 270 Hz sine waves with an amplitude of ±3 A on each phase and the DC load current is swept within the range of 5 to 25 A.

[0029] Figure 5 1 is a diagram showing an example of a two-phase magnetic coupling boost circuit of the U phase and the V phase.

[0030] Figure 6 1 is a diagram showing an example of a two-phase magnetic coupling step-down circuit of U phase and V phase.

[0031] Figure 7 1 is a diagram showing an example of a two-phase magnetic coupling buck-boost circuit of the U phase and the V phase.

[0032] Figure 8 This is a diagram showing an example of a current waveform when the current flowing through the coil of the reactor of the boost converter is in a continuous mode.

[0033] Fig. 9 This is a diagram showing an example of a current waveform when the current flowing through the coil of the reactor of the boost converter is in a discontinuous mode.

[0034] Fig.10 This is a diagram showing the relationship between the single-phase average current flowing through each coil of the reactor and the PWM duty ratio in the magnetic coupling converter.

[0035] Fig.11 This is a diagram showing an example of a time chart of a duty ratio, current, and voltage.

[0036] Fig.12 This is a diagram showing an example of a reactor current waveform when a converter including two-phase magnetic coupling reactors of the U phase and the V phase is driven in the same phase.

[0037] Fig.13 This is a diagram showing an example of a reactor current waveform when a converter including two-phase magnetic coupling reactors of the U phase and the V phase is driven with a phase difference of 180°.

[0038] Fig.14 This is a diagram showing an example of a reactor current waveform when the same-phase driving and 180° phase difference driving of a converter including two-phase magnetic coupling reactors of the U phase and the V phase are alternately performed.

[0039] Fig.15 This is a flowchart showing an example of control when measuring the AC impedance of a fuel cell in a fuel cell system including a converter having an n-phase magnetic coupling reactor.

[0040] Fig.16 This is a flowchart showing another example of control when measuring the AC impedance of a fuel cell in a fuel cell system including a converter having an n-phase magnetic coupling reactor.

[0041] Fig.17 This is a flowchart showing an example of control after measuring the AC impedance of a fuel cell in a fuel cell system including a converter having an n-phase magnetic coupling reactor.

[0042] Fig.18 It is a diagram showing a dead zone generation region in the discontinuous mode.

[0043] Fig.19 This is a flowchart showing an example of control when measuring the AC impedance of a fuel cell in a fuel cell system including a converter having a two-phase magnetic coupling reactor.

[0044] Description of Reference Numerals

[0045] 10… fuel cell; 20… boost converter; 21… reactor; 22… current sensor; 23… switch; 24… diode; 25… capacitor; 50… external load. DETAILED DESCRIPTION

[0046] The fuel cell system disclosed in the present invention is characterized in that the above-mentioned fuel cell system includes a fuel cell and a converter that performs at least one selected from the group consisting of stepping up and stepping down the output voltage of the fuel cell, the above-mentioned converter includes an inductor that magnetically couples coils of n phases to each other, switches of n phases that are respectively connected to the above coils, and a control unit, wherein n is an integer greater than or equal to 2, the above-mentioned control unit performs on-off control of the above-mentioned switches of n phases, the above-mentioned control unit monitors the current value of the above coils, the above-mentioned control unit measures the AC impedance of the above-mentioned fuel cell based on the current waveform and voltage waveform of the above-mentioned fuel cell, the above-mentioned control unit operates the above-mentioned switches of n phases at different phases, and operates the above-mentioned switches of n phases at the same duty ratio, and the above-mentioned control unit operates the above-mentioned switches of n phases at the same phase when it is determined that the following condition 1 is satisfied, and the condition 1 is an operating condition that the above-mentioned switches connected to the coil of at least one other phase are switched from on to off when the current flowing in the above coil is in a discontinuous mode and the current value flowing to the above coil of one phase is maintained at zero after the above-mentioned switches of n phases are respectively operated at different phases.

[0047] In a magnetic coupling converter, a dead zone is sometimes generated in which the output current value of the fuel cell does not change even if the duty cycle of the switch is changed. In this case, the AC impedance of the fuel cell cannot be properly measured. In the present disclosure, the dead zone, which is a phenomenon unique to the magnetic coupling inductor, is controlled in a manner to minimize the frequency of occurrence, and the AC impedance of the fuel cell is measured. As a specific control, under the condition of generating the dead zone, the coils of the magnetically coupled phases are switched with a common duty cycle and the same phase, thereby avoiding the dead zone and properly measuring the AC impedance of the fuel cell. By switching the coils of the magnetically coupled phases with a common duty cycle and the same phase, at least one phase is out of the range of the dead zone, so that the phase out of the dead zone can change the current. As a result, the AC impedance of the fuel cell can be properly measured, so that the operating state of the fuel cell can be optimally controlled.

[0048] Figure 1 FIG. 1 is a diagram showing an example of a circuit configuration of a fuel cell system including a boost converter and peripheral components.

[0049] Figure 1 The fuel cell system shown is mounted on a vehicle, for example, and a driving motor of the vehicle is connected via an inverter as an external load 50. In addition, although not shown, a battery may be provided in parallel with the fuel cell 10 and the boost converter 20. The output power of the fuel cell 10 is further converted from direct current to alternating current by the inverter after being boosted by the boost converter 20, and is supplied to the motor.

[0050] The boost converter 20 includes six-phase boost circuits connected in parallel. Figure 1 , a configuration including a 6-phase boost circuit is shown, but the number of phases is not particularly limited.

[0051] The boost circuit includes a reactor 21, a current sensor 22, a switch 23, a diode 24, and a capacitor 25. The boost circuit may include an input voltage sensor and an output voltage sensor.

[0052] In the 6-phase boost circuit, each two phases can share the core of one reactor 21 and be magnetically coupled to each other. In each boost circuit, if the switch 23 is turned on, the current flowing in the reactor 21 increases, and if the switch 23 is turned off, the current flowing in the reactor 21 decreases, and when the current reaches zero, it is maintained at zero. The current sensor 22 obtains the current value flowing in the reactor 21.

[0053] The switch 23 is turned on and off by a control unit (not shown) to control the voltage step-up ratio in the converter 20 and the output current value from the fuel cell 10 .

[0054] The output power of the fuel cell 10 varies greatly depending on the requirements of the vehicle (speed, acceleration, load, and road slope, etc.), and the output current also varies greatly accordingly. When the output current of the fuel cell 10 is large, if the current is allowed to flow to a boost circuit, the heat generation increases and the power conversion efficiency of the boost converter 20 decreases. In addition, when only a small current flows in the boost circuit that can withstand a large current, the energy loss also increases and the power conversion efficiency of the boost converter 20 decreases.

[0055] Therefore, the boost converter 20 includes a multi-phase boost circuit (in Figure 1 The boost converter 20 switches the number of phases to be driven according to the output current value of the fuel cell 10. For example, when the output current value of the fuel cell 10 is 0 to 150A, it is driven with two phases, when it is 150 to 300A, it is driven with four phases, and when it is 300 to 600A, it is driven with six phases. The efficiency of the boost circuit varies depending on the current flowing, and by changing the number of drive phases, it can operate with the best efficiency in each current range.

[0056] The fuel cell system includes a fuel cell and a converter.

[0057] A fuel cell may have only one unit cell, or may be a fuel cell stack in which a plurality of unit cells are stacked.

[0058] The converter performs at least one of a step-up and a step-down on the output voltage of the fuel cell. The converter may be a step-up converter, a step-down converter, or a step-up / step-down converter.

[0059] The converter may also be a DC / DC converter.

[0060] The converter includes a reactor for magnetically coupling coils of n phases (n is an integer greater than or equal to 2), an n-phase switch connected to each coil, and a control unit. The converter may also include a diode, a current sensor, a voltage sensor, a photocoupler, a capacitor, and the like.

[0061] The reactor has a coil and an iron core.

[0062] The core may be wound with coils of n phases (n is an integer greater than or equal to 2). If n is greater than or equal to 2, the upper limit is not particularly limited and may be less than or equal to 10, less than or equal to 5, less than or equal to 4, or less than or equal to 3.

[0063] The core and coil of the reactor may be the same as those used in a conventionally known converter.

[0064] In the present disclosure, a reactor having an iron core around which an independent coil is wound is referred to as a non-magnetic coupling reactor. In the present disclosure, a converter having a non-magnetic coupling reactor is referred to as a non-magnetic coupling converter. In the present disclosure, a reactor having an iron core around which two or more independent coils are wound is referred to as a magnetic coupling reactor. In the present disclosure, a converter having a magnetic coupling reactor is referred to as a magnetic coupling converter.

[0065] In the present disclosure, an independent coil refers to a coil including one or more scroll parts and two terminal parts.

[0066] The switch (switching element) is composed of a transistor and a protective diode. The transistor is an npn type transistor, and may be, for example, an IGBT (Insulated Gate Bipolar Transistor), a MOSFET (Metal Oxide Semiconductor Field Effect Transistor), or the like.

[0067] As the diode, a diode used in a conventionally known converter may be used.

[0068] The current sensor is not particularly limited as long as it can obtain the current value flowing in the coil of the reactor (sometimes referred to as the reactor current value), and a conventionally known ammeter can be used. In addition, in the present disclosure, the reactor average current refers to the average current flowing in the coil during the switching cycle of the duty cycle control.

[0069] The voltage sensor is not particularly limited as long as it can obtain an input voltage output from the fuel cell and input to the converter, and a conventionally known voltmeter or the like can be used.

[0070] The control unit may be an electronic control unit (ECU: Electric Control Unit) etc. The ECU is configured to include a CPU (Central Processing Unit), a memory, and an input / output buffer.

[0071] The control unit detects the value of the current flowing through the coil of the n-phase of the reactor based on the signal from the current sensor, and monitors the current value of the coil.

[0072] The control unit performs on-off control of the n-phase switches. The control unit may operate the magnetically coupled n-phase switches at a constant frequency.

[0073] The control unit may also perform on-off control of the switch by periodically switching between an on command and an off command to the switch, thereby controlling the duty ratio of the n-phase switch, thereby controlling the output current value from the fuel cell.

[0074] In a boost converter, the voltage is boosted by repeatedly performing the switching operation of accumulating and releasing the electric energy in the reactor. The duty ratio determines the proportion of the storage period in which the switch element is turned on and the electric energy is stored in the reactor during the switching cycle of the boost operation. When the switching cycle of the boost operation in the boost converter is T, and the period during which the switch element is turned on is T ON , and set the off period to T OFF When the duty cycle D is expressed as D(-)=T ON / T. For convenience, the duty cycle may be expressed as a percentage (%). In this case, D(%) = T ON / T×100.

[0075] In the present disclosure, a switching cycle (switching cycle) refers to a period from a time point when a switch is switched from OFF to ON to a time point when the switch is switched from OFF to ON again.

[0076] The control unit operates the switches of the n phases at different phases, and operates the switches of the n phases at the same duty ratio.

[0077] When operating the switches of the n phases at different phases, the control unit may operate the switches of the n phases at a phase difference of (360 / n)°.

[0078] In the case of two-phase switching and the two-phase switches are operated at different phases, the control unit may operate the two-phase switches at a phase difference of 30 to 180 degrees, or may operate at a phase difference of 180 degrees, i.e., opposite phases, from the perspective of improving power conversion efficiency. In the case of operation at opposite phases, a phase error may be generated within the range of -5 degrees to +5 degrees.

[0079] When determining that the following condition 1 is satisfied, the control unit operates the switches of the n phases in the same phase.

[0080] Condition 1: When the switches of the n phases are operated in different phases respectively, when the current flowing in the coil is in a discontinuous mode and the current value flowing to the coil of one phase is maintained at zero, the switch connected to the coil of at least one other phase is switched from on to off.

[0081] In the case of a magnetically coupled inductor with more than three phases, the operating condition in which the current value flowing to the above-mentioned coil of one phase is zero and the switch connected to the other coils is switched from on to off can also be the operating condition in which the switch of at least any one of the remaining phases is switched from on to off.

[0082] When the above-mentioned switches of n phases are operated at different phases, and when the n phases magnetically coupled to each other are driven with a phase difference of (360° / n), condition 1 may be a discontinuous mode and the duty ratio becomes less than (100-100 / n)%.

[0083] The control unit may store converter operating conditions corresponding to condition 1 as a data group in advance, and compare the current converter operating conditions with the data group to determine whether the condition 1 is satisfied. This avoids entering the dead zone, and improves the accuracy of the AC impedance measurement of the fuel cell.

[0084] The control unit may be configured such that, after operating the switches of the n phases at the same phase and measuring the AC impedance of the fuel cell, it determines whether an average current value flowing through the coil of the reactor is equal to or greater than a predetermined threshold value.

[0085] The configuration may be such that, when it is determined that the average current value flowing through the coil of the reactor is equal to or greater than a predetermined threshold value, the switches of the n phases are operated at different phases.

[0086] When the switches of the n-phases that are magnetically coupled to each other are operated in the same phase, if the output current of the fuel cell increases, it moves to the continuous mode. If the phase is switched from the same phase to a different phase immediately thereafter, it enters the range of condition 1. In addition, if the continuous mode is driven in the same phase, the decompression conversion efficiency is poor. Therefore, it can also be configured such that when it is determined that the output current of the fuel cell increases and the average current value flowing to the coil of the reactor is above a predetermined threshold value, the switches are operated in different phases respectively. Thereby, the generation of dead zones can be suppressed and the voltage conversion efficiency can be improved.

[0087] The predetermined threshold value of the average current value flowing through the coil of the reactor may be, for example, 3 to 15A.

[0088] [Dead Zone Description]

[0089] The "dead zone" is a region where the average current flowing to the coil of the reactor hardly changes even if the PWM duty cycle of the converter is increased or decreased. The negative current caused by the mutual inductance of magnetic coupling appears in a part of the discontinuous mode region with a duty cycle of less than 50% in a two-phase magnetically coupled unidirectional step-up (step-down) circuit.

[0090] Figure 2 It means that the input voltage Vfc is 200V and the output voltage V H The current changes of the average current value (single-phase average current value) flowing to each coil of the reactor when the voltage is 350V, the step-up ratio is constant, and the two-phase coils are driven to gradually increase the duty cycle. Figure 2 In the example, the self-inductance of the coil is 96.4μH, the mutual inductance of the coil is 62.7μH, and the driving frequency of the switch is 20kHz.

[0091] According to the state equation for calculating the average current value flowing to the coil of the reactor (described later), Figure 2 As shown by the dotted line, the average current value flowing to the coil of the reactor should increase monotonically as the duty cycle increases, but in fact, as shown by the solid line, the current increases in steps, and there is a "dead zone" in the portion surrounded by the dotted line where the average current value flowing to the coil of the reactor does not increase even if the duty cycle is increased. When the average current value flowing to the coil of the reactor is in the dead zone, the average current value flowing to the coil of the reactor, that is, the output current value of the fuel cell, does not change even if the duty cycle is changed, and the AC impedance of the fuel cell cannot be properly measured.

[0092] Figure 3 : is a diagram showing the current waveforms of the two-phase coils that are magnetically coupled in the region where the dead zone occurs. Figure 3 In the example, the current of the U phase is marked as L1, and the current of the V phase is marked as L2.

[0093] During the period from time t0 to t1, the switch of the V phase is on, and the L2 current gradually increases. On the other hand, the switch of the U phase is off, and the L1 current is maintained at 0. At time t1, the switch of the V phase is switched from on to off, and the L2 current of the V phase decreases. Here, the switch of the U phase is maintained off, but after time t1, the L1 current flowing to the U phase decreases and increases alternately due to the interaction of the V phase.

[0094] Here, for example, even if the switch of the U phase is turned on when the L1 current of the U phase increases as at time t2, the switch-on signal is not recognized because the current has already increased. Thereafter, at time t3, the L1 current of the U phase decreases and the switch-on of the U phase is recognized for the first time, and then the L1 current increases until the switch of the U phase is turned off. Since the timing of not recognizing the switch-on signal and the timing of recognizing the switch-on signal appear alternately, Figure 2 As shown, dead zones occur repeatedly.

[0095] Furthermore, as described above, when the current value of the coil of one phase being magnetically coupled is 0, the switch of the other phase is switched from on to off, the current increases and decreases in the coil of one phase, resulting in a dead zone.

[0096] Therefore, as in condition 1, when the above-mentioned switches of the n phases are driven in different phases respectively, the current flowing in the above-mentioned coil is in a discontinuous mode and the current value of the coil of one phase that is magnetically coupled is maintained at 0, and when the operating condition is such that the switch of the other phase is switched from on to off, it can be said that a dead zone is generated.

[0097] In the case of two-phase magnetic coupling, the phase difference between the two phases can also be 180°. In the case of driving the two phases with a phase difference of 180°, if the discontinuous mode is used and the duty cycle is less than 50%, a dead zone is generated when the average current value of the coil of one phase to be magnetically coupled is maintained at 0 and the switch of the other phase is switched from on to off.

[0098] Figure 4 This is a diagram showing the current waveform when the two-phase coils that are magnetically coupled to each other are driven to superimpose a 270 Hz sine wave with an amplitude of ±3A on each phase and sweep the DC load current within the range of 5 to 25A. Figure 4 In, with Figure 2 The same, the input voltage Vfc is 200V, the output voltage V H The voltage is 350V, the step-up ratio is constant, the self-inductance of the coil is 96.4μH, the mutual inductance of the coil is 62.7μH, and the drive frequency of the switch is 20kHz. Figure 4In the case of , it can be considered that the relationship between the duty ratio and the average current value flowing to the coil of the reactor is along the Figure 2 The dotted line shown in FIG. 1 is used to indicate the duty cycle amplitude. However, as mentioned above, the actual current value changes as Figure 2 The solid line shown is step-like, so even in Figure 2 In the dead zone indicated by the dotted line, the current will not change even if the duty cycle is increased or decreased. Figure 4 As shown by the dotted line, there is a region in the current where the sine waves cannot overlap, and the AC impedance of the fuel cell cannot be measured in this region. Figure 4 When a sudden change in the DC load occurs in a specific region indicated by a dotted line, the output response of the fuel cell may be delayed, and the load on the battery or the like may increase.

[0099] Therefore, as condition 1, when the current waveform does not overlap a sine wave when the switches of the n phases are operated at different phases and the switches of the n phases are operated at the same duty ratio, it can be determined that the current has entered the dead zone. That is, by determining whether the current waveform overlaps a sine wave when the switches of the n phases are operated at different phases and the switches of the n phases are operated at the same duty ratio, it can be determined whether the current has entered the dead zone.

[0100] Considering Figure 4 , the control unit may also increase or decrease the duty ratio of the n-phase switch in such a manner that the amplitude of the coil current value is the normal amplitude, and determine that condition 1 is satisfied when the amplitude of the measured coil current value is smaller than the expected value. When the amplitude of the coil current value is smaller than the expected value, it is equivalent to a state where the sine wave is not superimposed, and it can be determined that it has entered the dead zone.

[0101] Figure 5 : is a diagram showing an example of a two-phase magnetic coupling boost circuit of the U phase and the V phase. Figure 5 In, V L Indicates the input voltage (voltage before boost), I U Indicates the U phase current, I V Represents V phase current, V H represents the output voltage (voltage after boosting), D represents a diode, S represents a switch, M represents mutual inductance, L represents self-inductance, and r represents internal resistance.

[0102] Formula (1) is the state equation of the current I vector. By solving this state equation, we can obtain Figure 2 The current curve is shown by the dotted line in the equation (1). The voltage V vector in equation (1) is shown as in Table 1. Table 1 shows the voltage V vector in equation (1) and the dead zone generation conditions (shaded part).

[0103] Cause Figure 5The negative current caused by the mutual inductance of the magnetic coupling represented by M in the formula (1) appears as a dead zone in a part of the discontinuous mode region where the duty ratio is less than 50% in a two-phase magnetically coupled unidirectional step-up (step-down) circuit, as shown in the diagonal portion of Table 1, when the switches of the two phases are in the off state, that is, when the current flowing to one coil is 0A and the current flowing to the other coil is positive, the coil on the 0A side generates an electromotive force due to the mutual inductance and generates a negative current. Thereafter, the electromotive force of the coil disappears due to the increase of the negative current and immediately turns to decrease, but during the period until it returns to 0A, the switch becomes a pseudo-on state as shown in the diagonal portion of Table 1, so whether the switch on the negative current side is on or off, it cannot affect the current waveform, thus becoming a control dead zone.

[0104] For example, the dead zone generation condition in the case of a three-phase magnetic coupling boost circuit of the U phase, the V phase, and the W phase assumes that all switches of the three phases are off, I U >0,I V ≤0, IW≤0, I U >0,I V >0, IW≤0, etc.

[0105] [Formula 1]

[0106]

[0107] [Table 1]

[0108]

[0109] Figure 6 1 is a diagram showing an example of a two-phase magnetic coupling step-down circuit of U phase and V phase.

[0110] Figure 7 1 is a diagram showing an example of a two-phase magnetic coupling buck-boost circuit of the U phase and the V phase.

[0111] The problem of current control dead zone exists not only in boost circuits, but also in buck circuits and buck-boost circuits, just like boost circuits. By driving the same phase when dead zone occurs, the dead zone can be avoided or suppressed. In addition, for bidirectional circuits without discontinuous mode, it can be considered that there is no dead zone.

[0112] [Explanation of continuous mode and discontinuous mode]

[0113] Figure 8 This is a diagram showing an example of a current waveform when the current flowing through the coil of the reactor of the boost converter is in a continuous mode.

[0114] Fig. 9 This is a diagram showing an example of a current waveform when the current flowing through the coil of the reactor of the boost converter is in a discontinuous mode.

[0115] like Figure 8 As shown in FIG. 1 , the current (reactor current) flowing to the coil of the boost converter reactor becomes a triangular wave along with the switching action, and the central value of the triangular wave is the reactor average current (hereinafter referred to as average current). Here, if the duty is reduced to reduce the average current, the lowest point of the triangular wave reaches 0A. If the average current is further reduced, since the boost converter is a unidirectional circuit, Fig. 9 As shown, a period in which the reactor current becomes zero begins to be generated. Such an operation in which the current flowing to the coil of the reactor is zero during the switching cycle of the duty cycle control of the converter is called a discontinuous mode, and an operation in which the current flowing to the coil of the reactor is not zero during the switching cycle of the duty cycle control is called a continuous mode.

[0116] Fig.10 : is a diagram showing the relationship between the single-phase average current flowing to each coil of the reactor and the PWM duty ratio in the magnetic coupling converter. Fig.10 In the figure, the single-phase average current value flowing to each coil of the reactor when the pre-boost voltage is changed and the duty cycle is changed relative to the post-boost voltage of 350 V. When the pre-boost voltage is a lower voltage below 140 V, a "dead zone" appears around the duty cycle of 35-30% where the single-phase average current value does not increase even if the duty cycle is increased.

[0117] Fig.11 This is a diagram showing an example of a time diagram of a duty ratio, current, and voltage.

[0118] like Fig.11 As shown in FIG. 1 , in the dead zone, even if the duty ratio is increased or decreased with a constant amplitude, the FC output current value and the FC output voltage value will hardly change. For example, in the dead zone, the current can be increased or decreased by making the amplitude of the duty ratio larger than normal, but this may vary due to differences in products. Fig.10 When the characteristics deviate, there is a possibility that the amplitude of the current becomes excessively large and the controllability deteriorates.

[0119] For example, consider the case where the FC voltage value is 200V, the converter output voltage is 350V, and the FC current value is 20A. If driven in a single phase, the current fluctuation to the outside becomes large, so if two-phase 180-degree phase staggered driving is performed, for example Fig.10 In that case, the 10A dead zone may be captured, and the current control accuracy is reduced.

[0120] Fig.12 This is a diagram showing an example of a reactor current waveform when a converter including two-phase magnetic coupling reactors of the U phase and the V phase is driven in the same phase.

[0121] Fig.13 This is a diagram showing an example of a reactor current waveform when a converter including two-phase magnetic coupling reactors of the U phase and the V phase is driven with a phase difference of 180°.

[0122] Fig.14 This is a diagram showing an example of a reactor current waveform when the same-phase driving and 180° phase difference driving of a converter including two-phase magnetic coupling reactors of the U phase and the V phase are alternately performed.

[0123] Here, if Fig.13 By setting the same phase drive area according to the logic, the currents of the two phases are consistent, and the "area where only one of the two phases becomes zero current when both are disconnected" which is a condition for the dead zone can be avoided, and the influence of the dead zone can be improved. Since there is no dead zone, even if a sine wave is applied to the FC output current for impedance measurement, the converter can be stably controlled.

[0124] like Fig.12 As shown, for the driving waveform with the same phase, Fig.13 The current oscillation caused by the reverse current generated when the UV phase is disconnected and the current remains in a single phase, which was observed in the case of 180° phase difference driving, disappears.

[0125] [Measurement of AC impedance]

[0126] The control unit measures the AC impedance of the fuel cell based on the current waveform and the voltage waveform of the fuel cell.

[0127] It can also be configured as follows: the control unit confirms whether there is a requirement for measuring the AC impedance of the fuel cell, and when it is confirmed that there is a requirement for measuring the AC impedance of the fuel cell, determines whether condition 1 is satisfied, and when it is determined that condition 1 is satisfied, causes the switches of the n phases to operate in the same phase respectively, and measures the AC impedance of the fuel cell.

[0128] The control unit may also be configured such that, when it is determined that condition 1 is satisfied, the control unit checks whether there is a request for measuring the AC impedance of the fuel cell, and when it is confirmed that there is a request for measuring the AC impedance of the fuel cell, the switches of the n phases are operated in the same phase to measure the AC impedance of the fuel cell.

[0129] In order to understand the state of the electrolyte membrane of the fuel cell and the state of gas supply, the control unit performs AC impedance measurement of the fuel cell at a predetermined frequency during the operation of the fuel cell.

[0130] The control unit causes the PWM duty cycle of the converter to increase or decrease and switch periodically, obtains the output voltage and output current values ​​of the fuel cell when a load current containing a certain frequency component is applied as time series waveform data with a wavelength or longer, performs discrete Fourier transform on the waveform data, divides the discrete Fourier transform result of the voltage signal by the discrete Fourier transform result of the current signal, and thereby calculates the AC impedance of the fuel cell.

[0131] In a low load region called a discontinuous mode, if the duty ratio is increased, the average current value flowing through the coil of the reactor also increases.

[0132] In order to obtain the AC impedance of the fuel cell, the duty ratio may be controlled so that the output current value becomes a sine wave with an amplitude of about ±3A, for example.

[0133] In addition, the output voltage value of the fuel cell can be obtained by the voltage of the entire fuel cell stack or the voltage of each single cell. If the voltage value of the entire fuel cell stack is used, the AC impedance of the entire fuel cell stack can be obtained, and if the voltage value of each single cell is used, the AC impedance of each single cell can be obtained.

[0134] Furthermore, if the voltage is obtained for each of a plurality of cells (for example, every two cells, every four cells, etc.), the AC impedance of each arbitrary cell block can be obtained.

[0135] The single cell area is several hundred cm 2 In the case of about 1.5 Hz, the component of 200 Hz or more of the AC impedance obtained during power generation mainly represents the proton transfer resistance and contact resistance of the electrolyte membrane, and the component of several tens of Hz represents the gas diffusion resistance.

[0136] In addition, the method for measuring the AC impedance of the fuel cell in the present disclosure is not particularly limited, and a known method can be applied, and for example, the same method as that described in Japanese Patent Application Laid-Open No. 2008-098134 may be used.

[0137] Fig.15 This is a flowchart showing an example of control when measuring the AC impedance of a fuel cell in a fuel cell system including a converter having an n-phase magnetic coupling reactor.

[0138] like Fig.15As shown, the control unit operates the switches of the n phases at different phases and the same duty ratio. Thereafter, the control unit determines whether condition 1 is satisfied. If condition 1 is satisfied, the control unit changes the operation of the switches of the n phases from the operation at the same duty ratio and different phases to the operation at the same duty ratio and the same phase, and measures the AC impedance of the fuel cell. On the other hand, if condition 1 is not satisfied, the control unit operates the switches of the n phases at the same duty ratio and different phases, and measures the AC impedance of the fuel cell.

[0139] In the case of a two-phase magnetically coupled inductor, when the fuel cell system is in operation and enters the dead zone corresponding to condition 1, the two phases are controlled with the same phase when the coils of the two phases that are magnetically coupled to each other are driven to avoid the two phases entering the dead zone at the same time. Figure 2 The dead zone shown enables at least one phase to maintain a state where it can escape the dead zone and be controlled, and sin wave superposition can be performed.

[0140] In the case of a two-phase magnetic coupling reactor, the region corresponding to the dead zone generated in condition 1 may also be when the discontinuous mode and the duty ratio are less than 50%. Therefore, for example, it may be configured such that when the required output to the fuel cell decreases from the discontinuous mode and the duty ratio being 50% or more to less than 50%, the operation of the two-phase switches is changed from the operation with the same duty ratio and different phases to the operation with the same duty ratio and the same phase.

[0141] Fig.16 This is a flowchart showing another example of control when measuring the AC impedance of a fuel cell in a fuel cell system including a converter having an n-phase magnetic coupling reactor.

[0142] like Fig.16 As shown, the control unit operates the switches of the n phases at different phases and the same duty cycle. Thereafter, the control unit determines whether condition 1 is satisfied. When it is determined that condition 1 is satisfied, the control unit determines whether there is a requirement for the AC impedance measurement of the fuel cell. When it is determined that there is a requirement for the AC impedance measurement of the fuel cell, the control unit changes the operation of the switches of the n phases from the operation at the same duty cycle and different phases to the operation at the same duty cycle and the same phase, and performs the AC impedance measurement of the fuel cell. On the other hand, when the control unit determines that there is no requirement for the AC impedance measurement of the fuel cell, the control unit operates the switches of the n phases at the same duty cycle and different phases.

[0143] Therefore, only when condition 1 is satisfied and the AC impedance measurement of the fuel cell is implemented, the control unit changes the operation of the n-phase switch from the operation with the same duty ratio and different phases to the operation with the same duty ratio and the same phase. In other cases, the n-phase switches can also be operated with the same duty ratio and different phases. By driving the same phase only during impedance measurement, the influence of the dead zone can be improved, and the reduction in voltage conversion efficiency caused by increased fluctuations and the increase in capacitor temperature can be suppressed. In addition, it can also be configured as follows: the control unit determines whether there is a request for AC impedance measurement of the fuel cell, and when it is determined that there is a request for AC impedance measurement of the fuel cell, it determines whether condition 1 is satisfied.

[0144] Fig.17 This is a flowchart showing an example of control after measuring the AC impedance of a fuel cell in a fuel cell system including a converter having an n-phase magnetic coupling reactor.

[0145] like Fig.17 As shown, the control unit changes the operation of the n-phase switches from the operation with the same duty cycle and different phases to the operation with the same duty cycle and the same phase, and after measuring the AC impedance of the fuel cell, determines whether the average current value flowing in the coil of the inductor is above a specified threshold.

[0146] When it is determined that the average current value flowing in the coil of the reactor is greater than a predetermined threshold, the control unit changes the operation of the n-phase switches from the operation with the same duty ratio and the same phase to the operation with the same duty ratio and different phases. On the other hand, when it is determined that the average current value flowing in the coil of the reactor is less than the predetermined threshold, the control unit causes the n-phase switches to operate with the same duty ratio and the same phase. Thereby, the generation of the dead zone can be suppressed and the voltage conversion efficiency can be improved.

[0147] Fig.18 It is a diagram showing a dead zone generation region in the discontinuous mode.

[0148] The present researchers have found that, in a boost converter in which two phases are magnetically coupled and L>M is satisfied, the region surrounded by diagonal lines as the dead zone generation region is the region that satisfies condition 1 and generates a dead zone.

[0149] In addition, L is the self-inductance of the reactor, and M is the mutual inductance, which are inherent values ​​determined by the physical properties of the reactor. L is the input voltage of the boost converter (voltage before boost), V H is the output voltage of the boost converter (voltage after boost).

[0150] In the case of two-phase magnetic coupling (n=2), the two phases can be driven in different phases, and the phase difference can also be 180°. In the case of driving the two phases in different phases, if it is a discontinuous mode and the duty cycle is less than 50% (D<0.5), the operating condition becomes that the switch of the other phase is switched from on to off while the average current value of the coil of one magnetically coupled phase is maintained at 0.

[0151] From the viewpoint of accurately determining the operating conditions that cause the dead zone, in the case of two-phase magnetic coupling (n=2), when it is determined that the dead zone is generated, Fig.18 When the condition for the dead zone generation area shown is "when the switches of the two phases that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode, and the duty cycle (D) of the two-phase switches is less than 50%, and either the following formula (A) or formula (B) is satisfied", the control unit determines that condition 1 is satisfied.

[0152] Formula (A): D<{(1 / 2)(L-M)(V H -V L )} / (LV L +MV L -MV H )

[0153] Formula (B): D<(1 / 2){1-(V L / V H )}

[0154] [In equations (A) and (B), L is the self-inductance of the reactor, M is the mutual inductance of the reactor, V H is the converter output voltage, V L is the converter input voltage and D is the duty cycle.]

[0155] When (each) two phases are magnetically coupled, one of the conditions 1 is that "when the switches of the two phases that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode and the duty ratio (D) of the switches of the two phases is less than 50%."

[0156] On the other hand, when (each) n phase is magnetically coupled and the mutually magnetically coupled n phases are driven with a phase difference of (360° / n), the condition 1 is replaced by the condition "when the switches of the two phases that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode and the duty cycle (D) of the two-phase switches is less than 50%", and is changed to "when the switches of the n phases that are magnetically coupled to each other are operated with a phase difference of (360° / n), the current flowing in the coil is in a discontinuous mode and the duty cycle of the n-phase switches is less than (100-100 / n)%".

[0157] (specific example)

[0158] In the case of two-phase magnetic coupling (n=2), the control unit can also determine that condition 1 is satisfied when it is determined that "when the switches of the two phases that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode, and the duty cycle (D) of the switches of the two phases is less than 50% (D<0.5), and either equation (A) or equation (B) is satisfied."

[0159] Fig.19 This is a flowchart showing an example of control when measuring the AC impedance of a fuel cell in a fuel cell system including a converter having a two-phase magnetic coupling reactor.

[0160] The control unit operates the two-phase switches at a phase difference of 180 degrees and the same duty ratio to determine whether there is a request for the AC impedance measurement of the fuel cell. If it is determined that there is no request for the AC impedance measurement of the fuel cell, the control unit ends the control.

[0161] On the other hand, when it is determined that there is a requirement for measuring the AC impedance of the fuel cell, the control unit monitors the current value of the coil, and determines as condition 1 whether "when the switches of the two phases that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil (driven phase) is in a discontinuous mode, and the duty cycle (D) of the two-phase switches is less than 50% (D < 0.5), and either equation (A) or equation (B) is satisfied."

[0162] When it is determined that condition 1 is not satisfied, the control unit causes the operation of the two-phase switches to be operated at a phase difference of 180° and the same duty ratio to measure the AC impedance of the fuel cell. On the other hand, when it is determined that condition 1 is satisfied, the control unit changes the operation of the two-phase switches from the operation at a phase difference of 180° and the same duty ratio to the operation at the same phase and the same duty ratio to measure the AC impedance of the fuel cell.

[0163] It can also be constructed as: Fig.18 As shown in FIG. 1 , the “dead zone” is when “the current flowing in the coil is in a discontinuous mode, the duty ratio (D) of the two-phase switch is less than 50% (D < 0.5), and either equation (A) or equation (B) is satisfied”. Therefore, for example, when the operating conditions shift to the dead zone, Fig.18 At the timing of the dead zone region shown, the switching operations of the two phases are changed from operations with a phase difference of 180° and the same duty ratio to operations with the same phase and the same duty ratio.

[0164] The means for determining whether condition 1 for generating a dead zone is satisfied is not limited to the above-mentioned case, and may be, for example, configured as follows: a range that satisfies the dead zone is predefined based on the input voltage, the output voltage, and the duty cycle, and when the input voltage and the duty cycle enter or approach the range that satisfies the dead zone at a specified output voltage, it is determined that condition 1 is satisfied.

[0165] As described above, by predefining a range that conforms to the dead zone, when entering the range, the actions of the two-phase switches are changed from actions with a phase difference of 180° and the same duty cycle to actions with the same phase and the same duty cycle, thereby enabling the AC impedance of the fuel cell to be quickly obtained when it is desired to obtain the AC impedance.

[0166] exist Fig.19 In the embodiment, condition 1 is "when the switches of the two phases that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode, and the duty cycle of the switches of the two phases is less than 50%, and either equation (A) or equation (B) is satisfied", but the condition of "when either equation (A) or equation (B) is satisfied" can also be removed.

[0167] That is, as condition 1, "when the switches of the two phases that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode, and the duty ratio of the two-phase switches is less than 50%". In this case, for example, it is also possible to configure such that when the required output to the fuel cell decreases, the current flowing in the coil is in a discontinuous mode, and the duty ratio of the two-phase switches is 50% or more, the current flowing in the coil is in a discontinuous mode, and the duty ratio of the two-phase switches is less than 50%, the operation of the two-phase switches is changed from the operation with a phase difference of 180° and the same duty ratio to the operation with the same phase and the same duty ratio.

[0168] like Fig.18 As shown, according to formula (A) and formula (B), the area removed from the range of changing from different phases to the same phase is not large. Therefore, it can be considered that even if the condition of "satisfying either formula (A) or formula (B)" is removed, the effect on the increase of the battery life is limited. On the other hand, by removing the condition of "satisfying either formula (A) or formula (B)", the control can be simplified and the AC impedance of the fuel cell can be measured appropriately.

[0169] In addition, as a variation example, condition 1 may also be "after the two-phase switches that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode, and the duty cycle of the two-phase switches is less than 50%, and equation (A) is satisfied", or "after the two-phase switches that are magnetically coupled to each other are operated with a phase difference of 180°, the current flowing in the coil is in a discontinuous mode, and the duty cycle of the two-phase switches is less than 50%, and equation (B) is satisfied".

Claims

1. A fuel cell system, characterized in that: The fuel cell system includes a fuel cell and a converter for performing at least one of a step-up and a step-down on an output voltage of the fuel cell. The converter includes a reactor for magnetically coupling n-phase coils to each other, n-phase switches connected to the coils, respectively, and a control unit, wherein n is an integer greater than or equal to 2. The control unit performs on-off control of the switches of the n phases, The control unit monitors the current value of the coil, The control unit measures the AC impedance of the fuel cell based on the current waveform and the voltage waveform of the fuel cell. The control unit operates the switches of the n phases at different phases and operates the switches of the n phases at the same duty ratio. The control unit operates the switches of the n phases in the same phase when it is determined that the following condition 1 is satisfied. Condition 1 is an operating condition in which when the switches of the n phases are operated in different phases, the current flowing in the coil is in a discontinuous mode and the current value flowing to the coil of one phase is maintained at zero, and the switch connected to the coil of at least one other phase is switched from on to off.

2. The fuel cell system according to claim 1, characterized in that: When the control unit operates the switches of the n phases at different phases, the switches of the n phases are operated at a phase difference of (360 / n)°.

3. The fuel cell system according to claim 1 or 2, characterized in that: The control unit stores in advance the operating conditions of the converter corresponding to the condition 1 as a data group, The control unit compares the current operating condition of the converter with the data group to determine whether the condition 1 is satisfied.

4. The fuel cell system according to any one of claims 1 to 3, characterized in that: The control unit confirms whether there is a request for measuring the AC impedance of the fuel cell when it is determined that the condition 1 is satisfied. When confirming that there is a request to measure the AC impedance of the fuel cell, the control unit operates the switches of the n phases in the same phase to measure the AC impedance of the fuel cell.

5. The fuel cell system according to any one of claims 1 to 4, characterized in that: The control unit operates the switches of the n phases at the same phase and measures the AC impedance of the fuel cell, and then determines whether an average current value flowing through the coil of the reactor is equal to or greater than a predetermined threshold value. When it is determined that the average current value flowing through the coil of the reactor is equal to or greater than a predetermined threshold value, the switches of the n phases are operated at different phases.

6. The fuel cell system according to any one of claims 1 to 5, characterized in that: When n=2 and the switches of the two phases magnetically coupled to each other are operated with a phase difference of 180°, when the current flowing in the coil is in a discontinuous mode and the duty ratio of the switches of the two phases is less than 50% (D<0.5), and when either the following formula A or the following formula B is satisfied, the control unit determines that the condition 1 is satisfied. Formula A: D < {(1 / 2)(L - M)(V H - V L )} / (LV L + MV L - MV H ) Formula B: D<(1 / 2){1-(V L / V H )} In equations A and B, L is the self-inductance of the reactor, M is the mutual inductance of the reactor, V H is the converter output voltage, V L is the input voltage of the converter and D is the duty cycle (-).

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