Introduction

With the advancement of materials science and manufacturing technology, semiconductor lasers have gained widespread adoption in communications, medicine, industry, and military applications due to their compact size, light weight, and broad emission spectrum1,2,3,4. High-power semiconductor lasers5,6,7 are particularly significant in critical scenarios such as laser weapon systems and industrial processing, offering broad application prospects. Their driver power supplies typically comprise a boost conversion stage and a discharge driving stage, with the former delivering energy through a spatial secondary distributed power supply to provide the required high-voltage input for the latter. The secondary distributed power supply system commonly employs DC/DC converters for voltage regulation, ensuring that the semiconductor laser operates under a stable and efficient current drive.

For boost converters operating in continuous conduction mode (CCM)8,9, the inductor current ripple is relatively small. However, the switching devices operate under hard-switching conditions, and a larger inductance is required, which increases system size and losses. In contrast, the critical conduction mode (CRM) boost converter achieves zero-current turn-on (ZCS) of the power switch and zero-current turn-off (ZCD) of the diode10,11,12,13, effectively eliminating reverse recovery issues and allowing the use of smaller inductors. Nevertheless, CRM operation results in higher inductor current peaks, which can introduce significant electromagnetic interference (EMI) and place stricter requirements on the input filter. To maintain the advantages of a single-phase CRM boost converter while increasing output power and reducing current ripple, multiphase interleaving14,15 is widely adopted. This approach not only expands the output capacity but also significantly reduces both input and output current ripples, thereby minimizing EMI filter size and improving electromagnetic compatibility. However, multiphase interleaving typically requires multiple magnetic components, increasing system size and cost. As a result, magnetic integration has emerged as an effective means to improve core utilization, reduce volume, and lower cost. Depending on the winding coupling method, magnetic integration structures can be classified into direct coupling16,17 and inverse coupling18,19,20,21,22,23,24,25.

Existing research on inversely coupled inductors has primarily focused on coupled inductor design and performance comparison20,21,22,23, small-signal modeling24, and control strategy development25. In magnetic circuit and parameter modeling, conventional approaches typically treat the coupled inductor as a single equivalent parameter, without distinguishing its dynamic characteristics under different operating conditions. To address this limitation, this paper introduces the concepts of “equivalent steady-state inductance” and “equivalent transient inductance” for a two-phase interleaved inversely coupled inductor boost converter operating in critical conduction mode. The former reflects the equivalent behavior of the coupled inductor during steady-state energy transfer, primarily influencing the input current ripple, while the latter represents the effective inductance during dynamic response, directly determining the rate of change of the inductor current.

In multiphase interleaved parallel converters, phase synchronization accuracy and current-sharing performance are critical to overall efficiency and stability. However, in CRM operation, the switching frequency varies with load changes, which increases the difficulty of phase control. Existing methods often rely on external controllers (e.g., DSPs, FPGAs) or high-precision phase detection modules26,27,28 to achieve 180° phase interleaving and current sharing through real-time feedback, but such solutions require additional voltage/current sensors and high-speed processing units, resulting in greater hardware complexity and longer response delays. Although model predictive control (MPC)29,30 offers excellent phase-locking and current-sharing performance, its implementation depends on high-performance processors and complex algorithms, making it unsuitable for resource-constrained applications. To address this challenge, this paper proposes a dynamic on-time compensation strategy based on the static operating point VTon. This approach enables rapid on-time correction without additional sensors or extensive computation, achieving precise 180° phase interleaving and current sharing while retaining fast response capability. At the same time, it significantly reduces hardware complexity and control cost, making it particularly suitable for applications such as space-based laser power supplies, where high reliability and simplified design are essential.

Interleaved CRM boost converter with the coupled inductor

Equivalent inductances derivation

Figure 1 shows the interleaved CRM boost converter with an inversely coupled inductor, where \({V_{{\text{in}}}}\) is the input voltage and \({V_o}\) is the output voltage. The converter operates in an interleaved manner, and inverse coupling allows a higher equivalent inductance to be achieved with a smaller inductor volume.

Fig. 1
figure 1

CRM-mode coupled-inductor topology.

Full size image

It is assumed that the two windings of the coupled inductor are identical, \({L_1}={L_2}={L_{\text{c}}}\). The coupling coefficient is defined as \(k={M \mathord{\left/ {\vphantom {M {{L_{\text{c}}}}}} \right. \kern-0pt} {{L_{\text{c}}}}}\). The voltage relation for inverse coupling can be expressed as follows

$${V_1}+k{V_2}=(1 - {k^2}){L_{\text{c}}}\frac{{d{i_{L1}}}}{{dt}}$$
(1)

The converter operates in four distinct stages determined by the switching states of Q1 and Q2. When Q1 is turned on, the voltage cross winding N1 is \({V_1}={V_{{\text{in}}}}\), the current \({i_{L1}}\) increase. If Q2 is off, the voltage across winding N2 is \({V_2}={V_{{\text{in}}}} - {V_o}\), if Q2 is turned on, \({V_2}={V_{{\text{in}}}}\).

Substituting \({V_1}={V_{{\text{in}}}}\) and \({V_2}={V_{{\text{in}}}} - {V_o}\) into (1) yields

$${L_{{\text{eq}}1}}=\frac{{(1 - D)(1 - {k^2})}}{{1 - D - kD}}{L_{\text{c}}}$$
(2)

Substituting \({V_1}={V_{{\text{in}}}}\) and \({V_2}={V_{{\text{in}}}}\) into (1) yields

$${L_{{\text{eq2}}}}=(1 - k){L_{\text{c}}}$$
(3)

When Q1 is turned off, \({V_1}={V_{{\text{in}}}} - {V_o}\), and the current \({i_{L1}}\) decreases. If Q2 is off, \({V_2}={V_{{\text{in}}}} - {V_o}\), the equivalent inductance is \({L_{{\text{eq2}}}}\). if Q2 is turned on, \({V_2}={V_{{\text{in}}}}\).

Substituting \({V_1}={V_{{\text{in}}}} - {V_o}\) and \({V_2}={V_{{\text{in}}}}\) into (1) yields

$${L_{{\text{eq}}3}}=\frac{{D(1 - {k^2})}}{{D - k+kD}}{L_{\text{c}}}$$
(4)

Similarly, the equivalent inductance for winding N2 can be derived. The equivalent inductances of windings N1 and N2 under different switching states are summarized in Table 1.

Table 1 Equivalent inductance values for each stage.
Full size table

Based on the equivalent inductance expressions and the voltages applied on the windings, the key waveforms of the interleaved CRM boost converter with a coupled inductor are shown in Fig. 2. When D < 0.5, It is apparent that the inductor current ripple is determined by \({L_{{\text{eq}}1}}\), whereas for D > 0.5, it is determined by \({L_{{\text{eq}}3}}\). The parameter \({L_{{\text{eq2}}}}\) governs the input current ripple. In this study, \({L_{{\text{eq}}1}}\) and \({L_{{\text{eq}}3}}\) are regarded as transient equivalent inductances, representing the rate of change of inductor current during transient conditions, while \({L_{{\text{eq2}}}}\) is defined as the steady-state equivalent inductance, describing steady-state energy transfer and ripple characteristics. Based on this classification, this study further investigates the influence of the equivalent inductances on the inductor current ripple, switching frequency, and input current ripple.

Fig. 2
figure 2

Key waveforms of the interleaved CRM boost converter with an inversely coupled inductor in (a) D < 0.5 and (b) D > 0.5.

Full size image

Inductor current ripple

As shown in Fig. 2a, within one cycle, the rate of change of the inductance current is the same during the periods [t1, t2] and [t3, t4], and the time intervals are equal. Therefore, the area enclosed by the inductance current and the time axis is equal to the shaded region. As in the non-coupled case, the average current value of each inductance winding is equal to half of its peak value. Defining \({P_{{\text{in}}}}\) as the total input power, the input power per phase is \({{{P_{{\text{in}}}}} \mathord{\left/ {\vphantom {{{P_{{\text{in}}}}} 2}} \right. \kern-0pt} 2}\), so the peak current of each winding of the coupled inductance can be expressed as follows:

$${i_{L\_{\text{peak}}}}=2{i_{L\_{\text{avg}}}}=\frac{{{P_{{\text{in}}}}}}{{{V_{{\text{in}}}}}}$$
(5)

\({i_{L\_{\text{peak}}}}\) represents the maximum inductor current and \({i_{L\_{\text{avg}}}}\) represents the average inductor current, Therefore, we can conclude that the boost converter operates at CRM and that the peak inductor current is determined by the input power and voltage, regardless of coupling. The inductor current ripple of each phase, which is equal to the inductor current peak value, can be expressed as follows:

$$\Delta {i_{L\_{\text{c}}}}=\left\{ \begin{gathered} \frac{{{V_{in}}}}{{{L_{eq1}}}}D{T_s} \hfill \\ \frac{{{V_{in}}}}{{{L_{eq3}}}}(1 - D{T_s}){\kern 1pt} \hfill \\ \end{gathered} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}} {\frac{{D\left( {1 - D - kD} \right)}}{{1 - {k^2}}}\frac{{{V_{\text{o}}}{T_{\text{s}}}}}{{{L_{\text{c}}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D \leqslant 0.5)} \\ {\frac{{(1 - D)\left( {D+kD - k} \right)}}{{1 - {k^2}}}\frac{{{V_{\text{o}}}{T_{\text{s}}}}}{{{L_{\text{c}}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D>0.5)} \end{array}} \right.$$
(6)

Equation (6) shows that the pulsation values of the inductance currents for D < 0.5 and D > 0.5 are determined by the \({L_{{\text{eq}}1}}\) and \({L_{{\text{eq}}3}}\), respectively. The relationships between \({L_{{\text{eq}}1}}\), \({L_{{\text{eq}}3}}\) and D is shown in Fig. 3. Leq1 increases with D, while Leq3 decreases with D. Both \({L_{{\text{eq}}1}}\) and \({L_{{\text{eq}}3}}\) reach their maximum values when D = 0.5. It can be concluded that when the duty cycle approaches 0.5, the coupling effect increases the equivalent transient inductance, resulting in a larger rate of change in the inductance current and faster dynamic response speed. In CRM, with fixed inductance current pulsation, the larger the transient inductance, the lower the converter switching frequency.

Fig. 3
figure 3

Equivalent inductances vary with the duty cycle in (a) D < 0.5 and (b) D > 0.5.

Full size image

Switching frequency

As illustrated in Fig. 2 and expressed in Eq. (6), transient equivalent inductance \({L_{{\text{eq}}1}}\) governs the charging process of the inductor current and \({L_{{\text{eq}}3}}\) governs the discharging process. The switching on-time and off-time can be expressed as follows:

$$\left\{ \begin{gathered} {t_{on}}=\frac{{{L_{{\text{eq1}}}}{i_{L\_{\text{peak}}}}}}{{{V_{{\text{in}}}}}} \hfill \\ {t_{{\text{off}}}}=\frac{{{L_{{\text{eq3}}}}{i_{L\_{\text{peak}}}}}}{{{V_o} - {V_{{\text{in}}}}}} \hfill \\ \end{gathered} \right.\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D \leqslant 0.5)} \\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D>0.5)} \end{array}$$
(7)

Substituting (5) into (7) yields:

$$\left\{ \begin{gathered} {t_{{\text{on}}}}=\frac{{{P_{{\text{in}}}}{L_{{\text{eq1}}}}}}{{V_{{{\text{in}}}}^{2}}} \hfill \\ {t_{{\text{off}}}}=\frac{{1 - D}}{D}\frac{{{P_{{\text{in}}}}{L_{{\text{eq3}}}}}}{{V_{{{\text{in}}}}^{2}}} \hfill \\ \end{gathered} \right.\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D \leqslant 0.5)} \\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D>0.5)} \end{array}$$
(8)

Therefore, the switching frequency can be expressed as follows:

$${f_s}=\frac{1}{{{t_s}}}=\left\{ \begin{gathered} \frac{1}{{{{{t_{on}}} \mathord{\left/ {\vphantom {{{t_{on}}} D}} \right. \kern-0pt} D}}}=\frac{{DV_{{{\text{in}}}}^{{\text{2}}}}}{{{P_{{\text{in}}}}{L_{{\text{eq1}}}}}} \hfill \\ \frac{1}{{{{{t_{off}}} \mathord{\left/ {\vphantom {{{t_{off}}} {\left( {1 - D} \right)}}} \right. \kern-0pt} {\left( {1 - D} \right)}}}}=\frac{{DV_{{{\text{in}}}}^{{\text{2}}}}}{{{P_{{\text{in}}}}{L_{{\text{eq3}}}}}} \hfill \\ \end{gathered} \right.\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D \leqslant 0.5)} \\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D>0.5)} \end{array}$$
(9)

From Eq. (8), when \(D \leqslant 0.5\) and \(D>0.5\), the conduction time of the switch is associated with \({L_{{\text{eq1}}}}\) and \({L_{{\text{eq3}}}}\), respectively, and both \({L_{{\text{eq1}}}}\) and \({L_{{\text{eq3}}}}\) vary with the duty cycle D over half of the line cycle. With \({L_{\text{c}}}\) and k fixed, and for given input power and input voltage, Fig. 4 shows the relationship between switching frequency and duty cycle for different ranges of coupling coefficients, a properly designed coupled inductor results in the switching frequency varying with D over half of the line cycle, the conduction time of the switches is not constant.

Fig. 4
figure 4

Switching frequency varies with the duty cycle indifferent coupling coefficients ranges.

Full size image

Input current ripple

As shown in Fig. 2, the input current is the sum of the two inductor currents, and the input current ripple can be expressed as follows:

$$\Delta {i_{{\text{in\_c}}}}=\left\{ {\begin{array}{*{20}{c}} {2\frac{{{V_o} - {V_{{\text{in}}}}}}{{{L_{{\text{eq2}}}}}}(0.5 - D){T_s}} \\ {{\kern 1pt} {\kern 1pt} 2\frac{{{V_{{\text{in}}}}}}{{{L_{{\text{eq2}}}}}}(D - 0.5){T_s}} \end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}} {\Delta {i_{L1\_2}}+\Delta {i_{L2\_2}}=\frac{{D(1 - 2D)}}{{1 - k}}\frac{{{V_o}{T_s}}}{{{L_{\text{c}}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D \leqslant 0.5)} \\ {{\kern 1pt} \Delta {i_{L1\_1}}+\Delta {i_{L2\_1}}={\kern 1pt} \frac{{(1 - D)(2D - 1)}}{{1 - k}}\frac{{{V_o}{T_s}}}{{{L_{\text{c}}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D>0.5)} \end{array}} \right.$$
(10)

From Eq. (10), \({L_{{\text{eq2}}}}\) is related to k and \({L_{\text{c}}}\) and determines the input current ripple. The relationship between the input current ripple and the ratio of the input current ripple to the ripple of each inductor current is illustrated in Fig. 5. For the same coupling coefficient, the input current ripple is smaller than the ripple of each inductor current when D is close to 0.5, while it becomes larger when D deviates from 0.5. This is because, as the duty cycle approaches 0.5, the ripple waveforms of the two inductor currents become nearly out of phase, resulting in greater cancellation and thus a smaller input current ripple. For the same duty cycle, a higher coupling coefficient leads to a smaller steady-state equivalent inductance \({L_{{\text{eq2}}}}\), which in turn increases the input current ripple.

Fig. 5
figure 5

Ratio of the input ripple current and the inductor ripple current in different coupling coefficients.

Full size image

Reasonable coupling coefficient selection

The coupling coefficient \(\:k\) has a crucial influence on both the magnetic and electrical performance of the converter. Stronger coupling can reduce the number of turns and thmagnetic volume, while the switching frequency also varies with the coupling strength. As analyzed previously, the coupling coefficient affects both the steady-state equivalent inductance \({L_{{\text{eq}}2}}\) and the transient equivalent inductances \({L_{{\text{eq1,}}3}}\). With a fixed duty cycle, an increase in k reduces \({L_{{\text{eq2}}}}\), leading to larger input-current ripple; meanwhile, stronger coupling increases the transient inductances, accelerating the current slew rate and improving the dynamic response. Therefore, the coupling coefficient introduces a trade-off between current-ripple suppression and dynamic performance, Simulation results indicate that \(k=0.3\) represents a good practical balance between current ripple and transient response. From an implementation perspective, commonly used coupled-inductor structures—such as EE or EI cores with three identical air-gap legs (as shown in Fig. 6)—correspond to an effective coupling coefficient of approximately \(k={1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}\), which provides a good balance between performance and manufacturability. In practical applications, this configuration is generally preferred, therefore, it is ultimately selected as the design parameter in this paper.

The optimal value of k should be determined according to specific design objectives: moderate coupling is preferred when EMI performance and ripple suppression are prioritized, while stronger coupling is desirable when faster transient response and higher magnetic compactness are required.

Fig. 6
figure 6

Coupled inductor structure and its equivalent magnetic circuit model. (a) Physical structure and magnetic flux distribution. (b) Equivalent magnetic circuit.

Full size image

The expressions for the self-inductance \({L_c}\), mutual inductance M, and coupling coefficient k are as follows:

$$\left\{ \begin{gathered} {L_{\text{c}}}=\frac{{{N_{cp}}^{2}\left( {{R_{\text{s}}}{\text{+}}{R_{\text{c}}}} \right)}}{{{R_{\text{s}}}^{{\text{2}}}{\text{+2}}{R_{\text{s}}}{R_{\text{c}}}}}=\frac{{{N_{cp}}^{2}}}{{{R_{\text{s}}}{\text{(1+k)}}}} \hfill \\ M=\frac{{{N_{cp}}^{2}{R_{\text{c}}}}}{{{R_{\text{s}}}^{{\text{2}}}{\text{+2}}{R_{\text{s}}}{R_{\text{c}}}}} \hfill \\ k=\frac{M}{L}=\frac{{{R_c}}}{{{R_{\text{s}}}+{R_c}}} \hfill \\ \end{gathered} \right.$$
(11)

From Eq. (11), the reluctances of the outer and center legs can be expressed as follows:

$$\left\{ {\begin{array}{*{20}{c}} {{R_s}=\frac{1}{{(1+k)}}\frac{{{N_{cp}}^{{\text{2}}}}}{L}} \\ {{R_c}=\frac{k}{{1 - {k^2}}}\frac{{{N_{cp}}^{{\text{2}}}}}{L}} \end{array}} \right.$$
(12)

The reluctance can also be expressed as follows:

$$\left\{ \begin{gathered} {R_s}={\delta _s}/({\mu _0}{A_e}) \hfill \\ {R_c}={\delta _c}/(2{\mu _0}{A_e}) \hfill \\ \end{gathered} \right.$$
(13)

From Eqs. (12) and (13), the air gap lengths of the outer and center legs of the core can be derived as follows:

$$\left\{ {\begin{array}{*{20}{c}} {{\delta _s}=\frac{{{\mu _0}{A_e}{N_{cp}}^{2}}}{{L(1+k)}}} \\ {{\delta _c}={\delta _s}\frac{{2k}}{{1 - k}}} \end{array}} \right.$$
(14)

New control strategy

In a two-phase interleaved CRM-Boost converter, the introduction of coupled inductors is susceptible to interference from interlinking magnetic flux, which can lead to imbalance between the inductor current of the slave channel and that of the master channel. The master–slave control method is a commonly adopted approach. However, as shown in Eq. (9) and Fig. 4, the switching frequency varies with duty cycle perturbations. This makes it impractical for the slave channel to generate the turn-on signal using ZCD2 detection and to determine the turn-off signal with a fixed on-time. Furthermore, deriving the slave channel gating signals by shifting the master channel turn-on and turn-off signals by 180° introduces significant complexity. This paper proposes a new balanced control strategy for a two-phase, interleaved CRM-Boost converter with coupled inductors, in which a static operating point \({V_{{\text{Ton}}}}\) is introduced to achieve dynamic compensation of the turn-on time. The proposed dynamic on-time compensation control is implemented using an analog comparator and a reference voltage. This analog-based implementation eliminates the need for digital processors or temperature-sensitive components, resulting in a simple structure with strong noise immunity and minimal temperature drift, which is particularly suitable for aerospace power applications.

Fig. 7
figure 7

Control schematic of the balanced interleaved parallel topology.

Full size image

The control principle of the proposed method is illustrated in Fig. 7. In the diagram, the master channel (hereafter referred to as “phase A”) and the slave channel (hereafter referred to as “phase B”) are shown, with the control signal of phase A defined as ZCD1 and denoted at different instances as ZCD11, ZCD12, and ZCD13. The control methods of phases A and B are symmetrical, and the ZCD2 signals generated in phase B are denoted as ZCD21 and ZCD22. The comparison voltage \({V_{{\text{comp}}}}\) is determined by the static operating point voltage \({V_{{\text{Ton}}}}\) and \({V_{{\text{PI}}}}\). For phase A, the comparison voltage is \({V_{{\text{comp}}}}\), while for phase B, the comparison point is \({V_{{\text{comp}}}}+{V_e}\).

The control waveforms are shown in Fig. 8, where \({V_{1\_\hbox{max} }}\) and \({V_{2\_\hbox{max} }}\) represent the peak values of \({V_{{\text{ramp1}}}}\) and \({V_{{\text{ramp2}}}}\), respectively, and are used to characterize the time interval between adjacent zero-crossing detection signals of the two phases. When \({V_{1\_\hbox{max} }}<{V_{2\_\hbox{max} }}\), it indicates that ZCD21 has shifted from the midpoint between ZCD11 and ZCD12 toward ZCD11. In this case, the phase difference between phase B and phase A is \(\theta <{180^\circ }\), therefore \({V_{\text{e}}}={V_{2{\text{to}}1}} - {V_{1{\text{to}}2}}>0\), the comparison voltage of phase B increases, and the conduction time of the phase-B switch in the next cycle is extended, moving ZCD22 closer to the midpoint between ZCD12 and ZCD13. Conversely, the opposite adjustment occurs. Through this negative feedback mechanism, phase B tracks the phase of phase A, achieving a 180° phase shift.

Fig. 8
figure 8

Waveforms obtained using the improved control method when θ < 180°.

Full size image

Simulation and experimental results

Simulation analysis

The simulation study is implemented utilizing PSIM, a simulation model is constructed for the proposed topology. The input DC voltage \({V_{{\text{in}}}}\) is 100 V, the output capacitor \({C_{\text{o}}}\) is 440 µF, and the converter delivers a regulated DC output voltage of 300 V to a full-load resistance \({R_{\text{o}}}\) of 90 Ω, corresponding to a full-load output power of 1 kW. The self-inductance \({L_{\text{c}}}\) is 115 µH, the mutual inductance M is 33 µH, and the coupling coefficient is \(k={1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}\).

The PSIM simulation results are shown in Fig. 9. In Fig. 9a, the current waveforms include the input current (blue), the two-phase inductor currents (green and magenta), and the output current (red). The voltage waveforms show the input DC voltage (blue) and the output DC voltage (red). Figure 9b presents the two-phase inductor current waveforms along with the corresponding gate-drive signals. Figure 9c and d illustrate the inductor current waveforms and corresponding zero-current detection (ZCD) signals under full-load and light-load conditions, respectively. Experimental results verify that the proposed dynamic on-time compensation control strategy ensures stable CRM operation across the entire load range. Under an input voltage of 100 V, the output voltage is well regulated at 300 V. At full load, the converter operates at a relatively low switching frequency, and the two inductor currents have nearly identical amplitudes with a 180° phase shift, achieving effective current sharing. The input-current ripple is significantly reduced, and no waveform distortion or saturation is observed. Under light-load conditions, the system automatically increases the switching frequency to maintain the critical conduction characteristic. As the load current decreases, the energy transferred per cycle becomes smaller, resulting in lower inductor-current peaks and shorter on-time. Consequently, the ZCD signals are triggered at higher frequency and almost simultaneously, causing the two inductor currents to operate nearly in phase. Nevertheless, the converter still achieves zero-current switching at turn-on and zero-current detection at turn-off, maintaining stable CRM operation. These simulation results are in strong agreement with the theoretical analysis, confirming the effectiveness and feasibility of the proposed control strategy.

Fig. 9
figure 9

Simulation waveforms of coupled-inductor interleaved converter. (a) Inductor, Input & Output Current/Voltage Waveforms, (b) Two-Phase Inductor Currents and Drive Signals. (c) Inductor current waveforms and ZCD switching signals under full-load condition (1 kW), (d) Inductor current waveforms and ZCD switching signals under light-load condition (0.2 kW).

Full size image

Experimental results

In order to validate the previous simulation results, as well as the proposed modulation strategy, an experimental prototype of the coupled-inductor interleaved parallel converter was developed, while a discrete-inductor interleaved parallel converter was also constructed for comparison. The experimental setup is shown in Fig. 10.

Fig. 10
figure 10

Experimental prototype of discrete and coupled-inductor interleaved converters.

Full size image
Fig. 11
figure 11

Experimental waveform. (a) Switching Device Gate Drive Waveform; (b) Boost Converter Coupled-Inductor Topology; (c) Load Transient Response Waveforms; (d) Output Voltage Step Response.

Full size image

The experimental results of the inversely coupled inductor CRM-Boost interleaved parallel converter are shown in Fig. 11. In Fig. 11a, the gate–source drive voltage of the switches remains stable at 15 V, with the two gate signals exhibiting a 180°phase shift. Under full-load operation at an input voltage of 100 V (Fig. 11b), the output voltage is regulated at 301 V with a voltage stability of 0.33%, efficiency of 97.5%, an average input current of 9.57 A, and average inductor currents of 4.5 A for both phases. The low ripple factor meets the requirements of laser power supplies for low-ripple, high-efficiency secondary distributed power outputs. Figure 11c demonstrates that the converter exhibits strong load disturbance rejection and good output stability under load operation; under load variations, the transient recovery time is significantly reduced, the undershoot overshoot is minimal (approximately 5 V), and the recovery time is less than 20 ms. As shown in Fig. 11d, the output voltage remains steady at 300 V during normal operation; upon power-off, it drops sharply to 100 V, and the recovery process is smooth, fast, and precise.

To further evaluate performance, the efficiencies of both the discrete-inductor and coupled-inductor topologies were compared under identical conditions, with the measured data plotted as efficiency curves in Fig. 12. With the input DC voltage maintained at 100 V, the efficiency curve rises to a peak before slightly decreasing as the load increases. Under light-load conditions, the converter still operates in CRM mode and maintains zero-current turn-on. However, as the load current decreases, the on-time becomes shorter and the switching frequency rises sharply. In this region, frequency-dependent losses—such as gate-drive loss, output-capacitance charging and discharging loss, and core loss—dominate, leading to a reduction in overall efficiency.

At medium load, the converter operates at its optimal efficiency point, where conduction loss and frequency-related losses (including gate-drive and core losses) reach a dynamic balance. Compared with light load, the reduced switching frequency lowers frequency-related losses, while conduction loss remains relatively small, resulting in minimum total loss and peak efficiency.

Under heavy-load conditions, the effective current increases significantly, and the conduction loss of MOSFETs together with the winding resistance loss of the inductor becomes the main source of power dissipation, causing a slight efficiency drop. Compared with the discrete inductor design, the coupled-inductor configuration achieves higher magnetic-energy utilization and lower leakage-related losses, thereby delivering slightly higher efficiency and better performance across the entire load range.

Fig. 12
figure 12

Efficiency comparison between discrete and coupled inductors at full load.

Full size image

Conclusion

This paper has proposed an optimized magnetic integration scheme for a CRM interleaved Boost converter with SiC devices. By introducing the concepts of equivalent steady-state inductance and equivalent transient inductance, the analysis framework distinguishes between energy transfer in steady state and dynamic behavior, offering more practical guidelines for inductor design and coupling coefficient selection. Furthermore, a dynamic on-time compensation method based on a static operating point has been proposed. This approach enables precise 180° phase interleaving and current sharing without the need for external high-speed controllers. Experimental validation confirms that the proposed method not only reduces hardware complexity but also significantly shortens PI parameter tuning. Consequently, the converter delivers higher output accuracy, better current sharing, and improved implementation feasibility.