The proposed hybrid Si-SiC three phase inverter topology, as shown in Figure 4, consists of a low-frequency power conversion cell (LFPC-C),a high-frequency power conversion cell (HFPC-C) and a three-phase five-column medium-frequency step-down transformer. The input side is parallel and shared by a common DC source, and the outputs are coupled in series through the transformer. The LFPC-C employs Si IGBT devices as switching devices denoted as S1 ∼ S6, operating at a low switching frequency to deal with the main power of the proposed three-phase inverter. The HFPC-C uses SiC MOSFET devices as switching devices denoted as Q1 ∼ Q6, operating at a high switching frequency to handle fractional power. The transformer turns ratio is denoted as k, with the primary side parallel-coupled to the output filtering capacitor of the HFPC-C and the secondary side series-coupled and connected to the LFPC-C.

This topology realizes power sharing in two power conversion cells. Under the same DC side voltage, this topology reduces the switching losses of Si IGBT devices in the LFPC-C. Additionally, the step-down transformer ensures that the working current in the HFPC-C is less than the output current in the LFPC-C, reducing the current stress on SiC MOSFET devices and lowering on-state losses in the HFPC-C. The switching losses are concentrated on SiC MOSFET devices, and leveraging their excellent characteristics helps reduce the system’s overall switching losses.

The LFPC-C is responsible for the energy output of the system. However, due to its operation at a low frequency, it can lead to lower electrical energy quality in the final output of the system. Therefore, the HFPC-C needs to compensate and eliminate some of the harmonics present in the LFPC-C t to enhance the overall electrical energy quality of the system. The final output voltage of the system is composed of the voltage in the LFPC-C and the voltage on the secondary side of the transformer. The voltage in the LFPC-C is composed of the fundamental voltage and the harmonic voltage, while represents the output voltage of the HFPC-C. The relationship between these variables can be expressed as follows V o u t = u h − v S i C u h = u h 1 + u h h v S i C = V S i C / k (1)

The bridge arm output characteristics of the LFPC-C significantly impact the overall system performance. Specific Harmonic Elimination Pulse Width Modulation (SHEPWM) technology can eliminate particular harmonics, and the switching angles can be calculated through computations. SHEPWM operates at a low switching frequency, which helps in reducing the switching losses of the Si-IGBT devices. When using traditional SHEPWM to eliminate harmonics in the mid to low frequency range, it results in a large number of switching angles and necessitates a higher switching frequency. By serially connecting the HFPC-C and the transformer to the output circuit of the LFPC-C, certain harmonics in the bridge arm output of the LFPC-C can be eliminated. This approach ensures a reduction in the switching frequency of Si-IGBT devices in the LFPC-C, thus reducing switching losses while maintaining the total amount of eliminated output harmonics. In the HFPC-C, a Hybrid Sinusoidal Pulse Width Modulation (Hybrid SPWM) is employed. Hybrid SPWM refers to modulating the waveform, which is not a single-frequency sine wave but is determined by the superposition of multiple harmonic waves.

The collaborative elimination of mid to low-frequency harmonics by the LFPC-C and the HFPC-C involves segmenting the mid to low-frequency harmonics. Two technical approaches can be considered based on the harmonic distribution characteristics in the LFPC-C:

Method 1: The LFPC-C eliminates mid-frequency and high-frequency harmonics, while the HFPC-C eliminates low-frequency harmonics.

Method 2: The LFPC-C eliminates low-frequency and high-frequency harmonics, and the HFPC-C eliminates mid-frequency harmonics.

According to the residual harmonic distribution characteristics of SHEPWM (Cheng, 2021), both methods merely shift the harmonic energy to other frequency bands. However, when employing Method 2, where the LFPC-C eliminates low-frequency and high-frequency harmonics, and the mid-frequency harmonics are eliminated by the HFPC-C, it results in the HFPC-C handling harmonics excessively. The equivalent harmonic frequency within one power cycle can reach several kilohertz, demanding higher requirements for the control system and switching frequency of SiC MOSFET devices in the HFPC-C, thus increasing the switching losses of SiC MOSFET devices. Therefore, Method one is adopted, where the LFPC-C eliminates mid-frequency and high-frequency harmonics while the HFPC-C eliminates low-frequency harmonics. As illustrated in Figure 5A, the LFPC-C is responsible for system energy output, using SHEPWM to remove mid-frequency harmonics. The low-frequency harmonics are eliminated by the HFPC-C, and the high-frequency harmonics are eliminated by the LFPC-C’s filter. The HFPC-C compensates and eliminates low-frequency harmonics in the LFPC-C.

When employing the first technical approach, the specific division of mid to low-frequency harmonics and the modulation degree of the hybrid modulated waves m x in the HFPC-C, as well as the selection of transformer turns ratio k, pose constraints. The effective division and treatment of mid to low-frequency harmonics in the LFPC-C and determining the appropriate modulation degree in the HFPC-C are essential for achieving effective harmonic elimination. Additionally, the choice of transformer turns ratio is crucial as it impacts the coupling between the high-frequency and LFPC-C, thereby influencing the overall harmonic elimination performance and system efficiency. Careful consideration and proper optimization of these parameters are necessary to ensure optimal performance in mitigating harmonics and achieving efficient energy conversion in the system.

The low-frequency harmonics to be eliminated in the LFPC-C represent a modulation signal composed of multiple harmonics for the HFPC-C. In SHEPWM modulation, once the fundamental modulation degree and switching angles are determined, the remaining harmonic superposition waveform becomes fixed. The LFPC-C uneliminated low-frequency harmonic family can be represented as u h h _ l o w .

When a three-phase inverter is connected to a balanced load on the output side, The third harmonic and its multiples cancel each other out in the line-to-line voltage, hence we only need to focus on eliminating 6k±1 harmonics k = 1 , 2 , 3 , … . . .

In a three-phase half-bridge configuration, peak value of the output phase voltage U p h m is: U p h m = 0.5 m S i C t U d (1a)

The variable m S i C t represents the modulation index for the hybrid sinusoidal carrier modulation in the HFPC-C. The expression for the hybrid modulated wave m S i C t is: m S i C t = m 5 ⁡ sin 5 ω t + θ 5 + ⋯ + m i ⁡ sin i ω t + θ i (2)

The transformer is a crucial coupling component connecting the LFPC-C and the HFPC-C. Through its turns ratio k, a relationship between Eqs 1, 2 is established, yielding the following correlation k ≤ m S i C t U d 2 u h h _ l o w (3)

The modulation index of the HFPC-C and the transformer turns ratio constrain each other. If the turns ratio k is chosen to be too large, it can lead to over-modulation in the HFPC-C. This over-modulation can introduce harmonics from other frequency bands into the output voltage of the LFPC-C, thereby degrading the quality of the output waveform. On the other hand, if the turns ratio k is chosen to be too small, it can cause the current in the HFPC-C to approach the output current in the main circuit, increasing the current cost of SiC MOSFET devices.

The LFPC-C in this paper is responsible for eliminating harmonics from the 17th to the 41st order. Meanwhile, the HFPC-C compensates and eliminates the fifth, seventh, 11th, and 13th order harmonics.

The bridge arm output voltage waveform of the LFPC-C using SHEPWM is shown in (Supplementary Figure S1).

The output waveform in Figure 6 is symmetric about a 1/4 period, and it can be represented using a Fourier series as follows u ω t = ∑ n = 1 , 17 ⋯ 41 a n ⁡ sin ⁡ n ω t (4)

In the equation, a n represents the amplitude of harmonics. The expressions for the fundamental and the 17th to 41st harmonic components are as follows: a 1 = 2 V d n π ∑ i = 1 10 − 1 i + 1 ⁡ cos ⁡ n α i = U 1 a 17 = 2 V d n π ∑ i = 1 10 − 1 i + 1 ⁡ cos ⁡ n α i = 0 a 19 = 2 V d n π ∑ i = 1 10 − 1 i + 1 ⁡ cos ⁡ n α i = 0 ⋮ ⋮ a 41 = 2 V d n π ∑ i = 1 10 − 1 i + 1 ⁡ cos ⁡ n α i = 0 (5)

In the equation, α i represents the switching angle. The modulation ratio m is defined as (Dong et al., 2024) m = U 1 U d / 2 (6)

The HFPC-C employs hybrid sinusoidal pulse width modulation technique, where the modulation wave is composed of fifth, seventh, 11th, and 13th harmonic sinusoidal waves. The expression is: m S i C t = m 5 ⁡ sin 5 ω t + θ 5 + m 7 ⁡ sin 7 ω t + θ 7 + m 11 ⁡ sin 11 ω t + θ 11 + m 13 ⁡ sin 13 ω t + θ 13 (7)

Taking the modulation index m = 0.97 as an example, the schematic diagram of the modulated wave and the triangular carrier wave for the HFPC-C is shown in (Supplementary Figure S2).

The mathematical model is as shown in Figure 6, and there are two switch degrees of freedom, denoted as d 1 s and d 2 s , in the overall control circuit. and L x 1 、 L x 2 respectively refer to the leakage inductance on the secondary and primary sides of the transformer.

The relationship between V S i C s , d 2 s and i 1 s can be deduced from the block diagram. V S i C s = U d 1 + L 2 C 2 s 2 d 2 s + L 2 s 1 + L 2 C 2 s 2 − L x 2 s i 1 s k (8) H s can be represented as: H s = L 2 s 1 + L 2 C 2 s 2 − L x 2 s k (9) and v S i C s = V S i C s k + s L x 1 i 1 s (10)

The relationship between v S i C s , d 2 s and i 1 s can be derived as follows: v S i C s = U d k 1 + L 2 C 2 s 2 d 2 s + H s k + L x 1 s i 1 s (11)

If the secondary-side leakage inductance is attributed to the LFPC-C filter inductance, L 1 ′ = L 1 + L x 1 , If we consider v S i C s as an input for the LFPC-C, we can create an equivalent system diagram.

Let d 2 s and i o s be 0 separately, and determine the relationship between V o u t s and d 1 s V o u t s = U d d 1 s 1 + C 1 H s k s + L 1 ′ C 1 s 2 (12)

Let d 1 s and i o s be 0 separately, and determine the relationship between V o u t s and d 2 s V o u t s = − U d d 2 s k 1 + L 2 C 2 s 2 1 + C 1 H s k s + L 1 ′ C 1 s 2 (13)

Let d 1 s and d 2 s be 0 separately, and determine the relationship between V o u t s and i o s V o u t s = − L 1 ′ s + H s k 1 + L 1 ′ C 1 s 2 + H s C 1 k s i o s (14)

Finally, we can obtain the following expression: V o u t s = G s U d d 1 s − d 2 s k 1 + L 2 C 2 s 2 − L 1 ′ s + H s k i o s (15) G s can be expressed as G s = 1 1 + L 1 ′ C 1 s 2 + H s C 1 k s (16)

From Eqs 13, 14, it can be seen that the circuit has two degrees of freedom, d 1 s and d 2 s . The choice of closed-loop control target is also related to the level of control difficulty. If the overall output voltage V o u t of the system is chosen as the closed-loop target voltage, we need to consider not only d 2 s but also the impact of d 1 s . Here, d 1 s represents the SHEPWM modulation of the LFPC-C, which has drawbacks such as real-time complex computation and poor dynamic adjustment performance. Having two input variables significantly increases the difficulty of closed-loop control.

Since the control objective is to compensate for the secondary-side voltage of the transformer, fundamentally, the closed-loop control aims to control the secondary-side voltage of the transformer. Therefore, in designing the closed-loop control circuit, the effect of the LFPC-C current on the HFPC-C can be considered as a disturbance. We propose a coordinated control method, “LFPC-C open-loop, HFPC-C closed-loop”. In this approach, the LFPC-C uses open-loop SHEPWM control to adjust the voltage by only adjusting the fundamental modulation ratio, reducing the control difficulty for the HFPC-C. The HFPC-C achieves precise voltage regulation, compensating for the limitations of SHEPWM.

When processing the fifth, seventh, 11th, and 13th harmonic components in the HFPC-C and extracting the control system’s reference signal, the following two points need to be considered:

1) The physical sampling point is located between the inductance of the LFPC-C and the secondary side of the transformer. The sampled voltage still contains the uneliminated high-frequency harmonics, which are not the target of the HFPC-C tracking control.

Considering the above two points, Recursive Discrete Fourier Transform is used to meet the aforementioned requirements. The expression is as follows: G R D F T i = ∑ i = 5 、 7 、 11 、 13 2 N 1 − z − N 1 − z − 1 ⁡ cos ⁡ 2 π i N 1 − z − 1 e j 2 π i N 1 − z − 1 e − j 2 π i N (17)

Amplitude-frequency response and phase-frequency response of G R D F T i s is shown in (Supplementary Figure S3).

The above equation yields the reference signal v r e f for the closed-loop control of the HFPC-C, which is then compared with the output voltage v S i C of the transformer secondary side, and obtain the error signal Δ v .The control loop consists of voltage and current double loops. Due to the presence of multiple harmonic voltages in the output voltage and the need for high precision, a multiple Quasi-Proportional Resonant (QPR) voltage outer loop is employed to process the error signal Δ v , the transfer function is given by: G Q P R s = K p v + ∑ i = 5 、 7 、 11 、 13 2 K i v w c s s 2 + 2 w c s + w i 2 (18) ω i is the angular frequency of the 5th to 13th harmonic components, ω c is the damping coefficient. K i v is the resonance coefficient, K p v is the proportional coefficient.

Obtaining the current inner-loop reference signal i r e f , sampling the capacitor current to obtain i c , The inner loop adopts proportional control of the capacitor current to improve

The response speed. Controlling the inner loop with the capacitor current as the control target can increase system damping, suppress resonance, and reduce the difficulty of voltage outer loop control. G i s = K i (19)

The transfer function of the closed-loop control for the HFPC-C is G v s = K i U d L 2 C 2 s 2 + C 2 K i U d s + 1 G Q P R s (20)

The overall control policy is shown in Supplementary Figure S4.

When considering voltage compensation by the HFPC-C for the LFPC-C, it is necessary to establish a steady-state voltage ripple model. The primary side of the series-coupled transformer is connected in parallel across the filtering capacitor of the HFPC-C. Neglecting the fundamental component in the circuit, the ripple in the primary-side voltage is equivalent to a voltage source Δ U 2 on the secondary side, As shown in (Supplementary Figure S5), there are two voltage sources in the LFPC-C circuit, and the corresponding ripple currents are shown in Figure 7, the ripple current generated by the output voltage U S i and the inductance L 1 in the LFPC-C bridge arm is denoted by Δ I 3 p k − p k (Mao et al., 2009), and its expression is as follows: Δ I 1 p k − p k = U d T s 1 L 1 1 − D S i t D S i t (21)

In the equation, T s 1 represents the switching period, D S i t denotes the average duty cycle. Due to the adoption of SHEPWM modulation in the LFPC-C, by calculating for different switching signal sequences, D S i t for one complete switching action can be obtained. D S i t = ∑ i = 1 N − 1 α i + 1 − α i 2 π T s (22) Δ I 3 p k − p k represents the ripple current generated in the LFPC-C circuit under the excitation of Δ U 2 , and its expression is: Δ I 3 p k − p k = Δ U 2 z z = w x L − 1 w x C (23)

The total ripple current Δ I 5 p k − p k in the LFPC-C is constituted by the two equivalent ripple current sources Δ I 1 p k − p k and Δ I 3 p k − p k : Δ I 5 p k − p k = Δ I 1 p k − p k + Δ I 3 p k − p k (24)

The expression for the total output ripple voltage Δ U o is given by: Δ U o = T s 1 3 U d ω x L 1 C 1 − 1 2 π − ∑ i = 1 N − 1 α i + 1 − α i ∑ i = 1 N − 1 α i + 1 − α i + 4 π 2 L 1 C 1 ω x Δ U 2 32 L 1 C 1 f s 1 π 2 ω x 2 L 1 C 1 − 1 (25)

The ripple analysis for the HFPC-C is illustrated in Figure 10.

The parameter Δ I 2 p k − p k is determined by the output voltage U S i C of the HFPC-C bridge arm, the inductance L 2 , and the amplitude m S i C of the non-sinusoidal fundamental modulation wave. The expression is as follows: Δ I 2 p k − p k = U d T s 2 L 2 1 − m S i C sin ⁡ ω t m S i C sin ⁡ ω t (26)

The ripple in the output current of the LFPC-C bridge arm is represented by Δ I 5 p k − p k , and it can be equivalently modeled as Δ I 6 p k − p k on the primary side of the series-coupled transformer. Δ I 6 p k − p k = Δ I 5 p k − p k k (27)

The total ripple current Δ I 4 p k − p k in the HFPC-C is formed by the combination of the two equivalent ripple current sources Δ I 2 p k − p k and Δ I 6 p k − p k . Δ I 4 p k − p k = Δ I 2 p k − p k + Δ I 6 p k − p k (28)

The ripple voltage on the filter capacitor of the HFPC-C is represented by Δ U 1 Δ U 1 = Δ I 4 p k − p k 8 C 2 f s 2 (29)

Based on Δ U 2 = Δ U 1 / k , Δ U 2 can be determined as follows: Δ U 2 = k U d T s 2 ω x 2 L 1 C 1 − 1 1 − m S i C sin ⁡ ω t 8 L 2 C 2 f s 2 k 2 ω x 2 L 1 C 1 − 1 − ω x C 1 m S i C sin ⁡ ω t (30)

Substituting into Eq. 26, Δ U o is determined as follows: Δ U o = Y • 8 L 2 C 2 f s 2 k 2 ω x 2 L 1 C 1 − 1 − ω x C 1 + 4 π 2 L 1 C 1 ω x k U d T s 2 ω x 2 L 1 C 1 − 1 1 − m S i C sin ⁡ ω t m S i C sin ⁡ ω t 32 L 1 C 1 f s 1 π 2 ω x 2 L 1 C 1 − 1 • 8 L 2 C 2 f s 2 k 2 ω x 2 L 1 C 1 − 1 − ω x C 1 Y = T s 1 3 U d ω x L 1 C 1 − 1 2 π − ∑ i = 1 N − 1 α i + 1 − α i ∑ i = 1 N − 1 α i + 1 − α i (31)

Supplementary Figure S7, Figure 8 in Additional files illustrates the ratio of output voltage ripple to DC voltage on the direct current (DC) side based on Eq. 31 for different parameters chosen for L₁ and L₂.

As shown in Figure 5B above, the filters in the low-frequency and HFPC-C filter different ranges of harmonics. Additionally, the compensation of voltage ripples in the HFPC-C and the final circuit output ripples in the LFPC-C, as analyzed in the previous section, are closely related to the selection of passive components L₁、L₂、C₁and C₂ in the circuit. Considering the constraints mentioned above, frequency domain constraints also need to be taken into account.

For the LFPC-C, as shown in Figure 5B above, the harmonics from the 5th to the 13th order are eliminated through the HFPC-C, and the harmonics from the 17th to the 41st order are already eliminated through SHEPWM. The remaining harmonics in the high-frequency range are eliminated through a second-order low-pass LC filter. The cutoff frequency f S i of the second-order filter should be set in the mid-frequency range, specifically between 650 Hz and 2050 Hz: f S i = 1 2 π L 1 C 1 650 H z < f S i < 2050 H z (32)

Simultaneously, to prevent resonance peaks in the LFPC-C filter and to amplify the fifth, seventh, 11th, and 13th harmonics, increasing the compensation difficulty for the HFPC-C, requirements are imposed on the damping coefficient of the filter ζ : ζ = 1 2 R 1 L 1 C 1 ζ ≤ 0.7 (33)

The HFPC-C compensates for the highest harmonic frequency at 650 Hz. The selected switching frequency is 30 kHz, and the upper limit for the filter cutoff frequency f S i C is set to 3000 Hz, with a lower limit of 1000 Hz. This setting ensures that the highest compensating harmonic (650 Hz) can pass through without attenuation, providing an allowance: f S i C = 1 2 π L 2 C 2 1000 H z < f S i C < 3000 H z (34)

Due to the non-sinusoidal periodic components of the input terminal voltage injected into the transformer, which is a superposition of fifth, seventh, 11th, and 13th harmonic voltages, a three-phase five-column medium-frequency transformer is used. This ensures that the high-order harmonic flux can circulate smoothly, and harmonics can flow in the independent magnetic circuits of the three-phase five-column medium-frequency transformer (Leung et al., 2010). When the circuit operates normally, the highest harmonic frequency allowed to pass through is 650 Hz. This necessitates the transformer to have a relatively high passband. Additionally, due to the higher frequency of voltage polarity conversions, losses will increase. Therefore, thin silicon steel sheets are used for the transformer core. Thin silicon steel offers advantages such as high saturation magnetic flux density, ideal loss performance, and low noise. Transformer model is displayed in (Supplementary Figure S9). The length of the transformer is 45cm, the height is 25 cm and the width is 19 cm.

To ensure minimal voltage distortion, it is crucial to maintain the magnetic flux of the transformer in a non-saturated state and operate within the linear region (Li et al., 2011). The magnetic flux density B i corresponding to each frequency component is: B i = U i 4.44 f i N A c (35) U i and f i represent the effective values and frequencies of each component, N is the turns of the transformer, and A c is the effective magnetic area of the magnetic circuit. According to the superposition principle, the composite flux density B max at its maximum can be obtained as follows: B max = B 5 + B 7 + B 11 + B 13 (36)