# Carrier Synchronization Method using High-Speed Sampling for Parallel Inverters without Communication

Takumi Iwamoto, Hiroki Watanabe, Yuki Nakata, Jun-Ichi Itoh

► To cite this version:

Takumi Iwamoto, Hiroki Watanabe, Yuki Nakata, Jun-Ichi Itoh. Carrier Synchronization Method using High-Speed Sampling for Parallel Inverters without Communication. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0101. hal-05088983

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# Carrier Synchronization Method using High-Speed Sampling for Parallel Inverters without Communication

Takumi Iwamoto, Hiroki Watanabe, Yuki Nakata and Jun-ichi Itoh

NAGAOKA UNIVERSITY OF TECHNOLOGY

1603-1, Kamitomioka-machi, Nagaoka, Niigata, Japan

Tel.: +81 / (258) - 47.9533.

E-Mail: s225057@stn.nagaoaut.ac.jp, hwatanabe@vos.nagaokaut.ac.jp, ynakata@vos.nagaokaut.ac.jp, itoh@vos.nagaokaut.ac.jp

URL: http://itohserver01.nagaokaut.ac.jp/itohlab/jp/index.html

## Acknowledgements

This work was supported by the Council for Science, Technology and Innovation (CSTI), the Cross-ministerial Strategic Innovation Promotion Program (SIP), and the 3rd period of SIP "Smart energy management system" Grant Number JPJ012207 (Funding agency: JST).

## Keywords

«Parallel operation», «Inverter», «Single-phase system», «Synchronization», «High-speed sampling».

## Abstract

This paper proposes a novel carrier-synchronized control method for the parallel operation of two inverters using high-speed sampling. The proposed method focuses on the active power component at the switching frequency. When carrier synchronization is achieved, the active power at the switching frequency becomes zero for inverters connected in parallel. The proposed method calculates the active power at the switching frequency using high-speed sampling, switching states, and the fast Fourier transform (FFT). The proposed method requires only DC voltage and inductor current measurement without additional sensors. Furthermore, the proposed method is implemented in a single-phase inverter and does not need communication between units. Experimental results demonstrate that the circulating current is successfully suppressed to less than 66% of the rated current, thereby, validating the efficacy of the proposed control method.

## Introduction

In recent years, renewable energy applications and electric vehicles have been promoted from the perspective of achieving carbon neutrality. Accordingly, the application range of power converters has expanded, and elemental technologies such as circuit topology, control methods, and design methods have been developed for each application [1-3]. The development of power converters requires specialized and multifaceted expertise, including circuit design, software development, and thermal management, which in turn results in significant time and cost requirements. In order to accommodate a wide range of demands for power converters in the future, it is important to develop a power conversion system rapidly and cost-effectively.

One approach under consideration to reduce development time is the use of modular structures, such as a Power Electronics Building Block (PEBB) [4-5]. The PEBB concept enables the realization of a power conversion system by combining multiple modules, each of which is optimized for high power density. Since the main circuit design is already included in the module, the overall development time can be significantly shortened. Moreover, maintenance is simplified, as faulty modules can be easily replaced. However, while the main circuit and certain auxiliary circuits can be modularized, the controller and detection circuits still need to be developed separately, which adds time and cost to the process.

Fig. 1 shows the configuration of a power conversion system using Universal Smart Power Module (USPM) concept. The USPM has been proposed to further reduce the development times and reduce costs within the PEBB concept [6-8]. The USPM modularizes all components of the
power converter, including not only the main power circuit but also the controller and detection circuits. By simply selecting and combining USPMs, a power conversion system can be quickly designed, thus greatly reducing the development time. In this configuration, each USPM performs a specific role, and a master controller manages the overall system. The USPM has high-speed controller and responds quickly to commands from the master controller to achieve power conversion. Furthermore, the USPM can be connected in series and parallel, enabling their applications to high-voltage and high-current systems. However, circulating currents caused by carrier asynchrony in parallel connected autonomous power converters, such as the power conversion system using USPM. Therefore, carrier synchronization is necessary.

The conventional carrier synchronization method [9] requires a signal line to the master controller or the other converter. On the other hand, the USPM system relies on wireless communication, which is slower and less reliable, between the master controller and each USPM. Therefore, the USPM systems require carrier synchronization that does not require communication with the master controller or other USPMs. The carrier synchronization methods without communication have been proposed in [10] and [11]. While these methods remove the dependency on signal transmission, their robustness against noise and external disturbances remains a significant concern, potentially limiting their practical applicability in noisy operating environments.

This paper proposes a carrier-synchronized control method for two inverters connected in parallel using high-speed sampling. The new contribution of this paper is the establishment of a carrier synchronization method that is resistant to noise and does not require high-speed communication with other converters or controllers. As a result, the circulating current is suppressed by carrier synchronization of the inverters. The proposed method synchronizes the carrier by focusing on the active power of the switching frequency component being zero when the carrier phase difference is 0°. In particular, the active power of the switching frequency component is calculated using high-speed sampling and Fast Fourier Transform (FFT), and the carrier phase is shifted so that the active power is zero. The proposed method uses FFT and integral calculation to effectively minimize the impact of noise. The rest of this paper is organized

<figure><img src="image_2.png" /><figcaption>Fig. 1. Power conversion system using USPM. Each USPM operates autonomously, and communication between USPMs is via a master controller.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Single-phase inverter with two units connected in parallel. $\omega$ is the angular frequency of the grid.</figcaption></figure>

as follows: first, the circulating current in the parallel connection of a single-phase inverter is derived; next, the proposed carrier synchronization method is explained; finally, the effectiveness of the control method is verified through the verification of the simulation and the experiment.

## Circuit configuration and derivation of circulating current

Fig. 2 shows the circuit diagram of two inverters connected in parallel. This circuit is assumed to be grid-connected. The two inverters operate with bipolar modulation, and each inverter has its own current controller. The current command value is transmitted from the master controller and operates at the same value. Circulating currents due to carrier asynchrony are a common issue in autonomous distributed power converters. In this paper, for simplicity, the circulating current is derived from the
fundamental component of the switching frequency.

Fig. 3 shows the equivalent circuit at the fundamental component of the carrier. $\omega_{sw}$ is the angular frequency of the switching frequency component and $\delta$ is the phase difference of the carrier between two inverters. The currents $i_1$ and $i_2$ flowing through in each inverter are given by

$$i_1 = \frac{L_1 + L_2}{\omega_{sw}(L_1^2 + 2L_1L_2)} V_{DC} \sin\left(\omega_{sw}t - \frac{\pi}{2}\right) - \frac{L_2}{\omega_{sw}(L_1^2 + 2L_1L_2)} V_{DC} \sin\left(\omega_{sw}t - \frac{\pi}{2} + \delta\right) \dots(1),$$

$$i_2 = -\frac{L_2}{\omega_{sw}(L_1^2 + 2L_1L_2)} V_{DC} \sin\left(\omega_{sw}t - \frac{\pi}{2}\right) + \frac{L_1 + L_2}{\omega_{sw}(L_1^2 + 2L_1L_2)} V_{DC} \sin\left(\omega_{sw}t - \frac{\pi}{2} + \delta\right) \dots(2).$$

The circulating current is given by $i_1$ and $i_2$ and by

$$i_1 - i_2 = 2 \sin\left(-\frac{\delta}{2}\right) \frac{L_1 + 2L_2}{\omega_{sw}(L_1^2 + 2L_1L_2)} V \sin\left(\omega_{sw} + \frac{\delta}{2}\right) \dots(3).$$

From (3), the amplitude of the circulating current contains a sine function with carrier phase difference $\delta$. Therefore, the circulating current becomes zero when the carrier phase difference is zero.

Next, the relationship between active power and carrier phase difference is investigated. Equation (1) is simplified and is given by

$$i_1 = I_1 \sin(\omega_{sw}t - \phi) \dots(4),$$

where the amplitude $I_1$ and phase difference $\phi$ between the carrier frequency component of the inverter output voltage and the inductor current of $i_1$ are given by

$$I_1 = \frac{V_{DC}}{\omega_{sw}(L_1^2 + 2L_1L_2)} \sqrt{L_1^2 + 2L_2(L_1 + L_2)(1 - \cos \delta)} \dots(5),$$

$$\phi = \arccos\left(\frac{-L_2 \sin \delta}{\sqrt{L_1^2 + 2L_2(L_1 + L_2)(1 - \cos \delta)}}\right) \dots(6).$$

From (6), the current phase difference $\phi$ is 90°

<figure><img src="image_4.png" /></figure>

Fig. 3. Equivalent circuit of the carrier frequency components for a single-phase inverter with two units connected in parallel. $i_1$ and $i_2$ are the carrier frequency components of the inductor current, and $\delta$ is the carrier phase difference between $v_1$ and $v_2$.

<figure><img src="image_5.png" /></figure>

Fig. 4. Carrier phase difference versus current phase and active power. When the power supply $v_1$ is phase lag versus $v_2$ ($\delta > 0°$), the active power of $v_1$ is positive and the inductor current $i_1$ is phase lag by more than 90° versus $v_1$ ($\phi > 90°$). On the other hand, when $v_1$ is phase lag versus $v_2$ ($\delta < 0°$), the active power is negative and $i_1$ phase lag is less than 90° versus $v_1$ ($\phi < 90°$). (a) Active power. (b) Current phase.

when the carrier phase difference $\delta$ is 0° or 180°. Also, the instantaneous power is given by

$$p_1 = \frac{1}{2} V_{DC} I_1 \{\cos \phi - \cos(2\omega_{sw}t + \phi)\} \dots(7),$$

where substituting $\phi=90°$, the active power (average power) is zero. Thus, (6) and (7) show that during carrier synchronization, the power factor of the carrier frequency component is zero, and the active power is zero. Therefore, carrier synchronization is achieved by controlling the carrier phase angle of the inverter so that the active power is zero.

Fig. 4 shows the current phase and the active power versus the carrier phase difference. The
<figure><img src="image_6.png" /><figcaption>Fig. 5. Control block of proposed control. Inductor current control section is configured by PI control.</figcaption></figure>

active power is zero when the carrier phase difference $δ$ is 0° and 180°. Therefore, it is essential to consider the direction of the carrier phase shift. From Fig. 4, carrier synchronization is achieved by leading the carrier phase when the active power is positive and lagging the carrier phase when the active power is negative.

## Carrier synchronization method

Fig. 5 shows the control block diagram of the proposed carrier synchronization method. The inductor current control section uses a PI controller to control the inverter current $i_{L1}$ so that the power factor with the grid voltage is unity.

In the proposed carrier synchronous control, first, the amplitude and phase of the carrier frequency components are obtained from the high-speed sampled inductor current $i_{L1}$ and inverter voltage $v_{inv1}$ by FFT [12]. Next, the instantaneous power is calculated from the acquired amplitude and phase of the fundamental components $i_1$ and $v_{inv1}$ and integrated. The active power is calculated from the product of the integral and the switching frequency. The calculation period of FFT and the active power should be an integer multiple of the carrier period in order to reduce quantization errors. Note that $v_{inv1}$ is not detected and given from $V_{DC}$ and the state of the switches by

$$ \begin{cases} v_{inv1} = V_{DC} & (S_{1\_1} = 1) \\ v_{inv1} = -V_{DC} & (S_{1\_1} = 0) \end{cases} \quad (8). $$

The carrier phase difference is shifted by PI control so that the calculated active power is set to zero. The control period of the PI control depends on the active power calculation period.

This control uses only the switching state of the autonomous operated inverter and the detection of the input voltage $V_{DC}$, grid voltage $v_{ac}$ and inductor current. Therefore, the proposed

<table><caption>Table 1 Simulation condition.</caption><thead><tr><th>Quantity</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>DC voltage</td><td>$V_{DC}$</td><td>400 V</td></tr><tr><td>Grid voltage</td><td>$v_{ac}$</td><td>200 V<sub>rms</sub></td></tr><tr><td>Grid frequency</td><td>$f$</td><td>50 Hz</td></tr><tr><td>Rated current of one unit</td><td>$i_{ac}$</td><td>20 A<sub>rms</sub></td></tr><tr><td>Rated power of one unit</td><td>$P_1, P_2$</td><td>4 kW</td></tr><tr><td>Inverter side inductance</td><td>$L_1$</td><td>640 μH<br>(%Z=2%)</td></tr><tr><td>Grid side inductance</td><td>$L_2$</td><td>320 μH<br>(%Z=2%)</td></tr><tr><td>Switching frequency</td><td>$f_{sw}$</td><td>100 kHz</td></tr></tbody></table>

method does not require information from the other inverter and achieves carrier synchronization without communication with the master controller or other inverters. Detection noise is also a significant problem in high-speed sampling. However, the proposed control uses FFT calculation to obtain the switching frequency component of the inductor current. Therefore, the noise of the sampling frequency component is eliminated due to white noise.

## Simulation Verification

Table 1 shows the simulation conditions. In this paper, a USPM controller is assumed, and the inductor current is sampled at 10 MHz. FFT and the active power are performed in the range of 10 cycles of the switching frequency in order to reduce quantization error. In addition, the inductor current control operates in the switching cycle.

Fig. 6 shows the grid voltage, inductor current, the circulating current, the active power of the switching frequency component, and the carrier phase difference $δ$ during rated power operation in the simulation. The RMS value of the
circulating current before carrier synchronization is 410 mA (0.021p.u.), and that after synchronization is reduced to 32 mA (0.0016p.u.). This result clearly demonstrates that the circulating current is suppressed by the carrier synchronization. In addition, the circulating current before carrier synchronization contains a component that fluctuates at a period of the grid frequency of 50 Hz and a component that fluctuates at a carrier frequency of 100 kHz. The circulating current due to carrier asynchrony is affected by the duty cycle. In addition, the duty cycle varies with the grid voltage. Therefore, the circulating currents due to carrier asynchrony include the grid frequency component. In this paper, the carrier synchronization method focuses only on the circulating current of the carrier frequency component. However, the circulating current of the 50 Hz component is suppressed after carrier synchronization because the 50 Hz component is also caused by carrier asynchrony. In addition, the active power occurs before carrier synchronization. The active power is kept nearly zero by turning on carrier synchronization control. As a result, the carrier phase difference $δ$ is also nearly 0°. The maximum phase difference after carrier synchronization is 1.4°.

Fig. 7 shows the inverter output voltage and inductor current before and after carrier synchronization. The initial phase difference of the inverter output voltage is set to 30°, and the inductor current is not a triangular wave before carrier synchronization. On the other hand, and the inductor current is a triangular wave after carrier synchronization. In addition, before carrier synchronization, a potential difference occurs due to the circulating current. This increases the current ripple and the current RMS value. On the other hand, after carrier synchronization, a potential difference does not occur because there is no circulating current. Therefore, only the load current flows, and the current RMS value decreases.

Fig. 8 shows the circulating currents versus sampling frequency and FFT calculation period. The circulating current remains nearly constant when the sampling frequency exceeds 20 times the switching frequency. Therefore, the sampling frequency should be set between 20 and 30 times the switching frequency. This adjustment allows the switching frequency to be set from 3 to 5 times higher compared to cases where the sampling frequency is set to 100 times the switching frequency. In addition, the quantization error is reduced by increasing the calculation period of

<figure><img src="image_7.png" /></figure>

Fig. 6. The output voltage, the inductor current, the circulating current, the active power of switching frequency component, and the carrier phase difference. The carrier phase difference before carrier synchronization is set to 30°, and the carrier synchronization control is turned on in 20 ms.

<figure><img src="image_8.png" /></figure>

Fig. 7. Magnified waveforms of inverter voltage and inductor current. The carrier phase difference before carrier synchronization is set to 30°. (a) Before carrier synchronization. (b) After carrier synchronization.

FFT. The circulating current is reduced to less than 0.0014p.u. (at 500 cycles FFT) when the
sampling frequency is 100 times the switching frequency.

## Experimental verification

Table 2 shows the experimental conditions. In this experiment, low carrier frequency was used to verify the effect of the proposed method. In addition, the same controller is used in the two inverters to set the initial phase difference. Note that information is not shared between the two inverters. Besides, the current phase is compensated beforehand for the detection delay of the current sensor.

Fig. 9 shows the waveforms before and after carrier synchronization. Fig. 10(a) shows the waveforms before carrier synchronization. The initial phase difference $δ$ is set to 15°. The circulating current has a DC component. The circulating current is evaluated in terms of the AC component, as the carrier synchronization method affects only the AC component of the circulating current. This is due to inverter detection errors and imbalance of the inductance values. The RMS value of the AC component in the circulating current before carrier synchronization is 740 mA. The circulating current is caused by the carrier phase difference. In addition, similar to the simulation results, the circulating current contains a component that fluctuates at a grid frequency of 50 Hz and a component that fluctuates at a carrier frequency of 10 kHz. In contrast, Fig. 10(b) shows the waveforms after carrier synchronization by the proposed method. The phase difference $δ$ becomes approximately 0.37° (0.1 µs). The carrier phase difference is not 0° due to errors in the current phase calculation. This is due to quantization errors. The maximum error of the phase due to quantization error is 1/2 of the sampling period (2us). The phase difference after carrier synchronization is less than the maximum error of the phase due to quantization error. Thus, the proposed method achieves carrier synchronization. In addition, the RMS value of the AC component of the circulating current after carrier synchronization becomes 250 mA. The circulating current is suppressed to less than 66% of the rated current by carrier synchronization.

Fig. 10 shows the waveforms of the inductor current and circulating current near zero crossing of the grid voltage. The initial phase difference $δ$ is set to 15°. The inductor current is not a triangular wave, and the current ripple is 4.4 A before carrier synchronization. Fig. 11(b) shows

<figure><img src="image_9.png" /><figcaption>Fig. 8. Circulating currents versus sampling frequency and FFT calculation period. The circulating current is suppressed as the sampling frequency is larger and the FFT computation period is longer.</figcaption></figure>

<table><caption>Table 2 Simulation condition.</caption><thead><tr><th>Quantity</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>DC voltage</td><td>V<sub>DC</sub></td><td>200 V</td></tr><tr><td>Grid voltage</td><td>v<sub>ac</sub></td><td>100 V<sub>rms</sub></td></tr><tr><td>Grid frequency</td><td>f</td><td>50 Hz</td></tr><tr><td>Rated current of one unit</td><td>i<sub>ac</sub></td><td>10 A<sub>rms</sub></td></tr><tr><td>Rated power of one unit</td><td>P<sub>1</sub>,P<sub>2</sub></td><td>1 kW</td></tr><tr><td>Grid side inductance</td><td>L<sub>1</sub></td><td>240 μH<br>(%Z=3.6%)</td></tr><tr><td>Inverter side inductance</td><td>L<sub>2</sub></td><td>440 μH<br>(%Z=2.8%)</td></tr><tr><td>Switching frequency</td><td>f<sub>sw</sub></td><td>10 kHz</td></tr><tr><td>Sampling frequency</td><td>f<sub>samp</sub></td><td>250 kHz</td></tr><tr><td>FFT calculation period</td><td></td><td>0.05 s</td></tr></tbody></table>

the waveforms after carrier synchronization by the proposed method. The inductor current after carrier synchronization is almost a triangular wave. In addition, the inductor current ripple is 4.1A. Therefore, the RMS value of the inductor current is also reduced by carrier synchronization.

## Conclusion

This paper proposed a carrier synchronized control of two inverters connected in parallel using high-speed sampling. In the proposed control, carrier synchronization is achieved by calculating the active power using fast sampling and adjusting the carrier phase to minimize the active power to zero. This control does not require information from the other inverters. Thus, the signal lines to the other inverter and the master
<figure><img src="image_10.png" /><figcaption>Fig. 9. The grid voltage, inductor currents, and circulating current. The circulating current is suppressed by carrier synchronization. (a) Before carrier synchronization. (b) After carrier synchronization.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Inductor currents and the circulating current near zero crossing of grid voltage. Inductor currents before carrier synchronization are not triangular, but after carrier synchronization they are almost triangular. (a) Before carrier synchronization. (b) After carrier synchronization.</figcaption></figure>

controller are not required. In addition, the circulating current is suppressed by setting the sampling frequency and FFT period appropriately to suppress the effect of quantization errors. Experimental verification showed that the proposed method suppressed the circulating current to less than 66% of the rated current. These results verified the effectiveness of the proposed method. In the future, the proposed method will be verified under higher frequency conditions using the USPM controller.

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