# Digital Implementation of LCC Resonant Converters for X-ray Generator with Optimal Trajectory Startup Control

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## Abstract

**:**

## 1. Introduction

## 2. Analysis of State Trajectory-Based Startup for LCC Converter

_{maxN}in the first switching cycle. ±i

_{maxN}is the limit value of the resonant current in the state trajectory control, as shown in Figure 4. At the end of converter startup, the frequency value compensated by PI control is added to the frequency value predicted by state trajectory control [15]. V

_{in}and V

_{o}are sampled for the calculation of state trajectory control. As described in Figure 4, the LCC converter works in the 3rd operating mode with S1 and S4. In the 3rd operating mode, the state trajectory is arc AB, and the state trajectory equation is given as:

_{LrN}(t) is the normalized resonant current value, V

_{CrN}(t) is the normalized voltage value of capacitor C

_{r}, V

_{CpN}(t) is the normalized voltage value of capacitor C

_{p}, V

_{CprN}(t) is the sum of V

_{CrN}(t) and V

_{CpN}(t). i

_{LrN}(t

_{2}) and V

_{CprN}(t

_{2}) are the initial values of i

_{LrN}(t) and V

_{CprN}(t) in the 3rd operating mode.

_{LrN}(t

_{5}) and V

_{CprN}(t

_{5}) are the initial values of i

_{LrN}(t) and V

_{CprN}(t) in the 6th operating mode.

_{T0}and ρ

_{T1}. The central angle of arc AB is defined as θ

_{0}, and the central angle of arc BCD is defined as θ

_{1}+ θ

_{2}. The process of solving the central angle can be obtained as:

_{T0}and ρ

_{T1}can be expressed as:

_{0}of S

_{1}and S

_{4}and T

_{1}of S

_{2}and S

_{3}can be derived as:

_{0}is the double-element resonance impedance [19], Z

_{1}is the triple-element resonance impedance, i

_{LrN}(t

_{0}) and V

_{CprN}(t

_{0}) are the initial values of i

_{LrN}(t) and V

_{CprN}(t) in the 1st operating mode, i

_{LrN}(t

_{1}) and V

_{CprN}(t

_{1}) are the initial values of i

_{LrN}(t) and V

_{CprN}(t) in the 2nd operating mode.

_{LrN}(t

_{3}) and V

_{CprN}(t

_{3}) are the initial values of i

_{LrN}(t) and V

_{CprN}(t) in the 4th operating mode, i

_{LrN}(t

_{4}) and V

_{CprN}(t

_{4}) are the initial values of i

_{LrN}(t) and V

_{CprN}(t) in the 5th operating mode.

_{A}is the value of V

_{CprN}(t) at point A, V

_{B}is the value of V

_{CprN}(t) at point B, V

_{C}is the value of V

_{CprN}(t) at point C, V

_{D}is the value of V

_{CprN}(t) at point D, R is the radius of state trajectory arc in the 3rd operating mode, r is the radius of state trajectory arc in the 4th operating mode.

_{1}, B

_{1}, and C

_{1}can be expressed as (14).

_{in}and all currents with the current factor V

_{in}/Z

_{0}, the following expressions can be derived as [20]:

_{E}is the value of V

_{CprN}(t) at point E, V

_{F}is the value of V

_{CprN}(t) at point F, V

_{C}is the value of V

_{CprN}(t) at point C, V

_{G}is the value of V

_{CprN}(t) at point G.

_{1}, b

_{1}, and c

_{1}can be expressed by (19)

## 3. Implementation of State Trajectory Control Based on FPGA

_{1}, B

_{1,}and C

_{1}can be calculated in the 1st, 2nd, and 3rd calculation of multiplication correspondingly. θ

_{0}can be calculated in the calculation of arcsine. θ

_{1}can be calculated in the 3rd calculation of arc cosine. θ

_{2}is the difference between the 2nd and 1st calculation of arc cosine.

_{1}can be calculated in the 1st calculation of multiplication, and c

_{1}is the sum of the 4th and 6th calculations. θ

_{3}+ θ

_{5}can be calculated when the 3rd calculation of arc cosine is completed. θ

_{4}can be calculated in the 4th calculation of arc cosine.

## 4. Experiment Results

_{o}is 100 kV, i

_{max}is set as 200 A, the resonant tank current achieves the set value of 200 A during the first cycle, and no extra voltage or current overshoot is injected into the resonant circuit. A fast rise time is achieved during the startup process, where (10–90%) of V

_{o}is 350 μs. At the same time, for conventional PFM control, the experiment waveforms at startup are presented in Figure 10b A fast rise time is achieved during the startup process. Based on the same circuit parameters and classical PI control, the rise time of the output voltage of the generator is 1.182 ms.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 10.**Experimental waveforms: (

**a**) Waveforms of startup with OTC; (

**b**) Waveforms of startup with conventional PFM.

**Figure 11.**Experimental waveforms of switching process of different kV: (

**a**) From 80 kV to 120 kV; (

**b**) From 120 kV to 80 kV; (

**c**) From 80 kV to 140 kV; (

**d**) From 140 kV to 80 kV.

Base Value | ||
---|---|---|

Input voltage (V) | V_{in} | |

Output voltage (V) | V_{o} | |

L_{r}, C_{r} | Impedance (Ω) | ${Z}_{0}=\sqrt{{L}_{r}/{C}_{r}}$ |

Frequency (Hz) | ${\omega}_{0}=1/\sqrt{{L}_{r}{C}_{r}}$ | |

L_{r}, C_{r}, C_{p} | Impedance (Ω) | ${Z}_{1}=\sqrt{{L}_{r}\left({C}_{r}+{C}_{p}\right)/{C}_{r}\xb7{C}_{p}}$ |

Frequency (Hz) | ${\omega}_{1}=1/\sqrt{{L}_{r}{C}_{r}{C}_{p}/\left({C}_{r}+{C}_{p}\right)}$ |

Parameters | Value |
---|---|

Output Power (P) | 42 kW |

Input voltage (V_{in}) | 500 ± 50 V |

Output voltage (V_{o}) | 60 kV~140 kV |

Output current (I_{o}) | 10 mA~350 mA |

Load (R_{L}) | 512 kΩ |

Transformer Ratio | 15:301 (6) |

Resonant inductor (L_{r}) | 30 μH |

Series Resonant capacitor (C_{r}) | 0.66 μF |

Parallel Resonant capacitor (C_{p}) | 0.266 μF |

Switching frequency (f_{sw}) | 70~120 kHz |

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**MDPI and ACS Style**

Zhao, Z.; Zhang, S.; Li, L.; Fan, S.; Wang, C.
Digital Implementation of LCC Resonant Converters for X-ray Generator with Optimal Trajectory Startup Control. *World Electr. Veh. J.* **2022**, *13*, 71.
https://doi.org/10.3390/wevj13050071

**AMA Style**

Zhao Z, Zhang S, Li L, Fan S, Wang C.
Digital Implementation of LCC Resonant Converters for X-ray Generator with Optimal Trajectory Startup Control. *World Electric Vehicle Journal*. 2022; 13(5):71.
https://doi.org/10.3390/wevj13050071

**Chicago/Turabian Style**

Zhao, Zhennan, Shanlu Zhang, Lei Li, Shengfang Fan, and Cheng Wang.
2022. "Digital Implementation of LCC Resonant Converters for X-ray Generator with Optimal Trajectory Startup Control" *World Electric Vehicle Journal* 13, no. 5: 71.
https://doi.org/10.3390/wevj13050071