船舶甲板感应加热矫平机电能变换及能量转换系统

船舶甲板感应加热矫平机电能变换及能量转换系统 Electrical Energy Conversion and Energy Transformation System of Ship Deck Induction Heating Leveling Machine 1 first-author 李俊南（1994—），男，硕士，主要从事逆变焊接电源研究。 李俊南（1994—），男，硕士，主要从事逆变焊接电源研究。 1 corresp 方臣富（1954—），男，教授，博士生导师，主要从事逆变焊接电源和舰船电源的教学与研究。E-mail：13952846832@163.com。 方臣富（1954—），男，教授，博士生导师，主要从事逆变焊接电源和舰船电源的教学与研究。E-mail：13952846832@163.com。 13952846832@163.com 1 1 1 2 江苏科技大学 材料科学与工程学院,江苏 镇江 212003 江苏科技大学 材料科学与工程学院,江苏 镇江 212003 School of Materials Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China School of Materials Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China 江苏自动化研究所,江苏 连云港 222006 江苏自动化研究所,江苏 连云港 222006 Jiangsu Automation Research Institute,Lianyungang 222006, China Jiangsu Automation Research Institute,Lianyungang 222006, China 船厂以往均采用火焰加热工艺矫平工艺平整甲板，容易引发火灾且平整效率较低。感应加热矫平工艺采用感应加热矫平机将三相正弦波高压小电流交流电变换成单相高频矩形波低压大电流交流电，并通过电磁、磁电转换在钢板中产生涡流热能和磁滞热能，快速、精准、可控地加热甲板特定区域。感应加热矫平机由固定式调功器、移动式调频器和小车式感应加热器通过电缆串联组成，突破了不能高效长距离输送高频低压大电流交流电的技术瓶颈；研发了变频器频率跟踪、调节及锁相技术，保证了输出回路为准纯电阻性态，高效利用电网电能；设计了小车式感应加热器，可按需移至任意区域施行矫平工艺，参数可就近调节。实验室测试和造船厂应用表明：船舶甲板感应加热矫平机电能变换及能量转换系统工作稳定可靠，可始终保证矫平机工作在准谐振状态，使感应加热器输出功率因数接近为1。 Shipyards have previously used flame heating processes to level decks, which are prone to fires and have low leveling efficiency. The induction heating leveling process adopts an induction heating leveling machine to convert three-phase sine wave high-voltage small-current alternating current(AC) into single-phase high-frequency rectangular wave low-voltage high-current AC, and through electromagnetic and magnetoelectric conversion to produce eddy current heat and hysteresis heat in the steel plate to quickly, accurately and controllably heat specific areas of the deck. The induction heating leveling machine is composed of a fixed power regulator, a mobile frequency regulator and a trolley-type induction heater connected in series through cables, which breaks through the technical bottleneck that high-frequency, low-voltage, and high-current AC power cannot be efficiently transmitted over long distances. The frequency tracking, regulation and phase locking technology of the frequency converter was developed, which ensures the quasi-pure resistive state of the output circuit so as to efficiently utilize the grid power. A trolley type induction heater was designed, which can be moved to any area as required to carry out the leveling process and the parameters can be adjusted nearby. Laboratory tests and shipyard applications show that the electrical energy conversion and energy transformation system of ship deck induction heating leveling machine is stable and reliable, and can always ensure that the leveling machine works in a quasi-resonant state and the output power factor of the induction heater is close to one. 船舶甲板 感应加热 矫平 准纯电阻性态 准谐振状态 ship deck induction heating leveling quasi-pure resistive state quasi-resonant state 前言 1 集装箱货船、滚装船及邮轮等船舶甲板通常由4~8 mm较薄钢板拼焊而成，焊接产生的收缩变形、失稳波浪变形等残余变形会导致甲板不平整，造船厂以往均采用火焰加热矫平工艺平整甲板。火焰加热矫平工艺使用氧乙炔等可燃气体容易引发火灾、矫平质量主要依赖操作工人经验，且平整效率较低。感应加热矫平工艺采用感应加热矫平机将三相正弦波高压小电流交流电变换成单相高频矩形波低压大电流交流电，并通过电磁、磁电转换机制在钢板中产生涡流热能和磁滞热能，快速、精准、可控地加热甲板特定区域，使焊接接头应力松弛，重新调整和改善应力分布，以消除因焊接过程不均匀加热钢板产生的残余应力变形。感应加热矫平方法相比于传统的火焰加热矫平方法，具有加热速度快、高效清洁、安全无污染、易实现自动化等优势 ［ 1 1 - 7 7 1-7 ］ 。韩飞飞 ［ 8 8 ］ 针对功率传输电缆、同轴变压器、感应加热器及谐振电路四个关键部件进行深入分析与设计，并通过仿真结合实际测试验证了设计合理性。高宇 ［ 9 9 ］ 在理论分析的基础上设计了感应加热矫平机调功器，采用Buck软开关拓扑结构实现了矫平机输出功率调节。马瑞祥 ［ 10 10 ］ 设计了感应加热矫平机调频器逆变主电路和控制电路，感应加热器工作于准谐振状态。 本文针对感应加热矫平机电路结构、调功器、变频器、感应加热器的原理进行介绍，设计了一套船舶甲板感应加热矫平机电能变换及能量转换系统。实际应用结果证明该系统达到了预期要求。 感应加热矫平机电路结构 1 考虑到船厂应用环境的复杂性和工作范围大的特点，感应加热矫平机需要足够大的工作半径和长距离大功率输电能力。船舶甲板感应加热矫平机采用分体式、模块化结构，主要由研制的固定式调功器、移动式调频器和小车式感应加热器通过电缆串联组成电能变换及能量转换系统，突破了不能高效长距离输送高频低压大电流交流电的技术瓶颈。感应加热矫平机由三级电能变换及能量转换系统组成，其电路结构框图如 图1 图1 所示。 感应加热矫平机主要由以下几部分组成： （1）固定式调功器：包括三相整流滤波电路、全桥软开关逆变电路以及全波整流电路。调功器位于矫平机的最前端，电能变换过程是将三相网电整流滤波为直流电，再经过全桥软开关逆变电路变换为高频方波交流电，然后经过全波整流变换为可调节的低压大电流直流电，通过控制调功器的直流输出电压，来调节矫平机输出功率。然后通过同轴电缆远距离安全传输低压直流电至变频器模块。 （2）移动式变频器：包括全桥逆变电路、中频变压器和谐振匹配电路。变频器位于矫平机的中部，作用是将调功器输出可调直流电经过全桥逆变电路变换为高频交流电，通过变压器降压后再经谐振匹配电路将电能传输至感应加热线圈；其安全和便于移动的特点有效扩大了矫平机整机工作范围；利用频率跟踪技术实现逆变器输出电压和输出电流近似同频同相，使感应加热器工作于准谐振状态，实现高功率因数输出。 （3）小车式感应加热器：包括同轴变压器和感应加热头。位于矫平机的末端，按照甲板感应加热矫平工艺要求，通过操控小车式感应加热器完成甲板矫平。工作原理为：经同轴变压器将变频器输出的高频交流电变换为电压更低、电流更大的高频交流电；将感应加热器置于甲板待加热区域上方，高频交流电产生交变磁场，磁场内磁力线穿过甲板加热区域时，被切割而产生涡流热能和磁滞热能，完成电能—磁能—电能—热能的转化，实现加热矫平。 调功器 1 调功器是感应加热矫平机电能变换及能量转换系统的前级，用于功率调节，其主要功能是将三相正弦波高压小电流交流电变换为电压可控的低压直流电，实现远距离安全输电。 主电路 2 调功器功率主电路采用饱和电感式全桥逆变软开关拓扑结构，其主电路如 图2 图2 所示。调功器主电路由EMI滤波电路、整流滤波电路、全桥软开关逆变电路、中频变压器与全波整流电路组成。EMI滤波电路由X电容，Y电容与共模电感组成，作用是防止开关电源形成的高频杂波污染电网，提升设备电磁兼容性。整流滤波电路由三相整流桥与多个铝电解电容C n 并联而成，R 1 为电解电容的放电电阻。全桥软开关逆变部分主要由两组4个IGBT开关管、换流电感L r 与谐振电容C x 组成。全波整流部分主要由输出整流二极管VD 5 、VD 6 ，RC缓冲电路和输出电抗器L f 组成。 全桥软开关电路工作方式 2 全桥逆变软开关电路主要波形如 图3 图3 所示。其电能变换过程描述如下： t 0 ~ t 1 时段，开关管IGBT 1 、IGBT 4 导通，由于电感处于初始状态，有很大阻抗，可以抑制原边电流 I p 增大，因此IGBT 1 、IGBT 4 实现零电流开通，原边电流不够大而次级输出电压为0。 t 1 ~ t 2 时段，电感L r 处于完全饱和状态，可近似于一段导线，其两端电压为0，谐振电容电压 U Cx 保持不变，次级整流输出电压 U s 突变为输入电压与谐振电容电压的和除以匝比，二极管VD 5 关断，在此时间内， I p 上升而 U Cx 线性增加。 t 2 ~ t 3 时段，由于C 1 存在，IGBT 1 集电极发射极电压不能突变为零电压关断，此阶段 I p 不变。 t 3 ~ t 4 时段，C 3 两端电压下降为0，此时IGBT 3 的反向并联二极管VD 3 自然导通，IGBT 3 接收开通信号即可完成零电压开通。 t 4 ~ t 5 时段， I p 线性下降， U Cx 保持不变，初级电流 I p 折算到次级电流逐渐减小，最终小于输出电流 I o ，VD 5 与VD 6 同时导通，L f 处于供电状态。 t 5 ~ t 6 时段， I p 下降至很低的水平，此时IGBT 4 近似为零电流关断。后续导通过程按照以上规律往复进行。 变频器 1 变频器采用频率跟踪、调节及锁相技术，保证了感应加热矫平机输出回路瞬时阻抗始终为准纯电阻性态，极大提高了高频低压大电流交流电的功率因数，可将电网提供的电能最大化转换成钢板中的涡流热能和磁滞热能。 主电路 2 变频器主电路采用全桥串联谐振逆变拓扑结构，电路原理如 图4 图4 所示 ［ 11 11 ］ 。逆变主电路由输入滤波电路、RCD缓冲吸收电路、全桥逆变电路组成。输入滤波电路由R 1 、C 1 ~C 4 组成，R 1 是大电解电容C 1 ~C 4 的泄放电阻，该电路用于滤除进入全桥逆变电路中的交流分量，并联的大电容使逆变环节输入为更稳定的直流电压，可等效为恒压源。RCD缓冲电路由R 2 、C 5 、VD 1 以及R 4 、C 10 、VD 2 组成，用于限制功率主开关管开通和关断瞬间的电流上升率与电压上升率，从而保护开关管，延长使用寿命。IGBT模块IGBT 1 /IGBT 2 、IGBT 3 /IGBT 4 构成全桥逆变电路，实现直流到方波交流电的电能变换过程。 感应加热谐振槽路动态分析 2 感应加热头的等效磁场分布情况如 图5 图5 所示。由于待矫平钢板温度升高，磁导率降低，加热线圈耦合到待矫平钢板的磁力线发生变化，等效到谐振槽路的电感量随之发生变化，导致整个谐振槽路的谐振频率变化。另外，加热线圈与钢板的距离远近、串联谐振槽路的寄生电感和电容大小、变压器的漏感等都会影响谐振槽路的谐振频率 ［ 4 4 ］ 。 串联谐振槽路等效电路如 图6 图6 所示，R 2 是负载电阻对逆变器输出的等效电阻， <math id="M1"><mi mathvariant="normal">Δ</mi><msub><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045908&type= 5.75733376 2.62466669 是变化的加热线圈等效到逆变器输出端的电感，C 2 是谐振电容对逆变器输出的等效电容。在整个串联谐振等效电路中，影响谐振腔谐振频率主要因素是谐振电感L 2 。 将逆变器输出方波电压 <math 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id="M9"><msub><mrow><mi>f</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045958&type= 10.49866676 7.61999989 设负载谐振角频率为 <math id="M10"><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045944&type= 4.23333359 2.62466669 ，谐振频率为 <math id="M11"><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045965&type= 2.62466669 3.38666677 ，则： <math display="block" id="M12"><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><mi>L</mi><mi>C</mi></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045969&type= 17.27199936 8.97466660 在加热过程中，结合串联谐振槽路等效电路，品质因数为 <math id="M13"><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045949&type= 2.70933342 3.38666677 ，则： <math display="block" id="M14"><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><mfenced separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045976&type= 32.08866501 11.43000031 <math display="block" id="M15"><mi>Q</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045980&type= 44.53466415 10.07533360 谐振槽路等效电抗 <math id="M16"><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045985&type= 3.72533321 2.62466669 可表示为： <math display="block" id="M17"><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045987&type= 50.71533585 9.82133293 现假设逆变器工作角频率 <math id="M18"><msub><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046001&type= 3.55599999 2.62466669 和谐振角频率 <math id="M19"><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045992&type= 4.23333359 2.62466669 比值为 <math id="M20"><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29045996&type= 4.06400013 2.62466669 ，根据 式（5） 式（5） 、 式（6） 式（6） 可知： <math display="block" id="M21"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Q</mi><mo>-</mo><mfrac><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046005&type= 43.77266693 12.10733318 逆变器输出电流为： <math display="block" id="M22"><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>U</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">π</mi><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Q</mi><mo>-</mo><mfrac><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>t</mi><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046017&type= 62.14533615 17.01799965 式中　 <math id="M23"><msub><mrow><mi>φ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046021&type= 3.64066648 3.38666677 为谐振槽路的阻抗角，有如下表达式： <math display="block" id="M24"><msub><mrow><mi>φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046025&type= 54.86399841 13.54666710 由 式（9） 式（9） 可以得出结论，在感应加热过程中，等效谐振电感发生变化时，谐振槽路的谐振频率也会发生变化，如果逆变器输出电压频率不变，则其与谐振槽路的谐振频率不一致，导致串联谐振电流相位与逆变器输出电压相位不一致，最终致使输出功率因数无法达到最大值。因此，在加热过程中，需要同时检测输出电压与输出电流的频率及相位，通过频率跟踪控制策略，使逆变器输出电压与输出电流同频同相位。 变频器频率跟踪设计方案 2 在感应加热输出控制过程中采用频率跟踪技术，通过适当的闭环反馈控制电路，使逆变器开关管的工作频率 <math id="M25"><msub><mrow><mi>f</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046027&type= 2.03200006 3.38666677 始终跟随负载固有频率 <math id="M26"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046031&type= 2.20133328 3.38666677 的变化而变化，即 <math id="M27"><msub><mrow><mi>f</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>≈</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046036&type= 7.45066643 3.38666677 ，此时电路工作于准谐振状态，电压与电流相位近似相同，电路的输出功率因数接近1，达到最大功率输出。同时逆变电路的功率开关器件也工作在软开关状态，大大降低了开关损耗，简化了开关管保护吸收电路。 频率跟踪通常实现方式是锁相环技术，采用专用集成锁相芯片以及PWM控制芯片来实现相位锁定与频率同步。模拟锁相环为一个闭环负反馈控制系统，基本组成结构为：鉴相器，环路滤波器，压控振荡器，结构原理框如 图7 图7 所示 ［ 12 12 ］ 。 模拟锁相环的基本工作原理为：锁相电路在正常情况下有两种工作状态，首先，当回路中没有输入信号时，压控振荡器工作在自由振荡状态，此时的振荡频率称为中心频率 <math id="M28"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046065&type= 2.11666679 3.38666677 ， <math id="M29"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046065&type= 2.11666679 3.38666677 的值由压控振荡器的外接电容 <math id="M30"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046067&type= 3.21733332 2.62466669 和外接电阻 <math id="M31"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046057&type= 3.04800010 2.62466669 决定；其次，当回路中有输入信号 <math id="M32"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">i</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046076&type= 5.50333309 2.62466669 时，鉴相器开始工作，通过比较输入信号 <math id="M33"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">i</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046090&type= 5.50333309 2.62466669 和由压控振荡器输出端采样的反馈信号 <math id="M34"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">o</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046082&type= 5.92666674 2.62466669 的相位，产生一个误差电压信号 <math id="M35"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">d</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046084&type= 5.92666674 2.62466669 ，送入环路滤波器滤除高频谐波分量并进行降噪处理，产生平滑的直流电压信号 <math id="M36"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">c</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046104&type= 5.75733376 2.62466669 。 <math id="M37"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">c</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046098&type= 5.75733376 2.62466669 作为压控振荡器调节输出信号频率的依据，控制压控振荡器的振荡频率，在其作用下，锁相环输出信号频率 <math id="M38"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046107&type= 2.20133328 3.38666677 与输入信号频率 <math id="M39"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046102&type= 1.77800000 3.38666677 的差值逐渐减小，最终趋于一致。此时回路中；固有的反馈特性使得压控振荡器输出信号 <math id="M40"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">o</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046112&type= 5.92666674 2.62466669 与回路输入信号 <math id="M41"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">i</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046124&type= 5.50333309 2.62466669 同步或锁定， <math id="M42"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046192&type= 2.20133328 3.38666677 等于 <math id="M43"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046130&type= 1.77800000 3.38666677 ，锁定后输入信号与反馈信号（输出信号）之间的相位差表现为一个固定的电压值。 假设输入信号表示为： <math display="block" id="M44"><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046135&type= 32.17333603 5.16466665 式中　 <math id="M45"><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046149&type= 2.62466669 2.62466669 为输入信号的振幅； <math id="M46"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046151&type= 3.38666677 2.62466669 为输入信号的角频率； <math id="M47"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046153&type= 2.53999996 2.62466669 为输入信号的初始相位，作为鉴相器的比较初始值使用，为常数。 输出信号表示为： <math display="block" id="M48"><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046142&type= 37.76133347 5.16466665 式中　 <math id="M49"><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046146&type= 3.04800010 2.62466669 为输出信号的振幅； <math id="M50"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046157&type= 3.80999994 2.62466669 为输出信号的角频率； <math id="M51"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046162&type= 3.04800010 2.62466669 为输出信号的初始相位，其值随着时间 <math id="M52"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046386&type= 1.10066664 2.62466669 不断变化。 则输入与输出信号的相位差可表示为： <math display="block" id="M53"><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mi>t</mi><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046171&type= 37.16867065 3.38666677 当锁相环频率锁定时，输入信号与输出信号的差值为定值，则有： <math display="block" id="M54"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046176&type= 41.48666382 7.61999989 相位对时间的导数为瞬时频率，则输出信号的瞬时频率 <math id="M55"><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046180&type= 2.11666679 3.38666677 可表示为： <math display="block" id="M56"><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046194&type= 56.13399887 9.05933285 由此可见，锁相环实现了上文提到的输出信号与环路输入信号的相位差恒定，频率相同。此时， <math id="M57"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046189&type= 1.77800000 3.38666677 相当于负载回路的固有频率， <math id="M58"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046192&type= 2.20133328 3.38666677 送入PWM芯片产生相同频率的PWM控制信号，控制开关管的开通与关断，从而实现开关管工作频率实时跟随负载固有频率变化，两者在感应加热的任一时刻近似相等，电路工作在谐振状态，输出功率达到最大化。 变频器实现频率追踪 2 图8 图8 为最大功率条件下对应的变频器输出电压与负载电流波形。逆变系统输出电压为方波，电流波形为正弦波，是串联谐振式逆变感应加热电源的典型输出波形。逆变器输出电压与电流相位基本相同，达到谐振或准谐振状态，功率因数接近于1，获得最大功率输出。 感应加热器 1 感应加热器主要有同轴变压器和感应加热头两个能量转换部分，作为整个矫平机系统中电、磁、热能量转换的承载机构，其设计好坏直接决定了最终对钢板感应加热矫平的效果，连接示意如 图9 图9 所示。 同轴变压器 2 同轴变压器结构如 图10 图10 所示，最大特点是初级绕组与次级绕组共轴心，初级线圈沿着磁芯轴向绕制；次级是由前端盖板、后端盖板、导电板、穿过磁芯的轴心铜管和套在磁芯外侧的铜筒组成，初级次级共轴心。同轴变压器的优点是功率密度大：共轴心结构使原边副边紧密耦合，大大降低漏感；所用环形磁芯特点是磁芯损耗低，饱和磁通密度大，可以承载更高功率而不产生磁饱和现象；封闭水冷结构内部通两路冷却水，大大提升同轴变压器功率密度，与风冷结构相比受外部粉尘影响小，故障率低。 磁芯选型 2 同轴变压器磁芯为环型磁芯，是一种闭合磁路磁芯，漏磁小，制造工艺简单成熟，已得到广泛应用。 根据法拉第电磁感应定律： <math display="block" id="M59"><mi>U</mi><mo>=</mo><mi>N</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046204&type= 14.90133286 7.28133297 式中　 <math id="M60"><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046238&type= 2.70933342 2.62466669 为电压， <math id="M61"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046274&type= 2.53999996 2.62466669 为匝数，磁通 <math id="M62"><mi mathvariant="normal">Φ</mi><mo>=</mo><mi>B</mi><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046222&type= 10.15999985 2.62466669 。磁芯选择采用面积乘积法，其计算公式为： <math display="block" id="M63"><mi>A</mi><mi>P</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">0.53</mn><mi>f</mi><mo>⋅</mo><mi mathvariant="normal">Δ</mi><mi>B</mi><mo>⋅</mo><mi>J</mi><mo>×</mo><mn mathvariant="normal">0.01</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046224&type= 48.00600052 8.04333401 式中　 <math id="M64"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046226&type= 2.28600001 2.62466669 为变压器额定输出功率（单位：W）； <math id="M65"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046376&type= 1.10066664 3.38666677 为工作频率（单位：Hz）； <math id="M66"><msub><mrow><mi>A</mi></mrow><mrow><mi>w</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046270&type= 3.80999994 2.62466669 为磁芯窗口面积，即乘填充系数后的面积（单位：cm 2 ）； <math id="M67"><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046315&type= 3.38666677 2.62466669 为磁芯有效截面积（单位：cm 2 ）；等 式（15） 式（15） 两边在 <math id="M68"><mn mathvariant="normal">0</mn><mo>∼</mo><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046234&type= 8.38199997 2.62466669 内积分，由于 <math id="M69"><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046238&type= 2.70933342 2.62466669 是常数，对双极性变换器，磁通密度 <math id="M70"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046237&type= 2.28600001 2.62466669 由 <math id="M71"><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046254&type= 6.01133299 2.62466669 到 <math id="M72"><mo>-</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046256&type= 6.01133299 2.62466669 产生 <math id="M73"><mn mathvariant="normal">2</mn><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046258&type= 5.75733376 2.62466669 变化， <math id="M74"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046260&type= 3.89466691 2.62466669 为磁通密度最大值，达到磁通密度最大值意味着磁饱和，为保证磁芯工作状态，一般取磁通密度最大值的1/4~1/3作匝数计算，这里取 <math id="M75"><mn mathvariant="normal">0.6</mn><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046262&type= 8.55133343 2.62466669 并用 <math id="M76"><mi mathvariant="normal">Δ</mi><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046240&type= 4.57200003 2.62466669 表示取值0.6T； <math id="M77"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046242&type= 1.69333339 2.62466669 为电流密度（单位：A/mm 2 ），取为3 A/mm 2 ；0.53为经验系数。 根据设计要求开关频率为30 kHz，额定输出功率为30 kW，考虑输出损耗同轴变压器裕度取1.2，即变压器额定输出功率为36 kW。把数据代入 式（16） 式（16） 可得： <math display="block" id="M78"><mi>A</mi><mi>P</mi><mo>=</mo><mn mathvariant="normal">125.8</mn><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046244&type= 21.84399986 2.62466669 圆环磁芯有效横截面积 <math id="M79"><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046315&type= 3.38666677 2.62466669 为： <math display="block" id="M80"><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mi>N</mi><mfrac><mrow><mfenced separators="|"><mrow><mi>D</mi><mo>-</mo><mi>d</mi></mrow></mfenced><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046248&type= 24.63800049 7.61999989 式中　 <math id="M81"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046274&type= 2.53999996 2.62466669 为叠加磁芯个数； <math id="M82"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046252&type= 2.70933342 2.62466669 为外径； <math id="M83"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046269&type= 1.86266661 2.70933342 为内径； <math id="M84"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046266&type= 1.86266661 2.62466669 为高度。 磁芯窗口面积 <math id="M85"><msub><mrow><mi>A</mi></mrow><mrow><mi>w</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046270&type= 3.80999994 2.62466669 为： <math display="block" id="M86"><msub><mrow><mi>A</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi><msup><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0.09</mn><mi mathvariant="normal">π</mi><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046283&type= 29.80266571 7.45066643 环型磁芯内径与外径之比为0.6，高度与内径之比为0.4，则有 <math id="M87"><mi>d</mi><mo>=</mo><mn mathvariant="normal">0.6</mn><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046286&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046285&type= 12.44600010 2.70933342 ， <math id="M88"><mi>h</mi><mo>=</mo><mn mathvariant="normal">0.4</mn><mi>d</mi><mo>=</mo><mn mathvariant="normal">0.2</mn><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046287&type= 22.26733208 2.70933342 ，代入 式（18） 式（18） 得： <math display="block" id="M89"><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>≈</mo><mn mathvariant="normal">0.05</mn><mi>N</mi><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046289&type= 19.38866615 2.62466669 再代入 式（16） 式（16） 可得： <math display="block" id="M90"><mi>A</mi><mi>P</mi><mo>=</mo><mn mathvariant="normal">4.5</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup><mi mathvariant="normal">π</mi><mi>N</mi><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046291&type= 30.05666542 2.62466669 根据磁芯种类选取磁芯外径 <math id="M91"><mi>D</mi><mo>=</mo><mn mathvariant="normal">5.7</mn><mtext> </mtext><mi mathvariant="normal">c</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046293&type= 15.91733265 2.70933342 ，因为 <math id="M92"><mi>A</mi><mi>P</mi><mo>></mo><mn mathvariant="normal">121.7</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046295&type= 16.17133331 2.62466669 ，解得 <math id="M93"><mi>N</mi><mo>></mo><mn mathvariant="normal">8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046297&type= 7.61999989 2.62466669 ，取 <math id="M94"><mi>N</mi><mo>=</mo><mn mathvariant="normal">9</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046299&type= 7.61999989 2.62466669 。综合磁芯材料与尺寸，最终选择9个型号为CS572060铁硅铝磁环叠放作为磁芯，其饱和磁通密度为1.05 T，外径57.2 mm，内径35.6 mm，高度14 mm，代入 式（18） 式（18） 、 式（19） 式（19） 计算得 <math id="M95"><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">13.6</mn><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046301&type= 19.55800056 2.70933342 ， <math id="M96"><mi>A</mi><mi>P</mi><mo>=</mo><mn mathvariant="normal">135.45</mn><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046303&type= 24.46866608 2.70933342 符合设计要求。 绕组匝数设计 2 设加到同轴变压器一次绕组的电压峰值为 <math id="M97"><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046305&type= 7.28133297 2.62466669 ，绕组匝数为 <math id="M98"><msub><mrow><mi>N</mi></mrow><mrow><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046307&type= 3.64066648 2.62466669 ，代入 式（15） 式（15） 可得： <math display="block" id="M99"><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>p</mi></mrow></msub><mfrac><mrow><mtext> </mtext><mi mathvariant="normal">d</mi><mi mathvariant="normal">Φ</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046309&type= 20.23533440 7.28133297 由于磁通 <math id="M100"><mi>Φ</mi><mo>=</mo><mi>B</mi><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046311&type= 10.24466610 2.62466669 ， <math id="M101"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046313&type= 1.86266661 2.62466669 是磁芯有效截面积，即为前述 <math id="M102"><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046315&type= 3.38666677 2.62466669 。 得到高频变压器原边匝数 <math id="M103"><msub><mrow><mi>N</mi></mrow><mrow><mi>P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046317&type= 3.89466691 2.62466669 的计算公式为： <math display="block" id="M104"><msub><mrow><mi>N</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>B</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046319&type= 19.21933365 8.63599968 式中　 <math id="M105"><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046321&type= 7.28133297 2.62466669 为考虑网压波动系数（一般为1.1）后变压器原边输入电压峰值为197 V； <math id="M106"><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046323&type= 3.21733332 2.62466669 为最大导通时间（单位：s）。 代入数据得： <math display="block" id="M107"><msub><mrow><mi>N</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">197</mn><mo>×</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">30</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mrow><mrow><mn mathvariant="normal">1.05</mn><mo>×</mo><mn mathvariant="normal">0.6</mn><mo>×</mo><mn mathvariant="normal">13.6</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>≈</mo><mn mathvariant="normal">7.7</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046325&type= 50.37666702 11.68400002 设计同轴变压器变比为9∶1，计算得副边匝数约为0.85匝，综上所述，原边取9匝，副边取1匝。 绕组线径及铜筒厚度选择 2 电流渗透深度是指流过交变电流的导体其内层涡流强度数值等于表面涡流强度数值的0.368倍时，该处离导体表面的距离。电流渗透深度可由下式确定： <math display="block" id="M108"><mi>δ</mi><mo>=</mo><mroot><mrow><mfrac><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">π</mi><mi>f</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>μ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046327&type= 19.47333336 9.99066734 式中　 <math id="M109"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046329&type= 1.77800000 2.62466669 为导体中的电流渗透深度（单位：mm）； <math id="M110"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046331&type= 1.10066664 3.38666677 为交变电流频率（单位：Hz）； <math id="M111"><mi>ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046333&type= 1.77800000 3.38666677 为铜导体的电阻率，根据实际取值， <math id="M112"><mi>ρ</mi><mo>=</mo><mn mathvariant="normal">1.75</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>⋅</mo><mi mathvariant="normal">m</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mi mathvariant="normal">m</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046335&type= 39.53932953 3.38666677 ； <math id="M113"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046337&type= 3.04800010 3.38666677 为真空磁导率， <math id="M114"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">4</mn><mi mathvariant="normal">π</mi><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">7</mn></mrow></msup><mi>H</mi><mo>/</mo><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046339&type= 24.72266769 3.38666677 ； <math id="M115"><msub><mrow><mi>μ</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046341&type= 2.87866688 3.38666677 为导体的相对磁导率，为1。 当铜线流过电流频率为30 kHz时，带入 式（25） 式（25） 可得渗透深度为： <math display="block" id="M116"><mi>δ</mi><mo>=</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">1.75</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi mathvariant="normal">π</mi><mo>⋅</mo><mn mathvariant="normal">4</mn><mi mathvariant="normal">π</mi><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">7</mn></mrow></msup><mo>×</mo><mn mathvariant="normal">30</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot><mo>≈</mo><mn mathvariant="normal">0.38</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046343&type= 58.33533096 9.48266602 所选绕组线径需小于0.76 mm。同轴变压器特点是外部铜筒作为变压器次级参与功率传输。铜筒内部通冷却水实现同轴变压器绕组和磁芯的散热，同时大大提高了铜导体可承受的电流密度，变压器副边设计过程中铜筒厚度的选择优先考虑结构强度而不是渗透深度或电流密度。综合同轴变压器前端感应加热器重量，选择2 mm厚的铜筒作为变压器外层壳体。 感应加热头设计 2 设计的感应加热器如 图11 图11 所示。感应加热器呈矩形，与移动加热器底盘形状吻合，易于装配。通过上方的连接机构与同轴变压器相连，下端线圈底面与待矫平甲板平行，方便移动加热矫平。上下进水口和出水口分别与同轴变压器进出水口两端连接，感应加热器中形成水冷回路，装配时在接口处加入防漏垫圈，避免线圈漏水。 在感应加热过程中，由于存在尖角效应，磁力线极易集中于尖角处，高频交流电在铜管拐角处聚集，导致热量积聚而无法散失，甚至引发铜管开裂。所以在设计感应加热器时为了避免发生尖角效应，在铜管拐角处设置倒角。 设计的感应加热器耦合器采用扁长状铜材料作为工作线圈，其内部通一定压力的冷却水，通过两条进水通道与一条出水通道实现冷却水循环，在提升功率密度的同时保证铜材料工作时的结构强度。其扁长状加热线圈长度依照船舶甲板矫平工艺设计，保证不与变形甲板接触，避免短路故障。 工艺试验与结果分析 1 能量转换过程分析 2 实现感应加热的原理如 图12 图12 所示，感应加热矫平机将网电经整流滤波、逆变、整流、电能传输、逆变、变压与谐振匹配后，再经过变压后变换为高频交流电使待加热钢板中产生涡流热能与磁滞热能，使焊接接头应力松弛，应力分布重新调整和改善，以消除因焊接过程不均匀加热钢板产生的残余应力变形。 感应加热能量转换过程如 图13 图13 所示，感应加热器流过单相高频矩形波低压大电流交流电，根据安培定律，产生交变磁场；待加热甲板区域成为磁路一部分，根据电磁感应定律，钢板上形成变化的电场；变化的电场形成涡流；涡流效应最终在特定深度产生热量。 根据法拉第电磁感应定律，感应电动势 <math id="M117"><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046354&type= 1.69333339 2.62466669 为： <math display="block" id="M118"><mi>e</mi><mo>=</mo><mo>-</mo><mi>N</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046356&type= 16.00200081 7.28133297 式中　 <math id="M119"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046274&type= 2.53999996 2.62466669 为线圈匝数；假如交变磁通 <math id="M120"><mi mathvariant="normal">Φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046275&type= 2.70933342 2.62466669 是按照正弦规律变化的，则有： <math display="block" id="M121"><mi>Φ</mi><mo>=</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>ω</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046358&type= 18.71133423 2.62466669 式中　 <math id="M122"><msub><mrow><mi>Φ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046278&type= 4.48733330 2.62466669 为交变磁通最大值，可得出感应电动势为： <math display="block" id="M123"><mi>e</mi><mo>=</mo><mo>-</mo><mi>N</mi><msub><mrow><mi>Φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>ω</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>ω</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046280&type= 25.31533241 2.62466669 因此感应电动势 <math id="M124"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046282&type= 2.28600001 2.62466669 的有效值为： <math display="block" id="M125"><mi>E</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>N</mi><mi>f</mi><mtext> </mtext><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac><mo>=</mo><mn mathvariant="normal">4.44</mn><mi>N</mi><mi>f</mi><mtext> </mtext><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046374&type= 38.01533508 9.22866726 式中　 <math id="M126"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046376&type= 1.10066664 3.38666677 为交变电流的频率；工件中涡流的有效值 <math id="M127"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046378&type= 1.26999998 2.62466669 为： <math display="block" id="M128"><mi>I</mi><mo>=</mo><mfrac><mrow><mi>E</mi></mrow><mrow><mi>Z</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>E</mi></mrow><mrow><mroot><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046380&type= 28.19400024 9.82133293 式中　 <math id="M129"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046382&type= 2.28600001 2.62466669 为涡流回路等效电阻， <math id="M130"><msub><mrow><mi>X</mi></mrow><mrow><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046384&type= 3.55599999 2.62466669 为涡流回路等效感抗。涡流对工件进行加热，其时间 <math id="M131"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046386&type= 1.10066664 2.62466669 内产生的焦耳热 <math id="M132"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046388&type= 1.86266661 3.38666677 为： <math display="block" id="M133"><mi>q</mi><mo>=</mo><mn mathvariant="normal">0.24</mn><msup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>R</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29046390&type= 18.03400040 3.38666677 由式（27）~ 式（32） 式（32） 可以得出结论：感应电动势的大小与频率、磁场强度和线圈匝数成正比。使工件获得更大热量而快速升温的方法是增大涡流，而增大涡流一般有增大感应线圈上的电流和提高电源频率两个途径 ［ 13 13 ］ 。 横向收缩量试验 2 试验材料为800 mm×600 mm钢板若干，将感应线圈放置在距离钢板下边缘50 mm的中心位置，并且在感应加热器线圈加热位置两侧相距30 mm的位置设定7个测试点。在感应加热前后分别测试两侧测试点距离。试验测试点分布如 图14 图14 所示。实验结果如 图15 图15 所示。 变化规律总结为： （1）钢板在感应加热过程中，在未达到热态涡流透入深度前，加热方式都为透入式，并且在电流频率一定的条件下，其加热层透入深度是一定的。随着加热时间的延长，加热深度达到热态涡流透入深度后，加热方式变为热传导，热量从感应加热层传导到钢板其他未加热的部分。在整个加热过程中，钢板各部分存在的温度差导致热应力变形，加热时间长，钢板温度差变大，热应力变形就逐渐增大。随着加热时间变长，钢板的横向收缩量变大，如 图15 图15 所示。 （2）在感应加热过程中，被加热的矩形工件存在着端部效应，即矩形工件的端部温度低于中心温度。在本实验中小车式感应加热器两端附近的横向收缩量明显小于中间部分，这是由于感应加热器线圈端部温度低于中部温度而产生的热应力分布不均现象。 温度变化试验 2 在空气间隙5 mm、电流频率30 kHz、功率25 kW 条件下进行感应加热实验，利用手持式测温仪记录测试点在加热180 s的实验过程中每10 s的实时温度变化，如 图16 图16 所示，温升曲线大致呈线性。 船厂应用试验 2 在船厂实际应用场地使用样机按照矫平工艺对5 mm甲板进行矫平工艺试验。矫平机设备实物如 图17 图17 所示，船厂实际应用如 图18 图18 所示，移动式调频器扩大了矫平机工作范围，小车式感应加热器可按需移动至甲板区域施行感应加热矫平工艺，方便收纳、移动和作业。 结论 1 （1）研究了调功器的基本原理，设计了全桥逆变软开关电路，实现了高效功率调节，工作可靠性高，达到设计要求。 （2）研究了变频器的基本原理，设计了全桥逆变主电路拓扑结构；分析常用频率跟踪控制策略及工作原理，确定频率跟踪总体方案的同时设计了基于锁相环频率跟踪系统，设计的频率跟踪电路能有效地实现频率锁定与频率同步。 （3）研究了感应加热器的基本原理，设计了具有高功率密度通水冷却的同轴变压器和适应于甲板感应加热矫平工艺的感应加热头。 （4）实验室测试和造船厂应用表明，船舶甲板感应加热矫平机电能变换及能量转换系统工作稳定可靠，采用检测输出高频低压大电流交流电的电压、电流相位差，动态调整调频器逆变频率的跟踪、调节及锁相技术，可以保证矫平机始终工作在准谐振状态，小车式感应加热器输出高频低压大电流交流电的功率因数接近1。 参考文献 何翔 . 船舶及海洋工程结构感应加热校平机电源技术研究 ［D］. 江苏 ： 江苏科技大学 ， 2017 . 船舶及海洋工程结构感应加热校平机电源技术研究 He Xiang . 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