复杂倍频腔的电磁模型研究
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摘要
原子钟具有体积功耗小、环境适应性强、频率稳定性和漂移率等电性能指标高的优点，因此广泛用于通信、导航、时间频率计量等军民用工程。近年人们根据工程任务对体积、重量、功耗的需要，提出了小型化低噪声高稳定倍频腔的要求，所以建立倍频腔的电磁模型对于提升原子钟的性能是至关重要的。
原子钟系统中的倍频腔在轴向和幅向均具有不均匀性，根据其几何结构特点，把它分解为三个不同的区域分别分析。作者首先采用Tai C.T提出的Gm方法，结合散射叠加法，推导出了本文所需的第二类电型并矢格林函数显式表达式，建立磁场积分方程(MFIE)。接着依据等效原理，采用模式匹配方法(MMT)得到在轴向不连续面上场的匹配方程，解这个方程，不需要知道缝隙上的源分布就能得到本征谐振频率，推广Marek Jaworski的方法，应用广义傅立叶展开的贝塞尔级数在求解匹配方程时大大提高了运算精度，最后能够获得腔内电磁场分布的求解。本文给出的是一种严格的解析分析方法，适用于其他同类腔体的分析。
为了运用矩量法(MM)求解更广泛的函数类型，必须扩展原来算子的定义域，这在多变量问题中(多维空间的场)显得特别重要。本文研究了一种不依赖边界条件的基函数选择方法——扩展算子法。
With the advantages of low unit volume power consumption, strong circumstance adaptability and high stability of frequency, atomic clocks have been widely used in military and civil applications, such as communication, navigation and frequency standard. These days, however, various applications have put forward new challenges to the atomic clocks: small volume, light weight and low power consumption. Therefore a miniature, low-noise and stable frequency-doubling cavity is absolutely necessary to meet these demands. Accordingly, it is crucial to formulate the electromagnetic model of this cavity, so as to improve the performances of the atomic clock.
The cavity, working as a kernel part in an atomic clock, has axial and angular discontinuity. Making use of its geometry characteristic, the fields in the cavity can be analyzed separately in three different regions. Firstly, the Gm method introduced by Tai. C. Tai is adopted, and then combining the scatter superposition theorem, the analytic formulation for the electric dyadic green function of the second kind is derived, which is necessary for the analysis, to construct the magnetic field integral equations (MFIE). Based on the principle of equivalence, the matching equations of the fields on the discontinuous axial surfaces can be obtained by the mode matching technology (MMT). The equation which extends Marek Jaworski's method, once being solved, can give out the eigen-frequency without knowing the source distribution in the aperture. Finally, the application of Generalized Fourier Series on solving the matching equation greatly improves the operation precision and gives analytic solution of electromagne
tic fields in the cavity. This work presents a rigorous and analytical formulation, which can be applied to analyze other homogenous cavities as well.
In order to. solve many general functions by moment method (MM), the definition domain of the original operator must be extended, which is very important when it comes to multivariate problems (fields in multidimensional space). This paper also studies a method to choose base functions without considering the boundary conditions-extended operator.
原子钟系统中的倍频腔在轴向和幅向均具有不均匀性，根据其几何结构特点，把它分解为三个不同的区域分别分析。作者首先采用Tai C.T提出的Gm方法，结合散射叠加法，推导出了本文所需的第二类电型并矢格林函数显式表达式，建立磁场积分方程(MFIE)。接着依据等效原理，采用模式匹配方法(MMT)得到在轴向不连续面上场的匹配方程，解这个方程，不需要知道缝隙上的源分布就能得到本征谐振频率，推广Marek Jaworski的方法，应用广义傅立叶展开的贝塞尔级数在求解匹配方程时大大提高了运算精度，最后能够获得腔内电磁场分布的求解。本文给出的是一种严格的解析分析方法，适用于其他同类腔体的分析。
为了运用矩量法(MM)求解更广泛的函数类型，必须扩展原来算子的定义域，这在多变量问题中(多维空间的场)显得特别重要。本文研究了一种不依赖边界条件的基函数选择方法——扩展算子法。
With the advantages of low unit volume power consumption, strong circumstance adaptability and high stability of frequency, atomic clocks have been widely used in military and civil applications, such as communication, navigation and frequency standard. These days, however, various applications have put forward new challenges to the atomic clocks: small volume, light weight and low power consumption. Therefore a miniature, low-noise and stable frequency-doubling cavity is absolutely necessary to meet these demands. Accordingly, it is crucial to formulate the electromagnetic model of this cavity, so as to improve the performances of the atomic clock.
The cavity, working as a kernel part in an atomic clock, has axial and angular discontinuity. Making use of its geometry characteristic, the fields in the cavity can be analyzed separately in three different regions. Firstly, the Gm method introduced by Tai. C. Tai is adopted, and then combining the scatter superposition theorem, the analytic formulation for the electric dyadic green function of the second kind is derived, which is necessary for the analysis, to construct the magnetic field integral equations (MFIE). Based on the principle of equivalence, the matching equations of the fields on the discontinuous axial surfaces can be obtained by the mode matching technology (MMT). The equation which extends Marek Jaworski's method, once being solved, can give out the eigen-frequency without knowing the source distribution in the aperture. Finally, the application of Generalized Fourier Series on solving the matching equation greatly improves the operation precision and gives analytic solution of electromagne
tic fields in the cavity. This work presents a rigorous and analytical formulation, which can be applied to analyze other homogenous cavities as well.
In order to. solve many general functions by moment method (MM), the definition domain of the original operator must be extended, which is very important when it comes to multivariate problems (fields in multidimensional space). This paper also studies a method to choose base functions without considering the boundary conditions-extended operator.
引文
[1] 王秉中编著，计算电磁学[M]，北京：科学出版社，2002，7。
[2] 徐萃薇编，计算方法引论[M]，北京：高等教育出版社 1999，6。
[3] T. Sphicopoulos, F. Gardiol. slotted tube cavity:a compact resonator with empty core[J], IEE Proceedings, Vol.134, Pt.H, No.5, Oct. 1987, 405-410。
[4] R.F.Harrington, Time-harmonic Electromagnetic fields[M], McGraw-Hill, New York, 1961。
[5] Marek Jaworski.On the Resonant Frequency of a Reentrant Cylindrical Cavity[J], IEEE Trans. On MTT-26, No. 4, April. 1978,256-260.
[6] Tai,C.T.Dyadic Green's functions for coaxial line[J],IEEE TRANS ON AP-31, NO.2,MARCH 1983,pp.355-358。
[7] 戴振铎，鲁述．电磁理论中的并矢格林函数[M]．武汉：武汉大学出版社／1995：99-103。
[8] 姚端正，梁家宝．数学物理方法(第二版)[M]．武汉：武汉大学出版社／1997。
[9] W.W.Hansen,on the resonant frequencies of closed concentric lines[J], J.App.Phys., Vol. 10,pp.38-45,1939。
[10] A.C.Willianson, The Resonant Frequency and Tuning Characteristics of a Narrow-gap Reentrant Cavity [J],IEEE Trans. MTT, vol.24,pp. 182-187,Apr. 1976。
[11] Roger F.Harrington .Field Computation by Moment Method[M] New York: The Macmillan Company. 1968。
[12] 关肇直等著 线性泛函分析入门[M] 上海：上海科技出版社，1979。
[13] C.T.Tai,Dyadic Green's Funcion in Electromagnetic Theory[M], Intext EducationPublishers, Scratton, Pa. 1971。
[14] 宋文淼，并矢格林函数和电磁场的算子理论[M]，合肥：中国科技大学出版社，1991。
[15] 宋文淼著，矢量偏微分算子[M]，北京：科学出版社，1999．4。
[16] Collin R.E. Field Theory of Guided Wave[M]. New York: McGraw-Hill, 1960。
[17] Tai C.T. Eigenfunction Expansion of Dyadic Green's Functions, Math Note 28, Weapons
Systems Laboratory, Kirtland Air Force Base ,Albuguerque, 1973.
[18] Tai C T．专题文选．上海：上海师范大学微波教研室，1979。
[19] 龚书喜，电磁理论中的广义本征函数展开问题：(学位论文)，西安：西安交通大学，1987。
[20] Collin R.E, Zucker F.J. Antenna Theory[M]. NewYork: Mcgraw-Hill, 1969:438-506.
[21] Hadidi A,Hamid M. Electric and Magnetic Dyadic Green's Functions of Bounded Regions[J]. Can. J. Phys,1988, 66: 249-257.
[22] 张善杰．同轴腔的并矢格林函数[J]．电子学报／1984，12(1)：107—112。
[23] 褚庆昕，梁昌洪。并矢格林函数的对偶关系[J]．西安电子科技大学学报／1994，21(2)：204-209。
[24] 宋文淼．关于同轴腔中的并矢格林函数问题[J]．电子学报／1987,15(6)：116-118。
[25] Kut Yuen Chow, Kwok Wa Leung. Theory and Experiment of the Cavity-backed Slot-exicited Dielectric Resonator Antenna[J].IEEE Trans. Electromagnetic Compatibility,2000,42:290-297。
[26] 洪伟，计算电磁学研究进展[J]，东南大学学报(自然科学版) 2002年5月，Vol．32No．3，335-339。
[27] Steinberg B.Z.,Leviatan Y. On the use of wavelet expansions in the method of moments[J]. IEEE Trans.AP.,1993,4(5):610-619.
[28] Wang G.F., One the utilization of periodic wavelet expansions in the method of moments[J].IEEE Trans. MTT., 1995,43(10):2495-2498.
[29] Goswami J.C.,Chan A.K, Chui C.K. On solving first-kind integral equations using wavelets on bounded interval [J].IEEE Trans .AP., 1995,43(7):614-622.
[30] Wager R.L,Chew W.C., A study of wavelets for the solution of electromagnetic integral equations[J],IEEE Trans. AP., 1995,43(8):802-810.
[31] 柯熙政 吴贵臣，子波矩量法初步探讨[J]，电子科技第2期(总第32期) 1995年6月。
[32] W.W.Hansen, on the resonant frequencies of closed concentric lines[J], J.App.Phys.,Vol.10,pp.38-45, 1939,
[32] A.G.Willianson, the resonant frequency and tuning characteristics of a narrow-gap reentrant cavity[J],IEEE Trans. MTT, vol.24,pp. 182-187,Apr. 1976。
[33] W.W.Hanson. A New Type of Expansion in Radiation Problem[J], Phys. Rev. 47:139-143,1935。
[34] 赵中华，王岩青．预处理子空间迭代法[J]，东南大学学报(自然科学版)，2003，7vol．33No．4，pp511-513。
[35] (美)D．C斯廷逊著王昌曜刘天惠译，电磁学中的数学[M]，国防工业出版社，1982。
[36] 黄华，李正红，江金生相对论速调管单重入谐振腔的解析分析[J]，强激光与例子束，2000年8月，pp501—504。
[37] Kaneyuki Kurokawa, the Expansions of Electromagnetic Fields in Cavities[J], IRE Transactions on Microwave Theory and Technology, April, 1958,pp178-187。
[38] 周忠石，王亮，郭鹏翔铷原子频率标准的小型化研究[J]，宇航计测技术，vol．22,No．1,2002,pp 1-7。
[39] 陈银平，Green函数在电磁场中的应用，武汉大学硕士学位论文，2002年5月。
[2] 徐萃薇编，计算方法引论[M]，北京：高等教育出版社 1999，6。
[3] T. Sphicopoulos, F. Gardiol. slotted tube cavity:a compact resonator with empty core[J], IEE Proceedings, Vol.134, Pt.H, No.5, Oct. 1987, 405-410。
[4] R.F.Harrington, Time-harmonic Electromagnetic fields[M], McGraw-Hill, New York, 1961。
[5] Marek Jaworski.On the Resonant Frequency of a Reentrant Cylindrical Cavity[J], IEEE Trans. On MTT-26, No. 4, April. 1978,256-260.
[6] Tai,C.T.Dyadic Green's functions for coaxial line[J],IEEE TRANS ON AP-31, NO.2,MARCH 1983,pp.355-358。
[7] 戴振铎，鲁述．电磁理论中的并矢格林函数[M]．武汉：武汉大学出版社／1995：99-103。
[8] 姚端正，梁家宝．数学物理方法(第二版)[M]．武汉：武汉大学出版社／1997。
[9] W.W.Hansen,on the resonant frequencies of closed concentric lines[J], J.App.Phys., Vol. 10,pp.38-45,1939。
[10] A.C.Willianson, The Resonant Frequency and Tuning Characteristics of a Narrow-gap Reentrant Cavity [J],IEEE Trans. MTT, vol.24,pp. 182-187,Apr. 1976。
[11] Roger F.Harrington .Field Computation by Moment Method[M] New York: The Macmillan Company. 1968。
[12] 关肇直等著 线性泛函分析入门[M] 上海：上海科技出版社，1979。
[13] C.T.Tai,Dyadic Green's Funcion in Electromagnetic Theory[M], Intext EducationPublishers, Scratton, Pa. 1971。
[14] 宋文淼，并矢格林函数和电磁场的算子理论[M]，合肥：中国科技大学出版社，1991。
[15] 宋文淼著，矢量偏微分算子[M]，北京：科学出版社，1999．4。
[16] Collin R.E. Field Theory of Guided Wave[M]. New York: McGraw-Hill, 1960。
[17] Tai C.T. Eigenfunction Expansion of Dyadic Green's Functions, Math Note 28, Weapons
Systems Laboratory, Kirtland Air Force Base ,Albuguerque, 1973.
[18] Tai C T．专题文选．上海：上海师范大学微波教研室，1979。
[19] 龚书喜，电磁理论中的广义本征函数展开问题：(学位论文)，西安：西安交通大学，1987。
[20] Collin R.E, Zucker F.J. Antenna Theory[M]. NewYork: Mcgraw-Hill, 1969:438-506.
[21] Hadidi A,Hamid M. Electric and Magnetic Dyadic Green's Functions of Bounded Regions[J]. Can. J. Phys,1988, 66: 249-257.
[22] 张善杰．同轴腔的并矢格林函数[J]．电子学报／1984，12(1)：107—112。
[23] 褚庆昕，梁昌洪。并矢格林函数的对偶关系[J]．西安电子科技大学学报／1994，21(2)：204-209。
[24] 宋文淼．关于同轴腔中的并矢格林函数问题[J]．电子学报／1987,15(6)：116-118。
[25] Kut Yuen Chow, Kwok Wa Leung. Theory and Experiment of the Cavity-backed Slot-exicited Dielectric Resonator Antenna[J].IEEE Trans. Electromagnetic Compatibility,2000,42:290-297。
[26] 洪伟，计算电磁学研究进展[J]，东南大学学报(自然科学版) 2002年5月，Vol．32No．3，335-339。
[27] Steinberg B.Z.,Leviatan Y. On the use of wavelet expansions in the method of moments[J]. IEEE Trans.AP.,1993,4(5):610-619.
[28] Wang G.F., One the utilization of periodic wavelet expansions in the method of moments[J].IEEE Trans. MTT., 1995,43(10):2495-2498.
[29] Goswami J.C.,Chan A.K, Chui C.K. On solving first-kind integral equations using wavelets on bounded interval [J].IEEE Trans .AP., 1995,43(7):614-622.
[30] Wager R.L,Chew W.C., A study of wavelets for the solution of electromagnetic integral equations[J],IEEE Trans. AP., 1995,43(8):802-810.
[31] 柯熙政 吴贵臣，子波矩量法初步探讨[J]，电子科技第2期(总第32期) 1995年6月。
[32] W.W.Hansen, on the resonant frequencies of closed concentric lines[J], J.App.Phys.,Vol.10,pp.38-45, 1939,
[32] A.G.Willianson, the resonant frequency and tuning characteristics of a narrow-gap reentrant cavity[J],IEEE Trans. MTT, vol.24,pp. 182-187,Apr. 1976。
[33] W.W.Hanson. A New Type of Expansion in Radiation Problem[J], Phys. Rev. 47:139-143,1935。
[34] 赵中华，王岩青．预处理子空间迭代法[J]，东南大学学报(自然科学版)，2003，7vol．33No．4，pp511-513。
[35] (美)D．C斯廷逊著王昌曜刘天惠译，电磁学中的数学[M]，国防工业出版社，1982。
[36] 黄华，李正红，江金生相对论速调管单重入谐振腔的解析分析[J]，强激光与例子束，2000年8月，pp501—504。
[37] Kaneyuki Kurokawa, the Expansions of Electromagnetic Fields in Cavities[J], IRE Transactions on Microwave Theory and Technology, April, 1958,pp178-187。
[38] 周忠石，王亮，郭鹏翔铷原子频率标准的小型化研究[J]，宇航计测技术，vol．22,No．1,2002,pp 1-7。
[39] 陈银平，Green函数在电磁场中的应用，武汉大学硕士学位论文，2002年5月。