Ch6-微波传输线

1 第六章第六章 微波传输线微波传输线 1.1. MicroWaveMicroWave Transmission LineTransmission Line 2.2. Metal Waveguide (rectangle/cylindrical/coaxial)Metal Waveguide (rectangle/cylindrical/coaxial) 3.3. Optical FiberOptical Fiber 4.4. MicroMicro--strip Line and its designstrip Line and its design MicroWaveMicroWave Transmission LinesTransmission Lines 6.0 6.0 前言前言((IntroductionIntroduction)) (1)(1)研究的对象研究的对象((What we study in ?What we study in ?)) (2)(2)研究的目的研究的目的((Why we study ?Why we study ?)) (3)(3)研究的内容研究的内容 ••MicroWaveMicroWave Transmission LineTransmission Line、、WaveguideWaveguide ••Waveguide/MicroWaveguide/Micro--strip Line/Designstrip Line/Design Bridge the gapBridge the gap between electromagnetic theory between electromagnetic theory and and MicroWaveMicroWave circuitscircuits.. ••多导体线：多导体线： CoxialCoxial LineLine、、Planar LinePlanar Line…….... ••单导体线：单导体线： RectangularRectangular、、CircularCircular…….... ••开路边界结构：电极、介质棒、开路边界结构：电极、介质棒、…….. ••MicroMicro--strip Linestrip Line 前言前言 •• 18931893亥维赛：亥维赛：““必须用两根导体来传输电磁能量必须用两根导体来传输电磁能量””((有误！有误！)) •• 18971897年年William: William: ““数学证明数学证明””、、““无穷多无穷多””、、““截止截止”” •• AT&T 1932AT&T 1932年年 ““实验实验””、、19361936年年 ““论文论文””，， MIT MIT ““论文论文”” •• 波导、平面传输线、截止；波导、平面传输线、截止； 6.1 TEM TE/TM6.1 TEM TE/TM波通解波通解 •• P112 P112 传输线的一般传输特性传输线的一般传输特性 •• 纵向场分量表示横向场分量纵向场分量表示横向场分量 自学：自学：P112~P120P112~P120 •• 平行板波导平行板波导 –– TEMTEM模、模、TEMTEM模相速模相速 –– TMTM模、波导波长、截止波长、相速、导波波长、功率传播、衰减系数模、波导波长、截止波长、相速、导波波长、功率传播、衰减系数 –– TETE模、波导波长、截止波长、相速、导波波长、功率传播、衰减系数模、波导波长、截止波长、相速、导波波长、功率传播、衰减系数 •• 矩形波导矩形波导 –– TMTM模：几个重要 参数 转速和进给参数表a氧化沟运行参数高温蒸汽处理医疗废物pid参数自整定算法口腔医院集中消毒供应 模：几个重要参数 –– TETE模：几个重要参数模：几个重要参数 •• P121 6.2.7 P121 6.2.7 大纲不要求大纲不要求 •• 法兰：法兰：““flangeflange”” 主要应用主要应用((同轴线、微带线同轴线、微带线)) 2 6.1.0 6.1.0 Transmission Line Transmission Line 与与 WaveGuideWaveGuide WaveGuideWaveGuide：：一般一般““空心空心””或者或者““填充介质填充介质””的的““腔体腔体””((CavityCavity)) ““理想理想””情况下：情况下： (1)(1)电磁场的幅值在横截面内的分布规律不随座标电磁场的幅值在横截面内的分布规律不随座标zz的变化的变化 (2)(2)场的幅度、相位沿场的幅度、相位沿zz轴的变化规律与横截面坐标无关轴的变化规律与横截面坐标无关 ( ) ( ) zevuEzvuE ⋅−⋅= γ,,, GG 利用利用““亥姆霍兹亥姆霍兹””方程方程 ( ) ( ) ( ) 0,, 222 =++∇ vuEkvuE GG γτ βαγλ πεμω jk TEM +==⋅⋅= 2 (3)(3)理想情况下理想情况下CutCut--off Wavelengthoff Wavelength((截止波长截止波长)) βγ j+= 0 因为因为““传输传输””，所以，所以 必是必是““正实数正实数”” 令令:: 222 γ+= kKC β 22 CKk −=β ““传输线传输线””举例：举例：((本质上：波导也是传输线本质上：波导也是传输线)) )( mμ10 (a) (b) (c) (d) (e) (f) (g) 6.1.1 6.1.1 Transmission LineTransmission Line 简化：简化：时谐场、传输方向均匀、无限长、时谐场、传输方向均匀、无限长、 ((理想理想))金属导体金属导体 入手：入手： x y dl o z ( ) ( )( , ) ( , ) j t zT z zE E x y e E x y e ω β−= +G G G� � ( ) ( ).... ... j t zH e ω β−= +G� 麦氏方程：麦氏方程： (*)(*)研究的方法研究的方法 Helmholtz EquationsHelmholtz Equations ————复数形式的波动方程复数形式的波动方程 ⎪⎩ ⎪⎨ ⎧ =+∇ =+∇ 0~~ 0~~ 22 22 HkH EkE GG GG 6.3 6.3 矩形波导矩形波导 自学：自学：P112~P120P112~P120 6.4 6.4 圆柱波导圆柱波导 ((稍后再学稍后再学)) 3 6.4 6.4 Ideal Coaxial Line Ideal Coaxial Line ((理想同轴线理想同轴线) ) 讲义讲义P126P126 (1(1--1)1)组成组成((StructureStructure))：： Perfect ConductorPerfect Conductor ((理想导体理想导体)) Lossless dielectricLossless dielectric ((理想介质理想介质)) HomogeneousHomogeneous((均匀的均匀的))、、““理想同心圆柱体理想同心圆柱体”” a b (1(1--2)2)组成示意图：组成示意图： (2)(2)电磁场分布情况电磁场分布情况((Field ConfigurationField Configuration))：： ( ) jkzrab jkz e r aVeEE −− == GGG ln ~ 0 0 ( ) jkzabz er aVEaH −⋅=×= ϕηη GGGG 1 ln ~~ 0 E G H G (3)(3)金属的表面电流：金属的表面电流： HaJ nS GGG ×= ...|~~ 1 =×= =arrS HaJ GGG内导体的内导体的““外外””表面表面:: ( ) ...|~~ 2 =×−= =brrS HaJ GGG外导体的外导体的““内内””表面表面:: (4)(4)功率流：功率流： ( ) drrdSSdHEP b a ⋅=⎟⎠ ⎞⎜⎝ ⎛ •⎟⎠ ⎞⎜⎝ ⎛ ×= ∫ π2~~Re21 * GGG ab (5)(5)特征阻抗：特征阻抗： 法一：基本定理法一：基本定理 法二：已有的结论法二：已有的结论 法一：定义法法一：定义法 ( ) ( ) )(ln 2 1 2 ln 0 ln 1 2 0 0 0 0 Ω⎟⎠ ⎞⎜⎝ ⎛⋅⋅=⋅ ⋅== a b C LZ a b a b ε μ ππε π μ 法二：已有的结论法二：已有的结论 zj C eaEdlHI ⋅−⋅⋅=== ∫ βϕ ηπ 02... zjb a r e a baEdrEU ⋅−⋅⎟⎠ ⎞⎜⎝ ⎛⋅⋅== ∫ βln0 )(ln60ln 2 1... 00 Ω⎟⎠ ⎞⎜⎝ ⎛⋅=⎟⎠ ⎞⎜⎝ ⎛⋅⋅=== a b a b I UZ rεε μ π Commonly used TCommonly used T…… lineline：：50ohm50ohm、、75ohm75ohm General KnowledgeGeneral Knowledge((常识常识))：： Example: 1Example: 1 An airAn air--filled coaxial transmission line has outer filled coaxial transmission line has outer and inner conductor radiiand inner conductor radii((radiusradius)) equal to 6cm and equal to 6cm and 3cm, respectively. Calculate the values of :3cm, respectively. Calculate the values of : (a)(a)inductance per unit length inductance per unit length ；； (b)(b)capacitance per unit length capacitance per unit length ；； (c)(c)characteristic impedance of the line characteristic impedance of the line .. ab 例题例题2 2 讲义讲义P130P130 4 电缆的损耗和频率的平方根成正比电缆的损耗和频率的平方根成正比 6.4.2 6.4.2 Higher Order Modes in coaxial lineHigher Order Modes in coaxial line 6.5.2.1 6.5.2.1 Background knowledgeBackground knowledge (1) TEM(1) TEM波波 ηεμ === yxHETEMZ TEMH E Z x y −= ⎟⎠ ⎞⎜⎝ ⎛ ×= Ea Z H z TEM GGG ~1~ 相速相速((波传播的速度波传播的速度)=)=““光速光速”” εμβ ω ⋅== 1TEMpv k G xy z E G H G (2) TM(2) TM波波 特点：传播方向上无磁场！特点：传播方向上无磁场！ 0~ =zH ωε βα ωε j j j T H E H E TM x y y xZ +==−== ⎟⎠⎞⎜⎝⎛ ×= EaZH zTM GGG ~1~ xy z E G H G (3) TE(3) TE波波 特点：传播方向上无电场！特点：传播方向上无电场！ 0~ =zE ...==−== TjHEHETE xyyxZ ωμ ⎟⎠ ⎞⎜⎝ ⎛ ×⋅−= HaZE zTE GGG ~~ xy z E G H G 6.4.2.2 6.4.2.2 出现的条件出现的条件————Sufficiently High frequencySufficiently High frequency TM ModeTM Mode：： TMTM0101 ( )abc −≈ 2λ TE ModeTE Mode：： TETE1111 ( )abc +≈πλ 波导的波导的““高通高通””特性：特性： cλλ < 第一第一““高次模高次模””：： TETE1111 波导的波导的““单模单模””传输条件传输条件?? 时时…….... λ)( ab +π)(2 ab − 11TE01TM TEM 6.4.5 6.4.5 Coaxial Cable DesignCoaxial Cable Design ••Single ModeSingle Mode ••Minimal AttenuationMinimal Attenuation ((Maximal Transmission PowerMaximal Transmission Power)) ••Maximal Endure VoltageMaximal Endure Voltage a bSU ~ 1 ••Single ModeSingle Mode ••Minimal AttenuationMinimal Attenuation ((Maximal Transmission PowerMaximal Transmission Power)) ••Maximal Endure Voltage Maximal Endure Voltage ((承受电压承受电压)) a bSU ~ 6.46.4 Coaxial Cable DesignCoaxial Cable Design (1) (1) 单模传输单模传输————““Single ModeSingle Mode”” λ )( ab+⋅π)(2 ab − 11TE01TM 11TEλλ > 为什么此时是为什么此时是““单模单模””传输呢？传输呢？ 问题：问题：波导波导((单模单模))传输一定是传输一定是““高通高通””么？么？ x y z a bSU ~ x y z TE10 TE01 波长 TE20 2a2b a TE11 TM11 ... IIIIII a bSU ~ λ )( ab +⋅π)(2 ab − 11TE 01TM TEM (2) (2) 最大传输功率最大传输功率——Minimal AttenuationMinimal Attenuation 传输损耗＝导体衰减＋介质损耗传输损耗＝导体衰减＋介质损耗 我们这里考虑：导体衰减我们这里考虑：导体衰减 导体衰减：导体衰减： ( ) )/(.... ln 2 12 2 0 2 1 2 1 0 mNP a b R Z R baS C = ⎟⎠ ⎞⎜⎝ ⎛⋅⋅⋅ +⋅=⋅≈ ⋅⋅ ε μ π α ππ ( ) ( ) )(ln 2 1 2 ln 0 ln 1 2 0 0 0 0 Ω⎟⎠ ⎞⎜⎝ ⎛⋅⋅=⋅ ⋅== a b C LZ a b a b ε μ ππε π μ 6.30 ≈⇒= a b da d Cα求极值：求极值： ““导体外径不变导体外径不变”” 空气同轴波导：空气同轴波导： ( ) )(776.3ln1202 1 0 Ω=⋅⋅= ππZ ⎟⎠ ⎞⎜⎝ ⎛⋅⋅⋅==⎥⎦ ⎤⎢⎣ ⎡ •⎟⎠ ⎞⎜⎝ ⎛ ×= ∫ abaESdHEP S ln....~~Re21 22 0* η πGGG ( ) ( )∫∫ •×=•⎟⎠⎞⎜⎝⎛ × ba zrS draaaSdHE πϕ 2.......~~ * GGG GGG ⎟⎠ ⎞⎜⎝ ⎛⋅⋅⋅= a baEP Maxm ln 22 η π ⎟⎠ ⎞⎜⎝ ⎛⋅ = a ba U E s ln 0∵ (3) (3) 最大耐压最大耐压——Maximal Endure Voltage Maximal Endure Voltage 若知：最大击穿电压若知：最大击穿电压 MaxV ⎟⎠ ⎞⎜⎝ ⎛⋅ ⋅= a b U P s ln 2 η π ⎟⎠ ⎞⎜⎝ ⎛⋅⋅= a baEV MaxMax ln 65.10 ≈⇒= a b da dPm求极值：求极值： ““导体外径不变导体外径不变”” 空气同轴波导：空气同轴波导： ( ) )(3065.1ln120 2 1 0 Ω=⋅⋅= ππZ 2 小结：小结： Maximal Endure VoltageMaximal Endure Voltage Minimal AttenuationMinimal Attenuation 6.3≈a b 65.1≈ a b CompromiseCompromise：： 3.2=a b 介质同轴波导：介质同轴波导： )(50...0 Ω==Z Example 6.2Example 6.2 The primary constants for a The primary constants for a coxialcoxial cable at 1GHz, are cable at 1GHz, are L=250nH/m, C=95pF/m, R=0.06ohm/m, and G=0.L=250nH/m, C=95pF/m, R=0.06ohm/m, and G=0. (a)(a)Determine the attenuation coefficient Determine the attenuation coefficient ;; (b)(b)the phase constant the phase constant ;; (c)(c)the phase velocity the phase velocity VVpp ;; (d)(d)relative permittivity relative permittivity ;; (e)(e)Power loss for a length of 10m, when the input power isPower loss for a length of 10m, when the input power is 500 Watts.500 Watts. α β rε P0 Pll 6.4 6.4 Unbalance Characteristic s of Coaxial LinesUnbalance Characteristic s of Coaxial Lines a bSU ~ 6.4.3.1 What is 6.4.3.1 What is ““Unbalance LineUnbalance Line”” ?? 1. Character1. Character:: The two Conductors The two Conductors are notare not Opposite potentialsOpposite potentials to the to the groundground .. a b SU ~ 2. Where we use them 2. Where we use them ??————Coaxial Lines Coaxial Lines CATV CableCATV Cable、、OscillographOscillograph ConnectorConnector、、…… 3 6.4.3.2 What is 6.4.3.2 What is ““Balance LineBalance Line”” ?? 1. Character1. Character:: The two Conductors The two Conductors are are Opposite potentialsOpposite potentials to the groundto the ground .. 2. Where we use them 2. Where we use them ??————Differential SignalDifferential Signal highhigh--speed digital comm. speed digital comm. 、、 Differential Differential AmplifierAmplifier、、…… −u +u u −u +u 6.4.3.3 6.4.3.3 ““Balance LineBalance Line”” ------““Unbalance LineUnbalance Line”” (1)(1)MismatchMismatch--MatchMatch (2)(2) BalanceBalance––Unbalance ConvertorUnbalance Convertor An Example: An Example: the transformer in our the transformer in our ““Network CardNetwork Card”” A new word: A new word: BalunBalun 44--Ports HUBPorts HUB 实际应用实际应用11 实际应用实际应用22：： RF Band RF Band ““BalBalunun”” 4 同轴线的连接器同轴线的连接器 •• 基本知识基本知识 –– BNC: 2GBNC: 2G以内以内 –– TNCTNC：：<1G<1G –– NN型：型：11~18GHz11~18GHz –– SMASMA：：18~25GHz18~25GHz –– APCAPC--77：：18GHz18GHz左右左右 –– 2.4mm2.4mm：可达：可达50GHz50GHz 1 6.3 6.3 Circular WaveguideCircular Waveguide 圆柱形波导圆柱形波导((讲义讲义 P121)P121) 1. TE Wave Solution1. TE Wave Solution 2. TM Wave Solution2. TM Wave Solution 3. Dominant Mode 3. Dominant Mode 4. Degenerate Mode4. Degenerate Mode 5. Attenuation in Circular Waveguide 5. Attenuation in Circular Waveguide PreviewPreview：：What we will learn this seminarWhat we will learn this seminar?? Cylindrical CoordinateCylindrical Coordinate z平面 r柱面 ϕx y z { }zr ,,ϕ { }zyx ,, x yϕ ra G ya G xa G ϕa G O 顶视图顶视图 a b a 6.3.F1 6.3.F1 CharacteristicCharacteristic A circular waveguide A circular waveguide isis a cylindrical hollow a cylindrical hollow metallic pipe metallic pipe withwith uniform circular crossuniform circular cross--section of a section of a finite radiusfinite radius 6.3.F2 6.3.F2 Wave equations in WaveguideWave equations in Waveguide CounterevidenceCounterevidence: : disconfirmdisconfirm If If ……....，，We getWe get…… Using boundary conditionUsing boundary condition：：…….. No TEM wave are possible in it !No TEM wave are possible in it ! a 6.3.F3 6.3.F3 Wave equations in WaveguideWave equations in Waveguide Symmetry Symmetry ÆÆ ( )( , )rE E r Z zφ φ= ⋅G G� � � Wave equationsWave equations 0 ~~ 22 =⋅+∇ EE GG εμω 222 zt ∇+∇=∇ ( )( ) ( ) ( ) 2 2 2 2 2 t z r t r z E E Z E Z Z E φ φ φ ∇ = ∇ +∇ ⋅ = ∇ ⋅ + ∇ ⋅ G G� � � G G� �� � ( )( ) ( )2 2 2 0t r z rE Z Z Eφ φω μ ε∇ + ⋅ ⋅ + ∇ ⋅ =G G� �� �⇒ ttransectionransection ( ) ( )2 2 2 0rt r z EE Z Zφφω μ ε∇ + ⋅ + ∇ ⋅ = G�G� � � ( ) ( )2 2 2 rt r z EE Z Zφφω μ ε− ∇ + ⋅ = ∇ ⋅ G�G� � � 2 ( ) ( )22 2 zt r rZE EZφ φω μ ε ∇− ∇ + ⋅ = ⋅ �G G� � � ( ) 22~ ~ TZZz =∇∃∴ 0~ ~ 2 2 2 =⋅− ZT dz Zd βα jT += zTzT eAeAZ ⋅+− ⋅− + ⋅+⋅=~ ( )2 2 2 0t r rE T Eφ φω μ ε∇ + ⋅ + ⋅ =G G� � ( )2 2 2 0t r rE T Eφ φω μ ε∇ + ⋅ + =G G� � Define:Define: 22222 TkTK +=+⋅= εμω ““截止波数截止波数”” ?|""0 0=⇒⇒= =TcKOffCutT In the Cylindrical Coordinate:In the Cylindrical Coordinate: { }, ,r r zE E E Eφ φ=G� � � � 2 2 2 2 2 2 1 1r r r r r z ψ ψ ψψ φ ∂ ∂ ∂ ∂⎛ ⎞∇ = ⋅ + ⋅ +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ 2 2 0t r rE K Eφ φ∇ + ⋅ = G G� � { } *( , ), ( , ), ( , ) ( )r r z rE E r E r E r E E Z zφ φ φφ φ φ= ⎯⎯→ = ⋅G G G� � � � Three Three ScalarWaveScalarWave equations equations ⇒ ⎪⎩ ⎪⎨ ⎧ =⋅+∇ =⋅+∇ 0~~ ... 0~~ 22 22 zzt rrt EKE EKE 2 2 0t r rE K Eφ φ∇ + ⋅ = G G� � 6.3.F4 6.3.F4 TE TE 、、TM Wave SolutionsTM Wave Solutions ⎪⎩ ⎪⎨ ⎧ =⋅+∇ =⋅+∇ 0~~ ... 0~~ 22 22 zzt rrt EKE EKE First We want to get First We want to get EEzz oror HHzz 2 2( , ) ( , ) 0t z zE r K E rφ φ∇ + ⋅ =� � 2 2( , ) ( , ) 0t z zH r K H rφ φ∇ + ⋅ =� � ————((TMTM)) ————((TETE)) 2 2( , ) ( , ) 0t U r K U rφ φ∇ + ⋅ =� � ( , ) ( ) ( )U r R rφ φ= ⋅Φ� � � 0~~ 22 =⋅+∇ UKUt ““通用通用””方程：方程： 圆柱波导场求解小节圆柱波导场求解小节 (1)(1)排除排除TEMTEM模式模式 (2)(2)求解：求解：TETE、、TM ModesTM Modes ( )( , )rE E r Z zφ φ= ⋅G G� � � 0~~ 22 =⋅+∇ EE GG εμω ( ) ( )22 2 zt r rZE EZφ φω μ ε ∇− ∇ + ⋅ = ⋅ �G G� � � zTzT eAeAZ ⋅+− ⋅− + ⋅+⋅=~( ) 2 2 2 0t r rE T Eφ φω μ ε∇ + ⋅ + = G G� � { }( , ), ..., ...r r zE E r E Eφ φφ=G� 2 2 0t r rE K Eφ φ∇ + ⋅ = G G� � 2 2( , ) ( , ) 0t U r K U rφ φ∇ + ⋅ =� � ““通用通用””方程方程 ( , ) ( ) ( )U r R rφ φ= ⋅Φ� � � 2 2 2 2 1 0r d dR dr K r R dr dr dφ Φ⎛ ⎞⋅ + ⋅ + ⋅ =⎜ ⎟ Φ⎝ ⎠ ( ) 22~ ~ TZZz −=∇ 0~~ 22 =⋅+∇ UKUt 2 2 2 2 1 0r d dR dr K r R dr dr dφ Φ⎛ ⎞⋅ + ⋅ + ⋅ =⎜ ⎟ Φ⎝ ⎠ 2 2 2 2 2 1r d dR dr K r i R dr dr dφ Φ⎛ ⎞⋅ + ⋅ = − ⋅ = =⎜ ⎟ Φ⎝ ⎠ 常量 2 2 2 2 2 2 1 d i d r d dRr K r i R dr dr φ ⎧ Φ⋅ = −⎪ Φ⎪⎨ ⎛ ⎞⎪ ⋅ + ⋅ =⎜ ⎟⎪ ⎝ ⎠⎩ 2 2 2 2 2 2 1 1r r r r r z ψ ψ ψψ φ ∂ ∂ ∂ ∂⎛ ⎞∇ = ⋅ + ⋅ +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ 2 2 2 2 1 0r d dR dr K r R dr dr dφ Φ⎛ ⎞⋅ + ⋅ + ⋅ =⎜ ⎟ Φ⎝ ⎠ 2 2 2 1 d i dφ Φ⋅ = −Φ 222 irK dr dRr dr d R r =⋅+⎟⎠ ⎞⎜⎝ ⎛ ⋅∴ 2 2 2 0 d i dφ Φ + Φ = (2 ) ( )π φ φΦ + = Φ∵ ( ) ( )( ) sin cosn n nA n B nφ φ φΦ = +ni = [ ] 0222222 =⋅−⋅+⋅+⋅ RnrKdrdRrdrRdr [ ] 022222 =⋅−+⋅+⋅ RnududRuduRdu 0=i 0 0( ) Aφ φ φΦ = ⋅ + nin =∴ 0≠i ( ) ( )( ) sin cosn n n n nA i B iφ φ φΦ = ⋅ + ⋅ rKu ⋅= 3 [ ] 022222 =⋅−+⋅+⋅ RnududRuduRdu nnth Order th Order BesselBessel FunctionFunction First Class First Class ( ) ( )uNAuJAR nn ⋅+⋅= 21 ( ) ( ) ( )rKNArKJArR nn ⋅⋅+⋅⋅= 21 ( )uJn Second Class Second Class ( )uNn rKu c ⋅= BesselBessel函数函数 [ ] )()( 1 xJxxJxdxd nnnn −⋅=⋅I ClassI Class：： xJnJJ nnn ⋅−= −1' nnn Jx nJJ ⋅=+ +− 211 nnn JJJ '211 =− +− Small x Small x ：： Large x Large x ：： 0 2! 1 2 ≠⎟⎠ ⎞⎜⎝ ⎛⋅≈ nx n Jn ⎟⎠ ⎞⎜⎝ ⎛ +−⋅⋅≈ 4 12cos12 nx x Jn πII ClassII Class：：………… Small x Small x ：： Large x Large x ：： 0 2 )!1( 2 ≠⎟⎠ ⎞⎜⎝ ⎛⋅−−≈ − nxnNn π ⎟⎠ ⎞⎜⎝ ⎛ +−⋅⋅≈ 4 12sin12 nx x Nn π ( ) )(1)( xJxJ nnn ⋅−=− 第一类第一类 BesselBessel函数函数 nnn Jx nJJ ⋅=+ +− 211 nnn JJJ '211 =− +− ( ) )(1)( xJxJ nnn ⋅−=− ( ) ( ) ( ) ( ) ...2! 11... 2!3 1 2!2 1 2 1)( 2 2 6 2 4 2 2 0 +⎟⎠ ⎞⎜⎝ ⎛⋅⋅−++⎟⎠ ⎞⎜⎝ ⎛⋅−⎟⎠ ⎞⎜⎝ ⎛⋅+⎟⎠ ⎞⎜⎝ ⎛−= k k x k xxxxJ ( ) ... 2)!1(! 11... 2!3!2 1 2!2!1 1 2 )( 1253 1 +⎟⎠ ⎞⎜⎝ ⎛⋅+⋅⋅−++⎟⎠ ⎞⎜⎝ ⎛⋅⋅−⎟⎠ ⎞⎜⎝ ⎛⋅⋅−⎟⎠ ⎞⎜⎝ ⎛= +k k x kk xxxxJ ....1,0=k 2 4 6 8 10 12 -0.4 -0.2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 -0.2 0.2 0.4 0.6 First Class Bessel FunctionFirst Class Bessel Function( )uJn 2 4 6 8 10 -1.5 -1 -0.5 0.5 Second Class Bessel FunctionSecond Class Bessel Function n=0n=0 n=1n=1 n=3n=3 ( )uNn Boundary ConditionBoundary Condition--22 0r a Eφ= ⇒ =� Why ?Why ? ......0 ∴∞≠= Er G∵ ( ) ( )rKJArR n ⋅⋅= 1 Boundary ConditionBoundary Condition--11 a 4 ( ) ( )( ) sin cosn n nA n B nφ φ φΦ = + ( ) ( )rKJArR n ⋅⋅= 1 ( ) ( )( )0 cos sin T z z n n E E J K r e n φ φ − ⋅= ⋅ ⋅ ⋅ ⋅� ( ) ( )( ) ( ) ( )( ) 0 0 cos sin cos sin j z z n j z z n n E E J K r e n n H H J K r e n β β φ φ φ φ − − ⎧ = ⋅ ⋅ ⋅ ⋅⎪⎪⎨⎪ = ⋅ ⋅ ⋅ ⋅⎪⎩ � � 圆柱波导中场的解圆柱波导中场的解 z平面 r柱面 ϕx y z { }zr ,,ϕ { }zyx ,, 6.3.4 6.3.4 TE Wave SolutionTE Wave Solution ( ) ( )( ) ( ) ( )( ) 0 0 cos sin cos sin j z z n j z z n n E E J K r e n n H H J K r e n β β φ φ ϕ ϕ − − ⎧ = ⋅ ⋅ ⋅ ⋅⎪⎪⎨⎪ = ⋅ ⋅ ⋅ ⋅⎪⎩ � � TE Mode TE Mode 0~ ≡zE zHK jH ~1 ~ 2 ττ β ∇−= G ( )zz HaKjE ~1~ 2 ττ ωμ ∇×= GG 22222 βεμω −=+⋅= kTK { } { } , , , ,0 r z r H H H H E E E φ φ ⎧ =⎪⇒ ⎨⎪ =⎩ G� � � � G� � � | 0r aEφ = =� 0|)( =⋅⇒ =arn dr rKdJ Page 70Page 70 0|)( =⋅ =arn dr rKdJ ,...2,1' == i a xK ni 0)'(' =nn xJ TE Mode TE Mode …… …… …… …… KKcc 1.64a1.64a3.8323.832TETE01010 10 1 TETE3131 TETE2121 TETE1111 TETEnini 3 13 1 2 12 1 1 11 1 n=? .i=?n=? .i=? ………….. 2.06a2.06a3.0543.054 3.412a3.412a1.8421.842 λλcc=2=2ππ//KKc c = = 22ππa/a/xx’’ninixx’’nini 1.1. Cut off frequency Cut off frequency 2.2. Phase VelocityPhase Velocity 3.3. Guide WavelengthGuide Wavelength 4.4. Wave Impedance Wave Impedance 5.5. Single Mode TransmissionSingle Mode Transmission Something Else About TE ModeSomething Else About TE Mode 6.3.5 6.3.5 TM Wave SolutionTM Wave Solution ( ) ( )( ) ( ) ( )( ) 0 0 cos sin cos sin j z z n j z z n n E E J K r e n n H H J K r e n β β φ φ ϕ ϕ − − ⎧ = ⋅ ⋅ ⋅ ⋅⎪⎪⎨⎪ = ⋅ ⋅ ⋅ ⋅⎪⎩ � � TM Mode TM Mode 0~ ≡zH zz EaK jH ~1 ~ 2 ττ ωε ∇×−= G G zEK jE ~1 ~ 2 ττ β ∇−= G 22222 βεμω −=+⋅= kTK { } { } , ,0 , , r r z H H H E E E E φ φ ⎧ =⎪⇒ ⎨⎪ =⎩ G� � � G� � � � | 0r aEφ = =� 0|)( =⋅⇒ =arn rKJ Page 83Page 83 0|)( =⋅ =arcn rKJ ,...2,1== i a xK nic 0)( =nn xJ TM Mode TM Mode …… …… …… …… KKcc 1.22a1.22a5.1355.135TMTM21212 12 1 TMTM0202 TMTM1111 TMTM0101 TMTMnini 0 20 2 1 11 1 0 10 1 n=? .i=?n=? .i=? 1.14a1.14a5.5205.520 1.64a1.64a3.8323.832 2.61a2.61a2.4052.405 λλcc=2=2ππ//KKc c = 2= 2ππa/a/xxninixxnini 5 1.1. Cut off frequency Cut off frequency 2.2. Phase VelocityPhase Velocity 3.3. Guide WavelengthGuide Wavelength 4.4. Wave Impedance Wave Impedance 5.5. Single Mode TransmissionSingle Mode Transmission Something Else About TM ModeSomething Else About TM Mode Single Mode TransmissionSingle Mode Transmission …… …… …… …… KKcc 1.64a1.64a3.8323.832TETE01010 10 1 TETE3131 TETE2121 TETE1111 TETEnini 3 13 1 2 12 1 1 11 1 n=? .i=?n=? .i=? ………….. 2.06a2.06a3.0543.054 3.412a3.412a1.8421.842 λλcc=2=2ππ//KKc c = 2= 2ππa/a/xx’’ninixx’’nini …… …… …… …… KKcc 1.22a1.22a5.1355.135TMTM21212 12 1 TMTM0202 TMTM1111 TMTM0101 TMTMnini 0 20 2 1 11 1 0 10 1 n=? .i=?n=? .i=? 1.14a1.14a5.5205.520 1.64a1.64a3.8323.832 2.61a2.61a2.4052.405 λλcc=2=2ππ//KKc c = 2= 2ππa/a/xxninixxnini Dominant Mode Dominant Mode The lowest order cutThe lowest order cut--off frequency is :off frequency is : …… …… KKcc TMTM1111 TMTM0101 TMTMnini 1 11 1 0 10 1 n=? .i=?n=? .i=? 1.64a1.64a3.8323.832 2.61a2.61a2.4052.405 λλcc=2=2ππ//KKc c = 2= 2ππa/a/xxninixxnini …… …… KKcc TETE2121 TETE1111 TETEnini 2 12 1 1 11 1 n=? .i=?n=? .i=? 2.06a2.06a3.0543.054 3.412a3.412a1.8421.842 λλcc=2=2ππ//KKc c = 2= 2ππa/a/xx’’ninixx’’nini xx’’nini =1.842=1.842 λλc c =3.412a=3.412a TETE1111 Single Mode TransmissionSingle Mode Transmission TE11 波 长 TM01 3.412a2.06a 2.61a TE01 TM11 ... IIIIII TE21 1.64a In what conditionsIn what conditions，，it supports single mode transmissionit supports single mode transmission？？ 答案：答案： aa 41.361.2 << λ Which mode is dominant modeWhich mode is dominant mode?? Cylindrical SymmetryCylindrical Symmetry 6.4.5 Field Configuration6.4.5 Field Configuration TE ModeTE Mode Cylindrical SymmetryCylindrical SymmetryTM ModeTM Mode 6 6.4.6 6.4.6 Degenerate ModesDegenerate Modes 0|)( =⋅ =arn rKJ TM Mode TM Mode 0|)( =⋅ =arn dr rKdJ TE Mode TE Mode ( ) ( ) ( )xJxJxJ 110 ' −== − ii xx 10' =⇒ TETE0i 0i ∼∼ TMTM1i1i 注意：简并与场型无关！注意：简并与场型无关！ BesselBessel函数函数 [ ] )()( 1 xJxxJxdxd nnnn −⋅=⋅I ClassI Class：： xJnJJ nnn ⋅−= −1' nnn Jx nJJ ⋅=+ +− 211 nnn JJJ '211 =− +− Small x Small x ：： Large x Large x ：： 0 2! 1 2 ≠⎟⎠ ⎞⎜⎝ ⎛⋅≈ nx n Jn ⎟⎠ ⎞⎜⎝ ⎛ +−⋅⋅≈ 4 12cos12 nx x Jn πII ClassII Class：：………… Small x Small x ：： Large x Large x ：： 0 2 )!1( 2 ≠⎟⎠ ⎞⎜⎝ ⎛⋅−−≈ nxnNn π ⎟⎠ ⎞⎜⎝ ⎛ +−⋅⋅≈ 4 12sin12 nx x Nn π ( ) )(1)( xJxJ nnn ⋅−=− 小结小结(1)(1)：： DegenerateDegenerate ModesModes ZZww CircularCircular RectangularRectangular CoaxialCoaxial VVggVVppλλccDominantDominant ModeMode WaveguideWaveguide 小结小结(2)(2)传输线、波导传输线、波导 石英介质尺寸小带宽超宽光纤 开路式损耗大尺寸小、易集成微带线 难加工损耗大无色散、易集成带状线 有色散尺寸大高功率波导 TEM波难集成无色散同轴线 其他缺点优点项目 传输线的功率容限传输线的功率容限 传输线传输线：： (1)(1)同轴线同轴线 (2)(2)微带线微带线 (3)(3)圆柱波导圆柱波导 (4)(4)矩形波导矩形波导 ““什么位置场强最大什么位置场强最大””————击穿击穿 ““BreakdownBreakdown”” 圆柱波导求解过程圆柱波导求解过程((数学方法数学方法)) ( ) ( )( ) cos sin T z n n J K r e n φ φ − ⋅⋅ ⋅ ⋅ TETE、、TM Wave SolutionTM Wave Solution ••圆柱波导中圆柱波导中TETE波边界条件波边界条件 ••求解出求解出HHzz, , 进而得到：进而得到： ••利用边界条件：利用边界条件： ••圆柱波导中圆柱波导中TMTM波边界条件波边界条件 ••直接求解出直接求解出EEzz，，得到：得到： ••利用边界条件：