耦合传输线分析学习笔记(八)对称耦合微带线S参数矩阵推导与应用（上）

前要回顾

设四端口网络的端口电压和电流（定义所有电流流入网络为正）为：
Vp=U1,U2,U3,U4T,Ip=I1,I2,I3,I4T. \mathbf{V}_p = U_1, U_2, U_3, U_4^T, \quad \mathbf{I}_p = I_1, I_2, I_3, I_4^T. Vp=U1,U2,U3,U4T,Ip=I1,I2,I3,I4T.

四端口网络反射波和入射波定义为：
Bp=b1,b2,b3,b4T,Ap=a1,a2,a3,a4T.(1) \mathbf{B}p = b_1, b_2, b_3, b_4^T, \quad \mathbf{A}p = a_1, a_2, a_3,a_4^T.\quad(1) Bp=b1,b2,b3,b4T,Ap=a1,a2,a3,a4T.(1)
S11=ReflectedIncident=b1a1∣a2=0（输入端反射系数/输入匹配）S21=TransmittedIncident=b2a1∣a2=0（正向传输系数/增益或损耗）S22=ReflectedIncident=b2a2∣a1=0（输出端反射系数/输出匹配）S12=TransmittedIncident=b1a2∣a1=0（反向传输系数/隔离） \begin{aligned} S{11} &= \frac{\text{Reflected}}{\text{Incident}} = \left.\frac{b_1}{a_1}\right|{a_2=0} && \text{（输入端反射系数/输入匹配）} \\ S_{21} &= \frac{\text{Transmitted}}{\text{Incident}} = \left.\frac{b_2}{a_1}\right|{a_2=0} && \text{（正向传输系数/增益或损耗）} \\ S{22} &= \frac{\text{Reflected}}{\text{Incident}} = \left.\frac{b_2}{a_2}\right|{a_1=0} && \text{（输出端反射系数/输出匹配）} \\ S{12} &= \frac{\text{Transmitted}}{\text{Incident}} = \left.\frac{b_1}{a_2}\right|_{a_1=0} && \text{（反向传输系数/隔离）} \end{aligned} S11S21S22S12=IncidentReflected=a1b1 a2=0=IncidentTransmitted=a1b2 a2=0=IncidentReflected=a2b2 a1=0=IncidentTransmitted=a2b1 a1=0（输入端反射系数/输入匹配）（正向传输系数/增益或损耗）（输出端反射系数/输出匹配）（反向传输系数/隔离）

四端口S参数矩阵如下。
b1b2b3b4=S11S12S13S14S21S22S23S24S31S32S33S34S41S42S43S44a1a2a3a4 \begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{bmatrix} =\begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} \\ S_{21} & S_{22} & S_{23} & S_{24} \\ S_{31} & S_{32} & S_{33} & S_{34} \\ S_{41} & S_{42} & S_{43} & S_{44} \end{bmatrix} \begin{bmatrix} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \end{bmatrix} b1b2b3b4 = S11S21S31S41S12S22S32S42S13S23S33S43S14S24S34S44 a1a2a3a4

式中bib_ibi为反射波，aia_iai为入射波

奇偶模电压、电流定义为：
Vm=UeI,UeII,UoI,UoIIT,Im=IeI,IeII,IoI,IoIIT. \mathbf{V}_m = U_e\^I, U_e\^{II}, U_o\^I, U_o\^{II}^T, \quad \mathbf{I}_m = I_e\^I, I_e\^{II}, I_o\^I, I_o\^{II}^T. Vm=UeI,UeII,UoI,UoIIT,Im=IeI,IeII,IoI,IoIIT.

奇偶模入射波和反射波定义为：

Bm=beI,beII,boI,boIIT,Am=aeI,aeII,aoI,aoIIT.(2) \mathbf{B}_m = b_e\^I, b_e\^{II}, b_o\^{I},b_o\^{II}^T, \quad \mathbf{A}_m = a_e\^I, a_e\^{II}, a_o\^{I},a_o\^{II}^T.\quad(2) Bm=beI,beII,boI,boIIT,Am=aeI,aeII,aoI,aoIIT.(2)

 这里讨论的是如对称不均匀的耦合微带线，在图2中左右1和3端口、2和4端口满足对称条件

图1 端口顺序定义,推导S参数矩阵时每个端口端接对地特性阻抗Z0

参考前面的6篇文章中我们推出了如下公式
U1U2=111−1UeIUoI(3) \left\\begin{array}{l} U_{1} \\\\ U_{2} \\end{array}\\right=\left\\begin{array}{cc} 1 \& 1 \\\\ 1 \& -1 \\end{array}\\right\left\\begin{array}{l} U_{\\mathrm{e}}\^{I} \\\\ U_{\\mathrm{o}}\^{I} \\end{array}\\right \quad(3) U1U2=111−1UeIUoI(3)

U3U4=111−1UeIIUoII(4) \left\\begin{array}{l} U_{3} \\\\ U_{4} \\end{array}\\right=\left\\begin{array}{cc} 1 \& 1 \\\\ 1 \& -1 \\end{array}\\right\left\\begin{array}{l} U_{\\mathrm{e}}\^{II} \\\\ U_{\\mathrm{o}}\^{II} \\end{array}\\right \quad(4) U3U4=111−1UeIIUoII(4)

IeIIoI=12111−1I1I2(5) \left\\begin{array}{l} I_{\\mathrm{e}}\^{I} \\\\ I_{\\mathrm{o}}\^{I} \\end{array}\\right=\frac{1}{2}\left\\begin{array}{cc} 1 \& 1 \\\\ 1 \& -1 \\end{array}\\right\left\\begin{array}{l} I_{1} \\\\ I_{2} \\end{array}\\right \quad(5) IeIIoI=21111−1I1I2(5)

IeIIIoII=12111−1I3I4(6) \left\\begin{array}{l} I_{\\mathrm{e}}\^{II} \\\\ I_{\\mathrm{o}}\^{II} \\end{array}\\right=\frac{1}{2}\left\\begin{array}{cc} 1 \& 1 \\\\ 1 \& -1 \\end{array}\\right\left\\begin{array}{l} I_{3} \\\\ I_{4} \\end{array}\\right \quad(6) IeIIIoII=21111−1I3I4(6)

传输线ABCD矩阵可以由如下二式表示
Ue1Ie1=cos⁡θejZ0esin⁡θejY0esin⁡θecos⁡θeUeIIIeII(7) \begin{bmatrix} U_e^1 \\ I_e^1 \end{bmatrix} = \begin{bmatrix} \cos \theta_e & j Z_{0e} \sin \theta_e \\ j Y_{0e} \sin \theta_e & \cos \theta_e \end{bmatrix} \begin{bmatrix} U_e^{II} \\ I_e^{II} \end{bmatrix}\qquad(7) Ue1Ie1=cosθejY0esinθejZ0esinθecosθeUeIIIeII(7)
Uo1Io1=cos⁡θojZ0osin⁡θojY0osin⁡θocos⁡θoUoIIIoII(8) \begin{bmatrix} U_o^1 \\ I_o^1 \end{bmatrix} = \begin{bmatrix} \cos \theta_o & j Z_{0o} \sin \theta_o \\ j Y_{0o} \sin \theta_o & \cos \theta_o \end{bmatrix} \begin{bmatrix} U_o^{II} \\ I_o^{II} \end{bmatrix}\qquad(8) Uo1Io1=cosθojY0osinθojZ0osinθocosθoUoIIIoII(8)

（这里 Y0e=1/Z0e,Y0o=1/Z0oY_{0e} = 1/Z_{0e}, Y_{0o} = 1/Z_{0o}Y0e=1/Z0e,Y0o=1/Z0o）

 分析思路：采用类似上一节的方法，我们已经推出了耦合微带线的四端口A参数和Z参数与Y参数矩阵，下面我们需要将去耦后的传输线ABCD参数转换为二端口的S参数矩阵。

 全过程如下通过奇偶模分解法，将四端口网络分解为偶模和奇模两个独立的传输线二端口网络，然后通过前后两个转换矩阵，将端口波与奇偶模波联系起来组合得到整体S参数。

1. 端口奇偶模分解-变换矩阵

 将四端口网络分解为偶模和奇模两个独立的二端口网络，其电压电流关系由给定公式(3)-(8)描述：

U1U2\]=T\[UeIUoI\],\[U3U4\]=T\[UeIIUoII\],T=\[111−1\] \\begin{bmatrix} U_1 \\\\ U_2 \\end{bmatrix} = T \\begin{bmatrix} U_e\^I \\\\ U_o\^I \\end{bmatrix}, \\quad \\begin{bmatrix} U_3 \\\\ U_4 \\end{bmatrix} = T \\begin{bmatrix} U_e\^{II} \\\\ U_o\^{II} \\end{bmatrix}, \\quad T = \\begin{bmatrix} 1 \& 1 \\\\1 \& -1 \\end{bmatrix} \[U1U2\]=T\[UeIUoI\],\[U3U4\]=T\[UeIIUoII\],T=\[111−1

IeIIoI\]=TI\[I1I2\],\[IeIIIoII\]=TI\[I3I4\],TI=12\[111−1\] \\begin{bmatrix} I_e\^I \\\\ I_o\^I \\end{bmatrix} = T_I \\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix}, \\quad \\begin{bmatrix} I_e\^{II} \\\\ I_o\^{II} \\end{bmatrix} = T_I \\begin{bmatrix} I_3 \\\\ I_4 \\end{bmatrix}, \\quad T_I = \\frac{1}{2} \\begin{bmatrix} 1 \& 1 \\\\ 1 \& -1 \\end{bmatrix} \[IeIIoI\]=TI\[I1I2\],\[IeIIIoII\]=TI\[I3I4\],TI=21\[111−1

根据奇偶模分解原理，端口电压、电流与奇偶模电压、电流的线性变换关系为：

U1U2\]=\[111−1\]\[UeIUoI\],\[I1I2\]=\[111−1\]\[IeIIoI\] \\begin{bmatrix} U_1 \\\\ U_2 \\end{bmatrix} = \\begin{bmatrix} 1 \& 1 \\\\ 1 \& -1 \\end{bmatrix} \\begin{bmatrix} U_e\^I \\\\ U_o\^I \\end{bmatrix}, \\quad \\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix} = \\begin{bmatrix} 1 \& 1 \\\\ 1 \& -1 \\end{bmatrix} \\begin{bmatrix} I_e\^I \\\\ I_o\^I \\end{bmatrix} \[U1U2\]=\[111−1\]\[UeIUoI\],\[I1I2\]=\[111−1\]\[IeIIoI

在端口参考阻抗均为 Z0Z_0Z0 的前提下，定义端口入射波、反射波与奇偶模入射波、反射波（均相对于参考阻抗 Z0Z_0Z0归一化，方便导出一个简单的的变换矩阵关系，这样在奇偶模模态矩阵计算的时候也需要重新对参考阻抗Z0Z_0Z0归一化）为：

ai=12Z0(Ui+Z0Ii),bi=12Z0(Ui−Z0Ii) a_i = \frac{1}{2\sqrt{Z_0}}(U_i + Z_0 I_i), \quad b_i = \frac{1}{2\sqrt{Z_0}}(U_i - Z_0 I_i) ai=2Z0 1(Ui+Z0Ii),bi=2Z0 1(Ui−Z0Ii)
aeI=12Z0(UeI+Z0IeI),beI=12Z0(UeI−Z0IeI)aoI=12Z0(UoI+Z0IoI),boI=12Z0(UoI−Z0IoI)aeII=12Z0(UeII+Z0IeII),beII=12Z0(UeII−Z0IeII)aoII=12Z0(UoII+Z0IoII),boII=12Z0(UoII−Z0IoII) a_e^I = \frac{1}{2\sqrt{Z_0}}(U_e^I + Z_0 I_e^I), \quad b_e^I = \frac{1}{2\sqrt{Z_0}}(U_e^I - Z_0 I_e^I) \\ a_o^I = \frac{1}{2\sqrt{Z_0}}(U_o^I + Z_0 I_o^I), \quad b_o^I = \frac{1}{2\sqrt{Z_0}}(U_o^I - Z_0 I_o^I) \\ a_e^{II} = \frac{1}{2\sqrt{Z_0}}(U_e^{II} + Z_0 I_e^{II}), \quad b_e^{II} = \frac{1}{2\sqrt{Z_0}}(U_e^{II} - Z_0 I_e^{II}) \\ a_o^{II} = \frac{1}{2\sqrt{Z_0}}(U_o^{II} + Z_0 I_o^{II}), \quad b_o^{II} = \frac{1}{2\sqrt{Z_0}}(U_o^{II} - Z_0 I_o^{II}) aeI=2Z0 1(UeI+Z0IeI),beI=2Z0 1(UeI−Z0IeI)aoI=2Z0 1(UoI+Z0IoI),boI=2Z0 1(UoI−Z0IoI)aeII=2Z0 1(UeII+Z0IeII),beII=2Z0 1(UeII−Z0IeII)aoII=2Z0 1(UoII+Z0IoII),boII=2Z0 1(UoII−Z0IoII)

1.1 变换关系推导

由端口电压、电流与奇偶模电压、电流的线性关系，结合波定义式，得到：
a1=aeI+aoI,b1=beI+boI,a2=aeI−aoI,b2=beI−boI,a3=aeII+aoII,b3=beII+boII,a4=aeII−aoII,b4=beII−boII. \begin{aligned} a_1 &= a_e^I + a_o^I, \quad &b_1 = b_e^I + b_o^I, \\ a_2 &= a_e^I - a_o^I, \quad &b_2 = b_e^I - b_o^I, \\ a_3 &= a_e^{II} + a_o^{II}, \quad &b_3 = b_e^{II} + b_o^{II}, \\ a_4 &= a_e^{II} - a_o^{II}, \quad &b_4 = b_e^{II} - b_o^{II}. \end{aligned} a1a2a3a4=aeI+aoI,=aeI−aoI,=aeII+aoII,=aeII−aoII,b1=beI+boI,b2=beI−boI,b3=beII+boII,b4=beII−boII.

 在前面的(1)(2)中我们定义了Ap=a1,a2,a3,a4T\mathbf{A}_p = a_1, a_2, a_3, a_4^TAp=a1,a2,a3,a4T，Bp=b1,b2,b3,b4T\mathbf{B}_p = b_1, b_2, b_3, b_4^TBp=b1,b2,b3,b4T为端口波；Am=aeI,aeII,aoI,aoIIT\mathbf{A}_m = a_e\^I, a_e\^{II}, a_o\^I, a_o\^{II}^TAm=aeI,aeII,aoI,aoIIT，Bm=beI,beII,boI,boIIT\mathbf{B}_m = b_e\^I, b_e\^{II}, b_o\^I, b_o\^{II}^TBm=beI,beII,boI,boIIT 为奇偶模波。下面推导变换矩阵。

1.2 变换矩阵

将上述关系写成矩阵形式：

（Am→Ap\mathbf{A}_m \to \mathbf{A}_pAm→Ap，Bm→Bp\mathbf{B}_m \to \mathbf{B}_pBm→Bp）：
Ap=TAm, \begin{bmatrix} \mathbf{A}_p \\ \end{bmatrix} = \begin{bmatrix} T \end{bmatrix} \begin{bmatrix} \mathbf{A}_m \end{bmatrix}, Ap=TAm,
Bp=TBm, \begin{bmatrix} \mathbf{B}_p \end{bmatrix} = \begin{bmatrix} T \end{bmatrix} \begin{bmatrix}\mathbf{B}_m \end{bmatrix}, Bp=TBm,

也可以写成
ApBp=T00TAmBm, \begin{bmatrix} \mathbf{A}_p \\ \mathbf{B}_p \end{bmatrix} = \begin{bmatrix} T & 0 \\ 0 & T \end{bmatrix} \begin{bmatrix} \mathbf{A}_m \\ \mathbf{B}_m \end{bmatrix}, ApBp=T00TAmBm,

其中
T=101010−100101010−1. T = \begin{bmatrix} 1 & 0 & 1 & 0 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & -1 \end{bmatrix}. T= 110000111−100001−1 .
反过来也一样 （Ap→Am\mathbf{A}_p \to \mathbf{A}_mAp→Am，Bp→Bm\mathbf{B}_p \to \mathbf{B}_mBp→Bm）：
AmBm=T−100T−1ApBp, \begin{bmatrix} \mathbf{A}_m \\ \mathbf{B}_m \end{bmatrix} = \begin{bmatrix} T^{-1} & 0 \\ 0 & T^{-1} \end{bmatrix} \begin{bmatrix} \mathbf{A}_p \\ \mathbf{B}_p \end{bmatrix}, AmBm=T−100T−1ApBp,

其中
T−1=12110000111−100001−1. T^{-1} = \frac{1}{2} \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{bmatrix}. T−1=21 101010−100101010−1 .

 矩阵 TTT 将模态入射波/反射波线性组合为端口入射波/反射波。逆矩阵T−1T^{-1}T−1实现端口波到模态波的分解，用于奇偶模分析。接下来可以利用该变换和模态散射矩阵 Sm=diag⁡(Se,So)\mathbf{S}_m = \operatorname{diag}(\mathbf{S}^e, \mathbf{S}^o)Sm=diag(Se,So)，来得到整体散射矩阵Sp=TSmT−1\mathbf{S}_p = T \mathbf{S}_m T^{-1}Sp=TSmT−1。

2. S参数模态矩阵

 偶模和奇模传输线的ABCD矩阵由公式(7)-(8)给出,将偶模和奇模二端口网络的ABCD参数转换为S参数（两个端口电流均定义为流入网络）四个端口参考阻抗均为Z0Z_0Z0：

对奇偶模传输线我们有如下方程：

beIbeII\]=\[Se\]\[aeIaeII\] \\begin{bmatrix} b_e\^I \\\\ b_e\^{II} \\end{bmatrix} = \\begin{bmatrix} S\^e\\end{bmatrix} \\begin{bmatrix} a_e\^{I} \\\\ a_e\^{II} \\end{bmatrix} \[beIbeII\]=\[Se\]\[aeIaeII

boIboII\]=\[So\]\[aoIaoII\] \\begin{bmatrix} b_o\^I \\\\ b_o\^{II} \\end{bmatrix} = \\begin{bmatrix} S\^o\\end{bmatrix} \\begin{bmatrix} a_o\^{I} \\\\ a_o\^{II} \\end{bmatrix} \[boIboII\]=\[So\]\[aoIaoII

根据传输线A参数转换S参数公式奇偶模参考阻抗也为Z0Z_0Z0：
S=S11S12S21S22=A+BZ0−CZ0−DA+BZ0+CZ0+D2(AD−BC)A+BZ0+CZ0+D2A+BZ0+CZ0+D−A+BZ0−CZ0+DA+BZ0+CZ0+D \mathcal{S}=\left\\begin{array}{ll} S_{11} \& S_{12} \\\\ S_{21} \& S_{22} \\end{array}\\right=\left\\begin{array}{ll} \\displaystyle\\frac{A+\\frac{B}{Z_{0}}-C Z_{0}-D}{A+\\frac{B}{Z_{0}}+C Z_{0}+D} \& \\displaystyle\\frac{2(A D-B C)}{A+\\frac{B}{Z_{0}}+C Z_{0}+D} \\\\\[12pt \displaystyle\frac{2}{A+\frac{B}{Z_{0}}+C Z_{0}+D} & \displaystyle\frac{-A+\frac{B}{Z_{0}}-C Z_{0}+D}{A+\frac{B}{Z_{0}}+C Z_{0}+D} \end{array}\right] S=S11S21S12S22= A+Z0B+CZ0+DA+Z0B−CZ0−DA+Z0B+CZ0+D2A+Z0B+CZ0+D2(AD−BC)A+Z0B+CZ0+D−A+Z0B−CZ0+D

可以分别得到模态去耦 后的奇偶模的单根传输线S参数模态矩阵
So=j(Z00Z0−Z0Z00)sin⁡θo2cos⁡θo+j(Z00Z0+Z0Z00)sin⁡θo22cos⁡θo+j(Z00Z0+Z0Z00)sin⁡θo22cos⁡θo+j(Z00Z0+Z0Z00)sin⁡θoj(Z00Z0−Z0Z00)sin⁡θo2cos⁡θo+j(Z00Z0+Z0Z00)sin⁡θo S^o=\left\\begin{array}{cc} \\displaystyle\\frac{j\\left(\\frac{Z_{00}}{Z_{0}}-\\frac{Z_{0}}{Z_{00}}\\right) \\sin \\theta_o}{2 \\cos \\theta_o+j\\left(\\frac{Z_{00}}{Z_{0}}+\\frac{Z_{0}}{Z_{00}}\\right) \\sin \\theta_o} \& \\displaystyle\\frac{2}{2 \\cos \\theta_o+j\\left(\\frac{Z_{00}}{Z_{0}}+\\frac{Z_{0}}{Z_{00}}\\right) \\sin \\theta_o} \\\\\[12pt \displaystyle\frac{2}{2 \cos \theta_o+j\left(\frac{Z_{00}}{Z_{0}}+\frac{Z_{0}}{Z_{00}}\right) \sin \theta_o} & \displaystyle\frac{j\left(\frac{Z_{00}}{Z_{0}}-\frac{Z_{0}}{Z_{00}}\right) \sin \theta_o}{2 \cos \theta_o+j\left(\frac{Z_{00}}{Z_{0}}+\frac{Z_{0}}{Z_{00}}\right) \sin \theta_o} \end{array}\right] So= 2cosθo+j(Z0Z00+Z00Z0)sinθoj(Z0Z00−Z00Z0)sinθo2cosθo+j(Z0Z00+Z00Z0)sinθo22cosθo+j(Z0Z00+Z00Z0)sinθo22cosθo+j(Z0Z00+Z00Z0)sinθoj(Z0Z00−Z00Z0)sinθo

Se=j(Z0eZ0−Z0Z0e)sin⁡θe2cos⁡θe+j(Z0eZ0+Z0Z0e)sin⁡θe22cos⁡θe+j(Z0eZ0+Z0Z0e)sin⁡θe22cos⁡θe+j(Z0eZ0+Z0Z0e)sin⁡θej(Z0eZ0−Z0Z0e)sin⁡θe2cos⁡θe+j(Z0eZ0+Z0Z0e)sin⁡θe S^e=\left\\begin{array}{cc} \\displaystyle\\frac{j\\left(\\frac{Z_{0 e}}{Z_{0}}-\\frac{Z_{0}}{Z_{0 e}}\\right) \\sin \\theta_e}{2 \\cos \\theta_e+j\\left(\\frac{Z_{0 e}}{Z_{0}}+\\frac{Z_{0}}{Z_{0 e}}\\right) \\sin \\theta_e} \& \\displaystyle\\frac{2}{2 \\cos \\theta_e+j\\left(\\frac{Z_{0 e}}{Z_{0}}+\\frac{Z_{0}}{Z_{0 e}}\\right) \\sin \\theta_e} \\\\\[12pt \displaystyle\frac{2}{2 \cos \theta_e+j\left(\frac{Z_{0 e}}{Z_{0}}+\frac{Z_{0}}{Z_{0 e}}\right) \sin \theta_e} & \displaystyle\frac{j\left(\frac{Z_{0 e}}{Z_{0}}-\frac{Z_{0}}{Z_{0 e}}\right) \sin \theta_e}{2 \cos \theta_e+j\left(\frac{Z_{0 e}}{Z_{0}}+\frac{Z_{0}}{Z_{0 e}}\right) \sin \theta_e} \end{array}\right] Se= 2cosθe+j(Z0Z0e+Z0eZ0)sinθej(Z0Z0e−Z0eZ0)sinθe2cosθe+j(Z0Z0e+Z0eZ0)sinθe22cosθe+j(Z0Z0e+Z0eZ0)sinθe22cosθe+j(Z0Z0e+Z0eZ0)sinθej(Z0Z0e−Z0eZ0)sinθe
Bm=SmAm=beI,beII,boI,boIIT=SmaeI,aeII,aoI,aoIIT. \mathbf{B}_m =S_m \mathbf{A}_m= b_e\^I, b_e\^{II}, b_o\^{I},b_o\^{II}^T = S_ma_e\^I, a_e\^{II}, a_o\^{I},a_o\^{II}^T. Bm=SmAm=beI,beII,boI,boIIT=SmaeI,aeII,aoI,aoIIT.

合并得到四端口模态传输矩阵为

Sm=Se00So S_m = \begin{bmatrix} S^e & 0 \\ 0 & S^o \end{bmatrix} Sm=Se00So

对称耦合微带线四端口网络的整体 S 参数矩阵可通过奇偶模分解法推导，其中偶模和奇模二端口网络的 S 参数分别为：

Se=ΓeTeTeΓe,So=ΓoToToΓo \mathbf{S}^e = \begin{bmatrix} \Gamma_e & T_e \\ T_e & \Gamma_e \end{bmatrix}, \quad \mathbf{S}^o = \begin{bmatrix} \Gamma_o & T_o \\ T_o & \Gamma_o \end{bmatrix} Se=ΓeTeTeΓe,So=ΓoToToΓo

其中：

Γe=j(Z0eZ0−Z0Z0e)sin⁡θe2cos⁡θe+j(Z0eZ0+Z0Z0e)sin⁡θe,Te=22cos⁡θe+j(Z0eZ0+Z0Z0e)sin⁡θe \Gamma_e = \frac{j\left(\frac{Z_{0e}}{Z_{0}}-\frac{Z_{0}}{Z_{0e}}\right) \sin \theta_e}{2 \cos \theta_e + j\left(\frac{Z_{0e}}{Z_{0}}+\frac{Z_{0}}{Z_{0e}}\right) \sin \theta_e}, \quad T_e = \frac{2}{2 \cos \theta_e + j\left(\frac{Z_{0e}}{Z_{0}}+\frac{Z_{0}}{Z_{0e}}\right) \sin \theta_e} Γe=2cosθe+j(Z0Z0e+Z0eZ0)sinθej(Z0Z0e−Z0eZ0)sinθe,Te=2cosθe+j(Z0Z0e+Z0eZ0)sinθe2

Γo=j(Z0oZ0−Z0Z0o)sin⁡θo2cos⁡θo+j(Z0oZ0+Z0Z0o)sin⁡θo,To=22cos⁡θo+j(Z0oZ0+Z0Z0o)sin⁡θo \Gamma_o = \frac{j\left(\frac{Z_{0o}}{Z_{0}}-\frac{Z_{0}}{Z_{0o}}\right) \sin \theta_o}{2 \cos \theta_o + j\left(\frac{Z_{0o}}{Z_{0}}+\frac{Z_{0}}{Z_{0o}}\right) \sin \theta_o}, \quad T_o = \frac{2}{2 \cos \theta_o + j\left(\frac{Z_{0o}}{Z_{0}}+\frac{Z_{0}}{Z_{0o}}\right) \sin \theta_o} Γo=2cosθo+j(Z0Z0o+Z0oZ0)sinθoj(Z0Z0o−Z0oZ0)sinθo,To=2cosθo+j(Z0Z0o+Z0oZ0)sinθo2

 Γe\Gamma_eΓeΓo\Gamma_oΓo 为偶奇模二端口网络的输入反射系数，定义为beI/aeIb_e^I / a_e^I\quadbeI/aeIboI/aoIb_o^I / a_o^IboI/aoI（输出端接匹配负载Z0Z_0Z0）。表达式分子正比于j(Z0eZ0−Z0Z0e)sin⁡θej\left(\frac{Z_{0e}}{Z_0} - \frac{Z_0}{Z_{0e}}\right) \sin\theta_ej(Z0Z0e−Z0eZ0)sinθe，反映了因模式特性阻抗 Z0eZ_{0e}Z0e（或Z0oZ_{0o}Z0o）与系统参考阻抗Z0Z_0Z0 不匹配而产生的反射，其幅度随电长度θe\theta_eθe（或 θo\theta_oθo）变化。

 当θe=90∘\theta_e = 90^\circθe=90∘（四分之一波长）时，cos⁡θe=0\cos\theta_e = 0cosθe=0，Γe=Z0e2−Z02Z0e2+Z02\Gamma_e = \frac{Z_{0e}^2 - Z_0^2}{Z_{0e}^2 + Z_0^2}Γe=Z0e2+Z02Z0e2−Z02，为实数反射系数。Γ=(Zin−Z0)/(Zin+Z0)\Gamma=(Z_{in} - Z₀) / (Z_{in} + Z₀)Γ=(Zin−Z0)/(Zin+Z0)其中 Zine=Z0e2/Z0Z_{ine} = Z₀ₑ² / Z₀Zine=Z0e2/Z0
 ToT_oToTeT_eTe为奇偶模二端口网络的向前传输系数，定义为boII/aoIb_o^{II} / a_o^I\quadboII/aoI beII/aeIb_e^{II} / a_e^IbeII/aeI（输出端接匹配负载 Z0Z_0Z0）。

 当 θe=0\theta_e = 0θe=0 时，显然Te=1T_e = 1Te=1；当 θe=90∘\theta_e = 90^\circθe=90∘ 时，Te=2j(Z0eZ0+Z0Z0e)T_e = \frac{2}{j\left(\frac{Z_{0e}}{Z_0} + \frac{Z_0}{Z_{0e}}\right)}Te=j(Z0Z0e+Z0eZ0)2，根据均值不等式在模式阻抗不等于参考阻抗时，其幅度小于 1，表明因阻抗不匹配产生反射损耗。

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3.整体S参数矩阵推导

3.1计算过程

此矩阵即为对称耦合微带线四端口网络的整体 S 参数矩阵，满足互易性和对称性，可用于分析定向耦合器、滤波器等器件。

端口散射矩阵Sp\mathbf{S}_pSp 满足Bp=SpAp\mathbf{B}_p = \mathbf{S}_p \mathbf{A}_pBp=SpAp。结合变换关系：
Bp=TBm=TSmAm=TSmT−1Ap, \mathbf{B}_p = T \mathbf{B}_m = T \mathbf{S}_m \mathbf{A}_m = T \mathbf{S}_m T^{-1} \mathbf{A}_p, Bp=TBm=TSmAm=TSmT−1Ap,

故
Sp=TSmT−1. \mathbf{S}_p = T \mathbf{S}_m T^{-1}. Sp=TSmT−1.

计算 SmT−1\mathbf{S}_m T^{-1}SmT−1（略去常数因子1/21/21/2）

令 T~−1\tilde{T}^{-1}T~−1为T−1T^{-1}T−1去掉因子1/21/21/2 的矩阵，即：
T~−1=110000111−100001−1. \tilde{T}^{-1} = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{bmatrix}. T~−1= 101010−100101010−1 .

计算 SmT~−1\mathbf{S}_m \tilde{T}^{-1}SmT~−1：

X=ΓeTe00TeΓe0000ΓoTo00ToΓo110000111−100001−1=ΓeΓeTeTeTeTeΓeΓeΓo−ΓoTo−ToTo−ToΓo−Γo. \begin{aligned} \mathbf{X} &= \begin{bmatrix} \Gamma_e & T_e & 0 & 0 \\ T_e & \Gamma_e & 0 & 0 \\ 0 & 0 & \Gamma_o & T_o \\ 0 & 0 & T_o & \Gamma_o \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{bmatrix} \\ &=\begin{bmatrix} \Gamma_e & \Gamma_e & T_e & T_e \\ T_e & T_e & \Gamma_e & \Gamma_e \\ \Gamma_o & -\Gamma_o & T_o & -T_o \\ T_o & -T_o & \Gamma_o & -\Gamma_o \end{bmatrix}. \end{aligned} X= ΓeTe00TeΓe0000ΓoTo00ToΓo 101010−100101010−1 = ΓeTeΓoToΓeTe−Γo−ToTeΓeToΓoTeΓe−To−Γo .

计算 TSmT~−1T \mathbf{S}_m \tilde{T}^{-1}TSmT~−1 并乘以1/21/21/2即得到Sp\mathbf{S}_pSp

Sp=12Γe+ΓoΓe−ΓoTe+ToTe−ToΓe−ΓoΓe+ΓoTe−ToTe+ToTe+ToTe−ToΓe+ΓoΓe−ΓoTe−ToTe+ToΓe−ΓoΓe+Γo \mathbf{S}_p = \frac{1}{2} \begin{bmatrix} \Gamma_e + \Gamma_o & \Gamma_e - \Gamma_o & T_e + T_o & T_e - T_o \\ \Gamma_e - \Gamma_o & \Gamma_e + \Gamma_o & T_e - T_o & T_e + T_o \\ T_e + T_o & T_e - T_o & \Gamma_e + \Gamma_o & \Gamma_e - \Gamma_o \\ T_e - T_o & T_e + T_o & \Gamma_e - \Gamma_o & \Gamma_e + \Gamma_o \end{bmatrix} Sp=21 Γe+ΓoΓe−ΓoTe+ToTe−ToΓe−ΓoΓe+ΓoTe−ToTe+ToTe+ToTe−ToΓe+ΓoΓe−ΓoTe−ToTe+ToΓe−ΓoΓe+Γo

3.2形象理解上述矩阵运算的含义（一个计算实例）

 考虑 端口1激励 （入射波 a1≠0a_1\neq0a1=0,），其余端口匹配（没有反射即没有反向入射波a2=a3=a4=0a_2=a_3=a_4=0a2=a3=a4=0）。该激励可分解为偶模和奇模的叠加（参考阻抗为(Z0Z_0Z0)）：

a1=aeI+aoIa2=aeI−aoIa3=aeII+aoIIa4=aeII−aoII \begin{aligned} a_1 &= a_e^I + a_o^I \\ a_2 &= a_e^I - a_o^I \\ a_3 &= a_e^{II} + a_o^{II} \\ a_4 &= a_e^{II} - a_o^{II} \end{aligned} a1a2a3a4=aeI+aoI=aeI−aoI=aeII+aoII=aeII−aoII

 考虑仅在端口1施加激励，其余端口均接匹配负载（无反射），因此端口入射波满足：

a1≠0,a2=a3=a4=0 a_1 \neq 0, \quad a_2 = a_3 = a_4 = 0a1=0,a2=a3=a4=0
{a1=aeI+aoI(1)0=aeI−aoI(2)0=aeII+aoII(3)0=aeII−aoII(4) \begin{cases} a_1 = a_e^I + a_o^I & \quad (1) \\ 0 = a_e^I - a_o^I & \quad (2) \\ 0 = a_e^{II} + a_o^{II} & \quad (3) \\ 0 = a_e^{II} - a_o^{II} & \quad (4) \end{cases} ⎩ ⎨ ⎧a1=aeI+aoI0=aeI−aoI0=aeII+aoII0=aeII−aoII(1)(2)(3)(4)

由方程(2)得aeI=aoIa_e^I = a_o^IaeI=aoI，代入方程(1)得：

aeI=aoI=a12 a_e^I = a_o^I = \frac{a_1}{2} aeI=aoI=2a1

由方程(3)和(4)联立得：

aeII=0,aoII=0 a_e^{II} = 0, \quad a_o^{II} = 0 aeII=0,aoII=0

因此，得到模态入射波：

aeI=aoI=a12,aeII=aoII=0 a_e^I = a_o^I = \frac{a_1}{2}, \quad a_e^{II} = a_o^{II} = 0aeI=aoI=2a1,aeII=aoII=0

利用模态 S 参数矩阵计算反射波：

偶模： beI=ΓeaeI=Γea12b_e^I = \Gamma_e a_e^I = \frac{\Gamma_e a_1}{2}\qquadbeI=ΓeaeI=2Γea1beII=TeaeI=Tea12b_e^{II} = T_e a_e^I = \frac{T_e a_1}{2}beII=TeaeI=2Tea1

奇模：boI=ΓoaoI=Γoa12b_o^I = \Gamma_o a_o^I = \frac{\Gamma_o a_1}{2}\qquadboI=ΓoaoI=2Γoa1boII=ToaoI=Toa12b_o^{II} = T_o a_o^I = \frac{T_o a_1}{2}boII=ToaoI=2Toa1

端口反射波为模态反射波的叠加：
b1=beI+boI=12(Γe+Γo)a1b2=beII+boII=12(Te+To)a1b3=beI−boI=12(Γe−Γo)a1b4=beII−boII=12(Te−To)a1 \begin{aligned} b_1 &= b_e^I + b_o^I = \frac{1}{2}(\Gamma_e + \Gamma_o)a_1 \\ b_2 &= b_e^{II} + b_o^{II} = \frac{1}{2}(T_e + T_o)a_1 \\ b_3 &= b_e^I - b_o^I = \frac{1}{2}(\Gamma_e - \Gamma_o)a_1 \\ b_4 &= b_e^{II} - b_o^{II} = \frac{1}{2}(T_e - T_o)a_1 \end{aligned} b1b2b3b4=beI+boI=21(Γe+Γo)a1=beII+boII=21(Te+To)a1=beI−boI=21(Γe−Γo)a1=beII−boII=21(Te−To)a1

由此得第一列 S 参数：
S11=b1a1=Γe+Γo2,S21=b2a1=Te+To2,S31=b3a1=Γe−Γo2,S41=b4a1=Te−To2 S_{11} = \frac{b_1}{a_1} = \frac{\Gamma_e + \Gamma_o}{2}, \quad S_{21} = \frac{b_2}{a_1} = \frac{T_e + T_o}{2}, \quad S_{31} = \frac{b_3}{a_1} = \frac{\Gamma_e - \Gamma_o}{2}, \quad S_{41} = \frac{b_4}{a_1} = \frac{T_e - T_o}{2} S11=a1b1=2Γe+Γo,S21=a1b2=2Te+To,S31=a1b3=2Γe−Γo,S41=a1b4=2Te−To

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由于网络对称且互易，其他端口激励可类似推导，最终得到与上述对称耦合微带线四端口 S 参数矩阵一样的形式