第四章 射频网络分析

这里写目录标题

• 4.1 阻抗和导纳矩阵
• 4.1.2 互易网络
• 4.1.3 无耗网络
• 4.2 传输矩阵（ABCD矩阵）
• 4.3 混合参数矩阵[H] 教材100页 PDF116页
• 各种参数矩阵之间的关系★★
• 4.4 散射参量
• 4.4.1 散射参量的定义
• S参数的工程表示
• 网络传输过程与散射参量的关系
• 4.4.2 散射参量的物理意义
• 4.4.3 链形散射矩阵
• 4.5 信号流图
• 4.5.1 信号流图的分解
• 串联法则★
• 并联法则★
• 自闭环法则★
• 剖分法则★
• 作业

将网络本身看成黑匣子，网络有N个端口，每个端口都有电压和电流（vj、ik）两个变量。假设网络是线性网络，则变量之间的关系可以用线性方程组表示。

$\left[ {\begin{array}{l} {{v_1}}\\ {{v_2}}\\ \vdots \\ {{v_N}} \end{array}} \right] = \left[ {\begin{array}{l} {{z_{11}}}&{{z_{12}}}& \cdots &{{z_{1N}}}\\ {{z_{21}}}& \ddots &{}&{{z_{2N}}}\\ \vdots &{}& \ddots & \vdots \\ {{z_{N1}}}&{{z_{N2}}}& \cdots &{{z_{NN}}} \end{array}} \right] \left[ {\begin{array}{l} {{i_1}}\\ {{i_2}}\\ \vdots \\ {{i_N}} \end{array}} \right] \Leftrightarrow V = ZI$⎣⎢⎢⎢⎡​v1​v2​⋮vN​​⎦⎥⎥⎥⎤​=⎣⎢⎢⎢⎡​z11​z21​⋮zN1​​z12​⋱zN2​​⋯⋱⋯​z1N​z2N​⋮zNN​​⎦⎥⎥⎥⎤​⎣⎢⎢⎢⎡​i1​i2​⋮iN​​⎦⎥⎥⎥⎤​⇔V=ZI

$\left[ {\begin{array}{l} {{i_1}}\\ {{i_2}}\\ \vdots \\ {{i_N}} \end{array}} \right] = \left[ {\begin{array}{l} {{y_{11}}}&{{y_{12}}}& \cdots &{{y_{1N}}}\\ {{y_{21}}}& \ddots &{}&{{y_{2N}}}\\ \vdots &{}& \ddots & \vdots \\ {{y_{N1}}}&{{y_{N2}}}& \cdots &{{y_{NN}}} \end{array}} \right] \left[ {\begin{array}{l} {{v_1}}\\ {{v_2}}\\ \vdots \\ {{v_N}} \end{array}} \right] \Leftrightarrow I = YV$⎣⎢⎢⎢⎡​i1​i2​⋮iN​​⎦⎥⎥⎥⎤​=⎣⎢⎢⎢⎡​y11​y21​⋮yN1​​y12​⋱yN2​​⋯⋱⋯​y1N​y2N​⋮yNN​​⎦⎥⎥⎥⎤​⎣⎢⎢⎢⎡​v1​v2​⋮vN​​⎦⎥⎥⎥⎤​⇔I=YV

4.1 阻抗和导纳矩阵

二端口网络★

阻抗矩阵★
$\left[ {\begin{array}{l} {{v_1}}\\ {{v_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{z_{11}}}&{{z_{12}}}\\ {{z_{21}}}&{{z_{22}}} \end{array}} \right]\left[ {\begin{array}{l} {{i_1}}\\ {{i_2}} \end{array}} \right] \Leftrightarrow \begin{array}{l} {{v_1} = {z_{11}}{i_1} + {z_{12}}{i_2}}\\ {{v_2} = {z_{21}}{i_1} + {z_{22}}{i_2}} \end{array}$[v1​v2​​]=[z11​z21​​z12​z22​​][i1​i2​​]⇔v1​=z11​i1​+z12​i2​v2​=z21​i1​+z22​i2​​
${\left. {{z_{mn}} = \frac{{{v_m}}}{{{i_n}}}} \right|_{{i_k} = 0;k \ne n}}$zmn​=in​vm​​∣∣∣​ik​=0;k​=n​

导纳矩阵★
$\left[ {\begin{array}{l} {{i_1}}\\ {{i_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{y_{11}}}&{{y_{12}}}\\ {{y_{21}}}&{{y_{22}}} \end{array}} \right]\left[ {\begin{array}{l} {{v_1}}\\ {{v_2}} \end{array}} \right] \Leftrightarrow \begin{array}{l} {{i_1} = {y_{11}}{v_1} + {y_{12}}{v_2}}\\ {{i_2} = {y_{21}}{v_1} + {y_{22}}{v_2}} \end{array}$[i1​i2​​]=[y11​y21​​y12​y22​​][v1​v2​​]⇔i1​=y11​v1​+y12​v2​i2​=y21​v1​+y22​v2​​
${\left. {{y_{mn}} = \frac{{{i_m}}}{{{v_n}}}} \right|_{{i_k} = 0;k \ne n}}$ymn​=vn​im​​∣∣∣​ik​=0;k​=n​
如果上述矩阵是可逆的，则$Z = {Y^{ - 1}},Y = {Z^{ - 1}}$Z=Y−1,Y=Z−1
所有变量都属于复数域
各种参数矩阵之间的关系

4.1.2 互易网络

所谓两端口网络的互易性是指，互换两端口之间的激励源，其网络响应保持不变。

左边
$Vs = {z_{11}}{i_1} + {z_{12}}{I_P}$Vs=z11​i1​+z12​IP​
$0 = {z_{21}}{i_1} + {z_{22}}{I_P}$0=z21​i1​+z22​IP​
${I_P} = \frac{{{z_{12}}}}{{{z_{11}}{z_{22}} - {z_{12}}{z_{21}}}}Vs$IP​=z11​z22​−z12​z21​z12​​Vs
右边
$0 = {z_{11}}{I_P} + {z_{12}}i_2^{'}$0=z11​IP​+z12​i2′​
$Vs = {z_{21}}{I_P} + {z_{22}}i_2^{'}$Vs=z21​IP​+z22​i2′​
${I_P} = \frac{{{z_{21}}}}{{{z_{11}}{z_{22}} - {z_{12}}{z_{21}}}}Vs$IP​=z11​z22​−z12​z21​z21​​Vs

4.1.3 无耗网络

自身不消耗能量也不产生能量的网络称为无耗网络。
无耗网络在任何条件下满足$P = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{V^T}I_{}^*} \right\} = 0$P=21​Re{VTI∗​}=0★
${V^T}I_{}^* = {I^T}{Z^T}I_{}^* = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {{i_m}{z_{mn}}i_n^*} } = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {{i_m}i_n^*{z_{mn}}} }$VTI∗​=ITZTI∗​=n=1∑N​m=1∑N​im​zmn​in∗​=n=1∑N​m=1∑N​im​in∗​zmn​
由于各$i_n$in​是独立的，且在任何条件下满足上条件，因此各${z_{mn}}$zmn​均为虚数。（LC网络满足条件）
网络两端分别接恒流源，分别令恒流源{i1,i2}分别为{ 1,0 }、{ 0,1 }、{ 1,j }和{ j,1 }便可证明上述结论。

4.2 传输矩阵（ABCD矩阵）

$\begin{array}{l} {{v_1} = A{v_2} + B{i_2}}\\ {{i_1} = C{v_2} + D{i_2}} \end{array} \Leftrightarrow \left[ {\begin{array}{l} {{v_1}}\\ {{i_1}} \end{array}} \right] = \left[ {\begin{array}{l} A&B\\ C&D \end{array}} \right] \left[ {\begin{array}{l} {{v_2}}\\ {{i_2}} \end{array}} \right]$v1​=Av2​+Bi2​i1​=Cv2​+Di2​​⇔[v1​i1​​]=[AC​BD​][v2​i2​​]★
这种产生定义适于网络级联网络运算：
$\begin{array}{l} \left[ {\begin{array}{l} {{v_1}}\\ {{i_1}} \end{array}} \right] = \left[ {\begin{array}{l} {{A_1}}&{{B_1}}\\ {{C_1}}&{{D_1}} \end{array}} \right] \left[ {\begin{array}{l} {{v_2}}\\ {{i_2}} \end{array}} \right] \left[ {\begin{array}{l} {{v_2}}\\ {{i_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{A_2}}&{{B_2}}\\ {{C_2}}&{{D_2}} \end{array}} \right] \left[ {\begin{array}{l} {{v_3}}\\ {{i_3}} \end{array}} \right]\\ \left[ {\begin{array}{l} {{v_1}}\\ {{i_1}} \end{array}} \right] = \left[ {\begin{array}{l} {{A_1}}&{{B_1}}\\ {{C_1}}&{{D_1}} \end{array}} \right] \left[ {\begin{array}{l} {{A_2}}&{{B_2}}\\ {{C_2}}&{{D_2}} \end{array}} \right] \left[ {\begin{array}{l} {{v_3}}\\ {{i_3}} \end{array}} \right] \end{array}$[v1​i1​​]=[A1​C1​​B1​D1​​][v2​i2​​][v2​i2​​]=[A2​C2​​B2​D2​​][v3​i3​​][v1​i1​​]=[A1​C1​​B1​D1​​][A2​C2​​B2​D2​​][v3​i3​​]​
互易：AD-BC=1

ABCD矩阵 教材(4.10)的定义 PDF116页
$\left[ {\begin{array}{l} {{v_1}}\\ {{i_1}} \end{array}} \right] = \left[ {\begin{array}{l} A&B\\ C&D \end{array}} \right] \left[ {\begin{array}{l} {{v_2}}\\ {-{i_2}} \end{array}} \right]$[v1​i1​​]=[AC​BD​][v2​−i2​​]★
各种参数矩阵之间的关系

同教材107页表4.1 PDF123页★

4.3 混合参数矩阵[H] 教材100页 PDF116页

混合参数矩阵
$\left[ {\begin{array}{l} {{v_1}}\\ {{i_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{h_{11}}}&{{h_{12}}}\\ {{h_{21}}}&{{h_{22}}} \end{array}} \right] \left[ {\begin{array}{l} {{i_1}}\\ {{v_2}} \end{array}} \right]$[v1​i2​​]=[h11​h21​​h12​h22​​][i1​v2​​]★
常用于晶体管交流等效电路，一端口作为输入端口。
${{h_{11}}}$h11​表示输入阻抗
${{h_{12}}}$h12​表示反向电压增益
${{h_{21}}}$h21​表示正向电流增益
${{h_{22}}}$h22​表示输出导纳
各种参数矩阵之间的关系

各种参数矩阵之间的关系★★

各种参数矩阵之间的关系 统一电流方向为上图
阻抗矩阵
$\left[ {\begin{array}{l} {{v_1}}\\ {{v_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{z_{11}}}&{{z_{12}}}\\ {{z_{21}}}&{{z_{22}}} \end{array}} \right]\left[ {\begin{array}{l} {{i_1}}\\ {{i_2}} \end{array}} \right] \Leftrightarrow \begin{array}{l} {{v_1} = {z_{11}}{i_1} + {z_{12}}{i_2}}\\ {{v_2} = {z_{21}}{i_1} + {z_{22}}{i_2}} \end{array}$[v1​v2​​]=[z11​z21​​z12​z22​​][i1​i2​​]⇔v1​=z11​i1​+z12​i2​v2​=z21​i1​+z22​i2​​
导纳矩阵
$\left[ {\begin{array}{l} {{i_1}}\\ {{i_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{y_{11}}}&{{y_{12}}}\\ {{y_{21}}}&{{y_{22}}} \end{array}} \right]\left[ {\begin{array}{l} {{v_1}}\\ {{v_2}} \end{array}} \right] \Leftrightarrow \begin{array}{l} {{i_1} = {y_{11}}{v_1} + {y_{12}}{v_2}}\\ {{i_2} = {y_{21}}{v_1} + {y_{22}}{v_2}} \end{array}$[i1​i2​​]=[y11​y21​​y12​y22​​][v1​v2​​]⇔i1​=y11​v1​+y12​v2​i2​=y21​v1​+y22​v2​​
ABCD矩阵
$\left[ {\begin{array}{l} {{v_1}}\\ {{i_1}} \end{array}} \right] = \left[ {\begin{array}{l} A&B\\ C&D \end{array}} \right] \left[ {\begin{array}{l} {{v_2}}\\ {-{i_2}} \end{array}} \right]$[v1​i1​​]=[AC​BD​][v2​−i2​​]★
混合参数矩阵
$\left[ {\begin{array}{l} {{v_1}}\\ {{i_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{h_{11}}}&{{h_{12}}}\\ {{h_{21}}}&{{h_{22}}} \end{array}} \right] \left[ {\begin{array}{l} {{i_1}}\\ {{v_2}} \end{array}} \right]$[v1​i2​​]=[h11​h21​​h12​h22​​][i1​v2​​]★
由于各种矩阵都是由端口电压、电路作为变量构成，通过简单的移项，代入处理便可方便的转换
需要注意的是转换中电压电流的方向

教材109页PDF125页★
$\Delta X = {x_{11}}{x_{22}}{\rm{ - }}{x_{12}}{x_{21}}$ΔX=x11​x22​−x12​x21​

4.4 散射参量

前面定义的矩阵或线性方程组都以电流、电压为变量。
在求取矩阵参数时，不可避免涉及到电压为零或电流为零的条件
对应于电路而言，即端口的开路或短路
这在很多电路中是不可实现的
解决思路：
进行坐标变换，得到一组新的参量
新参量既满足物理可实现性又方便线性分析，还便于恢复电压电流参数
1、采用线性变换得到新变量，则新变量构成的系统仍然是线性系统
2、解决端口开短路的不可实现问题
回顾传输线方程，如果用入射波及反射波作为变量，则变量为零对应于某端口的阻抗匹配，而不是端口的开路或者短路
3、变量最好对功率归一化
$P = \frac{{{V^2}}}{R}{\rm{ }} \Rightarrow {\rm{ }}{V_X} = \frac{V}{{\sqrt R }}{\rm{ }} \Rightarrow {\rm{ }}P = V_X^2\\ P = R{I^2}{\rm{ }} \Rightarrow {\rm{ }}{I_X} = \sqrt R I{\rm{ }} \Rightarrow {\rm{ }}P = I_X^2$P=RV2​⇒VX​=R​V​⇒P=VX2​P=RI2⇒IX​=R​I⇒P=IX2​

4.4.1 散射参量的定义

传输线方程
$\left\{ \begin{array}{l} V(z) = {V_0}^ + {e^{ - \gamma z}} + {V_0}^ - {e^{\gamma z}}\\ I(z) = \frac{{{V_0}^ + }}{{{Z_0}}}{e^{ - \gamma z}} - \frac{{{V_0}^ - }}{{{Z_0}}}{e^{\gamma z}} \end{array} \right. \Rightarrow \left\{ \begin{array}{l} V(0) = {V_0} = {V_0}^ + + {V_0}^ - \\ I(0) = {I_0} = \frac{{{V_0}^ + }}{{{Z_0}}} - \frac{{{V_0}^ - }}{{{Z_0}}} \end{array} \right.${V(z)=V0​+e−γz+V0​−eγzI(z)=Z0​V0​+​e−γz−Z0​V0​−​eγz​⇒{V(0)=V0​=V0​++V0​−I(0)=I0​=Z0​V0​+​−Z0​V0​−​​
用终端电流电压表示终端的电压入射波和反射波
${V_0}^ + = ({{{V_0} + {Z_0}{I_0}}})/{2}\\ {V_0}^ - = ({{{V_0} - {Z_0}{I_0}}})/{2}$V0​+=(V0​+Z0​I0​)/2V0​−=(V0​−Z0​I0​)/2
再对电压入射波和反射波进行功率归一化，产生一组端口新变量

${a_n} = \frac{{{V_n} + {Z_0}{I_n}}}{{2\sqrt {{Z_0}} }}$an​=2Z0​​Vn​+Z0​In​​★(4.36)
${b_n} = \frac{{{V_n} - {Z_0}{I_n}}}{{2\sqrt {{Z_0}} }}$bn​=2Z0​​Vn​−Z0​In​​★
${V_n} = \sqrt {{Z_0}} \left( {{a_n} + {b_n}} \right)$Vn​=Z0​​(an​+bn​)★(4.37)
${I_n} = \frac{1}{{\sqrt {{Z_0}} }}\left( {{a_n} - {b_n}} \right)$In​=Z0​​1​(an​−bn​)★
${P_n} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{V_n}I_n^*} \right\} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {\left[ {\sqrt {{Z_0}} \left( {{a_n} + {b_n}} \right)} \right]{{\left[ {\frac{1}{{\sqrt {{Z_0}} }}\left( {{a_n} - {b_n}} \right)} \right]}^*}} \right\}\\ = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{a_n}a_n^* - {a_n}b_n^* + a_n^*{b_n} - b_n^*b_n^*} \right\} = \frac{1}{2}\left( {{{\left| a \right|}^2} - {{\left| b \right|}^2}} \right)$Pn​=21​Re{Vn​In∗​}=21​Re{[Z0​​(an​+bn​)][Z0​​1​(an​−bn​)]∗}=21​Re{an​an∗​−an​bn∗​+an∗​bn​−bn∗​bn∗​}=21​(∣a∣2−∣b∣2)★
其中：Z0为特征阻抗，实数。其它均为复数
${{V^ + } = \sqrt {{Z_0}} a{\rm{ }}}$V+=Z0​​a(4.39)
${{I^ + } = {a \mathord{\left/ {\vphantom {a {\sqrt {{Z_0}} }}} \right.} {\sqrt {{Z_0}} }}}$I+=a/aZ0​​Z0​​
${{V^ - } = \sqrt {{Z_0}} b}$V−=Z0​​b
${{I^ - } = {{ - b} \mathord{\left/ {\vphantom {{ - b} {\sqrt {{Z_0}} }}} \right.} {\sqrt {{Z_0}} }}}$I−=−b/−bZ0​​Z0​​
$\Gamma = \frac{{{V_{}}^ - }}{{{V_{}}^ + }} = \frac{b}{a}$Γ=V​+V​−​=ab​

$\left[ {\begin{array}{l} {{b_1}}\\ {{b_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{{\rm{S}}_{11}}}&{{{\rm{S}}_{12}}}\\ {{{\rm{S}}_{21}}}&{{{\rm{S}}_{22}}} \end{array}} \right] \left[ {\begin{array}{l} {{a_1}}\\ {{a_2}} \end{array}} \right]$[b1​b2​​]=[S11​S21​​S12​S22​​][a1​a2​​]★
a和b描述的是功率归一化的电压参量，与实际电压入射波、反射波只是比例不同
因此S参数矩阵也可表示为
${{{V_n^ + } \mathord{\left/ {\vphantom {{V_n^ + } {\sqrt {{Z_0}} }}} \right.} {\sqrt {{Z_0}} }} = {a_n}}\\ {{{V_n^ - } \mathord{\left/ {\vphantom {{V_n^ - } {\sqrt {{Z_0}} }}} \right.} {\sqrt {{Z_0}} }} = {b_n}}$Vn+​/Vn+​Z0​​Z0​​=an​Vn−​/Vn−​Z0​​Z0​​=bn​
$\left[ {\begin{array}{l} {V_1^ - }\\ {V_2^ - } \end{array}} \right] = \left[ {\begin{array}{l} {{{\rm{S}}_{11}}}&{{{\rm{S}}_{12}}}\\ {{{\rm{S}}_{21}}}&{{{\rm{S}}_{22}}} \end{array}} \right] \left[ {\begin{array}{l} {V_1^ + }\\ {V_2^ + } \end{array}} \right]$[V1−​V2−​​]=[S11​S21​​S12​S22​​][V1+​V2+​​]

教材PDF129页

入射波的方向指向网络内部
反射波指向网络外部
入射波网络外向网络内输入能量
反射波为网络向外输出能量

关于波与电压电流的关系问题
1、S参数方程中的波a、b必须由网络端口的两条线共同传输
不能认为是一条传输a而另一条传输b。
2、入射波表示向网络注入能量，反射波由网络向外输出能量
3、入射波与反射波的定义与电压电流方向的关系（两端口均为关联参考方向）
4、入射波与反射波的方向表示传输方向
虽然传输方向相反，但它们定义的端口电压方向相同

S参数的工程表示

矩阵公式中描述的S参数为复比例值
工程中常用dB表示幅度值
线性描述：${S_{mn}} = {S_{mnR}} + j{S_{mnI}} = \left| {{S_{mn}}} \right|{e^{j{\varphi _{mn}}}}$Smn​=SmnR​+jSmnI​=∣Smn​∣ejφmn​
dB描述：$X\angle {\varphi _{mn}}$X∠φmn​、$X = 20\log \left| {{S_{mn}}} \right|$X=20log∣Smn​∣
magnitude = sqrt(Re^2 + Im^2)
phase = arctan(Im / Re)
${S_{11}} = 0.5 + j0.5{\rm{ }} \Leftrightarrow {S_{11}} = {\rm{0}}{\rm{.707}}\angle {\rm{4}}{{\rm{5}}^ \circ }$S11​=0.5+j0.5⇔S11​=0.707∠45∘

网络传输过程与散射参量的关系

特征阻抗${Z_0}$Z0​
${\Gamma _S} = \frac{{{Z_S} - {Z_0}}}{{{Z_S}{\rm{ + }}{Z_0}}}, {\Gamma _{{\rm{in}}}} = \frac{{{Z_{{\rm{in}}}} - {Z_0}}}{{{Z_{{\rm{in}}}}{\rm{ + }}{Z_0}}}, {\Gamma _{{\rm{out}}}} = \frac{{{Z_{{\rm{out}}}} - {Z_0}}}{{{Z_{{\rm{out}}}}{\rm{ + }}{Z_0}}}, {\Gamma _{\rm{L}}} = \frac{{{Z_{\rm{L}}} - {Z_0}}}{{{Z_{\rm{L}}}{\rm{ + }}{Z_0}}}$ΓS​=ZS​+Z0​ZS​−Z0​​,Γin​=Zin​+Z0​Zin​−Z0​​,Γout​=Zout​+Z0​Zout​−Z0​​,ΓL​=ZL​+Z0​ZL​−Z0​​★
对于阻抗不匹配时，会产生来回的多次反射，这时的 ${a_i}$ai​、${b_i}$bi​是同方向的波的叠加，而不仅仅是某一端口的入射和反射的关系
因此不能直接用反射系数（a,b）计算S参数。

4.4.2 散射参量的物理意义

对于射频电路设计来讲，特征阻抗是一个标准值
所有电路、设备的输入输出阻抗，负载，电源输出阻抗等都要求与之相等
在网络连接标准电源和标准负载的条件下：
1、S11为网络输入反射系数，描述了输入阻抗与标准之间的差异。
${S_{11}} = {\Gamma _{in}} = \frac{{{Z_{in}} - {Z_0}}}{{{Z_{in}} + {Z_0}}}$S11​=Γin​=Zin​+Z0​Zin​−Z0​​★
回波损耗
$RL = - 20\log \left| {{S_{11}}} \right|$RL=−20log∣S11​∣★
2、S21表示网络的正向传输增益
3、S12表示网络的反向传输系数
4、S22表示网络的输出阻抗
${S_{22}} = {\Gamma _{out}} = \frac{{{Z_{out}} - {Z_0}}}{{{Z_{out}} + {Z_0}}}$S22​=Γout​=Zout​+Z0​Zout​−Z0​​★

功率问题：
我们研究电路通常研究负载上实际耗散的功率，而不是其存储的功率
换句话说我们对实功功率感兴趣。
如果信号为单频信号，用峰值描述，则端口接收的实功功率可表示为
${P_n} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{V_n}I_n^*} \right\} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {\left[ {\sqrt {{Z_0}} \left( {{a_n} + {b_n}} \right)} \right]{{\left[ {\frac{1}{{\sqrt {{Z_0}} }}\left( {{a_n} - {b_n}} \right)} \right]}^*}} \right\}\\ = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{a_n}a_n^* - {a_n}b_n^* + a_n^*{b_n} - b_n^*b_n^*} \right\} = \frac{1}{2}\left( {{{\left| a \right|}^2} - {{\left| b \right|}^2}} \right)$Pn​=21​Re{Vn​In∗​}=21​Re{[Z0​​(an​+bn​)][Z0​​1​(an​−bn​)]∗}=21​Re{an​an∗​−an​bn∗​+an∗​bn​−bn∗​bn∗​}=21​(∣a∣2−∣b∣2)★

4.4.3 链形散射矩阵

$\left\{ {\begin{array}{l} {{a_1}}\\ {{b_1}} \end{array}} \right\} = \left\{ {\begin{array}{l} {{T_{11}}}&{{T_{12}}}\\ {{T_{21}}}&{{T_{22}}} \end{array}} \right\}\left\{ {\begin{array}{l} {{b_2}}\\ {{a_2}} \end{array}} \right\}${a1​b1​​}={T11​T21​​T12​T22​​}{b2​a2​​}

$\left[ {\begin{array}{l} {{b_1}}\\ {{b_2}} \end{array}} \right] = \left[ {\begin{array}{l} {{{\rm{S}}_{11}}}&{{{\rm{S}}_{12}}}\\ {{{\rm{S}}_{21}}}&{{{\rm{S}}_{22}}} \end{array}} \right] \left[ {\begin{array}{l} {{a_1}}\\ {{a_2}} \end{array}} \right]$[b1​b2​​]=[S11​S21​​S12​S22​​][a1​a2​​]★
${T_{11}} = \frac{1}{{{S_{21}}}}{\rm{ ; }}{T_{12}} = - \frac{{{S_{22}}}}{{{S_{21}}}}{\rm{ ; }}{T_{21}} = \frac{{{S_{11}}}}{{{S_{21}}}}{\rm{ ; }}{T_{22}} = \frac{{ - \left( {{S_{11}}{S_{22}} - {S_{12}}{S_{21}}} \right)}}{{{S_{21}}}} = \frac{{ - \Delta S}}{{{S_{21}}}}$T11​=S21​1​;T12​=−S21​S22​​;T21​=S21​S11​​;T22​=S21​−(S11​S22​−S12​S21​)​=S21​−ΔS​★
${S_{11}} = \frac{{{T_{21}}}}{{{T_{11}}}}{\rm{ ; }}{S_{12}} = - \frac{{\Delta T}}{{{T_{11}}}}{\rm{ ; }}{S_{21}} = \frac{1}{{{T_{11}}}}{\rm{ ; }}{S_{22}} = - \frac{{{T_{12}}}}{{{T_{11}}}}$S11​=T11​T21​​;S12​=−T11​ΔT​;S21​=T11​1​;S22​=−T11​T12​​★

4.5 信号流图

4.5.1 信号流图的分解

串联法则★

${V_3} = {S_{32}}{V_2} = {S_{32}}{S_{21}}{V_1}$V3​=S32​V2​=S32​S21​V1​

并联法则★

${V_2} = {S_a}{V_1} + {S_b}{V_1} = ({S_a} + {S_b}){V_1}$V2​=Sa​V1​+Sb​V1​=(Sa​+Sb​)V1​

自闭环法则★

${V_2} = {S_{21}}{V_1} + {S_{22}}{V_2}$V2​=S21​V1​+S22​V2​
${V_3} = {S_{32}}{V_2}$V3​=S32​V2​
${V_3} = \frac{{{S_{32}}{S_{21}}}}{{1 - {S_{22}}}}{V_1}$V3​=1−S22​S32​S21​​V1​

剖分法则★

${V_4} = {S_{42}}{V_2} = {S_{21}}{S_{42}}{V_1}$V4​=S42​V2​=S21​S42​V1​

作业

《射频电路设计——理论与应用》
第四章中习题 4.3，4.6，4.13，4.15，4.19