Lec12 Microwave Network Analysis （IV）

4.4 THE TRANSMISSION (ABCD) MATRIX Many microwave networks consist of a cascade connection of two or more two-port networks. In this case it is convenient to define a 2 × 2 transmission, or ABCD, matrix, for each two-port network. The ABCD matrix of the cascade connection of two or more two-port networks can be easily found by multiplying the ABCD matrices of the individual two-ports. The ABCD matrix is defined for a two-port network in terms of the total voltages and currents Fig. a two-port network or 由方程推导ABCD参数的定义。

In the cascade connection of two two-port networks then The ABCD matrix of the cascade connection of the two networks is equal to the product of the ABCD matrices representing the individual two-ports.

ABCD Parameters of Some Useful Two-Port Circuits 求解过程。

Relation to Impedance Matrix The impedance parameters of a network can be easily converted to ABCD parameters. Note: the direction of I2 in ABCD matrix is different from that in Z matrix. If the network is reciprocal, then Z12 = Z21 and AD-BC=1. If the network is symmetric , then Z11 = Z22 and A=D. If the network is lossless, A and D are real, and B and C are imaginary.

Equivalent Circuits for Two-Port Networks conversions between two-port network parameters

Equivalent circuits for a reciprocal two-port network. An arbitrary two-port network can be described in terms of impedance parameters as or in terms of admittance parameters as If the network is reciprocal, then Z12 = Z21 and Y12 = Y21. These representations lead naturally to the T and π equivalent circuits

Equivalent circuits for a reciprocal two-port network. (a) T equivalent. (b) π equivalent. If the network is reciprocal, there are six degrees of freedom (the real and imaginary parts of three matrix elements), so the equivalent circuit should have six independent parameters. A nonreciprocal network cannot be represented by a passive equivalent circuit using reciprocal elements. If the network is lossless, the impedance or admittance matrix elements are purely imaginary. This reduces the degrees of freedom for such a network to three, and the T and π equivalent circuits can be constructed from purely reactive elements.

A coax-to-microstrip transition and equivalent circuit representations A coax-to-microstrip transition and equivalent circuit representations. (a) Geometry of the transition. (b) Representation of the transition by a “black box.” (c) A possible equivalent circuit for the transition Because of the physical discontinuity in the transition from a coaxial line to a microstrip line, electric and/or magnetic energy can be stored in the vicinity of the junction, leading to reactive effects. Characterization of such effects can be obtained by measurement or by numerical analysis

如何选择网络的矩阵表示 [Z]=[Z1]+[Z2]+….+[ZN] 问题：N端口网络可以用阻抗矩阵[Z], 导纳矩阵[Y], 散射矩阵[S]表示 二端口网络还可以用转移矩阵[ABCD]表示。如何选择？ 1. 对于多个网络的串联，采用阻抗矩阵比较方便。 两个双口网络串联，每个口的电流不变，电压分压。故 所以 一般地，N端口串联网络有 两个串联双口网络 [Z]=[Z1]+[Z2]+….+[ZN]

[Y]=[Y1]+[Y2]+….+[YN] 2. 对于多端口网络的并联，采用导纳矩阵比较方便。 两个双口网络并联，每个口的电压不变，电流分流。故 两个并联双口网络 因此 一般地，N端口并联网络有 [Y]=[Y1]+[Y2]+….+[YN]

2. 对于二端口级联网络，采用ABCD矩阵。 网络的级联即对应的ABCD矩阵顺序相乘。 3. 对于微波网络，很难定义电压和电流，开路和短路，适合用散射矩阵。 微波测量，可以测量入射波和反射波的大小和相位，从而得到散射参数

归一化的电压、电流和阻抗 输入阻抗 对波导等色散传输线，由于不能单值定义特性阻抗。因此，虽然反射系数可以由测量唯一确定，但Z以及V、I的值仍不能单值地确定，即使选择波阻抗或等效阻抗来代替特性阻抗，但仍带来相当大的人为性（随意性）。 为克服该困难，采用圆图中的方法，即阻抗归一化：定义归一化阻抗为 对应于归一化阻抗下的等效电压、等效电流称为归一化等效电压和等效电流

满足 此外，归一化等效电压、等效电流和未归一化等效电压、等效电流满足 1）功率相等条件 2）阻抗条件 得归一化电压电流表达式

归一化参量不论对非色散传输线还是色散传输线都严格成立，使微波传输线可应用双线传输线理论及圆图进行计算。 注意，归一化电压电流已不再具有电路中原来电压、电流的意义，不具有电压和电流的量钢，而且它们的量纲相同（ ）。 入射电压转化为归一化入射波： 反射电压转化为归一化反射波： 归一化电压、电流满足： 归一化的特性阻抗

归一化电压电流为 从而，入射功率和反射功率分别为 总功率：

归一化的矩阵参数 N端口网络中，端口常接传输线，其各端口传输线的特性阻抗可能会不同，常常需要归一化去掉端口特性阻抗对网络矩阵的影响，即矩阵参数的归一化。 1. 散射矩阵 [S] 如果各个端口的特性阻抗相同，是否对端口电压归一化不影响散射参数的值。 事实上，很多书籍定义散射参数为归一化反射波与归一化入射波的关系。

2.阻抗矩阵 由于 则有 其中 归一化阻抗矩阵与未归一化阻抗矩阵间的关系为 其矩阵参数

3.导纳矩阵 由于 则有 其中 归一化阻抗矩阵与未归一化阻抗矩阵间的关系为 其矩阵参数

4.转移矩阵 ABCD矩阵定义的电压电流关系 其中 归一化abcd矩阵定义的归一化电压电流关系 其中

归一化参量间的转化

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