陕西理工学院毕业设计
where Zij is given in (8.99).
We can analyze the lter characteristics of this circuit by calculating the image imped- ance (which is the same at ports 1 and 3), and the propagation constant. From Table 8.1, the image impedance in terms of the impedance parameters is
2Z11Z13Zi?Z?Z33211 (8.101)
?1(Z0e?Z0o)2csc2θ?(Z0e?Z0o)2cot2θ21(Z0e?Z0o) (8.102) 2When the coupled line section is λ/4 long (θ = π/2), the image impedance reduces to
Zi?which is real and positive sinceZ0e?Z0o. However, when θ → 0 orπ, Zi →?j?, indicating a stopband. The real part of the image impedance is sketched in Figure 8.43, where the cutoff frequencies can be found from (8.101) as
cosθ1??cosθ2?Z0e?Z0o .
Z0e?Z0oThe propagation constant can also be calculated from the results of Table 8.1 as
cosβ?Z11Z33Z11Z0e?Z0o??cosθ (8.103) 2Z13Z13Z0e?Z0owhich shows β is real for θ1?θ?θ2?π?θ1, where cosθ1?(Z0e?Z0o)(Z0e?Z0o).
Design of Coupled Line Bandpass Filters
Narrowband bandpass lters can be made with cascaded coupled line sections of the form shown in Figure 8.42c. To derive the design equations for lters of this type, we first show that a single coupled line section can be approximately modeled by the equivalent circuit shown in Figure 8.44. We will do this by calculating the image impedance and propagation constant of the equivalent circuit and showing that they are approximately equal to those
FIGURE 8.44 Equivalent circuit of the coupled line section of Figure 8.42c.
of the coupled line section for θ = π/2, which will correspond to the center frequency of the bandpass response.
The ABCD parameters of the equivalent circuit can be computed using the ABCD matrices for transmission lines from Table 4.1:
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?cosθ?AB???CD???jsinθ???Z0
jZ0sinθ?0??cosθ????jJ?cosθ?jJ???jsinθ?0???Z0jZ0sinθ??cosθ?? (8.104)
?1cos2θ?22j(JZ0sinθ?)??(JZ0?JZ)sinθcosθJ0???11?j(sin2θ?Jcos2θ)(JZ0?)sinθcosθ?2??JZ0?JZ0?The ABCD parameters of the admittance inverter were obtained by considering it as a quarter-wave length of transmission of characteristic impedance, 1J. From (8.27) the image impedance of the equivalent circuit is
BZi??C2JZ0sin2θ?(1J)cos2θ (8.105) 222(1JZ0)sinθ?Jcosθwhich reduces to the following value at the center frequency, θ = π/2:
2 (8.106) Zi?JZ0From (8.31) the propagation constant is
cosβ?A?(JZ0?1)sinθcosθ (8.107) JZ0Equating the image impedances in (8.102) and (8.106), and the propagation constants of (8.103) and (8.107), yields the following equations:
12(Z0e?Z0o)?JZ0, 2
Z0e?Z0o1 ?JZ0?Z0e?Z0oJZ0Where we have assumed sinθ?1forθnear π/2. These equations can be solved for the even- and
odd-mode line impedances to give
Z0e?Z01?JZ0?(JZ0)2 (8.108a) Z0o?Z01?JZ0?(JZ0)2 (8.108b)
????
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附录B 外文文献翻译
8.7 耦合线滤波器
在7.6节中讨论过的平行耦合传输线（对于定向耦合器），也能用于构建多种类型的滤波器。要制作带宽小于20%的微带线型多节带通或带阻耦合线滤波器实际上是容易办到的。更宽的带宽滤波器通常需要很紧密的耦合线，这在制造上是很困难的。我们首先研究耽搁四分之一波长耦合线段的滤波器的特性，然后说明如何用这些耦合线段设计贷通滤波器[7]。用耦合线设计的其他滤波器可以在参考文献[1]中找到。 8.7.1 耦合线段的滤波器特性
图8.42（a）显示了平行耦合线段，且带有端口电压和电流的定义。考虑图8.42（b）所示的耦模和奇模激励的叠加，可对这个四端口网络推导出开路阻抗矩阵。所以。电流i1和i3驱动该线的耦模，而i2和i4驱动该线的奇模。通过叠加，我们看到总端口电流I可以用耦模和奇模电流表示为:
I1 = i1 + i2, (8.87a)
I2 = i1 - i2, (8.87b)
I3 = i3 - i4, (8.87c)
I4 = i3 + i4. (8.87d)
首先考虑用i1电流源在耦模下驱动此线。假如其他端口开路，在端口1或端口2看到的阻抗为：
e Zin=-jZ0ecotβl (8.88)
在这两个导体上的电压可表示为
11va(z)?vb(z)?Ve?[e_j?(z_l)?ej?(zl)_?2Vcos?(lz)?e_ (8.89)
所以在端口1或端口2的电压是
1+e v1a(0)=vb(0)=2Vecosβl=i1Zin
利用这个结果和式（8.88），可把式(8.89)改写成用i1，表示为 va(z)=vb(z)=jZ0e11_cosβ(l_z)i (8.90)
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陕西理工学院毕业设计
（a） (b)
（c）
图8.42 关于耦合线滤波器节的定义：
(a)用端口电压和电流定义的平行耦合线段；(b)用耦模和奇模电流定义的平行耦合线段；(c)有带通响应的二端口耦
合线段
同样，用电流源i3驱动的线上偶模电压是 v(z)=v(z)=jZ0e3a3b_cosβzi (8.91) sinβl3现在考虑当电流i2驱动线上奇模时的情形。若其他端开路，在端口1或端口2看到的阻抗是
0_ Zin=jZ0ocotβl (8.92)
在每个导体上的电压可以表示为
22 va(z)=_vb(z)=V0+[e_jβ(z_l)+ejβ(zl)=2V0+cosβ(l_z) (8.93)
_则在端口1和端口2的电压为
220 va (0)=_vb(0)=2V0+cosβl=i2Zin用这个结果和式(8.92)，可把式(8.93)改写成用i2表示为 v(z)=v(z)=jZ0o2a_2b_cosβ(l_z)i2 (8.94)
sinβl同样，由电流源i4驱动线上奇模的电压是 va(z)=vb(z)=jZ0o现在在端口1的总电压是
234V1=v1a(0)+va(0)+va(0)+va(0)4_4_cosβzi (8.95) sinβl4=_j(Z0ei1+Z0ei2)cotθ_j(Z0ei3+Z0ei4)cscθ (8.96)
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使用公式(8.90),(8.91),(8.94)和(8.95)中的结果,且θ=β。可得: i1 =
1(I＋I2), (8.97a) 211i2 =(I1－I2), (8.97b)
21i3 =(I3＋I4), (8.97c)
21i4 =(I4－I3), (8.97d)
2_将这些结果带入式(8.96)得:
V1=
j(Z0eI1+Z0eI2+Z0oI1_Z0oI2)cotθ2 (8.98) _j=(Z0eI3+Z0eI4+Z0oI4_Z0oI3)cscθ2该结果可给出描述耦合线段的开路阻抗矩阵[Z]的第一行。根据对称性，一旦第一行已知，则所有其他矩阵元都能求出。于是该矩阵元是
_ Z11=Z22=Z33=Z44=j(Z+Z0o)cotθ (8.99a) 20e_Z12=Z21=Z34=Z43=j(Z0e__Z0o)cotθ (8.99b) 2j(Z0e__Z0o)cscθ (8.99c) 2j(Z0e+Z0o)cscθ (8.99d) 2_Z13=Z31=Z24=Z42=_Z14=Z41=Z23=Z32=一个二端口网络可以由藕合线段形成，方法是把四个端口中的两个端口终端开路或短路；有十种可能的组合，如表8.8所示。正如在表中所指明的那样，各种电路有不同的频率响应，包括低通、带通、全通和全阻。对于带通滤波器，我们最感兴趣的是图8.42(c)所示的情况，因为在制作上开路比短路容易。在这种情况下，I2=I4=0,所以四端口阻抗矩阵方程式简化为 V1?Z11I1?Z13I3 (8.100a)
V3?Z31I1?Z33I3 (8.100b)
式中，Zij由式(8.99)给出。
我们能通过计算镜像阻抗(在端口1和端口3是相同的)和传播常数来分析该电路的滤波特性。从表8.1看到用z参量表示的镜像阻抗是
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