Development and Investigation of Microstrip Directional Coupler with Phase Velocity Compensation Based on Sawtooth Configuration of Coupled Lines
基于耦合线锯齿结构的相速度补偿微带定向耦合器的研制与研究

Index Terms - Directional coupler, microstrip line, coupled lines, phase velocity.
索引术语 - 定向耦合器、微带线、耦合线、相速度。

THE DESIGN THEORY of couplers based on transmission lines with T-type wave is founded on the assumption that phase velocities are equal in symmetric and asymmetric excitation modes. However, inequality of dielectric permittivity values results in phase velocity difference, which causes a significant decrease in directionality and reduction of an operating frequency band.
基于T型波传输线的耦合器的设计理论是建立在对称和非对称激励模式下相速度相等的假设之上的。然而，介电常数值的不平等导致相速度差，这导致方向性显着下降和工作频带减小。

There is a lot of ways to compensate phase velocity difference. All methods may be conventionally divided into three large groups. The first group includes the methods based on equalization of effective dielectric permittivity in different excitation modes. This may be achieved through placement of dielectric layers above the microstrip structures [1] using anisotropic substrate [2] or making engraving in a substrate or on screened surface [3]. The main disadvantage of these methods is the complex manufacturing process that deprives microstrip structure of their advantage - ease of fabrication. In addition to this, there are not handy equations for calculation of parameters for such compensation. The design procedure is reduced to the multi-iterative numerical synthesis.
补偿相速度差的方法有很多种。所有方法通常可以分为三大组。第一组包括基于不同激励模式下有效介电常数均衡的方法。这可以通过使用各向异性基板 [2] 在微带结构 [1] 上方放置介电层或在基板或屏蔽表面 [3] 上进行雕刻来实现。这些方法的主要缺点是制造工艺复杂，从而剥夺了微带结构的优点——易于制造。除此之外，没有方便的方程来计算这种补偿的参数。设计过程简化为多次迭代数值综合。

The second group of compensation methods includes connection of reactive concentrated components [4], shunting of lines [5] or connection of components in series with one or several ports [6], [7]. In several cases such compensation methods allow achieving directionality of more than 70 dB , but within a very narrow frequency band [7]. Such compensation is characterized by narrow band. The capacitive compensation method should be put into a separate
第二组补偿方法包括无功集中元件连接[4]、线路分流[5]或元件与一个或多个端口串联[6]、[7]。在某些情况下，此类补偿方法可以实现超过 70 dB 的方向性，但频带范围非常窄 [7]。这种补偿的特点是频带窄。电容补偿方法应单独放入
category. It allows achieving high directionality in a broad frequency band. Two identical capacitances are included between coupled lines at their boundaries. These exterior capacitances increase length of equivalent electric transmission line in the asymmetric excitation mode and almost does not have any effect on wave propagation in the symmetric excitation mode. However, it is not always possible to ensure adequate field homogeneity in points of connection between capacitance and a microstrip. This, in combination with complexity in fabrication of the concentrated capacitance operating at high frequencies, restricts application of this compensation method at frequencies above several GHz .
类别。它允许在宽频带内实现高方向性。耦合线之间在其边界处包括两个相同的电容。这些外部电容增加了非对称激励模式下等效电传输线的长度，并且对对称激励模式下的波传播几乎没有任何影响。然而，并不总是能够确保电容和微带之间的连接点具有足够的场均匀性。这与在高频下工作的集中电容的制造的复杂性相结合，限制了该补偿方法在高于几GHz的频率下的应用。
The third group of compensation methods gives the better results that capacitive compensation and does not impose any limitations on frequency increase. Such technologies of full planar compensation proposed by A. Podell [8] are based on physical increase in wave path length in the asymmetric excitation mode, which increases the equivalent dielectric permittivity.
第三组补偿方法给出了比电容补偿更好的结果，并且对频率增加没有任何限制。 A. Podell [8] 提出的这种全平面补偿技术基于非对称激励模式下波程长度的物理增加，从而增加了等效介电常数。

Analysis of couplers on coupled lines is convenient to be conducted on the basis of symmetric and asymmetric excitation modes [9] using the plane of symmetry located between the primary and the second transmission lines. At symmetric excitation actuation signals of equal values and phases are transmitted to line inputs; at asymmetric excitation phases of actuation signals are opposite. In this case partial four-pole networks of symmetric and asymmetric excitation take the form of regular transmission lines with T-wave. Lengths of these line segments $l$$l$ll coincide with length of communication section. Wave impedances of such four-pole networks $Z_0^{+}$$Z_0^{+}$Z_(0)^(+)Z_{0}^{+}and $Z_0^{-}$$Z_0^{-}$Z_(0)^(-)Z_{0}^{-}(symmetric excitation, index + ; asymmetric excitation, index - ) depend on boundary condition ( $E=0$$E=0$E=0E=0 or $H=0$$H=0$H=0H=0 ) in the interface region. Fig. 1 shows the distribution of the electrical field at different excitation modes.
使用位于主传输线和第二传输线之间的对称面，可以方便地基于对称和非对称激励模式[9]对耦合线上的耦合器进行分析。在对称激励下，相等值和相位的驱动信号被传输到线路输入；在不对称激励下，驱动信号的相位相反。在这种情况下，对称和不对称激励的部分四极网络采用具有 T 波的规则传输线的形式。这些线段的长度 $l$$l$ll 与通信部分的长度一致。这种四极网络的波阻抗 $Z_0^{+}$$Z_0^{+}$Z_(0)^(+)Z_{0}^{+} 和 $Z_0^{-}$$Z_0^{-}$Z_(0)^(-)Z_{0}^{-} （对称激励，指数 + ；非对称激励，指数 - ）取决于边界条件 ( $E=0$$E=0$E=0E=0 或者 $H=0$$H=0$H=0H=0 ）在界面区域。图1显示了不同激励模式下的电场分布。
It follows from comparison of electrical field structures that in the asymmetric excitation mode the electrical field
通过电场结构的比较可以看出，在非对称激励模式下，电场
located in line edges is mainly concentrated in the air region unlike the symmetric mode, where the field is mainly concentrated in the dielectric substrate under microstrip lines. As a result, effective dielectric permittivity is higher in the latter case.
与对称模式不同，位于线路边缘的场主要集中在空气区域，对称模式中的场主要集中在微带线下方的介质基板中。结果，在后一种情况下，有效介电常数更高。

Fig. 1 - Distribution of the electrical field at different excitation modes.
图 1 - 不同激励模式下的电场分布。

Phase velocity in the dielectric medium without losses is determined by the following expression [10]:
无损耗电介质中的相速度由以下表达式确定[10]：

Where ${\epsilon }_0$${\epsilon }_0$epsi_(0)\varepsilon_{0} is dielectric permittivity of vacuum, $\mathcal{E}$$\mathcal{E}$E\mathcal{E} is relative dielectric permittivity, ${\mu }_0$${\mu }_0$mu_(0)\mu_{0} is magnetic constant, $\mu$$\mu$mu\mu is relative magnetic permittivity.
在哪里 ${\epsilon }_0$${\epsilon }_0$epsi_(0)\varepsilon_{0} 是真空的介电常数， $\mathcal{E}$$\mathcal{E}$E\mathcal{E} 是相对介电常数， ${\mu }_0$${\mu }_0$mu_(0)\mu_{0} 是磁常数， $\mu$$\mu$mu\mu 是相对磁介电常数。

Fig. 2 shows the geometry of the directional coupler with sawtooth compensation of phase velocities in different excitation modes [11].
图 2 显示了不同激励模式下具有锯齿波相速度补偿的定向耦合器的几何结构 [11]。

Fig. 2 Geometry of saw-tooth compensation.
图 2 锯齿波补偿的几何结构。
The sawtooth geometry of coupled lines may be considered as connection of additional distributed capacitance between coupled lines.
耦合线的锯齿几何形状可以被认为是耦合线之间的附加分布电容的连接。

Asymmetric excitation  不对称激发
Fig. 3. Static capacity parameters of coupled lines in different excitation modes.
图3 不同励磁方式下耦合线路的静态电容参数。

Fig. 3 represents the per-unit-length capacitance in the form of flat capacitor capacitance ( $Cp$$Cp$CpC p ) and two edge capacitances from each side $\left(C_f\right)$$\left(C_f\right)$(C_(f))\left(C_{f}\right) of a strip. Capacitances in symmetric and asymmetric excitation modes may be expressed as functions of ratios $\mathrm{w}/\mathrm{h}$$\mathrm{w}/\mathrm{h}$w//h\mathrm{w} / \mathrm{h} and $\mathrm{s}/\mathrm{h}$$\mathrm{s}/\mathrm{h}$s//h\mathrm{s} / \mathrm{h} and effective
图3表示扁平电容器电容形式的单位长度电容（ $Cp$$Cp$CpC p ）和每侧的两个边缘电容 $\left(C_f\right)$$\left(C_f\right)$(C_(f))\left(C_{f}\right) 条带的。对称和非对称激励模式下的电容可以表示为比率的函数 $\mathrm{w}/\mathrm{h}$$\mathrm{w}/\mathrm{h}$w//h\mathrm{w} / \mathrm{h} 和 $\mathrm{s}/\mathrm{h}$$\mathrm{s}/\mathrm{h}$s//h\mathrm{s} / \mathrm{h} 和有效的
dielectric constant $\left({\epsilon }_{eff}\right)$$\left({\epsilon }_{eff}\right)$(epsi_(eff))\left(\varepsilon_{e f f}\right) [10], [12]. Per-unit-length capacitance for symmetric mode may be written as follows:
介电常数 $\left({\epsilon }_{eff}\right)$$\left({\epsilon }_{eff}\right)$(epsi_(eff))\left(\varepsilon_{e f f}\right) [10]，[12]。对称模式的单位长度电容可写如下：

where $c_{fp}=c_f+c_p;c_f^{+}$$c_{fp}=c_f+c_p;c_f^{+}$c_(fp)=c_(f)+c_(p);quadc_(f)^(+)c_{f p}=c_{f}+c_{p} ; \quad c_{f}^{+}is edge per-unit-length capacitance of inner strip boundaries in the symmetric mode.
在哪里 $c_{fp}=c_f+c_p;c_f^{+}$$c_{fp}=c_f+c_p;c_f^{+}$c_(fp)=c_(f)+c_(p);quadc_(f)^(+)c_{f p}=c_{f}+c_{p} ; \quad c_{f}^{+} 是对称模式下带内边界的边缘单位长度电容。
For asymmetric excitation:
对于不对称激励：

where:  在哪里：

where $c_{fa}^{-}$$c_{fa}^{-}$c_(fa)^(-)c_{f a}^{-}is a component determined by boundary field in a slot at asymmetric excitation, $c_{fd}^{-}$$c_{fd}^{-}$c_(fd)^(-)c_{f d}^{-}is per-unitlength capacitance determined by the boundary field in a slot inside the dielectric at asymmetric excitation.
在哪里 $c_{fa}^{-}$$c_{fa}^{-}$c_(fa)^(-)c_{f a}^{-} 是由不对称激励下槽中的边界场确定的分量， $c_{fd}^{-}$$c_{fd}^{-}$c_(fd)^(-)c_{f d}^{-} 是由不对称激励下电介质内部槽中的边界场决定的单位长度电容。
As has been previously said, phase velocity in the asymmetric mode may be reduced, until achieving the equality to phase velocity in the symmetric mode, through implementation of sawtooth geometry on interior edges, as shown by Fig. 2.
如前所述，通过在内边缘上实现锯齿几何形状，可以降低非对称模式下的相速度，直到达到与对称模式下的相速度相等，如图2所示。
In the asymmetric mode without compensation per-unit-length capacitance is determined by expression (3). In case of compensation it can be written as:
在不带补偿的非对称模式下，单位长度电容由表达式（3）确定。如果是补偿的话，可以写成：

where $c_{fk}^{-}$$c_{fk}^{-}$c_(fk)^(-)c_{f k}^{-}is edge per-unit-length capacitance in case of compensation.
在哪里 $c_{fk}^{-}$$c_{fk}^{-}$c_(fk)^(-)c_{f k}^{-} 是补偿情况下的边缘单位长度电容。
Let us assume that implementation of the sawtooth structure has the effect only on capacitance between conductors. We must ensure the following equation for equalization of velocities in even and uneven mode:
让我们假设锯齿结构的实现仅对导体之间的电容有影响。我们必须确保均匀模式和非均匀模式下的速度均衡符合以下方程：

where ${\epsilon }^{+}$${\epsilon }^{+}$epsi^(+)\varepsilon^{+}is effective dielectric permittivity in the symmetric mode, ${\epsilon }_k^{-}$${\epsilon }_k^{-}$epsi_(k)^(-)\varepsilon_{k}^{-}is effective dielectric permittivity in the asymmetric mode in case of compensation.
在哪里 ${\epsilon }^{+}$${\epsilon }^{+}$epsi^(+)\varepsilon^{+} 是对称模式下的有效介电常数， ${\epsilon }_k^{-}$${\epsilon }_k^{-}$epsi_(k)^(-)\varepsilon_{k}^{-} 是补偿情况下非对称模式下的有效介电常数。
For achieving this equation capacitance in the uneven mode must be increased by ${\epsilon }^{+}/{\epsilon }_k^{-}$${\epsilon }^{+}/{\epsilon }_k^{-}$epsi^(+)//epsi_(k)^(-)\varepsilon^{+} / \varepsilon_{k}^{-}. By multiplying (3) by this coefficient and equating it to (4), we have the solution for $c_{fk}^{-}$$c_{fk}^{-}$c_(fk)^(-)c_{f k}^{-}:
为了实现这个方程，非均匀模式下的电容必须增加 ${\epsilon }^{+}/{\epsilon }_k^{-}$${\epsilon }^{+}/{\epsilon }_k^{-}$epsi^(+)//epsi_(k)^(-)\varepsilon^{+} / \varepsilon_{k}^{-} 。通过将（3）乘以该系数并使其等于（4），我们得到了解 $c_{fk}^{-}$$c_{fk}^{-}$c_(fk)^(-)c_{f k}^{-} :

Subtract the edge per-unit-length capacitance of noncompensated partial four-pole network $c_f^{-}$$c_f^{-}$c_(f)^(-)c_{f}^{-}(3) from the calculated capacitance $c_{fk}^{-}$$c_{fk}^{-}$c_(fk)^(-)c_{f k}^{-}to find the additional capacitance:
减去非补偿部分四极网络的边缘单位长度电容 $c_f^{-}$$c_f^{-}$c_(f)^(-)c_{f}^{-} (3)由计算电容 $c_{fk}^{-}$$c_{fk}^{-}$c_(fk)^(-)c_{f k}^{-} 找出附加电容：

The calculated additional capacitance may be used for compensation by the capacitance method described earlier.
计算出的附加电容可以用于通过前面描述的电容方法进行补偿。

For compensation by the sawtooth geometry method path length in the asymmetric mode must be increased by the multiplier $c_{fk}^{-}/c_f^{-}$$c_{fk}^{-}/c_f^{-}$c_(fk)^(-)//c_(f)^(-)c_{f k}^{-} / c_{f}^{-}.
为了通过锯齿几何方法进行补偿，非对称模式下的路径长度必须增加乘数 $c_{fk}^{-}/c_f^{-}$$c_{fk}^{-}/c_f^{-}$c_(fk)^(-)//c_(f)^(-)c_{f k}^{-} / c_{f}^{-} 。

It follows from the analysis of the geometry (Fig. 2) length of teeth may be determined as follows:
从几何形状（图2）分析可知，齿的长度可按下式确定：

Then height of a tooth is equal to:
那么牙齿的高度等于：

The expressions obtained for calculation of the compensation geometry are based on the assumption that the sawtooth structure has an effect only on edge per-unitlength capacitance in the asymmetric mode $c_f^{-}$$c_f^{-}$c_(f)^(-)c_{f}^{-}. However, this is not true. The sawtooth structure increases edge per-unit-length capacitance in the symmetric mode $c_f^{+}$$c_f^{+}$c_(f)^(+)c_{f}^{+}. This becomes especially evident at large distances between coupled lines $s$$s$ss [11]. This may be corrected with additional reduction of length in the asymmetric excitation mode at keeping the same tooth pitch $\mathrm{\Delta }l$$\mathrm{\Delta }l$Delta l\Delta l (see Fig. 2). The communication length will be determined as follows:
补偿几何计算所获得的表达式基于锯齿结构仅对非对称模式下的边缘单位长度电容有影响的假设 $c_f^{-}$$c_f^{-}$c_(f)^(-)c_{f}^{-} 。然而，事实并非如此。锯齿结构增加了对称模式下的边缘单位长度电容 $c_f^{+}$$c_f^{+}$c_(f)^(+)c_{f}^{+} 。这在耦合线之间的距离较长时变得尤其明显 $s$$s$ss [11]。这可以通过在保持相同齿距的情况下在不对称激励模式下额外减小长度来纠正 $\mathrm{\Delta }l$$\mathrm{\Delta }l$Delta l\Delta l （见图2）。通信长度将按下式确定：

where:  在哪里：

where $c_{fk}^{+}$$c_{fk}^{+}$c_(fk)^(+)c_{f k}^{+}edge per-unit-length capacitance in symmetric mode after compensation, ${\lambda }_k^{-}$${\lambda }_k^{-}$lambda_(k)^(-)\lambda_{k}^{-}is wavelength in the symmetric mode and ${\lambda }_k^{-}$${\lambda }_k^{-}$lambda_(k)^(-)\lambda_{k}^{-}is wavelength in the asymmetric mode in case of compensation.
在哪里 $c_{fk}^{+}$$c_{fk}^{+}$c_(fk)^(+)c_{f k}^{+} 补偿后对称模式的边缘单位长度电容， ${\lambda }_k^{-}$${\lambda }_k^{-}$lambda_(k)^(-)\lambda_{k}^{-} 是对称模式下的波长， ${\lambda }_k^{-}$${\lambda }_k^{-}$lambda_(k)^(-)\lambda_{k}^{-} 是补偿情况下非对称模式的波长。

The directional coupler was designed and manufactured on the basis of the theoretical calculation. It is shown in Fig. 4 and Fig. 5.
在理论计算的基础上设计并制作了定向耦合器。如图4、图5所示。

Fig. 4. Coupler topology  图 4. 耦合器拓扑

The coupler is manufactured on substrate with thickness of 1 mm with relative permittivity 2.8 . This coupler is used as measurement sensor in systems for measurement of complex reflection coefficient [13], [14]. Ports of the secondary line of the coupler were matched with microcircuit terminals of the log amplifier. This resulted in certain deterioration of coupler parameters.
耦合器在厚度为1 mm、相对介电常数为2.8 的基板上制造。该耦合器在复反射系数测量系统中用作测量传感器[13]、[14]。耦合器的副线端口与对数放大器的微电路端子相匹配。这导致耦合器参数的某些恶化。

Fig. 1. Directional coupler
图 1. 定向耦合器
Fig. 6 shows the results obtained from research of the directional coupler in AWR microwave Office.
图6所示为AWR微波办公室定向耦合器的研究结果。

Fig. 6. Scattering parameters of coupler (dB).
图 6. 耦合器的散射参数（dB）。
The developed coupler at frequency band from 300 MHz to 2.5 GHz has directionality of more than 30 dB . Matching level is not less than 40 dB .
所开发的耦合器在300 MHz至2.5 GHz频段具有超过30 dB的方向性。匹配电平不低于40dB。

The technology for compensation of phase velocity differences in symmetric and asymmetric modes in microstrip coupled lines based on implementation of sawtooth geometry allows achieving high directionality in a wide frequency band, without using additional concentrated components. This will significantly simplify design and fabrication of directional couplers.
基于锯齿几何实现的微带耦合线对称和非对称模式相速度差补偿技术允许在宽频带内实现高方向性，而无需使用额外的集中元件。这将显着简化定向耦合器的设计和制造。

[1] C. S. Kim, Y. T. Kim, S. C. S. Kim, Y. T. Kim, S. H. Song,W. S. Jung, K. Y. Kang, J. S. Park, and D. Ahn, “A design of microstrip directional coupler for high directivity and tight coupling,” Eur. Gallium Arsenide and Other Semiconduct. Applicat. Symp., pp. 126-129, Sep. 2001.
[1] CS Kim、YT Kim、SCS Kim、YT Kim、SH Song、WS Jung、KY Kang、JS Park 和 D. Ahn，“一种用于高方向性和紧耦合的微带定向耦合器设计”，Eur。砷化镓和其他半导体。申请。 Symp.，第 126-129 页，2001 年 9 月。
[2] Kobayashi, M. and Terakado, R., “Method for Equalizing Phase Velocities of Coupled Microstrip Lines by Using Anisotropic Substrate,” IEEE Trans. Microwave Theory and Tech., vol. 28, pp. 719 - 722, Jul. 1980.
[2] Kobayashi, M. 和 Terakado, R.，“使用各向异性基板均衡耦合微带线相速度的方法”，IEEE Trans。微波理论与技术，卷。 28，第 719 - 722 页，1980 年 7 月。