通信工程文献翻译——3D史密斯圆图的数学理论在黎曼球表面上实现了有源和无源微波电路设计。反射系数平面通过南极在单位球面的表面上进行立体映射。结果，包括被动负载的经典2D史密斯圆图被立体地映射到北半球，而具有负电阻（在经典平面史密斯圆图之外）的电路被映射到南半球。东半球是感应电路的地方，而西半球容纳电容电路。同时，格林威治子午线是纯电阻电路的轨迹（见图1，其中恒定电阻r和电抗x圆分别绘制为蓝色和红色）。

图1 2D（a）和3D（b）史密斯圆图的表示

3D史密斯圆图与先前尝试以基本方式推广平面2D史密斯圆图不同：无量纲的处理方式。前面的理论不能以简单和严格的方式合并主动和被动世界，因为他们提出了一个将无限区域映射到有限表面的经验解。这些方法变成复杂的变换微波问题，使得视觉和直观解释非常困难。

在本文中，首先描述了2D和3D史密斯图是如何处理无限区域的。接下来，使用3D史密斯圆图来表示有源和无源负载的优点通过两个示例来说明：放大器的稳定性圆和微波振荡器的阻抗。

史密斯圆图中的无限处理

归一化阻抗或z平面的无穷大区域是由于开路，因为输入阻抗的大小随着负载趋向于开路而增加到无穷大。由于在实际应用中，可以在靠近开路处找到阻抗，因此使用z平面来作负载的视觉表示是相当不常见的。

对于反射系数平面（plane），无穷区域对应于与线路的特征阻抗具有相同幅度和相反符号的负载（即无限不匹配或归一化阻抗......

The mathematical theory of the 3D Smith 

chart1 unifies active and passive microwave circuit design on the surface of a Riemann sphere. The reflection coefficient plane is mapped stereographically through the South Pole on the surface of a unit sphere. As a result, the classical 2D Smith chart including the passive loads2 is mapped stereographically into the North hemisphere, while the circuits with negative resistance (that are outside the classical planar Smith chart) are mapped into the South one. The East hemisphere is the place of inductive circuits, whereas the West hemisphere hosts the capacitive circuits. 

Meantime, the Greenwich meridian is the locus of pure resistive circuits (see Figure 1, where the constant resistance r and reactance x circles are drawn in blue and red, respectively).

The 3D Smith chart differs from previous attempts4 to generalize the planar 2D Smith chart in a fundamental way: the way in which infinity is treated. The preceding theories fail to merge the ctive and passive worlds in a simple and rigorous manner, since they propose an empirical solution to map an in-finite region into a finite surface. These approaches turned into complicated transforming equations, making the visual and intuitive interpretation of microwave problems very difficult.

In this article, how the 2D and 3D Smith charts deal with the infinite regions are first described. Next, the advantages of using the 3D Smith chart to represent both active and passive loads are illustrated with two examples: the stability circles of an amplifier and the impedance of a microwave......