**The Smith Chart is a standard tool in RF design. Developed by Phillip Smith in 1939, it has become the most popular graphic method for representing impedances and solving operations with complex numbers. Traditionally, the Smith Chart has been used as 2-D polar form, centered at an unit radius circle. However, the 2D format has some restrictions when the active impedances (oscillators) or stability circles (amplifiers) are represented, since these ones usually leave the polar chart. Last years, three-dimensional Smith Chart has become popular. Advances in 3D rendering software make it easy to use for design. In this post, I will try to show the handling of the three-dimensional Smith Chart and its application for a low-noise amplifier design.**

When Phillip Smith was working at Bell Labs, he have to match one antenna and he was looked for a way to solve the design graphically. By means of the mathematical expressions that define the impedances in the transmission lines, he got to represent the impedance complex plane by circles with constant resistances and reactances. These circles made it easier for him to represent any impedance in a polar space, with the maximum matching placed in the center of the chart and the outer circle representing the pure reactance. Traditionally, Smith’s Chart has been represented in polar form as shown below

The impedance is normalized calculating the ratio between the impedance and the generator impedance. The center of the chart is pure unit resistance (maximum matching) while the peripheral circle that limits the chart is the pure reactance. The left end of the chart represents the pure short circuit and the right end, the pure open circuit. The chart was then very popular to be able to perform calculations for matching networks with transmission lines using a graphical method. However, the design difficulties with the chart happened when active impedances were analyzed, studying amplifiers stability and designing oscillators.

By its design, the chart is limited to the impedances with positive real part, but it could represent, extending the complex plane through the Möbius transformation, impedances with negative real part [1]. This expanded chart, to the negative real part plane, can be seen in the following figure

However,this chart shows two issues: 1) although it allows to represent all the impedances, there is a problem with the complex infinity, so it remains limited and 2) the chart has large dimensions that make it difficult to us in a graphic environment, even in a computer-aided environment. However, the extension is needed when the amplifier stability circles are analyzing, since in most of cases the centers of these circles are located outside the passive impedance chart.

In a graphical computer environment, representing the circles is already performed by the software itself through the calculations, being able to limit the chart to the passive region and drawing only a part of the circle of stability. But with oscillators still have the problem of complex infinity, which could be solved through a representation in a Riemann’s sphere.

**RIEMANN’S SPHERE**

The Riemann’s sphere is a mathematical solution for representing the complete complex plane, including infinity. The entire complex surface is represented on a spherical surface by a stereographic projection of this plane.

In this graphic form the southern hemisphere represents the origin, the northern hemisphere represents infinity and the equator the circle of unitary radius. The distribution of complex values in the sphere can be seen in the following figure

So, it is possible to represent any complex number on a surface easy to handle.

**SMITH’S CHART ON A RIEMANN’S SPHERE**

Since Smith’s Chart is a complex representation, it can be projected in the same way to a Riemann’s sphere [2], as shown in the following figure

In this case, the northern hemisphere shows the impedances with positive resistance (passive impedances), in the southern hemisphere, the impedances with negative resistance (active impedances), in the eastern hemisphere, the inductive impedances, and in the western one the capacitive impedances. The main meridian shows the pure resistive impedance.

Thus, when we wish to represent any impedance, either active or passive, it can be represented at any point in the sphere, greatly facilitating its drawing. In the same way, we can represent the stability circles of any amplifier without having to expand the chart. For example, if we want to represent the stability circles for one transistor, which parameters S at 3GHz are the next

S_{11}=0,82/-69,5 S_{21}=5,66/113,8 S_{12}=0,03/48,8 S_{22}=0,72/-37,6

its representation in the conventional Smith’s Chart is

while in the three-dimensional chart it is

where both circles can be seen, a fraction in the northern hemisphere and the other one in the south. Thus, its representation has been greatly facilitated.

**A PRACTICAL APPLICATION: LOW NOISE AMPLIFIER**

Let’s see a practical application of the 3D chart matching the previous amplifier with the maximum stable gain and minimum figure of noise, at 3GHz. Using traditional methods, and knowing the transistor parameters which are the next

S_{11}=0,82/-69,5 S_{21}=5,66/113,8 S_{12}=0,03/48,8 S_{22}=0,72/-37,6

NF_{min}=0,62 Γ_{opt}=0,5/67,5 R_{n}=0,2

S-parameters are represented in the3D Smith’s chart and the stability circles are drawn. For a better representation 3 frequencies are used, with a 500MHz bandwidth.

It can be seen that S-parameters as well as the stability circles in both the conventional Smith’s chart and 3D one. In the conventional Smith’s chart, the stability circles leave the chart.

One amplifier is unconditionally stable when the stability circles are placed in the active impedance area of the chart, in the southern hemisphere, under two conditions: if the circles are placed in the active region and do not surround the passive one, the unstable impedances are located inside the circle. If the circles surround the passive region, the unstable impedances are located outside the circle.

.

In this case, since part of the circles enters on the passive impedances region, the amplifier is conditionally stable.Then the impedances that could unstabilize the amplifier are placed inside the circles. This is something that cannot be seen clearly in the three-dimensional chart yet, the app does not seem to calculate it and would be interesting to include in later versions, because it would greatly facilitate the design.

Let’s match now the input for the minimum noise. For this, it is needed to design a matching network to transform from 50Ω to reflection coefficient Γ_{opt}, being its normalized impedance Z_{opt}=0,86+j⋅1,07. In the app, opening the design window and writing this impedance

Using now the admittance, we translate in the circle of constant conductance until the real part of the impedance is 1. This is down by estimation and a 0,5 subsceptance is got. It should be increased 0,5 – (- 0,57) = 1.07 and this is a shunt capacitor, 1,14pF.

Now it is only needed to put a component that makes zero the reactance, when the resistance is constant. As the reactance is -1.09, the added value should be 1.09, so that the reactance is zero. This is equivalent to a series inductor, 2,9nH.

Once calculated the input matching network for the lower noise figure, we recalculate the S-parameters. Being an active device, the matching network transforms the S parameters, which are:

S_{11}=0,54/-177 S_{21}=8,3/61,1 S_{12}=0,04/-3,9 S_{22}=0,72/-48,6

and which are represented in the Smith’s chart to get the stability circles.

The unstable regions are the internal regions, so the amplifier remains stable.

Now the output matching network is got for maximum stable gain, and the ouput reflection coefficient S_{22}=0,72/-48,6 should be loaded by Γ_{L} (S_{22} conjugate), translating from 50Ω to Γ_{L}=0,72/48,6. This operation is performed in the same way that input matching network. By doing the complete matching , S parameters are recalculated, with input and oputput matching networks. These are

S_{11}=0,83/145 S_{21}=12/-7.5 S_{12}=0,06/-72,5 S_{22}=0,005/162

The gain is 20·log(S_{21})=21,6dB, and the noise figure, 0,62dB (NF_{min}). Now it is only represented these parameters in the three-dimensional chart to get the stability circles.

In this case, the stable region in the **input stability circle **is inside and in the **otuput stabiliy circle **is outside. Due to both reflection coefficients, **S _{11}** y

**S**are into the stable regions, then the amplifier is stable.

_{22}**CONCLUSIONS**

In this entry I had the first contact with the three-dimensional Smith’s chart. The object was to study its potential with respect the traditional chart in microwave engineering. New advantages are observed in this respect in that it is possible to represent the infinite values from the Möbius transform to a Riemann’s sphere and thus having a three-dimensional graphical tool where practically all passive and active impedances and parameters which can be difficult to draw in the traditional chart as stability circles.

In its version 1, the app, which can be found on the website 3D Smith Chart / A New Vision in Microwave Analysis and Design, shows some design options and configurations, although some applications should be undoubtedly added In future versions. In this case, one of the most advantageous applications for the chart, having studied the stability circles of an amplifier, is the location of the stability regions graphically. Although this can be solved by calculation, the visual image is always more advantageous.

The app has a user manual with examples explained in a simple way, so that the designer becomes familiar with it immediately. In my professional opinion, it is an ideal tool for those of us who are used to using Smith’s chart to perform our matching network calculations.

**REFERENCIAS**

- Müller, Andrei; Dascalu, Dan C; Soto, Pablo; Boria, Vicente E.;
*” The 3D Smith Chart and Its Practical Applications”*; Microwave Journal, vol. 5, no. 7, pp. 64–74, Jul. 2012 - Zelley, Chris;
*“A spherical representation of the Smith Chart”;*IEEE Microwave, vol. 8, pp. 60–66, July 2007 - Grebennikov, Andrei; Kumar, Narendra; Yarman, Binboga S.;
*“Broadband RF and Microwave Amplifiers”*; Boca Raton: CRC Press, 2016; ISBN 978-1-1388-0020-5