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## Using the Three-Dimensional Smith Chart

**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

## A 900Mhz Feedfordward Amplifier with MOSFET

*In this article, we are going to demonstrate a 900Mhz feedforward amplifier design. Feedforward is a linearization technique for the IM distortion, caused by the nonlinear feature of the active device. Lateral interference, generated by the IM distortion on both sides of the main frequency, affecting the Adjacent Channel. Decreasing this interference is the purpose of this entry*.

We are going start with * a LDMOS amplifier*, tuned to 900Mhz. The active device is a STMicroelectronics’ MOSFET, PD84001, It

operates at 8 V in common source mode at frequencies of up to 1 GHz. P

_{OUT}is 31dBm (I

_{DQ}=50mA) and Drain Efficiency, 60%. Once we have designed the amplifier, we’ll make the linearization of the IM products, using the feedforward technique.

**THE MOSFET AMPLIFIER**

The amplifier is designed in common source mode. The Operating Point is chosen as the optimal features of the manufacturer: V_{DS}=8V, I_{DQ}=50mA . At this OP, and at 900MHz, Z_{IN}=3,6+j·4,3Ω and Z_{OUT}= 3,9 + j·5,5Ω. Maximum power transfer is obtained with a conjugate matching network at the generator and load impedances, which are Z_{0}=50. Once the matching networks are calculated, the purposed schema for the amplifier is

Amplifier’s gain is * 34,3dB*, and its phase is

*. Input and Output Return Losses are respectively 30,7 and 39,8dB. Then, the amplifier is matched and the maximum power at 900MHz is 27dBm, for 1-Tone.*

**76,4deg**For a 2-Tone input, the IM distortion generates a power drop, caused by the Third Order Distortion. TOI (*Third Order Intercept*) is * 31,7dBm*, near of the maximum output power of the datasheet, and it causes the power drop.

Intermodulation products are * 12dB* below the carrier, and this value may cause interference on the Adjacent Channel. Therefore, we must reduce this value as much as possible, using a linearization technique.

There are many linearization techniques, but we are going to use the * feedforward technique*, because it is a technique that requires only the use of RF networks.

The amplifier gain, including the second and third order distortions, could be expressed by

Where the input signal P_{i} is a 2-Tone signal. In this case, * we will not take into consideration the second order distortion*, since the P

_{i }frequencies will be very close together.

**A bandpass filter could remove**the second order spurious.

The amplifier gain is complex, The coefficients **g _{1}** and

**g**could be expressed using the polar notation (in

_{3}*). Then, these are*

**mag/phase**These coefficients * are going to use to calculate the phase shifter of the first stage*. Now, we shall describe shortly the feedforward technique.

**FEEDFORWARD PRINCIPLE**

The Feedfordward Principle is based on * reducing the distortion by mixing in phase opposition with the same distortion*. In a RF amplifier, an output distorted signal is generated

*. It could be mixed with the input signal in phase opposition, adjusting the levels of both signals. So, we get the distorted signal on one port, and on the other port, only the distortion spurious.*

**due to the active device’s nonlinearity**Cancellation of the main signals on the second port is achieved * by placing a delay line* (

*), in the secondary network of the first stage. One sample of the signal output of the amplifier (*

**τ1***) is derived to combine with the secondary network, with a combiner. The levels of both signals are equalized by an inter-stage attenuator (*

**G1***). Then, both signals are combined. Then, the ouput signal of the amplifier is called*

**β***, and the combined signal,*

**MAIN***.*

**AUX**

** AUX** is now used as an error signal in the second stage, which

*(*

**is***mplified by an error amplifier***a***), while the*

**G2***(*

**MAIN is delayed with another delay line***). In this second stage, we want to get the same effect than the first stage: put both signals in phase opposition, and combine them. Then, the distortion is cancelled and reduced the interference on the Adjacent Channel.*

**τ2**The level at the output of the amplifier could be written as

and * P_{AUX1}* (the sample level before the error combiner) could be expressed by

The level * P_{AUX2}* at the secondary network is

In these expressions, * β* is the magnitude of the losses of the inter-stage attenuator and

*is its phase; and*

**θ**_{β}*is the phase of the delay line*

**θ**_{A2}*. It must be satisfied*

**τ1*** θ_{1}* is the phase of the linear gain of the amplifier. Then,

*, but also*

**not only the phases must be in phase opposition***. In magnitude, it must be satisfied*

**the delay time must be the same in every subnetworks****|**.

*β*·g_{1}|=1In the second stage, the gain of the amplifier must equalize the **g**** _{2}** and

**g**levels, and their phases must satisfy the same equations (absolute phase and delay time) of the first stage, to combine and cancel the distortions.

_{3}In RF designs, the adders must be replace by hybrid couplers or directional couplers, which have insertion and coupler losses. Using two hybrid couplers (3dB for insertion losses) to split * P_{i}* and combine

*and*

**P**_{AUX1}*, and two directional couplers (with*

**P**_{AUX2}*for coupler losses) to take the sample in the first stage and combine the error sample in the second stage, the expressions are now*

**C**at the first stage and

at the second stage.

In a narrowband amplifier, * delay time could not be considered*, because

*.*

**its phase slope will be smallest than the phase slope of a broadband amplifier****900MHz FEEDFORDWARD AMPLIFIER**

Now, we are going to design our feedforward amplifier, based in our two-stage LDMOS amplifier. In first, we must split the input signal in two outputs, one to the amplifier and the other to the phase shifter. We are going to use an * 180-deg hybrid coupler*, with 3dB of insertion losses. At this frequencies, hybrid couplers could be easily found in the market, as a

**S***(*

**urface Mounting Device****).**

*SMD**.*

**The designed amplifier is a narrowband amplifier**The output levels of the hybrid coupler are the same, in magnitude and phase. The phase of the amplifier gain was * 76,4deg* in linear mode, but in nonlinear mode, we have got a phase of

*, with 0dBm of input power. Taking a sample of the output level of the amplifier with a directional coupler, which introduces a*

**69,4deg***coupling phase, with 10dB of coupling level, we have got a sample level of 12dBm, with a phase of*

**90deg***.*

**159,4deg**Then, we are going to * combine with another hybrid coupler*, and as in the secondary network the level is -6dBm, we have to equalize both levels with the attenuator, whose attenuation must be ≈20dB. The phase shifter should be adjusted to a phase of ≈-12 deg.

Adjusting the phase and the level with the phase shifter and the attenuator, we are able to optimize the response for several input levels.

We are going to complete now the second stage amplifier, where an error amplifier increases the level of * AUX* spurious intermodulation to combine in phase opposition with the

*line. The error amplifier*

**MAIN***should not be a power amplifier, at this stage. A linear, general-purpose amplifier maybe used. The gain is calculated by the difference between the*

**G2***and*

**MAIN****IM spurious. This value is 45,5dB, because we are**

*AUX**in the*

**combining with a directional coupler, to reduce the insertion losses***line. Using an amplifier with a magnitude of*

**MAIN***and a phase of*

**45,5dB***, we have got a phase shifter with the same value, and after the coupler, the IM distortion decreases around*

**-145deg***. The output level is now*

**65dB***, and the TOI*

**31dBm***.*

**increases to 75dBm**The definitive amplifier is

**CONCLUSIONS**

With the amplifier designed we have achieved a significant improvement: increasing efficiency around * 40dB* for the same output level, on adjacent channel. Furthermore, the amplifier is very simple to realize

*. The design is*

**with a few RF devices***.*

**very easy and intuitive**However, the Feedforward has two serious disadvantages: on the PCB, * it needs a lot of surface*; and the input level

*above the input level that provides maximum output level of the MOSFET, because the distortion can be increased above the value we have corrected.*

**cannot be increased**In broadband we must take into consideration * not only the phase of the amplifiers but also the group delay*, because the phase slope of the amplifiers has to be compensated by the phase shifter. Then,

*, because it must be a delay line, too.*

**the phase shifter could have a larger surface dimensions****References**

- R. Cordell, “
*A MOSFET Power Amplifier with Error Correction*”; JAES, vol. 32, nr. 1/2, 1984 Jan/Feb - J. Vanderkooy, S.P. Lipshitz, “
*Feed-Forward Error Correction in Power Amplifiers*”, JAES, Vol. 28, Nr. 1/2, 1980 Feb - A.M. Sandman, “
*Reducing Distortion by ‘Error add-on*‘”, Wireless World, vol.79, p.32, 1974 Oct