# 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

Fig. 1 – Traditional Smith’s Chart

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

Fig. 2- Smith’s Chart expanded to active impedances

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.

Fig. 3 – Projection of the complex plane on a sphere

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

Fig. 4 – Distribution of complex values in the sphere

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

Fig. 5 – Projection of the Smith’s Chart on a Riemann’s sphere

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

S11=0,82/-69,5   S21=5,66/113,8   S12=0,03/48,8  S22=0,72/-37,6

its representation in the conventional Smith’s Chart is

Fig. 6 – Traditional representation for stability circles

while in the three-dimensional chart it is

Fig. 7 – Stability circles on the 3D chart

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

S11=0,82/-69,5   S21=5,66/113,8   S12=0,03/48,8  S22=0,72/-37,6

NFmin=0,62  Γopt=0,5/67,5 Rn=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.

Fig. 8 – S-parameters and stability circles for the transistor (S11 S21 S12 S22 Input Stability Circle Output Stability Circle)

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.

.

Fig. 9 – Possible cases for stability circles in the active region

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 Γopt, being its normalized impedance Zopt=0,86+j⋅1,07. In the app, opening the design window and writing this impedance

Fig. 10 – Representation of Γopt

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.

Fig. 11 – Translating to circle with real part 1.

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.

Fig. 12 – Source impedance matched to Γopt

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:

S11=0,54/-177   S21=8,3/61,1   S12=0,04/-3,9  S22=0,72/-48,6

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

Fig. 13 – Transistor with matching network to Γopt and 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 S22=0,72/-48,6 should be loaded by ΓL (S22  conjugate), translating from 50Ω to ΓL=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

S11=0,83/145   S21=12/-7.5   S12=0,06/-72,5  S22=0,005/162

The gain is 20·log(S21)=21,6dB, and the noise figure, 0,62dB (NFmin). Now it is only represented these parameters in the three-dimensional chart to get the stability circles.

Fig. 14 – Low noise amplifier and 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, S11 y S22 are into the stable regions, then the amplifier is stable.

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.

REFERENCES

1. 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
2. Andrei A. Muller, P. Soto, D. Dascalu, D. Neculoiu and V. E. Boria, “A 3D Smith Chart based on the Riemann Sphere for Active and Passive Microwave Circuits,” IEEE Microwave and Wireless Components. Letters, vol 21, issue 6, pp 286-288, june, 2011
3. Zelley, Chris; “A spherical representation of the Smith Chart”; IEEE Microwave, vol. 8, pp. 60–66, July 2007
4. 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. POUT is 31dBm (IDQ=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: VDS=8V, IDQ=50mA . At this OP, and at 900MHz, ZIN=3,6+j·4,3Ω and ZOUT= 3,9 + j·5,5Ω. Maximum power transfer is obtained with a conjugate matching network at the generator and load impedances, which are Z0=50. Once the matching networks are calculated, the purposed schema for the amplifier is

PD84001 Two-Stage Amplifier

Amplifier’s gain is 34,3dB, and its phase is 76,4deg. 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.

2 Tone Output Power and TOD vs. Input Power, at 900Mhz

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

$P_o= \hat{g} (P_i)=\hat{g}_1 P_i+\hat {g}_2 P_i^2+\hat {g}_3 P_i^3$

with

$\hat{g}_1=g_1e^{j \theta_1}$

$\hat{g}_2=g_2e^{j \theta_2}$

$\hat{g}_3=g_3e^{j \theta_3}$

Where the input signal Pi is a 2-Tone signal. In this case, we will not take into consideration the second order distortion, since the Pfrequencies will be very close together. A bandpass filter could remove the second order spurious.

The amplifier gain is complex, The coefficients g1 and g3 could be expressed using the polar notation (in mag/phase). Then, these are

$\hat{g}_1=2961,54 \cdot e^{j 76,4}$

$\hat{g}_3=11,43 \cdot e^{-j 95,3}$

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 due to the active device’s nonlinearity. 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.

Cancellation of the main signals on the second port is achieved by placing a delay line (τ1), in the secondary network of the first stage. One sample of the signal output of the amplifier (G1) is derived to combine with the secondary network, with a combiner. The levels of both signals are equalized by an inter-stage attenuator (β). Then, both signals are combined. Then, the ouput signal of the amplifier is called MAIN, and the combined signal, AUX.

Feedforward Principle

AUX is now used as an error signal in the second stage, which is amplified by an error amplifier (G2), while the MAIN is delayed with another delay line (τ2). 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.

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

$P_{MAIN}= \hat{g} \left( \dfrac {P_i}{2} \right)=g_1e^{j \theta_1} \dfrac {P_i}{2}+g_2e^{j \theta_2} \dfrac {P_i^2}{4}+g_3e^{j \theta_3} \dfrac {P_i^3}{8}$

and PAUX1 (the sample level before the error combiner) could be expressed by

$P_{AUX_1}=\beta g_1e^{j \theta_{A1}} \dfrac {P_i}{2}+\beta g_2e^{j \theta_{A2}} \dfrac {P_i^2}{4}+\beta g_3e^{j \theta_{A3}} \dfrac {P_i^3}{8}$

and $\theta_{Ai}=\theta_i + \theta_{\beta}$

The level PAUX2 at the secondary network is

$P_{AUX_1}=\dfrac {P_i}{2} e^{j{\theta}_{A2}}$

In these expressions, β is the magnitude of the losses of the inter-stage attenuator and θβ is its phase; and θA2 is the phase of the delay line τ1. It must be satisfied

$\theta_1+\theta_{\beta}=\theta_{A2} \pm 180$

$\dfrac {d \theta_1}{d \omega}+\dfrac {d \theta_{\beta}}{d \omega}=\dfrac {d \theta_{A2}}{d \omega}$

θ1 is the phase of the linear gain of the amplifier. Then, not only the phases must be in phase opposition, but also the delay time must be the same in every subnetworks. In magnitude, it must be satisfied |β·g1|=1.

In the second stage, the gain of the amplifier must equalize the g2 and g3 levels, and their phases must satisfy the same equations (absolute phase and delay time) of the first stage, to combine and cancel the distortions.

$\left| \beta G_2 \right|=1$

$\theta_{\beta}+\theta_{G_2}=\theta_{\tau 2} \pm 180$

$\dfrac {d \theta_{\beta}}{d \omega}+\dfrac {d \theta_{G_2}}{d \omega}=\dfrac {d \theta_{\tau 2}}{d \omega}$

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 Pi and combine PAUX1 and PAUX2, and two directional couplers (with C for coupler losses) to take the sample in the first stage and combine the error sample in the second stage, the expressions are now

$\left| \dfrac {\beta}{IL_{hyb} C_{coup}}g_1 \right|=1$

$\theta_1+\theta_{\beta}+\theta_{coup}=\theta_{A2} \pm 180$

$\dfrac {d \theta_1}{d \omega}+\dfrac {d \theta_{\beta}}{d \omega}+\dfrac {d \theta_{coup}}{d \omega}=\dfrac {d \theta_{A2}}{d \omega}$

at the first stage and

$\left| \dfrac {\beta}{IL_{hyb}^2 C_{coup}^2}G_2 \right|=1$

$\theta_{\beta}+2 \theta_{coup}+\theta_{G2}=\theta_{\tau 2} \pm 180$

$\dfrac {d \theta_{\beta}}{d \omega}+2\dfrac {d \theta_{coup}}{d \omega}+\dfrac {d \theta_{G2}}{d \omega}=\dfrac {d \theta_{\tau 2}}{d \omega}$

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 Surface 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 69,4deg, with 0dBm of input power. Taking a sample of the output level of the amplifier with a directional coupler, which introduces a 90deg coupling phase, with 10dB of coupling level, we have got a sample level of 12dBm, with a phase of 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.

First stage for AUX adjustment

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

AUX Main an Intermodulation Frequencies, vs Input Power

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 MAIN line. The error amplifier G2 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 MAIN and AUX IM spurious. This value is 45,5dB, because we are combining with a directional coupler, to reduce the insertion losses in the MAIN line. Using an amplifier with a magnitude of 45,5dB and a phase of -145deg, we have got a phase shifter with the same value, and after the coupler, the IM distortion decreases around 65dB. The output level is now 31dBm, and the TOI increases to 75dBm.

Output level and TOD, after the feedforward correction

The definitive amplifier is

Definitive Feedforward Amplifier

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 with a few RF devices. The design is very easy and intuitive.

However, the Feedforward has two serious disadvantages: on the PCB, it needs a lot of surface; and the input level cannot be increased above the input level that provides maximum output level of the MOSFET, because the distortion can be increased above the value we have corrected.

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, the phase shifter could have a larger surface dimensions, because it must be a delay line, too.

References

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