### What is a Smith Chart?

February 10, 2023## Introduction

The Smith chart is a useful graphical tool to convert between impedances and reflection coefficients. It may also be used to solve impedance-matching problems. It can be thought of as a polar reflection coefficient chart with overlaid impedance curves.

Figure 1 – Smith Chart

## Reflection Coefficients

The reflection coefficient of an impedance can be measured with a vector network analyzer.

Figure 2 – Measured Reflection Coefficient

That reflection coefficient will be:

where Z is the complex impedance R+jX and Z_{0} is the characteristic impedance of the source.

For Z = Z_{0}, the reflection has a magnitude of zero, a dot at the center of the smith chart. For Z = 0, the reflection is 1∠0 or Cartesian 1,0.

For Z = ∞, the reflection is 1∠180 or Cartesian -1,0.

For Z = 23+j75, the reflection coefficient is 0.762∠64 or 0.333+j0.685

These four reflections are plotted on the polar plot in Figure 3.

Figure 3 – Polar Reflection Coefficients

## Building the Smith Chart

These reflection coefficients were calculated from Equation 1. Using this equation, we could choose a real part of the impedance, the resistance, hold it constant, and then let the imaginary part, the reactance, go from -∞ to +∞ and draw the result on the polar plot.

With the resistances normalized to 50 ohms, we can draw these circles of constant resistance for 10, 20, 50, 100, and 200 ohms, or 0.2, 0.4, 1, 2, and 4 when normalized.

Figure 4 – Polar Chart with Resistance Circles

The circles are centered at R/(R+1),0, with a radius of 1/(R+1), where R is the normalized value.

We can then choose reactances and sweep the resistance from -∞ to +∞ to obtain arcs of constant reactance.

Figure 5 – Completed Smith Chart

After the circles of resistance and curves of reactance are added to the polar chart and labeled, one can locate the reflection coefficient of any impedance by inspection.

## Impedance Matching

You can use a Smith chart to assist in impedance matching with lumped elements or transmission lines. For instance, if we start at an impedance of 20+j35Ω and want to match to 50Ω at 500 MHz, we can use a shunt capacitor followed by a series capacitor. To make the graphical method simpler, a mirror image of the resistance and reactance circles and arcs must be added (in blue in Figure 6) to represent shunt conductance and susceptance.

The first step is to add a shunt term to provide enough susceptance to get to the constant 50Ω resistance circle. To get from 20+j35 to the constant 50 ohm resistance circle, we added 11.9 *10^{-3} mhos of capacitive susceptance, or 3.8 pF at 500 MHz. From there, a negative reactance of 39.8 ohms is needed to get to the center of the Smith chart, or 8.0 pF at 500 MHz. These manipulations can be done directly on a Smith chart, but it is much simpler to use the Smith program designed by Prof. Fritz Dellsperger at www.fritz.dellsperger.net. The demo version is free of charge, while the licensed version has more features. It is very instructive to see how the addition of series or shunt elements moves the chart’s impedance.

Figure 6 – Matching

This same match could be accomplished with a series capacitor to get to the 0.02 m-mho conductance circle (1/50Ω), followed by a shunt capacitor to get to the middle of the Smith chart. 30.1 pF and 7.8 pF respectively, as in Figure 7.

Figure 7 – Alternate Match

Finally, you could use a 5.3 pF series capacitor to get to the 0.02 m-mho (1/50Ω) conductance circle on the other side, and then use a shunt 13 nH inductor, as in Figure 8, to get to the middle of the Smith Chart. The possibilities are endless, but the matching bandwidth will become narrower and narrower as more distance is traversed on the chart.

Figure 8 – LC Match

## Transmission Line Matching

Transmission lines can also be used for impedance matching. In a special case, if it is necessary to match from one real impedance to another, such as 50Ω to 100Ω, a quarter wavelength transmission line with impedance of the square root of the product of the two, or 70.7Ω will work.

Figure 9 – Series Transmission Line Match

A Wilkinson power splitter is comprised of two, quarter wavelength 70.7Ω transmission lines, which are joined together at one end. Each 50Ω terminated line looks like 100Ω where they are joined together, resulting in an effective 50Ω impedance.

In general, a series transmission line will rotate any real impedance Z clockwise around impedance and a 50Ω transmission line will rotate an impedance around the center of the Smith chart.

For a large change in impedance, such as from 50Ω to a 2Ω LDMOS power amplifier gate, make the transition in two or more steps to maximize the match bandwidth. A quarter wavelength 6Ω line, followed by a quarter wavelength 30Ω line, is preferrable to a single quarter wavelength 10Ω line. The bandwidth is superior, as shown in Figure 11.

Figure 10 – Transmission Line Match

Figure 11- Two vs One Transmission Line Match

## Summary

The Smith chart is a very convenient tool to graphically calculate and visualize impedance transformations with lumped elements or transmission lines. There are many lumped element solutions for any given matching problem, and transmission lines—particularly quarter wavelength lines— may be used singly or in cascade to match two real impedances.

The chart is educational and helps provide an intuitive understanding of impedance transformations.

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After pressing the Confidence Check button in the User Interface, an approximately 20 dB attenuator is connected internally between the A and B ports of the ACM and a precision factory measurement of that same attenuator is loaded into the active trace memory. The memory trace will normally be covered by the active measurement, so it is best to subtract the live measurement from the memory trace by navigating to Display>Memory>Data Math>Data / Memory. Set the scale to a low value, such as 0.05 dB/Div, for a magnitude reading and then move the ACM from side to side to flex the test cables slightly. There should be very little movement of the displayed difference. If wild movement is seen, then the test cables are either loose or defective. Defective cables should be discarded immediately. Figure 3 - ACM Connection for 1-Port Calibration, Load Unused Port Figure 4 - ACM Connection for 2-Port Calibration Why Use an ACM? The measurement accuracy of the VNA depends on the quality of the user calibration. You can best understand the quality of calibration by examining the residual errors which remain after the calibration is performed. The 12-term error model is generally used to characterize these errors – six terms for forward measurements, such as S11 and S21, and six for the reverse, S22 and S12. These errors are depicted in the S-parameter network diagrams shown below. Figure 5 - 12 Term Error Model, Forward and Reverse The actual Device Under Test (DUT) S-parameters are shown in the grey box, and the error vectors appear on either side of it. These errors are: Table 1 - 12 Term Errors After calibration, these errors are greatly reduced but are not zero. The remaining errors are called residual errors. For instance, the raw directivity of the directional bridge in the VNA might be 18 dB, but after calibration with an ACM, it might be 46. 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Residual Errors and Reflection Uncertainty The other error terms are corrected to the much smaller residual values based on the quality of the calibration standards within the ACM. The uncertainty of VNA measurements depends on the values of these residual errors. A typical reflection uncertainty chart after ACM calibration is shown in Figure 7. Figure 7 - Reflection Uncertainty The floor for reflection measurement accuracy is determined by the residual directivity error. The quality of the calibration load is very important. To put it another way, 1-port reflection measurements are not noise limited, but rather interference limited by the residual directivity. The reflection uncertainty curve above is only valid for calibration with an ACM. If using a mechanical kit with a 30 dB return loss calibration load instead, the above curve would be shifted 16 dB to the right and the uncertainty for a -20 dB reflection measurement would be about ± 3 dB instead of ± 0.6. From EUROMET [1], the equation for the approximate uncertainty of a reflection measurement in terms of the residual errors is (in linear terms): Where Sii is the measured reflection T is the Residual Reflection Tracking M is the Residual Source Match L is the Residual Load Match And R accounts for random factors such as connector repeatability The Load match term, L, only applies for reflection measurements of a DUT, which is terminated on the other side by the opposite port of the VNA and insertion loss through the DUT is low (S12 = S21≈1). Residual reflection tracking and source match dominate for high reflection, and residual directivity dominates for low reflection measurements. Residual directivity, D, is approximately equal to the calibration load uncertainty; 30 dB for the expensive mechanical kit and 46 dB for the ACM. Residual source match is approximately equal to the weighted square root of the sum of the squares of the calibration load uncertainty and the angular phase error (radians) of the Open and Short calibration standards. Residual load match is approximately equal to the residual source match for a 2-port calibration with unknown thru. The uncertainties of the calibration standards determine the residual errors and how those residual errors affect the final reflection measurement uncertainty. A more thorough treatment of VNA metrology is given in [2]. Residual Errors and Transmission Uncertainty A typical transmission measurement uncertainty chart after ACM calibration (CMT model S5065) is shown in Figure 8. Figure 8 - Transmission Uncertainty, Model S5065 For transmission measurements, the approximate measurement uncertainty is given by: Where: T is the Residual Transmission Tracking error M is a residual error due to DUT mismatch I is the residual Isolation error R includes a number of random factors but is dominated by the noise floor for small reflections Residual transmission tracking is proportional to the product: Where: µ1 is Residual Source Match L2 is Raw Load Match µ2 is Residual Load Match L1 is Raw Source Match Raw source and load match are properties of the VNA hardware, and residual source and load match are approximately equal to the uncertainty of the calibration load standard. The raw source and load match of the VNA should be reasonable, but the two products above are dominated by the small residual values. 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Sometimes, adapters are added to the ACM to allow for calibration using test cables with non-matching connectors. User Characterization can be performed on the ACM to allow for this. As an example, use test cables that have 3.5mm male connectors and an ACM that has female N connectors. Place the N Male to 3.5mm female adaptors on the ACM and torque them appropriately. Calibrate the ends of the test cables using a 3.5mm calibration kit, another ACM, or a good mechanical kit. Make sure the frequency range is appropriate for the adapters and set the number of points to 1601. Attach the test cables to the ACM with adapters using proper torque. In the VNA User Interface under Calibration/AutoCal, select Characterization and change it from “factory” to one of three user characterizations. Press the Characterize AutoCal Module button, allow the calibration to complete, then fill out the information on the dialogue screen which will automatically appear. After User Characterization, the adapters should be left on the ACM. If they need to be removed and then re-attached, the user characterization should be repeated. Conclusion The accuracy of the VNA measurement greatly depends on the quality of the calibration kit in use, especially for 1-port reflection measurements. Because of its thermally compensated, data-based internal calibration standards, the ACM delivers consistently superior results as compared to calibration using even a very high-grade mechanical calibration kit. In addition, the Confidence Check feature provides a fast and easy method to verify the robustness of the calibration and test set-up. For precision VNA measurement, an ACM is required. 2 and 4-Port ACM modules can be found on the CMT website here. References [1] EURAMET, “Guidelines on the Evaluation of Vector Network Analyzers”, Euramet cg-12 Version 2.0, 3/2011. [2] Brian Walker, Copper Mountain Technologies, “Introduction to the Metrology of VNA Measurement”, May 31st, 2022.

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Everyone knows that a good VNA should have both excellent hardware performance and an easy to use software interface with useful post-processing capabilities. There are numerous VNAs in the market with different performance levels; some of them are economy grade, and others are truly laboratory test grade. What separates the two?