WEBVTT NOTE The Smith Chart 1 00:00:00.000 --> 00:00:06.736 What it is for. A Smith chart is a graphical calculator for transmission-line problems. 2 00:00:06.736 --> 00:00:18.040 It was invented in the 1930s to avoid complex arithmetic, and it survives because it makes the behaviour visible in a way the arithmetic does not. 3 00:00:18.040 --> 00:00:24.120 Question. Which of the following can be calculated using a Smith chart? 4 00:00:25.020 --> 00:00:29.404 Answer. Impedance along transmission lines. 5 00:00:30.004 --> 00:00:35.974 Question. Which of the following is often determined using a Smith chart? 6 00:00:36.874 --> 00:00:42.674 Answer. Impedance and SWR values in transmission lines. 7 00:00:43.274 --> 00:00:48.992 Question. Which of the following is a common use for a Smith chart? 8 00:00:49.892 --> 00:00:56.172 Answer. Determine the length and position of an impedance matching stub. 9 00:00:56.772 --> 00:01:10.747 That last one is the killer application. Finding stub length and position algebraically is tedious; on a Smith chart it is two arcs and a reading. The coordinate system. 10 00:01:10.747 --> 00:01:20.092 This question refers to one of the exam's circuit diagrams, which you will need to look at in the written lesson. 11 00:01:20.092 --> 00:01:26.024 Question. What type of coordinate system is used in a Smith chart? 12 00:01:26.924 --> 00:01:31.804 Answer. Resistance circles and reactance arcs. 13 00:01:32.404 --> 00:01:38.843 Question. What are the two families of circles and arcs that make up a Smith chart? 14 00:01:39.743 --> 00:01:43.731 Answer. Resistance and reactance. 15 00:01:44.331 --> 00:01:49.261 Question. What do the arcs on a Smith chart represent? 16 00:01:50.161 --> 00:01:54.721 Answer. Points with constant reactance. 17 00:01:55.321 --> 00:02:10.740 The features the pool asks you to name on the figure: Feature, The large outer circle; Name, reactance axis. Feature, The only straight line; Name, the resistance axis. 18 00:02:10.740 --> 00:02:24.047 Feature, Circles tangent to the outer circle; Name, constant resistance. Feature, Arcs meeting at the right-hand point; Name, constant reactance. 19 00:02:24.047 --> 00:02:31.298 Feature, Optional third family added when matching; Name, constant-SWR circles. 20 00:02:31.298 --> 00:02:48.001 The straight horizontal line is where reactance is zero, the resistance axis, and it runs from a short circuit at the left, through the prime centre, to an open circuit at the right. 21 00:02:48.001 --> 00:02:58.681 Question. What third family of circles is often added to a Smith chart during the process of designing impedance matching networks? 22 00:02:59.581 --> 00:03:03.941 Answer. Constant-SWR circles. 23 00:03:04.541 --> 00:03:21.466 Circles centred on the prime centre. Since SWR depends only on the magnitude of the reflection coefficient, constant SWR is a constant radius, which is why moving along a lossless line traces a circle on the chart. 24 00:03:21.466 --> 00:03:33.725 That single geometric fact is what makes the chart work. There is a diagram here. A Smith chart inside the unit circle of the reflection-coefficient plane. 25 00:03:33.725 --> 00:03:43.691 Constant-resistance circles for normalised r of 0, 0.5, 1, 2 and 5 are all tangent to the unit circle at its right-hand point. 26 00:03:43.691 --> 00:03:57.057 Constant-reactance arcs for x of plus and minus 0.5, 1 and 2 also converge on that same right-hand point, curving upward above the horizontal axis and downward below it. 27 00:03:57.057 --> 00:04:07.497 A dashed circle centred on the middle marks a constant standing-wave ratio of 3. A marked point plots the normalised load 1 plus j1. 28 00:04:07.497 --> 00:04:17.541 The left-hand edge is labelled short circuit, the centre is labelled matched, and the right-hand edge is labelled open circuit. 29 00:04:23.541 --> 00:04:35.614 The chart is a picture of one function. Normalise impedance to the system impedance, z equals r plus jx, and plot the reflection coefficient, which is written out in the lesson text. 30 00:04:35.614 --> 00:04:47.223 This is a Möbius transformation, a ratio of two linear functions, and Möbius maps have a property that does all the work here: they map circles and lines to circles and lines. 31 00:04:47.223 --> 00:05:02.480 (A line is just a circle through the point at infinity, so it is really one rule.) In the impedance plane, "constant resistance" and "constant reactance" are straight lines: vertical and horizontal rulings of the right half-plane. 32 00:05:02.480 --> 00:05:13.558 Push them through the formula in the text and they must come out as circles. Working out which: The formula for this is written out in the lesson text. Read those off. 33 00:05:13.558 --> 00:05:27.488 Every constant-r circle is centred on the real axis and passes through the formula in the text. Every constant-x circle is centred on the vertical line through the formula in the text and passes through it too. 34 00:05:27.488 --> 00:05:38.234 That common point is the open circuit, which is why the whole chart appears to be drawn around its right-hand edge, every curve on it goes through that one point. 35 00:05:38.234 --> 00:05:48.185 Two more consequences worth having: r equals 0 maps to the unit circle itself. A purely reactive load reflects everything, so the formula in the text. 36 00:05:48.185 --> 00:05:56.742 The outer boundary is not decoration; it is the r equals 0 member of the same family. The right half-plane maps to the unit disc. 37 00:05:56.742 --> 00:06:03.276 Any passive load has the formula in the text, so it lands inside. 38 00:06:03.276 --> 00:06:23.984 A point outside the chart would mean the formula in the text, more power coming back than went in, which is why a negative-resistance oscillator is drawn on an extended chart and a passive antenna never is. 39 00:06:23.984 --> 00:06:43.586 The plotted point is z equals 1 plus j1: the formula in the text, so the formula in the text and SWR equals 1.447 divided by 0.553 equals 2.62. It sits just inside the SWR = 3 circle, as it must. 40 00:06:43.586 --> 00:06:56.755 The circle centres and radii above are verified against scikit-rf's own mapping in scripts/tests/test_rf_numbers.py. Normalisation. 41 00:06:56.755 --> 00:07:00.843 Question. How is a Smith chart normalized? 42 00:07:01.743 --> 00:07:06.461 Answer. Reassign the prime center's impedance value. 43 00:07:07.061 --> 00:07:17.592 The chart's centre is 1.0, a perfect match, and every impedance is expressed as a multiple of the system impedance. 44 00:07:17.592 --> 00:07:36.181 Set the prime centre to 50 ohms and a 100-ohm load plots at 2.0; set it to 75 and the same load plots at 1.33. Normalisation is why one printed chart serves every system impedance. The wavelength scales. 45 00:07:36.181 --> 00:07:42.781 Question. In what units are the wavelength scales on a Smith chart calibrated? 46 00:07:43.681 --> 00:07:49.521 Answer. In fractions of transmission line electrical wavelength. 47 00:07:50.121 --> 00:08:05.412 Electrical wavelength, so velocity factor is already accounted for, and the distances you read off must be converted to physical length before you cut anything. The transmission-lines lesson has the conversion. 48 00:08:05.412 --> 00:08:18.081 One full trip round the chart is a half wavelength of line, not a whole one, which follows from the half-wave repeat rule: a half wavelength returns you to where you started. 49 00:08:18.081 --> 00:08:35.586 Why it is worth learning despite the calculators. A NanoVNA draws a Smith chart on its screen, so the chart has outlived the arithmetic it replaced. What it gives you is intuition: Moving along a line is rotation. 50 00:08:35.586 --> 00:08:43.557 Clockwise toward the generator. A high-SWR load is far from centre; matching means moving inward. 51 00:08:43.557 --> 00:08:58.514 A series reactance moves you along a constant-resistance circle; a shunt reactance moves you along a constant-conductance circle. That is the whole design procedure for an L-network. 52 00:08:58.514 --> 00:09:13.635 None of those are visible in a table of numbers. Check yourself. What does the only straight line on a Smith chart represent? A load plots at 2.0 + j0 on a chart normalised to 50 ohms. 53 00:09:13.635 --> 00:09:23.086 What is the actual impedance, and the SWR? How much line does one complete rotation around the chart correspond to? 54 00:09:29.086 --> 00:09:48.846 The resistance axis, where reactance is zero. 2.0 times 50 = 100 ohms resistive, and SWR = 100/50 = 2:1. A half wavelength of electrical length.