Smith chart of λ/4 Transmission Line - Page 1
Forum section: electronics / rf-microwave

author: promach
author_rank: Frequent Contributor
timestamp: May 05, 2019, 02:38:32 am
content: I came across thefollowing slide on power combiner circuitand I have questions about it.1. Why traverse across the upper half circle ? In other words, why is Zin located at the right side of Zout ?2. Why when translating an impedance towards the generator we move clockwise around the chart ?3. Why "ZT-line is the geometric mean of Zout and Zin." ?4. Why overlay the Smith chart scaled for ZT on top of the Z0 Smith chart ?5. Why is L=λ/4 located directly opposite to L=0 ?

author: radiolistener
author_rank: Super Contributor
timestamp: May 05, 2019, 03:32:53 am
content: Because Zin= Z0* (ZL+ j * Z0* tg( k*L )) / (Z0+ j * ZL* tg( k*L ))whereZin- input impedance of the transmission lineZ0- characteristic impedance of the transmission lineZL- load impedance on the transmission line outputk - wave number k = 2 * pi / lambdaL - length of transmission line in meters

author: promach
author_rank: Frequent Contributor
timestamp: May 05, 2019, 03:40:02 am
content: Quotehttps://en.wikipedia.org/wiki/Quarter-wave_impedance_transformer#Theory_of_operationYou have a typo in your expression. Please cross-check with the wikipedia article.

author: radiolistener
author_rank: Super Contributor
timestamp: May 05, 2019, 03:42:00 am
content: fixed

author: promach
author_rank: Frequent Contributor
timestamp: May 05, 2019, 04:00:54 am
content: Hmm... but how to directly link the wikipedia expression to my five questions above ?

author: T3sl4co1l
author_rank: Super Contributor
timestamp: May 05, 2019, 06:03:49 am
content: It's all just geometry. Recall the equations for circles and lines, and do the math for complex numbers, Z/Y/Γ, as phasor vectors. Finally, consider the coordinates on the Smith chart itself, and what that allows you to do with all these.Too high-level? What are you having trouble with? Definitions? Connecting the dots?Tim

author: 0culus
author_rank: Super Contributor
timestamp: May 05, 2019, 06:55:21 am
content: Quote from: radiolistener on May 05, 2019, 03:32:53 amBTW, you can use LaTeX in the forum for stuff like this:$$Z_{in} = \frac{Z_0 \cdot \left(Z_{L} + j \cdot Z_0 \cdot tg(k\cdot L)\right)}{Z_0 + j\cdot Z_L \cdot tg(k\cdot L)}$$

author: promach
author_rank: Frequent Contributor
timestamp: May 08, 2019, 03:00:21 pm
content: Do you have guys have any mathematicalPROOFfor each of the directions for parallel or series R, L or C ?

author: Andy Watson
author_rank: Super Contributor
timestamp: May 08, 2019, 05:59:21 pm
content: Quote from: promach on May 08, 2019, 03:00:21 pmThese articles are the best explanation I've found for explaining why the charts work the way they do. The first one is the most informative:https://www.americanradiohistory.com/Archive-Wireless-World/60s/Wireless-World-1960-01.pdfhttps://www.americanradiohistory.com/Archive-Wireless-World/60s/Wireless-World-1960-02.pdfhttps://www.americanradiohistory.com/Archive-Wireless-World/60s/Wireless-World-1960-03.pdfThe appendix at the end of the first article gives details for contructing the charts - try it - it becomes more apparent how they work.I really miss this format of Wireless World.

author: TheUnnamedNewbie
author_rank: Super Contributor
timestamp: May 09, 2019, 07:05:26 am
content: As I also said on reddit, there is no real sense in 'proving' this. The smith-chart is a constructed thing - a 'made up' way to represent impedances. It has been 'designed' in such a way that those things actually happen. You can look up a lot of math on how it works, starting from those equations governing reflection coefficients and impedance.Keep in mind that the first step for any of this is to always normalize everything! You divide everything by Zo!

author: KE5FX
author_rank: Super Contributor
timestamp: May 09, 2019, 07:52:28 am
content: Quote from: Andy Watson on May 08, 2019, 05:59:21 pmThose are great. I actually didn't have an hour to kill, but there it went...

author: Electro Fan
author_rank: Super Contributor
timestamp: May 11, 2019, 07:44:53 am
content: Quote from: KE5FX on May 09, 2019, 07:52:28 amthose are good articlesand an hour is pretty good; this hobby (or profession I’m guessing) could easily consume weeks, hours, yearsas an aside - on a related note, in an effort to get a 10 MHz reference I recently hooked up a GPSDO and Lady Heather; in LH there is a setting for turning on an hourly chime - it has kind of caught me off guard / surprised me how often I’ve heard the chime sound - time flies

author: promach
author_rank: Frequent Contributor
timestamp: July 10, 2019, 02:49:41 pm
content: I found that Richard Li's bookRF Circuit Design 2nd editionchapter 3 as well as its chapter appendix gives better explanation than the magazine links above.When I read the derivation proof of Smith Chart in the book, I found that there might be a mistake in expression (3.A.12)Could anyone else confirm my observation ?