Transmission Line Characteristic impedance - Page 1
Forum section: electronics / rf-microwave

author: dmowziz
author_rank: Contributor
timestamp: September 24, 2023, 11:37:17 am
content: Hi...Please where can we find the derivation of the inductance and capacitance values of high and low characteristic impedance?I mean, a series inductor can be modelled as a high characteristic impedance line. How can we derive it? (the equation for the inductance)Thanks

author: gf
author_rank: Super Contributor
timestamp: September 24, 2023, 12:33:41 pm
content: Quote from: dmowziz on September 24, 2023, 11:37:17 amDo you meanTelegrapher's Equation?

author: dmowziz
author_rank: Contributor
timestamp: September 24, 2023, 12:41:19 pm
content: No, not that.What I mean is about stepped-impedance filters. High Zo is a series inductor. Low Zo is a shunt capacitor.The equation relating Zo and inductance/capaictance

author: T3sl4co1l
author_rank: Super Contributor
timestamp: September 24, 2023, 01:25:37 pm
content: At low frequencies (much less than 1/4λ),\$L = l L_0\$, \$C = l C_0\$where \$l\$ is the length and \$L_0\$ and \$C_0\$ are the reactance rates of the line in question. Which are in turn (for typical materials with \$\mu_r = 1\$),\$L_0 = \frac{Z}{Z_0} \frac{ \mu_0}{c}\$, \$C_0 = \frac{Z_0}{Z} \frac{\epsilon_0}{c}\$for velocity factor \$c = \frac{1}{\sqrt{\epsilon_r}}\$, characteristic impedance \$Z\$, and impedance of free space \$Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} \approx 377\Omega\$.For example, Belden 8219, your average RG-58-like coax, specifies Z = 54Ω and c = 76%, and has \$C_0\$ of 85.3 pF/m. \$\epsilon_0\$ is 8.85 pF/m, so the calculation gives 81.2 pF/m, pretty close (probably several of these parameters are slightly in error -- transmission lines being better than 10% accuracy is kind of extraordinary (i.e. you'll pay a lot for precision much better than that), so this is within expected error.At least, that's how I always keep track of it. Having memorized these constants, I find it easiest to think in terms of ratios of them: note that \$\mu_0\$ and \$\epsilon_0\$ have units of reactance per length, so they're basically what we already want, just missing geometry and material factors.Further derivation from fields, or solving the field equation, is an exercise for the student.Note that, structures we commonly use for inductive components, arenotsimple prismatic nondispersive transmission lines: helical solenoids, and other more complicated structures, have a whole variety of additional effects. Of which the most important is the ability to get much higher \$Z_0\$ with realistic materials and geometry, at the expense of being dispersive (the resonant frequencies are anharmonic), having external fields, etc. Let alone the effect of magnetic loading (cores). You can read about some modeling of the simplest such case (helical waveguide) here:http://hamwaves.com/antennas/inductance.htmlTim

author: szoftveres
author_rank: Regular Contributor
timestamp: September 24, 2023, 03:10:46 pm
content: Quote from: dmowziz on September 24, 2023, 11:37:17 amZ = √(L/C)

author: dmowziz
author_rank: Contributor
timestamp: September 24, 2023, 05:21:10 pm
content: I'm not sure we are on the same page... Or are we?I attached the eqn I'm trying to derive (sadly I think it is easy but I can't do it)

author: dmowziz
author_rank: Contributor
timestamp: September 24, 2023, 05:50:43 pm
content: It looks unnecessarily confusing...For the inductor (that is, a high Zo can be approximated as a series inductor with reactance of) : X = (Zo) * sin(theta)Capacitor : X = (Zo)/ sin(theta)And I believe the starting point to derive these is the equation of the input impedance of a transmission line :theta is the (beta * L) in the equation

author: T3sl4co1l
author_rank: Super Contributor
timestamp: September 24, 2023, 05:59:35 pm
content: Ah, well that's amuchmore advanced, and specific, topic than you led on with!You might start with the impedance matrix (or transmission, or anything really) of a transmission line, and derive the LC equivalent circuit (and values) from that.There also exists a transformation (which either underlies, or is being led up to by, your present work, I would guess), between lumped-element and TL-segment networks, under suitable assumptions (fundamental mode only). We could basically just apply that and be done, though if you need to prove the theorem, working through some background problems (like this) would definitely be a help.Tim

author: dmowziz
author_rank: Contributor
timestamp: September 24, 2023, 06:07:23 pm
content: Quote from: T3sl4co1l on September 24, 2023, 05:59:35 pmYeah I think so..But before getting to this equation,"A capacitor can be represented as a short length(less than 90 degrees) Transmission line with an open termination"with X = (Zo)/ tan(theta)This is easy to prove from the Zin equation... Solving for Zin with (Zl approaching infinity)So I'm thinking the problem is along the same line, maybe trying to solve for Zin with (Zo approaching infinty) for the inductor reactance

author: Marsupilami
author_rank: Frequent Contributor
timestamp: September 24, 2023, 07:10:33 pm
content: Pozar - Microwave Engineeringhttps://a.co/d/3iArC4l8.6 Stepped-Impedance Low-Pass FiltersHTH

author: tggzzz
author_rank: Super Contributor
timestamp: September 24, 2023, 10:58:01 pm
content: Quote from: Marsupilami on September 24, 2023, 07:10:33 pmWhat is thebenefitof posting an obfuscated URL in preference to the obvious URL?https://www.amazon.com/dp/B008ACCCHOWould you get a referrer's fee from amazon, or something similar?

author: RFDx
author_rank: Regular Contributor
timestamp: September 25, 2023, 01:59:22 am
content: Quote from: dmowziz on September 24, 2023, 12:41:19 pmInductor:l = (L * c0 * v) / ZLCapacitor:l = C * c0 * v * ZLl = Transmission line length L = Inductance C = Capacitance c0 = 3 * 10E8 m/sv = Velocity factorZL = Transmission line impedanceValid for l << lambda/10.

author: Marsupilami
author_rank: Frequent Contributor
timestamp: September 25, 2023, 08:17:27 pm
content: Quote from: tggzzz on September 24, 2023, 10:58:01 pmTrust issues much?(Be careful, the link to this picture is obfuscated too. Don't look at it too long if you don't want me to be enriched.)

author: abeyer
author_rank: Frequent Contributor
timestamp: September 26, 2023, 10:10:56 pm
content: Quote from: tggzzz on September 24, 2023, 10:58:01 pmNo, in fact I believe that the "Copy Link" functionality that usually produces those is partly there specifically because it will strip out affiliate/referral/tracking codes from links that might be included if you just copied the url from your browser. Also I think they point to the "correct" local amazon site based on your country rather than getting a link to whatever country the person sharing it with you uses and then having to find the same item on your local one.