power line impedance matching
biff44 said:
[Edit missing removed quote]
You lost me on that one. Coaxial cable characteristic impedance is ~not frequency dependent. A 50 ohm cable at 100 MHz is, surprise, still 50 ohm cable at 1 KHz. Impedance has to do with √(L/C), where L is inductance per length, and C is capacitance per length. Does not change with frequency.
Ah, You also read and use textbooks simplified formula!!!
textbook have written mostly two simplified case:
for audio frequency 0 - ~20 kHz) ie. g << ωr and r >> ωl is simplified to
γ = α + jβ = √(jωrc)
α = β = √((ωrc)/2) or γ = √(ωrc) |_ 45 degree
Z = √(r/(jωc) or |Z| = √(r/(ωc)) and |_- 45 degree
α is attenuate in Neper per unit length
β i phase constant i radianer per sec and unit length
(this works up ~ 10 KHz on ordinary twisted pair but angle is not longer inside 44-45 degree range for frequency above 10 KHz if calculate with full telegraph formula down here)
For high frequency (>= 1 MHz) ie. g << ωc and r << ωl is simplified formula as:
α = (r/2 * √(c/l)) + (g/2 * √(l/c))
β = ω√(lc)
Z = √(l/c) and asumme is a real impedance
---
Above is simplified near value formule make from complete telegraph equation as:
γ = α + jβ = √((r + jwl) * (g + jωc))
Z = √((r + jωl) / (g + jωc))
Most Textbook make lot of paperspace to describe how to simplifie this formula to above simplified near vaule formula so people not need solve (to much) complex equtions...
But near value formula handle around 0-10 KHz and 1 MHz and up but still miss some case intressting range between 10 kHz to 1 MHz (ie ADSL frequency) - most textbook skip this part without comments...
---
If you have calculators can solve complex square root and easly handle +,-,/,* with complex number (ex. HP42S), is a no reason to use simplified formula first part in this text. I'm always using full telegraf equation to done this, is more easy to remember and give more exact result compare to near value formula. (unfortly, excel:s 'IMPOWER' cannot handle power of 0.5 (to make complex square root) or complex power (e^a+ib - using) on complex number, only allow integer... - why not written complete complex support in excel if still try written some complex functions, Microsoft??? )
For example simplified formula can make value, using in other formula, give cable allow signal going faster than speed of light without notice and indicate simplified formula used on outside valid range...
---
Value 'g' is near always missing or unknow - so you can try '0' on low frequency or if unknow plastic as cable insulation and higher frequency, you can try g = ω * c * δ and δ = 0.0003 or look up in table for used insulations material
δ =tan(d) = DF (Dissipations Factor) = 1/Q = esr/Xc
DF can written as 0.035 or 3.5 %
ex: Tan(0.01719 degree) = DF 0.0003 = DF 0.03%
you need also adjust value 'r' depend of frequency on high frequency depend of skin effect on conductor as r = constant * √(f) above > 50 KHz - constant is very depend of used cable geometry and used metal in conductor.