Originally Posted by chesebert /img/forum/go_quote.gif
Fundamental Understanding of the Transmission line
Metallic bond and its effect on signal propagation
The propagation of signal through an element is directly affect by the atomic makeup of that element. Atoms are made up of the nucleolus and a cloud of electrons. The cloud of electrons are usually represented by energy levels, where the electrons with the highest energies hang out in the outer layer while the weaker ones are closer to the core. Valance electron is the electron that hangs in the outer most rim of the electron cloud. In order to conduct current, which is the transfer of energy from one electron to another, or you can call it drift current, the valance electron must be able to move around. Metallic bound, unlike covalent or ionic bound, do not restrict the movement of their valance electrons. Although semiconductors are the exception with covalent bound (that's another topic all together).
So why is one metal a better conductor than the other? The simple answer is the more levels of energy a given metal has, the better it conducts electricity. The easier and less restrictive the movement of the electron the better it conduct electricity. One of the most important reasons is that when valance electrons are further from the core, there is less positive force pulling on it and since the valance electrons are usually the stronger ones that jumped from the level below, it has enough energy to 'swim' around the cloud. When an electric field is applied to the element, the energy is transferred from one electron to another and from one atom to another down the chain. Ag is a larger atom than Cu, but both have 2 valance, so they are pretty good conductors, with Ag being the better of the 2. Al, on the other hand, is pretty bad. It has 3 valance electrons and the atom is small. So the electric energy is freely passed in Ag and Cu, but is no so in Al.
In theory, the speed of propagation is c (speed of light, 3x10^8m/s), but there is loss in energy when one electron hand over the energy to another electron and to another electron. Thus, the propagation delay is material dependent. Cu has a theoretical propagation of 66.667%c or (2x10^8m/s). This, of course, does not count any boundary electron jump between bonding materials (solder).
So what is phase delay? Phase delay is a shift of the waveform in the time domain.
Voltage drop across transmission line
To calculate voltage drop across a transmission line, the propagation delay and the frequency which the signal is traveling at is important.
V1 = V0 cos(w(t-l/c)) where w= 2pif. And c is the speed which the energy travels. and l is the length of the cable
The determining factor in voltage drop is wl/c. By comparing theoretical c to the c of the copper, the power loss is measurable. One also need to taken into account the dispersive effects of the material, which for cu, I am not sure what that is. Dispersive effects are generally thought as different frequency propagate at different speed, so not only do you have phase delay of the superposed waveform, there is a phase delay in different frequency components as well!
How to properly calculate RLGC in Coaxial Cable
The Coaxial Cables are constructed with two coaxial conductors separated by dielectrics (of course conventional construction includes an outer layer of shielding).
R = (Rs/(2pi))(1/a+1/b) where a=2r(inner) and b=2r(outer), and Rs= sqrt(pi(f)(uc)(qc)) where uc = magnetic permeability and qc = electric conductivity (sorry no roman letters
As you can see, the resistance is a function of frequency and R is independent of V1 where V1 is the voltage drop due to propagation and again R is not dependent on phase delay and dispersion effects. Also notice the math does not involve any effect of the imperfect dielectric and electron deposition.
L = u/(2pi) x ln(b/a) Again no baring on phase delay
G = (2pi*q)/(ln(b/a))
C = (2pi(e))/ln(b/a)
Notice none of the RLGC is responsible for power loss, phase delay and dispersion effects and R is a function of frequency.
Now if you look the transmission line equation
-dV/dz = (R+jwl)I(z) and -dI(z)/dz = (G+jwC)V(z)
Now if differentiate both sides, you will arrive with (y) or complex propagation constant, which is y=alpha + jbeta
Alpha = Re(sqrt((R+jwL)(G+jwC))
Beta = Re(sqft((R+jwl)(G+jwC))
So basically, after doing all the math, the traditional RLC measurements are not only inaccurate, its down right faulty as RLC is a function of frequency at which the wave travels, and is dependent on the electrical permittivity, magnetic permittivity, and electrical conductivity of the individual material. This however does not even consider the power loss or dispersive effects.
I hope the above analysis answers some questions regarding why a manufacturer may want to optimize multiple areas of the cable to give it a lower propagation delay, optimize RLGC with different material and also optimize RLGC with the use of novel geometries. Of course you can ignore this entire discussion and just use your ear.
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