chesebert
18 Years An Extra-Hardcore Head-Fi'er
- Joined
- May 17, 2004
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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! The effect of short dispersive line is that higher order frequencies are effectively cut off. For example, if you pass a square wave through a short dispersive line, what you see on the scope is a square wave with the rise/fall edge fairly rounded, which indicated some higher order harmonics missing in its structure. Although common intuition tells us that we can't hear the difference, but audiophiles/musicians, unlike 'normal' people have tuned their ear to hear much more information and some are more sensitive than other to this effect.
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.
Dielectric in Coaxial Cable
In my previous discussion the element of dielectric was assumed to be theoretical but in real life that's not the case. To understand why dielectric behave differently from a conductor one has to look again at the elemental bonding, energy level, and the available valence electrons.
In dielectric material, the outermost shell is bond tightly to the atom. In the absence of electric field, the distribution of the outermost shell is uniform, which means the center of the cloud is where the nucleus is at. This is because the electric generated by the positively charged nucleus cancels out the electric field generated by the electrons.
However, when E(ext) is applied, although the energy normally would not be strong enough to detach any electron from the atom, the E(ext) can nevertheless polarize the atoms or molecules in the material by distorting the center of the cloud and the location of the nucleus, thereby creating a induced electric field or polarization field. One can express this relationship with D=e0E+P where D is the electric flux density (This should look familiar as a modification to one of Maxwell's equations). This equation is further complexed by whether the dielectric medium is either linear or isotropic. Thus P=e0XeE where Xe is the electric susceptibility. Combining the two equations yields D=e0E + e0xeE= eE or e=e0(1+Xe). You can substitute the new e in the C calculation of the lumped element model of the RLGC calculation.
I think with that I have covered the fundamentals of cables.
THIS IS NOT A GENERAL DISCUSSION OR A DISCUSSION ON MERE OPINIONS. PLEASE REFRAIN FROM POSTING IF YOUR REPLY DOES NOT DEAL DIRECTLY WITH ONE OF THE FOLLOWINGS:
Frequency Domain Measurement Techniques, Time Domain Measurement Techniques, Modeling Techniques Simulation Techniques for Interconnect, Structures, Electromagnetic Field Theory, Analysis and Modeling of Power Distribution Networks, Propagation Characteristics on Transmission Lines, Coupling Effects on Interconnects, Guided Waves on Interconnects, Radiation & Interference, Electromagnetic Compatibility, Power/Ground-Noise, Testing & Interconnects, Optical Interconnects
THANK YOU ALL
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! The effect of short dispersive line is that higher order frequencies are effectively cut off. For example, if you pass a square wave through a short dispersive line, what you see on the scope is a square wave with the rise/fall edge fairly rounded, which indicated some higher order harmonics missing in its structure. Although common intuition tells us that we can't hear the difference, but audiophiles/musicians, unlike 'normal' people have tuned their ear to hear much more information and some are more sensitive than other to this effect.
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.
Dielectric in Coaxial Cable
In my previous discussion the element of dielectric was assumed to be theoretical but in real life that's not the case. To understand why dielectric behave differently from a conductor one has to look again at the elemental bonding, energy level, and the available valence electrons.
In dielectric material, the outermost shell is bond tightly to the atom. In the absence of electric field, the distribution of the outermost shell is uniform, which means the center of the cloud is where the nucleus is at. This is because the electric generated by the positively charged nucleus cancels out the electric field generated by the electrons.
However, when E(ext) is applied, although the energy normally would not be strong enough to detach any electron from the atom, the E(ext) can nevertheless polarize the atoms or molecules in the material by distorting the center of the cloud and the location of the nucleus, thereby creating a induced electric field or polarization field. One can express this relationship with D=e0E+P where D is the electric flux density (This should look familiar as a modification to one of Maxwell's equations). This equation is further complexed by whether the dielectric medium is either linear or isotropic. Thus P=e0XeE where Xe is the electric susceptibility. Combining the two equations yields D=e0E + e0xeE= eE or e=e0(1+Xe). You can substitute the new e in the C calculation of the lumped element model of the RLGC calculation.
I think with that I have covered the fundamentals of cables.
THIS IS NOT A GENERAL DISCUSSION OR A DISCUSSION ON MERE OPINIONS. PLEASE REFRAIN FROM POSTING IF YOUR REPLY DOES NOT DEAL DIRECTLY WITH ONE OF THE FOLLOWINGS:
Frequency Domain Measurement Techniques, Time Domain Measurement Techniques, Modeling Techniques Simulation Techniques for Interconnect, Structures, Electromagnetic Field Theory, Analysis and Modeling of Power Distribution Networks, Propagation Characteristics on Transmission Lines, Coupling Effects on Interconnects, Guided Waves on Interconnects, Radiation & Interference, Electromagnetic Compatibility, Power/Ground-Noise, Testing & Interconnects, Optical Interconnects
THANK YOU ALL
