This document tries to clear out some details of transmission linesand cable inductance. This document is only a brief introduction tothose topics.If you expect to work much with transmission lines, coaxial orotherwise, then it will be worth your while to get a book on thatsubject. The ideal book depends on your background in phsics or electricalengineering, and in mathematics.
The basic idea is that a conductor at RF frequencies no longer behaves like aregular old wire. As the length of the conductor (wire) approaches about 1/10the wavelength of the signal it is carrying - good ol' fashioned circuitanalysis rules don't apply anymore. This is the point where things likecable impedance and transmission line theory enter the picture.
The key tenet of all transmission line theory is that the source impedancemust be equal to the load impedance in order to achieve maximum power transferand minimum signal reflection at the destination. In real world casethis generally means that the source impedance is the same as cable impedanceand the value of the receiver in another end of the cable has also thesame impedance.
Characteristic impedance of the cable ratio ofthe electric field strength to the magnetic field strength for wavespropagating in the cable (Volts/m / Amps/m = Ohms).
Ohm's Law states that if a voltage (E) is applied to a pair of terminals and a current (I) is measured in this circuit, the following equation can be used to determine the magnitude of the impedance (Z). The following formulawill hold truth:
Z = E / IThis relationship holds true whether talking about direct current (DC) or alternating current (AC).
Characteristic Impedance and is usually designated Zo or "Zed nought".When the cable is carrying RF power, without standing waves, Zo also equalsthe ratio of the voltage across the line to the current in flowing in theline conductors. So the characteristic impedance is definedwith the formula:
Zo = E / IThe voltages and currents depend on the inductive reactance andcapacitive reactance in the cable. So the characteristic impedanceformula can be written in the following format:
Zo = sqrt ( (R + 2 * pi * f * L ) / (G + j * 2 * pi * f * c) )Where:
Zo = sqrt ( R / (j * 2 * pi * f * C))If the capacitance does not vary with frequency, the Zo varies inversely with the square root of the frequency and has a phase angle which is -45o near DC and decreases to 0o as frequency increases. Polyvinyl chloride and rubber decrease somewhat in capacitance as frequency increases, while polyethylene, polypropylene, and Teflon* do not vary significantly.
When f becomes large enough, the two terms containing f become so large that R and G may be neglected and the resultant equation is:
Zo = sqrt ( (j * 2 * pi * f * L) / (j * 2 * pi * f * C) )Which can be simplified to the form:
Zo = sqrt ( L / C )
At the high frequencies you can't look at the cable as a usual cable. Onhigher frequency it works as a waveguide. Characteristic impedance isspecific resistance for electro-magnetic waves.So: It's the load the cable poses at high frequencies. Thehe highfrequency goes (dependent of cable of course) usually from 100kHz andup.
If you feed a sinusoidal electrical AC signal of reasonable frequencyinto one end of the cable, then the signal travels as an electrical wavedown the cable. If the cable length is an extremely large number ofwave-lengths at the frequency of that AC signal, and you measure theratio of AC Voltage to AC current in that traveling wave, then that ratio iscalled the characteristic impedance of the cable.
In practical cables the characteristic impedance is determined bycable geometry and dielectric. The cable length has no effect of it'scharacteristic impedance.
The coax is represented schematically by a series ofcapacitors and inductances, a sort of odd filter arrangement, theparticular values unique to the particular coax type. At a givenfrequency, if correctly chosen, that arrangement passes most of the signal;while at higher frequencies, that arrangment attentuates signal.
The length has nothing to do with a coaxial cable impedance.Characteristic impedance is determined by the size and spacing of theconductors and the type of dielectric used between them.For ordinary coaxial cable used at reasonable frequency, thecharacteristic impedance depends on the dimensions of the inner andouter conductors, and on the characteristics of the dielectric materialbetween the inner and outer conductors.
The following formula can be usedfor calculating the characteristic impedance of the coaxial cable:(formula taken from Reference Data for Radio Engineers book publishedby Howard W. Sams & Co. 1975, page 24-21)
impedance = (138 / e^(1/2)) * log (D/d)
Where:
In a nut shell the characteristic impedance of a coax cable is thesquare root of (the per unit length inductance divide by the per unitlength capacitance).For coaxial cables the characteristic impedance will be typicallybetween 20 and 150 ohms. The length ofthe cable makes no difference whatsoever in regard to the characteristic impedance.
If the frequency is much too high for the coaxialcable, then the wave can propagate in undesired modes (i.e., haveundesired patterns of electric and magnetic fields), and then the cabledoes not function properly for various reasons.
Characteristic impedance is determined by the size and spacing of theconductors and the type of dielectric used between them.Balanced pair, or twin lines, have a Zo which depends on the ratio of thewire spacing to wire diameter and the foregoing remarks still apply.For practical lines, Zo at high frequencies is very nearly, but not exactly,a pure resistance.
The following formula can be usedfor calculating the characteristic impedance of balanced pair near ground:(formula taken from Reference Data for Radio Engineers book publishedby Howard W. Sams & Co. 1975, page 24-22)
impedance = (276 / e^(1/2)) * log ((2D/d) * (1 + (D/2h)^2))^(1/2))
Where:
impedance = (276 / e^(1/2)) * log ((2D/d)
For twin line Zo will be typically between 75 and 1000 ohms depending onthe intended application. The impedance of typical old telephone pairin the telephone poles in the air has characteristic impedance ofaround 600 ohms. The telephone and telecommunication cables in usehave typically a characteristic impedance of 100 or 120 ohms.
If you know the imductance and capacitance of certain lenght ofcable you can use the following electrical model for it:
L L L / / L ---+uuuu+-+-+uuuu+-+-+uuuu+--/ ... /+uuuu+--- | | | / / | --+-- --+-- --+-- --+-- C --+-- C --+-- C--+-- C --+-- | | | / / | ----------+--------+------+-/ ... /------+--- / /For this model it is a beneficial to know an useful impedanceequation which described the relation of impedance, capacitanceand inducatance:
Z = sqrt ( L / C )
The equations and model are based on the fact that for "long" cables you can calculate the cable impedance with the following model:
L L L / / L ---+uuuu+-+-+uuuu+-+-+uuuu+--/ ... /+uuuu+-> | | | / / | --+-- --+-- --+-- --+-- C --+-- C --+-- C--+-- C --+-- | | | | Z = jwL + [(1/jwC) || {(jwL + [(1/jwC) || ... =Z Since the chain is infinite, the terms on the right are just equal to Z. You get a nice quadratic."long" isn't real restrictive so as to be in the wavelength or better ballpark.
Cable characteristic impedance is a cable characteristics whichis only valid for high frequency signals. Multimeters use DC currentfor resistance measurements, so you can't measure the cable impedanceusing your multimeter or other simple measurement equipments.It is usually best to check the cable type (usually printed on cable)and it's characteristics impedance from some catalogue instead oftrying to measure it.
A relationship exists which makes determination of Zo rather simple with the proper equipment. It can be shown that if, at a given frequency, the impedance of a length of cable is measure with the far end open (Zoc), and the measurement is repeated with the far end shorted (Zsc), the following equation may be used to determine Zo:
Zo = sqrt ( Zoc * Zsc )Where:
High frequency measurements of Zo are made by determining the velocity of propagation and capacitance of the cable or by reflectometry.
In order for a cable's characteristic impedance to make anydifference in the way the signal passes through it, the cable must beat least a large fraction of a wavelength long for the particularfrequency it is carrying.
Most wires will have a speed of travel for AC current of 60 to 70 percent ofthe speed of light, or about 195 million meters per second. An audio frequency of 20,000 Hz has awavelength of 9,750 meters, so a cable would have to be four or five*kilometers* long before it even began to have an effect on an audiofrequency. That's why the characteristic impedance of audio interconnectcables is not something most of us have anything to worry about.
Normal video signal rarely exceed 10 MHz. That's about20 meters for a wavelength. Those frequencies are getting close to beinghigh enough for the characteristic impedance to be a factor.High resolution computer video signals and fast digita signalseasily exceed 100 MHz so the proper impedance matching is neededeven in shor cable runs.
First, you want to drive the cable withan electrical source that has an output impedance equal to thecharacteristic impedance of the cable,so that all of the source's output power goes intothe cable, rather than being reflected from the cable's input end backinto the source. Second, you want the electrical load on the output endof the cable, to have an input impedance equal to the characteristicimpedance of the cable, so that all of the power goes into the loadrather than being reflected from the load back into the cable.
There are many exceptions to this normal driving method, but thoseare used for for special effects. You can choose an impedance match formaximum power transfer at low bandwidth, or mismatch the impedance fora flatter frequency response. It's the engineer's call, depending on whathe wants.
If you have mismatches between the source's output impedance,the cable's characteristic impedance, and load's input impedance, thenthe reflections can depend critically on the length of the cable. And ifyou distort the cable, as by crushing or kinking, or if you installconnectors improperly, then you can have reflections, with resulting powerloss. Andsometimes reflected power can damage the power source if lits of poweris sent to the cable (e.g., a radio transmitter).So you need to be careful of impedance mismatches.
An anomoly that is not in all text books is when antenna pushespower back (not a proper termination), it looks at the insideof the shield and the outside, which ever one is lowest gets the power .This means the RF can travel on the outside of the coax.The most difficult concept about coax is the XL,XC do not exist(to your transmitter) if cable is terminated.
Most common reasons for listing a cable impedance is that becauseof its reliable electrical characteristics, and that very impedancelisting. Coax is often is used to carry low level higher frequency signalsthat are separated. Separations are very expensive in terms of signal loss-- a perfect impedance match will cost you half the signal, and even aslight mismatch is very costly, particularly at antenna strength signals.Carefully matched carriers, like coax, are necessary to preserve signal atreduced noise.
Capacitance of the cable is nothing to do if the coax is terminated.The transmitter will see absolutely no capacitance nor inductance.
And this transmission line characteristic is used to hide capacitance inhigh frequnecy PCB's. Engineers can design the PCB traces so that they havethe proper capacitance and inductance values so that the transmitter willsee nothing but a proper impedance transmission line.
If a cable is terminated in its matching characteristic impedance you can't tell from the sending end that the cable is not infinitely long - all the signal that is fed into the cable is taken by the cable and the load.
If the impedances are not matches, part of the waves in the cablewill be reflected back on the cable connections distorting the outbound waves.When these reflected waves hit the wave generator, they are again reflected and mingle with the outbound waves so that it is difficult to tell which waves are original and which are re-reflections.
The same thing happens when pulses are sent down the cable - when they encounter an impedance other than the characteristic impedance of the cable, a portion of their energy is reflected back to the sending end. If the pulses encounter an open circuit or a short circuit, all of the energy is reflected (except for losses due to attenuation - another subject). For other terminations, smaller amounts of energy will be reflected.
This reflected energy distorts the pulse, and if the impedance of the pulse generator is not the same as the characteristic impedance of the cable, the energy will be re-reflected back down the cable, appearing as extra pulses.
If the coaxial cable is very short, the cable impedance does nothave much effect on the signal. Usually the beast way to transmitsignal through coaxial cable is to do the impedance matching,although there are some applications where the normal impedancematching on both ends is not done. In some special applicationsthe cable might be only impedance matched at only one endor intentionally mistached at both ends. Those applicationare special cases, where the cable impedance is take acountso that the combination of the cable and the terminationsat the ends of the cable produce the desired transmissioncharacteristics to the whole system. In this kind ofspecial application the cable is not considered as apassive transmission line, but a signal modifying componentin the circuit.
Velocity of propogation ratio percentage based on the speed of light invacuum. The percentage tells what is the speed of the signal in the cablecompared to the speed of light in vacuum.In coax cable, under reasonable conditions, the propagation velocitydepends on the characteristics of the dielectric material.
That usually is due mainly to the limited penetration of current intothe inner and outer conductors (the skin effect). With increasingfrequency, the current penetrates less deeply into the conductors, andthus is confined to a thinner region of metal. Therefore theresistance, hence attenuation, is higher. It also can be caused partlyby energy loss in the dielectric material.
For a line with fixed outer conductor diameter, and whose outerand inner conductors have the same resistivity, and assumingyou use a dielectric with negligible loss (such as polyethyleneor Teflon in the high-frequency range at least), then you getminimum loss in coax if you minimize the expression:
(1/d + 1)/ln(1/d)where d is the ratio of inner conductor diameter to outerconductor ID.A spreadsheet or calculator gets you closepretty quickly: D/d = 3.5911 is close.Thr formula was claimed to be derived from the formulafor coax impedance versus D/d and a formula for loss that you'llfind in "Reference Data for Engineers" published by Howard Sams,on pg. 29-13 in the seventh edition.
The interesting thing tonotice is that this minimum loss does not directly yield aline impedance: the line impedance depends on the dielectricconstant of the dielectric. For air insulated line, thecorresponding impedance is about 76.71 ohms, but if the lineis insulated with solid polyethylene, then minimum attenuationis at about 50.6 ohms. So, however it came to be, all theRG-58 we use for antenna feeds and test equipment connectionsis pretty close to minimum attenuation given the aboveconditions, and that the dielectric is polyethylene.
But if the line uses foam dielectric with a velocity factor of 0.8,then the impedance of minimum atten would be about 61 ohms.However, that minimum is a pretty broad one, and you don'tstart loosing a lot till you get more than perhaps 50% awayfrom the optimal impedance.
Note that foam-dielectric line with the same impedance andouter diameter as solid-dielectric line will have lower loss.That's because, to get the same impedance, the foam linewill have a larger inner conductor, and that larger conductorwill have lower RF resistance, and therefore lower loss.
The most typical coaxial cable impedances used are 50 and 75 ohm coaxialcables. 50 ohm coaxial cables might be the most commonly used coaxialcables and they are used commonly with radio transmitters, radioreceivers, laboratory equipments and in ethernet network.
Another commonly used cable type is 75 ohm ciaxial cable which isused in video applications, in CATV networks, in TV antenna wiringand in telecommunication applications.
600 ohms is a typical impednace for open-wire balanced lines fortelegraphy and telephony.A twisted pairs of 22 gage wire withreasonable insulation on the wires comes out at about 120 ohms for thesame mechanical reasons that the other types of transmission lineshave their own characteristic impedances.
Twin lead used in some antenna systema are 300 ohms to match to afolded dipole in free space impedance (However, when that folded dipoleis part of a Yagi (beam) antenna, the impedance is usually quite a bitlower, in the 100-200 ohm range typically.).
Standard coaxial line impedance for r.f. power transmission inthe U.S. is almost exclusively 50 ohms. Why this value was chosen isgiven in a paper presented by _Bird Electronic Corp._ Standard coaxial line impedance for r.f. power transmission inthe U.S. is almost exclusively 50 ohms. Why this value was chosen isgiven in a paper presented by Bird Electronic Corp.
Different impedance values are optimum for different parameters.Maximum power-carrying capability occurs at a diameter ratio of 1.65corresponding to 30-ohms impedance. Optimum diameter ratio for voltagebreakdown is 2.7 corresponding to 60-ohms impedance (incidentally, thestandard impedance in many European countries).
Power carrying capacity on breakdown ignores current densitywhich is high at low impedances such as 30 ohms. Attenuation due toconductor losses alone is almost 50% higher at that impedance than atthe minimum attenuation impedance of 77 ohms (diameter ratio 3.6).This ratio, however, is limited to only one half maximum power of a30-ohm line.
In the early days, microwave power was hard to come by and linescould not be taxed to capacity. Therefore low attenuation was theoverriding factor leading to the selection of 77 (or 75) ohms as astandard. This resulted in hardware of certain fixed dimensions. Whenlow-loss dielectric materials made the flexible line practical, theline dimensions remained unchanged to permit mating with existingequipment.
The dielectric constant of polyethylene is 2.3. Impedance of a77-ohm air line is reduced to 51 ohms when filled with polyethylene.Fifty-one ohms is still in use today though the standard for precisionis 50 ohms.
The attenuation is minimum at 77 ohms; the breakdown voltage is maximum at 60 ohmsand the power-carrying capacity is maximum at 30 ohms.
Another thing which might have lead to 50 ohm coax is thatif you take a reasonable sized center conductor and put ainsulator around that and then put a shield around that and choose allthe dimensions so that they are convenient and mechanically look good,then the impedance will come out at about 50 ohms. In order to raisethe impedance, the center conductor's diameter needs to be tiny withrespect to the overall cable's size. And in order to lower theimpedance, the thickness of the insulation between the inner conductorand the shield must be made very thin.Since almost any coax that *looks* good for mechanical reasonsjust happens to come out at close to 50 ohms anyway, there was anatural tendency for standardization at exactly 50 ohms.
Take a chunk of coax, connected to nothing. The center conductor andshield form a capacitor. If you charge that capacitor up to 100V,then short the shield to the center conductor, What is the currentflow?
It is not infinite (or determined by parasitic resistance ancreactance ) like a "normal capacitor" but it is determined by thecharacteristic impedance of the line. If it is 50 ohm line charged to100V then the current WILL be 2Amps. (100/50) It will be a squarepulse, and temporal width (time duration, pulse width whatever youchoose to call it) will be determined by the length of theline (around 1.5 nS/foot depending on line's velocity factor).
This method can be used for example to generate current pulses tosemicondictor lasers. To get the pulse lengths longer than easily availabewith practical coaxial lines you can use use lumped impedance near-equivilant.
If 50 ohm cable sees a 75 ohm load (the receiver), a substantial part of thesignal will be reflected back to the transmitter. Since the transmitter is also 75 ohm, thisrelected signal will be substantially reflected back to the receiver. Becauseof the delay, it will show up as a nasty ghost in the picture. Multiple ghostslike this look like ringing. Also, the reflections cause partial signalcancellations at various frequencies.
The cable impedance itself can't be converter unless you replacethe whole cable with new one which has the right impedance.If you absolutelu need to use the existing cable for your applicationthen there is one way to use the exiting cable: impedance converters.There are transformers which can make the cable look like different impedancecable when those are installed to both ends of the cable.
In some application it is possible to resistive adapers to convertthe cable impedances. Those adapters are simpler than transformers buttypically have a noticable signal loss in them (typically around 6 dbfor 75 ohm to 50 ohm conversion).
High speed signals can be routed on a circuit board if care is takento make the impedance of the traces match the source driver impedanceand the destination termination impedance. A microstrip line willexhibit a characteristic impedance if the thickness, width, and heightof the line above the ground plane are controlled.
Characteristic impedance formula:
Z = (87 / sqrt( Er + 1.41 )) * ln( (5.98*h)/(0.8*w + t))Where:
When routing circuit board traces, differential pairs should have thesame length trace. These trace lines should also be as short as possible.
If two cables with different impedances are connectedtogerther or a cable is connected to a source which has differentimpedance then some kind of impdance matching is needed toavoid the signal reflections in the place where the cablesare connected together.
The most classical method for matching different impedances isto use a matching transformer with proper impedance tranfer ratio.The impednace tranfer ratio of a transformer is determined byusing the formula:
Za / (Na^2) = Zb / (Nb^2)Where:
Zb = Za * (Nb/Na)^2From that equation you can see that Nb/Na is same as thetransformer voltage transferrign ratio betweenprimaty and secondary. This means that when you knowthat ratio you can use the equation without knowingthe exact turns ratio.
The matching network shown below can be used to match twounequal impedances, provided that Z1 is grater than Z2.
____ ----|____|---+--------- R1 | | | Z1 | | R2 Z2 |_| | -------------+----------
The resistor for this circuit can be calxulated using the followingequations:
R1 = Z1 - Z2*R2 / (Z2+R2) R2 = Z2 * sqrt(Z1) / (Z1-Z2)The table below will show some precalculated values for somemost common interfacing situations:
Z1 Z2 R1 R2 Attenuation (ohm) (ohm) (ohm) (ohm) (dB) 75 50 42,3 82,5 5,7 150 50 121 61,9 9,9 300 50 274 51,1 13,4 150 75 110 110 7,6 300 75 243 82.5 11,4
As you can see from the table the cost of simple resistorbased impedance matching is quite large signal level attenuationin the conversion process.
I have received the following comment on the cable impedance equation:
I have read your document, which is I must say very well written.I found a small mistake there though in "How does coaxial cable chacteristics define the impedance ?".Your formula is *impedance = (138 / e^(1/2)) * log (D/d)*, but this is only true for ideal conductor.Speed of wave in copper is less than in vacuum, and it equals to about 248827740 m/s; this means that it should be multiplied by a factor of 0.83.So the formula should look like this:
impedance = ( (138/e^1/2) * log(D/d) ) * 0.83
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