"Standard coaxial line impedance for r.f. power transmission in the U.S. is almost exclusively 50 ohms.
Different impedance values are optimum for different parameters. Maximum power-carrying capability
occurs at a diameter ratio of 1.65 corresponding to 30-ohms impedance. Optimum diameter ratio for
voltage breakdown is 2.7 corresponding to 60-ohms impedance (incidentally, the standard impedance in
many European countries).
Power carrying capacity on breakdown ignores current density which is high at low impedances such as 30
ohms. Attenuation due to conductor losses alone is almost 50% higher at that impedance than at the
minimum attenuation impedance of 77 ohms (diameter ratio 3.6). This ratio, however, is limited to only
one half maximum power of a 30-ohm line.
In the early days, microwave power was hard to come by and lines could not be taxed to capacity.
Therefore low attenuation was the overriding factor leading to the selection of 77 (or 75) ohms as a
standard. This resulted in hardware of certain fixed dimensions. When low-loss dielectric materials made
the flexible line practical, the line dimensions remained unchanged to permit mating with existing
equipment.
The dielectric constant of polyethylene is 2.3. Impedance of a 77-ohm air line is reduced to 51 ohms when
filled with polyethylene. Fifty-one ohms is still in use today though the standard for precision is 50 ohms.
The attenuation is minimum at 77 ohms; the breakdown voltage is maximum at 60 ohms and the power-carrying
capacity is maximum at 30 ohms.
Another thing which might have lead to 50 ohm coax is that if you take a reasonable sized center conductor
and put a insulator around that and then put a shield around that and choose all the dimensions so that they
are convenient and mechanically look good, then the impedance will come out at about 50 ohms. In order
to raise the impedance, the center conductor's diameter needs to be tiny with respect to the overall cable's
size. And in order to lower the impedance, the thickness of the insulation between the inner conductor and
the shield must be made very thin. Since almost any coax that *looks* good for mechanical reasons just
happens to come out at close to 50 ohms anyway, there was a natural tendency for standardization
at exactly 50 ohm.
"Someone hands you a 1000 foot roll of coax cable and says to you " that's 50 ohm coax use it wisely". So to check out this "50 Ohm" assertion you put your ohmmeter on one end from center conductor to outer conductor (no connection at other end) and to your surprise it reads near infinite impedance! You then short one end and measure the open end with the meter. It now reads near zero ohms! " How can this be!" you ask yourself , "I was assured it was 50 ohm cable!" The reason your meter did not tell you that the cable was 50 ohms is that it could NOT read the Instantaneous voltage/current ratio( V=IR) and its own internal resistance is so high that it causes a very large time constant in the reading. You cannot use a typical ohmmeter to measure characteristic impedance.
So how can we measure the characteristic impedance of your coax? Instead of trying to use an ohmmeter we will use the circuit of figure 1. The circuit allows us to generate a pulse of current by toggling the switch with the asterisk indicating where one would measure the current. Now what happens? At the moment the switch connects battery (+) to the center conductor of the coax cable it starts to "charge up" that piece of coax, sort of like charging a capacitor. Then we discharge the cable by shorting center conductor to shield or battery minus. This provides our "pulse" of current. If you measure the current in the center conductor during the pulse generation it will reach a maximum value of Imax=Vbat / Zo, where Zo is the characteristic impedance or equivalently called the surge impedance of the coax. The question arises what properties of coax l! imit the inrush current to the _expression_ given above? Or stated another way why doesn't the coax charge up "instantly"?
To answer that question lets examine the way an ideal capacitor would charge up vs. coax. In theory , an ideal , discharged, capacitor would see an infinite current for zero time if you connect it to a perfect source ( a perfect source has zero internal resistance). Stated another way it charges up "instantly" to the applied voltage of the source. There are two crucial differences in the way a piece of coax charges up and a ideal capacitor would charge up when connected to a battery. First an ideal capacitor has zero inductance in the current path which would limit current inrush rate. Second an ideal capacitor has zero physical length so no there is no propagation in space of current pulse. Coax cable does not charge up instantly. This is because it has a finite series inductance per unit length, a capacitance per unit length and it has a physical length which all contribute to a propagati! on distribution in time and space of the current surge. In short the series inductance resists the flow of current that wants to charge the capacitance therefore causing propagation delay of the current surge. This propagation delay causes current surge to spread in time. Simultaneously the physical length creates a propagation distribution in space of the current surge. So instead of a infinite "impulse" of current in zero time and zero space, as in the ideal capacitor, the current quickly rises to a maximum and starts propagating down the coax. The speed of propagation is always less than the speed of light and depends on the materials the coax is made from. The current surge method is not how one would normally measure coax cable characteristic impedance but it is a viable method and has an intuitive appeal. Another way to measure the characteristic impedance of coax cable is to measure its inductance and capacitance per unit! length , the square root of L divided by C will be in ohms(no! t farads or henrys) and will be equal to the characteristic impedance. So why do different cables have different characteristic impedances? Every coax cable or other transmission media has its own unique capacitance and inductance per unit length. For coax cables this will l be determined by inner/outer conductor ratios and the dielectric constant of the material between conductors for coax cables. For microstrip lines its primarily the trace width, dielectric constant of the pc board and thickness of PC board.
Okay so now maybe "50 Ohm" cable makes some sense and you are now a "50 ohm" systems zealot. You now strive for "perfect 50 ohms" in all your 50 ohm system work. You have become so unreasonable that you insist that all systems be EXACTLY 50 ohms. Now you are in trouble. In truth no coax, connector, amplifiers etc. is EXACTLY 50 ohms. The fact is that it is amazing how far off 50 Ohms you can be in your designs and not see that much degradation in performance! We need a way of expressing how close we are to 50 Ohms in our designs and systems. The most common way of doing this is VSWR or Voltage Standing Wave Ratio. So before we can discuss your fervent pursuit of "pure" 50 ohm systems we need to understand the concept of VSWR. In fact VSWR calculations will work for ANY characteristic impedance 50 ohms or otherwise.
Suppose you took your 100 foot roll of 50 ohm coax and cut a 20 foot chunk off it. Now connect one end to a real nice RF type signal generator that we will assume has perfect "50 ohm" impedance and leave the other end of the coax chunk open. Set the generator frequency to say 50 MHz , although just about any frequency will work, 50 MHz is a good spot for most coax. So now we have our "50 ohm " generator supplying a 50 MHz sine wave to a "50 Ohm" piece of coax with no connection at the other end. What happens?
Well here is what happens: The sine wave when FIRST applied to the cable, starts to "propagate" towards the open end of the cable. When the sine wave gets to the end of the cable it COMPLETELY "reflects" , turns around, and heads right back towards the generator! Once inside the generator it "dissipates" itself as heat in the generators internal 50 ohm resistor. Now we do the same experiment except we short the other end of the coax cable. Again we would see a complete reflection of the sine wave and total dissipation of reflected wave within the internal 50 ohms of the generator. So if the cable end is open or shorted we get TOTAL reflection of our applied sine wave. This is defined as a VSWR of "infinity to1" Now connect a "perfect" 50 ohm resistor at the end of the coax line. In this case we TERMINATED the cable in its characteristic impedance. The applied sine wave will COMPLETELY dissipate in! this termination and there will be zero reflection back towards the generator. Why? Because we fooled the sine wave into thinking it was traveling down a "infinite" piece of cable therefore no reflection."
