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ham_radio:impedance_along_matching_line [2013/01/08 19:00] (current)
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 +Dear Brian, Jerry & The uw Group,
 +INTRODUCTION -- Brian has asked an interesting and provocative question that gets down to some basics of a peculiar type of transmission line system (full question repeated below), "what is the VSWR and voltage on a 1/4 wave matching transformer that has a four 50 ohm loads?" ​ As usual, there is more going on than first meets the eye.
 +THE MATCHING SECTION -- The four loads of 50 ohms each combine to create an output load of 50/4 = 12.5 ohms.  We have often been told that a single section type of 1/4 wave matching transformer should have an impedance of Square-Root(Zin x Zout). ​ In this case that's (50 x 12.5)^0.5 = 25 ohms.
 +HOW DOES IT WORK? -- A 1/4 wave long matching section will act like an "​Impedance Inverter"​. ​ It is a 25 ohm line looking into a total load of
 +12.5 ohms (a 2:1 VSWR), therefore it presents an input impedance of twice it's Zo (25 ohms) = 50 ohms.
 +However, this is a 2:1 VSWR on a 25 ohm line, not a 50 ohm line -- that makes a difference.
 +WHAT VOLTAGE and CURRENT? -- Ordinarily, a 50 ohm line with a 2:1 VSWR would have a voltage (and current) that is 33% higher than normal [1.33 volts or amps relative] somewhere on that line, as well as 33% lower than normal [0.66 volts or amps relative] somewhere else, if it is at least 1/4 wave long, and resistively terminated. ​ The 2:1 VSWR is the result of the 1.33/0.66 voltage (or current) ratio = 2.0.
 +WHAT'S THE REAL VOLTAGE -- But, in this case the input voltage on the 25 ohm line will be 1.0 volts (relative), and the output voltage will be 0.5 volts relative -- still a 2:1 VSWR.  To be specific, if you are running a kilowatt CW on a 50 ohm line, the voltage [& current] will be 223.6 volts [4.5 amps]
 +(rms) on your 50 ohm transmission line; there will be 223.6 volts [4.5 amps] at the input end of the your 1/4 wave 25 ohm matching transformer,​ and there will be 111.8 volts [8.9 amps] at the output end.  The absolute magnitude of the voltage along the 25 ohm line will taper with a waveform that will look like a COSINE on a Pedestal; the upper portion will look like one half of the COSINE Wave from 0.0 degrees to 180 degrees. ​ The waveform is illustrated in Theodore Moreno, "​Microwave Transmission Design Data", Dover Publications,​ 1948, p 22, Fig 2-5, figure E.
 +IT'S EASIER THAN YOU THINK -- The bottom line of all of this is you do not have to protect the system from a high voltage condition. ​ You only have to consider that the Current on the 25 ohm line will be twice as high at the output load end.  But, that almost doesn'​t matter. ​ A 25 ohm line has a fatter center conductor, and it can easily tolerate twice the current. ​ By the way, you can synthesize a 25 ohm line by putting a pair of 1/4 wave long, 50 ohm lines, in parallel.
 +VOLTAGE IS NOT A PROBLEM -- My good friend Hank Keen, W2CTK (sk) has made this measurement many times: Pulses of RF applied to a Type N connector with a Polonium Source nearby (a source of ions) will not arc-over until you exceed 20 kW (1,000 volts rms).  Air at sea level is good for (has a Dielectric Strength of) 30 kV/cm = 76 kV/​inch. ​ Also, moisture in the air (high
 +humidity) does not change the Dielectric Strength. ​ Almost all connectors fail from current overheating the center pin, not from arc-over.
 +LOSS IS ALSO NOT A PROBLEM -- Your matching transformer is only 1/4 wave long, and your connectors are much-much shorter yet.  Unless you use resistive materials, or insulators that have a big "Loss Tangent",​ you can only create RF loss by using transmission line materials that have a significant length.  ​
 +There are many mis-informed amateurs who believe each UHF Connector has 0.5 dB of loss at 432 MHz -- this is crazy. ​ Yes, they may have a 1.15:1 VSWR.  ​
 +This really is not a "​Loss"​ mechanism, it a "​Reflection"​ mechanism. ​ The true
 +432 MHz insertion loss (of a good brand) is about 2 watts out of 1,000 -- that's about 0.009 dB each.
 +73 es Good High Power DX, \\
 +Dick, K2RIW
ham_radio/impedance_along_matching_line.txt ยท Last modified: 2013/01/08 19:00 (external edit)
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