First some information:

Transmission lines have currents that flows in two directions the vector sum of which should be zero. If the sum is not zero, there exists what is known as a common mode and this common mode will radiate and distort the pattern of the antenna connected to it.

That being sed there are some situations where transmission lines that are not matched are useful. As one moves away from a mismatched termination the impedance varies as one moves along the transmission line, however in no case will it ever be equal to the characteristic impedance of the transmission line. The impedance will vary in value from that of the termination to that given by

ZT=Z0*Z0/ZL

where
Z0 = the characteristic impedance of the transmission line,
ZL = the value of the termination and
ZT = the extreme impedance transformation.

The value of this impedance will repeat at electrical distances of 1/2 wavelength down the transmission line and will include complex values resulting in capacitive and inductive values. At an electrical distance of 1/4 wavelength from the termination, there will be a maximum transformation.

A practical use of this concept is as an impedance transformer, where a 1/4 wavelength of transmission line of an intermediate impedance may be used to match an antenna to a transmission line of a standard impedance. For instance a 1/4 wavelength of 75 ohm transmission line may be used to match a 100 ohm antenna to a 50 ohm transmission line.
A short piece of transmission line that is terminated in a short circuit at one end may be used as an inductor with the inductive reactance reaching a maximum when it is 1/4 of a wavelength long. Similarly, if the same piece of transmission line were terminated in an open circuit, it would behave as a capacitor with the capacitive reactance decreasing with length.

This concept of a transmission line impedance transformer is what we use to make it possible to feed the zeppelin antenna with 50 ohm coax.

Below is a basic zeppelin antenna showing the currents on the wires.
 

The overall length of the antenna is 3/4 of a wavelength long, now if the first 1/4 wavelength is balanced and no radiation is emitted from that portion of the antenna we end up with a radiating antenna that is 1/2 wavelength long. Yes this zeppelin is just a end feed 1/2 wavelength antenna.

Note the first 1/4 wavelength. The currents on the long wire and the short wire are equal and opposite in phase. This makes this 1/4 wavelength section of our antenna a transmission line which does not radiate. We hope!

This first 1/4 wavelength section is also our transmission line impedance transformer used to transform the approximately 2000 to 5000 ohm at the end of the 1/2 wavelength antenna to the desired 50 ohm impedance of our coax.
Lets assume the impedance of our 1/2 wave antenna presents 4500 ohms at the end ( the point at which the 1/4 wavelength balanced line will end) and we wish to feed the antenna with 50 ohm line then a little math will give us the needed balanced line impedance. Where did I get the 4500 ohms you say? Would you believe I have no idea, it is just a number I remember from days long past and it has served me well, and I have long forgotten the formula for calculating it, so there.

Using the formula  ZT=Z0*Z0/ZL from above rearranged just a little we can solve for our transformer.
Z0 = sqr(ZT*ZL)  =  sqr(50*4500)  =  sqr 225000 = 474.34  or about 474 ohms, there is always room to adjust and indeed you will need to adjust due to conditions anyway so don't sweat the small stuff for this type of antenna.

Now you say where do I get this 474 ohm balanced line? Well you make it!
This antenna is just 2 lengths of wire. One is 3/4 wavelengths long the other is 1/4 wavelengths long. The 1/4 wavelength portion of the antenna is the balanced line that we want to be 474 ohms and here is how we are going to do that.

First we select the wire for our antenna. It must be of sufficient size to hold itself up without breaking and be a good RF conductor, copper or aluminum are good choices. Copper  16 or 14 awg are good or 19 to 14 awg aluminum fence wire will work just fine also. That is the easy part now for the tuff part, more MATH.

The formula for open wire parallel line is
Zo = 276 * Log10(2*s / d)
where
Zo =  the impedance of the transmission line
Log10 = Log base 10
s = wire spacing center to center
d = wire diameter

Now a 1/4 wavelength of open wire parallel line impedance will be about 80 percent ( derived from the velocity factor for open wire line made from insulated house wire )  of the calculated surge impedance. That is if the line is 400 ohm ( calculated ) line then a 1/4 wavelength section will present an impedance of about 320 ohms.  So we are looking for about 474 ohms therefor we need to add 20 percent to 474 which will be about 569 ohms.
 

OK lets run one up the flag pole as they say. Using the above info lets start with these assumptions.

Lets use 14 awg insulated house wire ( dia. is 0.064 inches not including the insulation ) and  a spacing of 5 inches
Using
Zo = 276 * Log10(2*s / d)
we get
Zo = 276 Log10(10/0.064)  =    Zo = 276 Log10 (156.25)  =     276*2.1938 = 605.5 ohms (line surge impedance)
605.5 * 0.8 = 484 ohms (1/4 wavelength line impedance)

Remember we needed 474 ohms to match our 1/2 wavelength end fed antenna, well we missed it by 10 ohms, what to do? Just put the thing up and tune it that's what. You will need to adjust due to conditions anyway so don't sweat it. It will tune up!

The above math is not exact nor is it the correct way to calculate the complex  impedance for this antenna, however it is a way to get a working antenna with the least amount of head scratching and the end result is what counts. With very little effort this antenna will match 50 ohm coax with a less than 1.5:1 vswr.

If you look at the construction article for the 6 meter antenna you will notice the spacing of the parallel line that for antenna is not 5 inches but 3 inches!
Then I tell you the spacing is critical! So you say, first you say don't change the spacing then you change it from the caculations above what gives Ben?
Well folks 5 inch spacing is a little wide for a stable line so I just made it 3 inches to make the construction a little easier. This changes the impedance of the line but not enough to be a problem. If you do the math you will find the final value to be 435.4 ohms verses 484 ohms still well within the tuning range of the antenna transformer combination.

The statement ' the spacing is critical for that model in the link above is valid!' the length / spacing of the short wire will get you close to the desired final dimensions as presented. In other words if you change the spacing the length of both wires must also be changed.
The dimensions as presented will require the least cut and try on your part to tune the antenna. So there.
The antenna in the above link has been proven in the field by at least 5 people I know of to date, so have fun.

How to tune the antenna:
(1) Put the antenna up in the clear about head high, adjust the long wire length for the lowest vswr at the center frequency you designed it for, you are looking for the lowest vswr you can get, not necessarily the best.
(2) Adjust the short wire length for the lowest vswr, should be 1.5:1 or less.
(3) Hoist antenna to operating position and have fun.

That my friends is all there is to the antenna. No magic, smoke or mirrors just basically ohms law at work.

Let me say again, this is NOT the proper or correct way to design this or any other antenna or transmission line impedance transformer. It is however A way to get THIS antenna to work and work well with a minimum of effort.
 
 

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