I am working on designing a helical antenna and have been using the book Antennas for all Applications as my primary source of information. In the book, Dr. Kraus describes a method of converting the ~150 ohm impedance of the antenna to 50 ohms at the coaxial cable interface. This is done by gradually flattening the antenna's copper tube as it approaches the ground plane, then precisely spacing the flattened end of the tube from the ground plane with a dielectric. The coaxial feed line is inserted through a hole in this dielectric and soldered to the end of the flattened tube.

That sounded easy enough to do, but I have run into a roadblock. In the book, Dr. Kraus provides the following formula for calculating the thickness of the dielectric needed:

where h is the thickness of the dielectric, w is the width of the flattened tubing, ε_r is the relative permittivity of the dielectric sheet, and Z_0 is the characteristic impedance of the dielectric sheet.

I cannot figure out how to calculate the characteristic impedance of the dielectric sheet. I have only basic knowledge on this subject, and I am only really familiar with what impedance is and why impedance matching is important in antenna designs. Beyond that, though, I really have no clue what I'm doing. If it helps, here are the values I am using for the other variables:

w = 8.1966 mm
ε_r = 2.1

While my ultimate goal is getting the correct characteristic impedance so I can proceed with my design, I would also like to learn the proper way of calculating it in the first place. I would greatly appreciate any explanations as to how this is done!

 

The charactertistic impedance could be for the antenna or the cable or it could be a combination of the two. I don't have that specific book by Dr. Kraus, but I do have his original work on Antennas. I would read the section of the book closely to see if you can glean any more information. Taking a shot in the dark I would take the geometric mean of the two impedances {50, 150} and use that as a starting point. Then I would recompute with the values 50 and 150 to see how much difference there is in the theoretical value. You can do this easily with any spreadsheet program. I used LibreOffice.

I tried this experiment and I didn't get the result I expected. In fact for 150Ω I got a negative thickness. I'm going to revise my guess and say that Zo should be the 50Ω characteristic impedance of the coaxial cable.

For 50Ω I get 2.56 mm.
For 86.6Ω (The geometric mean of 50 and 150) I get 8.16 mm.

I conclude that this function for thickness is very sensitive to the value you plug in for Zo, and it is highly non-linear in the vicinity of 50Ω. I don't know if you can fabricate material to 0.01 mm tolerances but I guess your going to learn -- and fast.

 

I went back the section of the book I copied the equation from and lo and behold there was an example of a worked out problem right below it. The value used in the example was 50 ohms. So it would certainly appear that Dr. Kraus is referring to the impedance of the cable. That makes a lot more sense than what I was thinking, which was that I needed to calculate the impedance of the dielectric itself (I'm not even sure if that is possible without directly measuring it).

Looks like I was just getting confused over nothing. Thanks for pointing me in the right direction!

As far as tolerances go, I'm planning to get some 0.01" thick PTFE shim stock and layer that as necessary. Considering 2.56 mm is almost exactly 0.1", I should be able to get very close.

 

I went back the section of the book I copied the equation from and lo and behold there was an example of a worked out problem right below it. The value used in the example was 50 ohms. So it would certainly appear that Dr. Kraus is referring to the impedance of the cable. That makes a lot more sense than what I was thinking, which was that I needed to calculate the impedance of the dielectric itself (I'm not even sure if that is possible without directly measuring it).

Looks like I was just getting confused over nothing. Thanks for pointing me in the right direction!

As far as tolerances go, I'm planning to get some 0.01" thick PTFE shim stock and layer that as necessary. Considering 2.56 mm is almost exactly 0.1", I should be able to get very close.

I'm not sure layering will do what you want. No matter how you compress the stuff it is not the same has having a homogeneous piece. You should be able to mill a piece to that tolerance.

 

I'm not sure layering will do what you want. No matter how you compress the stuff it is not the same has having a homogeneous piece. You should be able to mill a piece to that tolerance.

That hadn't occurred to me. If layering the PTFE won't work then I'll just have to get a solid piece milled. Thanks for pointing that out!

 

That hadn't occurred to me. If layering the PTFE won't work then I'll just have to get a solid piece milled. Thanks for pointing that out!

How were you planning to hold the pieces together? I hope it was not glue!!

 

How were you planning to hold the pieces together? I hope it was not glue!!

Hahaha no not glue. I'm using four small bolts that will hold the rf connector and the dielectric to the ground plane. Here's an image of what I've got right now in Solidworks. The weird rectangle at the end of the tube is just my lazy way of approximating the tube being flattened at the end. I also haven't modeled in the actual nuts and bolts yet.

 

I think you are better off with a solid piece. Trying to drill several pieces accurately will be a challenge.

 

Well, the dielectric will be a solid piece now. That's the white piece in the top image. The two plates in the bottom image are going to be cut from aluminum left over from fabricating the ground plane and only serve to properly space the rf connector and establish electrical continuity between the ground plane and the body of the rf connector. They can be drilled at the same time (one plate on top of another) and then cut to size afterwards. They aren't going to have terribly tight tolerances anyway, and I should have enough scrap aluminum to make at least 8, so I'll have pleanty of tries to get it right. If it comes down to it I could have someone do them on a 3-axis CNC mill but I don't think that will be necessary.

 

Well, the dielectric will be a solid piece now. That's the white piece in the top image. The two plates in the bottom image are going to be cut from aluminum left over from fabricating the ground plane and only serve to properly space the rf connector and establish electrical continuity between the ground plane and the body of the rf connector. They can be drilled at the same time (one plate on top of another) and then cut to size afterwards. They aren't going to have terribly tight tolerances anyway, and I should have enough scrap aluminum to make at least 8, so I'll have pleanty of tries to get it right. If it comes down to it I could have someone do them on a 3-axis CNC mill but I don't think that will be necessary.

I was talking about using 100 or so .01" thick pieces to make a 0.1" dielectric and having to drill through such thin material one piece at a time or holding 100 of them together while trying to drill through all of them. I don't know if you have ever tried to drill thin material but it is extraordinarily challenging. I prefer to use a punch and a hammer. It takes some skill to make the punch out of tubing, but at least it doesn't tear up the material like a drill bit.

 

Tap the holes in your ground plane so you don't have to use nuts on the coil side. Have the screws protrude approx the thickness of your teflon dielectric and they will constrain the teflon. To machine teflon, you need to freeze it (get a can of freeze spray), then quickly do your milling/drilling operation.

I take it that this is an end-fire helix?

And Zo refers to the characteristic impedance you are shooting for. In this case, 50Ω.

 

I was talking about using 100 or so .01" thick pieces to make a 0.1" dielectric and having to drill through such thin material one piece at a time or holding 100 of them together while trying to drill through all of them. I don't know if you have ever tried to drill thin material but it is extraordinarily challenging. I prefer to use a punch and a hammer. It takes some skill to make the punch out of tubing, but at least it doesn't tear up the material like a drill bit.

Ah ok, that makes more sense. I haven't worked with plastics this thin before, so I don't know what to expect as far as machining goes. Usually when I am drilling really thin stuff I like to sandwich it between some sacrificial wood pieces to help prevent tearing, but with such thin plastic that might not actually work well. I like the idea of making punches to cut out the holes though. I'll try that out sometime if I do end up needing to put holes in thin plastic.

Tap the holes in your ground plane so you don't have to use nuts on the coil side. Have the screws protrude approx the thickness of your teflon dielectric and they will constrain the teflon. To machine teflon, you need to freeze it (get a can of freeze spray), then quickly do your milling/drilling operation.

I take it that this is an end-fire helix?

And Zo refers to the characteristic impedance you are shooting for. In this case, 50Ω.

The ground plane is only 0.05" thick so I'm not sure how well it will tap. It might make more sense to use countersunk screws in the PTFE. Is protrusion of metal above the PTFE going to have a significant impact on the antenna?

Yes, it is a standard end-fire helical antenna sized at 915 MHz with 5 coils.

 

The ground plane is only 0.05" thick so I'm not sure how well it will tap. It might make more sense to use countersunk screws in the PTFE. Is protrusion of metal above the PTFE going to have a significant impact on the antenna?.

The problem with that method is that teflon will flow, which will give you ground problems from loosening. You also want all the protrusions to be as small as possible, if not smaller. You could increase the ground plane thickness to make it possible to get good tapped holes.

 

The problem with that method is that teflon will flow, which will give you ground problems from loosening. You also want all the protrusions to be as small as possible, if not smaller. You could increase the ground plane thickness to make it possible to get good tapped holes.

Would it be possible to use thin hex nuts on the dielectric side, with the PTFE sandwiched in-between them? They should actually be slightly shorter than the height of the PTFE.

 

Would it be possible to use thin hex nuts on the dielectric side, with the PTFE sandwiched in-between them?

No. The PTFE will flow under pressure.

 

No. The PTFE will flow under pressure.

I think I explained that poorly. I don't mean to say that the dielectric will be between the nuts and bolts, but rather horizontally between the four mounting nuts. Like so:

 

I think I explained that poorly. I don't mean to say that the dielectric will be between the nuts and bolts, but rather horizontally between the four mounting nuts.

That will work just fine, but I would put the heads on the top and the nuts/lock washers on the bottom.

 

That will work just fine, but I would put the heads on the top and the nuts/lock washers on the bottom.

Ok, I'll do that. Thanks for all the help!