**Developing the master equation**

For a given frequency and placement of each *LC* loading element on the flattop, we expect *Z1 + Z2 = 0* in order to cancel the reactive component of impedance along the wire (thus making it resonant). As a result, we arrive at our basic equation:

Now all that's necessary is to expand this basic expression into a five-equation set — one equation for each design frequency. Then we solve simultaneously for the four variables (*L, C, θ1, and θ2* ). Let's say the design frequencies *f1* to *f5* are 3.5, 7, 14, 21, and 28 MHz, respectively, which correspond to those used in the Amateur Radio Service. At this point, the practicing engineer needs to realize that the electrical lengths of *θ1* and *θ2* will be directly proportional to the frequency of operation. Thus the five-band equation set becomes:

Now solving for four unknowns simultaneously in our five-equation set is still difficult to do by hand. In practice, we add a dummy variable *E* in the first equation, making the righthand side equal to 1 + *E* , to solve formally for five unknowns in five equations and to measure how close the nonlinear set of equations is to providing an exact solution, (assuming there is a solution for this case). For that part, we turn to a piece of $12 software: Systems of Nonlinear Equations, v1, by Numerical Mathematics.

**Completing the solution**

This technique adequately addresses antenna feedpoint impedance, though it can't be seen directly. We address it easily because we need to operate the antenna on an odd-multiple of a half wavelength at all frequencies (a half-wave at 3.5 MHz, 3/2 wave at 7 and 14 MHz, 5/2 wave at 21 MHz, and 7/2 wave at 28 MHz). Thus we virtually ensure that the dipole's feedpoint impedance will be in the general range of 75 ohms at its fundamental frequency and otherwise below a maximum 3:1 SWR at any design frequency, which traditionally defines adequate matching to meet the requirements of typical transmitter/receiver circuitry. So all that's left to do is solve for the unknown variables, consistent with a practical length for the antenna.

Thus, we solve for the unknowns consistent with the condition that *θ1* + *θ2* < 1.57 radians, which ultimately describes a quarter-wave transmission line segment (or a half-wave antenna) at 3.5 MHz. Now, we can directly confirm that, without loading elements, the absolute maximum antenna length 2(*l1* +* l2* ) will be about 139 feet. To find that, we set *θ1* and *θ2* = 1.57. (Also, note that radio amateurs still largely work in feet.) Alternatively, we can adopt the time-honored value of 135 feet that's been applied over the years for general long-wire antennas.

The numerical software evaluating the five-equation set and some practical tweaking yields *L* = 10 μH and C = 33 pF or so for a cw of about 500 ohms (the antenna would be about 30 feet off the ground), with *θ1* = 0.993 radians and *θ2* = 0.381 radians (total equals 1.374 radians). Thus the actual antenna length will be in the range of 118-122 feet, e.g., 135 times (1.374/1.57) = 118. The actual length after pruning is about 120 feet. The antenna's measured component and SWR values (though, curiously, not feedpoint impedance at the higher frequencies as directly measured in several of these final designs) is closely in agreement with the well-known EZNEC antenna program for cut-and-try. The antenna's on-the-air performance has been exactly as expected.

The aforementioned design method opens up the way for predicting by *physical inspection* if an antenna for a given set of frequencies is likely to work. You basically need to know the values of *L* and *C* and the antenna's total length. We'll leave that discussion for another time.

But the greatest lesson for a young engineer comes right now: Make certain you build and carefully check your design to verify it works. That seems obvious, but one surprise in this design example came from a rather scholarly paper for antenna design that addressed lumped-circuit techniques. The technique may ultimately succeed, but it fell short in its initial try, because its computational results matched the stated performance for an antenna that in reality never met its design claims.

for multiband trap designs.

**Related posts:**

Interesting series.

@Sadie!—Glad you like it

I have been following the three articles on the method, and I would say that it is quite interesting and it is been put in the simplest way possible for any person like me who is not a lover of calculation. Through your experience, the previous two articles have acted as steps to understanding the development of the master equation. The idea behind the method is applicable in any field of knowledge and understanding of the basics is a great help to application of real life situations.

Thank you very much Vincent for bringing out this equation and for explaining it into detail. It has been always in mind but I have not taken my time to explore it much. It is simple they way it is arranged considering the five-equations, the problem that was giving me a problem was how to solve for the four unknown given five equations. The idea of further introducing the fifth unknown is good and makes it sound simple.

Special gratitude for this post and for the revolutionary MacGyver`s methods that will help greatly in creating ease for young upcoming antenna engineers. This method, presents a perfect way of addressing the antenna feed point impedances. By addressing this issue, engineers get to do the appropriate matching in their design so that they can come up with designs that measure up to the standards of a regular transmitter or receiver circuitry. The ability to make predictions for the likelihood of varying sets of frequencies to operate has also been made very possible through this mathematical procedure.

@Vincent–as I mentioned in my comment to part 2, vehicular mobile radio antennas pose certain challenges of getting a reasonable pattern and gain and efficiency from a relatively short antenna on a car. Additional complexities come in wherein the trunk lid or roof acts as a ground plane, and at low enough frequencies it is too small, and the real behavior is more complex than expected.

@Vincent–it is hard to argue with a few pages of hand written equations plus a $12 solver versus a commercial package, like AWR Microwave Office, which might run $25,000 per seat (note–I'm not current on AWR pricing so this is just meant for a relative comparison). AWR and similar products do have some nice features like sliders to vary Ls and Cs in a design. For some “what if” days at your desk, a package like that could pay for itself. But if you need a basic wire design, it is overkill in most cases.

Hi Vincent–last comment on your series–it is a bit risky to design antennas with no way to effectively measure them. More than one budding EE has looked at a solution on a network analyzer and the VSWR looks good. Trouble is, there might be losses, or the pattern of the antenna may be very unsuitable, etc. In a lot of the antenna world these days (like for mobile devices and other small antenna environments) you cannot be a supplier without 3D chambers to measure pattern and efficiency, plus simulation results, VSWR charts, efficiency vs. frequency, and gain vs. pattern.

For your example, do you know what the antenna pattern is at the various harmonics? The antenna made from microstrip transmission line I mentioned in part 1 had multiple resonances as well, but some of them have undesireable patterns. In fact, my Sr. RF almost tossed the design in the trash because his initial simulation didn't show a good pattern at the target (lowest resonance) frequency. He gave me a stack of simulations across ~2 GHz to 6 GHz, including return loss and patterns. I happened to notice a higher resonance that had good gain and nice omni patterns. When I showed this to him, we had a working antenna very soon after. So one of your points is very important–most resonant circuits as well as intentional radiators will be resonant at more than one frequency. This is a degreee of freedom rarely exploited in most antenna designs.

Vincent has come up with a very good initiative by bringing up this equation. Breaking it down to make it easier to solve mathematics problems has made work much easier. The fact that you have developed the master equation and broken it down to a five band set equation makes everything to be so simplified. They are quite easy to solve when simplified. Addressing antenna feed point impendence has been achieved by easily completing the solution at hand. This has also helped adequately in predicting of the frequencies that can be used with specific wavelengths.

Many people are not used to deriving equations from a simple scratch formula to get to the solution may be because they get confused along the way or they are just lazy of doing. Due to technological improvement and various developments in the world of software engineering, calculations have been made easier. I think this may be helpful to some extent but I rather prefer manual derivation to that which ahs be simplified by apps.

@SunitaT0: Thank you. The value for the dummy variable, E, in this case was 0.63 or so. Thus there is no exact solution to our original equation set, as we might expect when we encounter nonlinear equations especially. This class of multiband antenna is often said to be a “compromise” antenna, and our value of E bears that statement out. Although we can say this antenna is a far better “compromise” than previous types, many of which did not function nearly as well as claimed.

As to the math problem you encountered, four unknowns in five equations implies one redundant equation, assuming of course the set has a solution. Thus in solving for the variables, we also selected several sets of four equations to solve for four unknowns. In many of those cases, the values of L, C, theta 1 and theta 2 in the “4-by-4” sets converged towards the same corresponding numbers as for the situation with five equations and five unknowns. How might one select the “right” four equations of five to end up with virtually the same answers as a “5-by-5” set? I leave that proof to an experienced mathematician.

“I think this may be helpful to some extent but I rather prefer manual derivation to that which ahs be simplified by apps.”@yalanand: Is that because of the fun and enjoyment you derive from the process od manual derivation? Or does it have to do with the fact that you simply don't trust the automatic derivation that does not involve you at all?

” For some “what if” days at your desk, a package like that could pay for itself. But if you need a basic wire design, it is overkill in most cases.”@eafpres1: I think that's a tricky situation. You can't always tell the level of sophistication of an experiment you're undergoing because the outcome often determines the worth. For instance, you may be randomly experimenting and the results might prove to be a huge success. Perhaps this success can at times be downplayed if you hadn't used a good software package while working with it.

@tzubair–you are of course correct. This is the challenge of applied R&D. The management wants it to pay for iteself. But the R part of R&D is inherently unpredicatable. In my experience, providing the teams with the best possible tools got more results. However, as Vincent has so clearly brought out here, just paying for expensive simulation tools does not help if the engineers using them don't really understand what is going on.

In my career I have been lucky to have many really smart engineers who could use simulation to extend their minds and come up with solutions impossible by other means.

@Vincent–I think you are stating a very important point–nonlinear systems can have stable points or they can be unstable. I use that term loosely. For example, it is possible to use advanced EM simulation to “design” wire antennas that have mult-band performance and good efficiency. However, I've seen very few cases where the engineers ran Monte Carlo simulations on top of the EM simulation to test the sensitivity of the design. I liken this to your dummy variable, or the alternate sets of equations. If you find, as you did, that things generally converge in the same direction then you probably have a stable design. But if you find that little changes in the dummy variable or the construction of the chosen set of equations lead to radically different designs, I would want to look a sensitivity of any of the solutions.

I've seen people publish “antennas” that look like a bunch of bent and tangled paper clips. The trouble is if you shorten one wire by 0.25 mm, or move the position of one wire by 0.5 mm, the thing may not work at all.

Thanks again for a very elegant series of articles.

@eafpres1—Yes, multi-frequency operation has its advantages and drawbacks. And beyond what antenna analysis programs like EZNEC provide, professionals and amateurs alike also face the task of making representative measurements. In the absence of an antenna range, which very few noncommercial users have access to, there are fortunately still decent evaluation techniques. Radio amateurs, for example, informally but usually effectively measure antenna response for various (unwanted) harmonics by contacting nearby amateurs 1-5 miles away to detect those outputs. Experience shows that if signals from, say, a 100-watt transmitter are no higher than S-1 or -2 at some harmonic, the amateur is not likely to radiate sufficient energy at those frequencies to interfere with communications at a significant distance.

Although a dipole antenna (as in this case) has a broad radiation pattern, we can also gauge overall pattern and efficiency by who you can hear and talk to. If you can work Egypt from New York with 35 watts and a simple wire and Spratly Island with 100 watts, even if their received signals are at S-2 or -3, odds are that your installation is very efficient. On the other hand, you'll know you're not as good as can be if you can't even hear those stations (assuming decent band conditions) and the antenna itself seems to generate more noise than your other dipole antennas. That indicates possible feedline radiation and a skewing of the radiation pattern, and thus the need for a choke balun at the antenna's feedpoint.

As you've noted, SWR measurements wouldn't likely alert you to the noise problem, and in a few cases the SWR measurement for the antenna itself might be in error if somehow the feedline added to the effective antenna length.

The one scenario that is highly likely to cause trouble in this multiband dipole is to drive it from a low-cost transmitter whose design uses frequency-doubling and -tripling stages. A transmit signal derived from, say, a 7 MHz oscillator stage for your intended operation at 21 MHz might well generate sufficient unwanted energy simultaneously at 14 and 28 MHz and be heard thousands of miles away.

It sounds like antenna performance would be changed depending on vehicle roof type. Location of antenna and design of antenna board including GPS and XM affects the performance. I guess that MacGyver's method would be extended for more antenna design.

@DaeJ–you are correct. For higher frequencies, such satellite radio (2.5 GHz), and GPS (1.5 GHz), the main things affecting the behavior of the dielectric loaded patch antenna is the module into which it becomes a part. Each module is tailored to exactly fit the roof curvature of the vehicle for which it is designed. Thus, the patch antennas are generally tuned to a fixture using the production-intent base plate and plastic housing etc.

For lower frequencies, especially FM Radio, the effects are so significant (due to the ground plane of the car being relatively small compared to the wavelength) those antennas are tuned to the car, incluing the matching and amplifier circuits. So if you moved a module w/short (say, < 300 mm) FM antenna to a different car model, it might not work as well.