In part 1, we introduced two simple problems that could each be solved with one basic equation vs. a lot of unnecessary mathematics and explanation. Now let's expand and use that idea to solve a real-world design of a resonant multiband single wire antenna. It's been a popular engineering textbook topic since at least the 1940s. But the books even now don't offer much in the way of a simple design technique.
Antenna work in general often leans on three field expressions relating to Maxwell's equations — one equation having to do with electric field intensity and the other two with magnetic field intensity. It takes a master practitioner to relate the various electromagnetic field variables directly to parameters familiar to the electrical engineer such as inductance (L ), capacitance (C ), and feedpoint impedance. As a result, the designer usually shifts to “cut and try” experimentation.
Still, the purists desire an elegant solution based on Maxwell's fundamental principles. But it's the hard way to do things, and despite some efforts and partial solutions in the form of infinite series equations that appeared in the 70s for this multiband antenna type, no complete start-to-finish technique emerged that I know of.
Today's single-wire designs largely use one or more pairs of L and C elements in parallel to resonate the antenna at up to five or six frequencies. Until recently, there remained four general flaws with present approaches: No simple, unified method to determine the value and positioning of the LC elements; very critical tuning for small changes in L, C , and specific location of those elements on the wire; feedline radiation issues often render incorrect measurements and misleading claims; and subsequent failure of the design even after extensive “cut and try” techniques to meet design objectives.
Two underlying drawbacks tie the designer's hands behind his back: The LC sections have most often been applied as “traps” to isolate half-wave sections of the antenna for a given frequency range of operation, thus killing the option to use them as general loading elements that can be placed anywhere along the antenna. Also, designers often resort to such band-aid measures as tacking on small lengths of wire along the antenna to get needed compensating capacitance or to use traps with special characteristics. But can, say, a five-band wire antenna work as designed using just two traditional LC circuits and no special construction?
Thinking on the same wavelength
Yes it can, for certain design-frequency combinations, and this technique will largely tell you what's possible and what's not. Here's how we reason it (see QEX Magazine, September/October 2010, pp. 28-31): The current and impedance distribution on a transmission line with standing waves is basically the same as on the antenna itself. Thus we can view the antenna as an extension of the transmission line — a common thought in antenna design over the years, but one rarely implemented in a simple way.
We therefore model the antenna wire (i.e., flattop) using the transmission-line stub equations. We use both open and shorted “stub” sections, and connect them to each side of the L and C loading elements. Then, at five spot frequencies, we equate the stubs' resultant impedance (a function of their length) to the frequency-dependent impedance of the LC “tuned circuit” to cancel the wire's reactance. From this basic equation, we find the required values of L and C, their relative position along the antenna wire, and the total length of the antenna.
The impedance of an inductor and capacitor in parallel is:
Second, we define the general impedance along the antenna wire as:
where cw = 138 log 4h/d is the characteristic impedance of a single wire over earth (i.e., the flattop modeled as a transmission line over ground). Now h is the wire's height above ground, d is the wire's diameter, and θ1 and θ2 is the line length in electrical degrees (or alternatively, radians). Looking at it from the “stub” perspective, cw j tan θ1 = cw j tan (2π l1 /λ) , where λ is the wavelength of operation, represents the reactance of a shorted section of transmission line extending from the antenna's feedpoint (considered the stub's shorted end) to the traps.
Similarly, –cw j cot θ2 = –cw j cot (2π l2 /λ) is the reactance of an open section of transmission line extending from the traps to the antenna's end insulators. Now l1 + l2 is the length of one side of the dipole, and so 2(l1 + l2) is its total length.
In Part 3 we will discuss the Master Equation development and we will complete the solution.