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.