Antenna analysis programs provide compound conductor loads. A series inductance and resistance realistically models a coil inductor at a spot frequency. However, coil inductance varies with frequency due to self-resonance. The variation may be small enough to ignore over a narrow frequency range, but it can cause large errors for wideband models. Adding a parallel capacitance widebands the RL model. The inductance exhibited by an RLC network can closely match that of a coil below self-resonance.
Each element of this trapped triband Yagi has four loads shown as green dots. Each load consists of a parallel coil and capacitor with components LTRAP, CTRAP, and RTRAP. Inner and outer traps differ. The yellow trace is current magnitude. The red dot is the feed point. Boom length is about 16 feet.
I used this example to compare wideband accuracy for RLC and RL coil models. An automatic optimizer maximized forward gain and F/R at 14.2, 21.3, and 28.5 MHz. Notation: RLC model, RL model. For the RLC model, LTRAP = L. C depends on L, but as long as CTRAP > C, both CTRAP and LTRAP can be optimized along with the antenna dimensions. If CTRAP < C, fix CTRAP somewhat greater than C. Optimizing LTRAP alone compromises antenna performance only slightly. Build the trap with capacitance CBUILD = CTRAP − C.
I modeled the antenna with the AO Antenna Optimizer using 1″ lossless elements. Each trap used a 3″ diameter coil of #12 copper wire. After optimizing the design with RLC coils, I modified the traps for RL coils by using LTRAP = L, CTRAP = CBUILD, and RTRAP = R (21.3 MHz values). This compares results for the two coil models.
--------------- RLC ------------- --------------- RL --------------
Freq 14.2 21.3 28.5 14.2 21.3 28.5 MHz
Z 24.9+j0.0 23.0+j0.2 25.1+j0.0 292+j1072 23.0-j0.1 19.9-j373 ohms
Gain 5.20 6.07 6.48 -0.36 6.07 -16.98 dBd
F/R 19.90 20.02 11.51 -0.19 20.18 -6.85 dB
Loss 0.21 0.23 0.19 0.21 0.23 11.23 dB
RL results are realistic only at 21.3 MHz. The following L graph for the inner coil shows why.
The following L and C graphs for the RLC model are nearly constant.
These graphs compare R and R:
First, automatically optimize the antenna design, including LTRAP and CTRAP, using an estimate for RTRAP. Next, adjust dimensions in COIL at 21.3 MHz while displaying RLC values to make L equal to LTRAP. Then use the COIL optimizer to minimize R while keeping L constant. Set RTRAP to R. If C > CTRAP, increase and fix CTRAP. Then iteratively reoptimize the antenna and coil designs until LTRAP is close to L. A couple iterations should do. Fix LTRAP as L, set RTRAP to R, and do a final antenna optimization.
R varies much less than R, but it is not constant. Determine final performance figures using the correct RTRAP values at 14.2 and 28.5 MHz. EZNEC can do this automatically by varying RTRAP as the square root of frequency.
The optimized traps resonated at 26.8 and 39.9 MHz. I expected something much closer to 21.3 and 28.5 MHz. I changed CTRAP to provide the expected resonant frequencies and then reoptimized the antenna design with LTRAP and CTRAP fixed. This compares results with the previous results for optimized traps:
--------- Optimized Traps ------- ----------- Fixed Traps ---------
Freq 14.2 21.3 28.5 14.2 21.3 28.5 MHz
Z 24.9+j0.0 23.0+j0.2 25.1+j0.0 24.4+j0.1 25.0+0.0 25.0+j0.0 ohms
Gain 5.20 6.07 6.48 4.41 5.55 6.94 dBd
F/R 19.90 20.02 11.51 19.55 24.34 12.14 dB
Loss 0.21 0.23 0.19 0.85 0.49 0.04 dB
Optimized traps provide significantly more gain at 14.2 and 21.3 MHz. The 1988 edition of the ARRL Antenna Book notes that some commercial antennas have trap resonances far outside the operating bands. Evidently the trick the optimizer discovered has been known for some time. The book says elements may resonate as an electrical 3 ∕ 2 or 5 ∕ 2 wavelength on the higher bands. The tribander element currents suggest that is the case.
Current in the outer trap is very small at 28.5 MHz. Still, this is above the coil self-resonant frequency, which may degrade model accuracy.
C can greatly increase when optimizing a coil. This may necessitate fixing CTRAP. Limit coil diameter to limit C.
Antenna programs model loads as geometric points, but real coils have nonzero length and diameter. A realistic antenna model must account for this.
Automatic optimization of LTRAP and CTRAP is possible only because CTRAP can assimilate C. A loading coil has no parallel capacitor to provide this function. Since C depends on L, independently optimizing either will degrade model accuracy. Instead use an iterative procedure similar to that for traps. Fix CCOIL and RCOIL and automatically optimize the antenna model, including LCOIL. Iteratively update CCOIL and RCOIL using optimized COIL results that maximize Q. Stop when LCOIL is close to L. Fix LCOIL as L, set CCOIL to C and RCOIL to R, and do a final antenna optimization.
88–108 MHz