Iterations on Boom Length
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Why? The longer an LPA, the greater its gain. Not linearly so, however. Plots below quantify the diminishing returns. Click on thumbnails to see full size.

• Antenna: Log Periodic Sawtooth Array
• Frequencies: Elements = 16.2-46 MHz; Range = 17.6-30.4 MHz; Analysis @ 24.89 MHz
• Impedance: 200 to 300 Ohms
• Length: 10m through 20m in 1m steps
• Elements: Count = 15; LHA^* = 10.344° through °; Diam = 1.628 mm (14 AWG)
• Boom: Spacing = 200 mm, parallel; Diam = 2.588 mm (10 AWG)
• Segments: 120/Lambda
• Analysis: NEC2 via Nec2Go, version 2.1.24
• AGT: (both FS and over S-N ground)
  • 1.00 @ 10m boom length
  • 0.98 @ 15m boom length
  • 0.96 @ 20m boom length
• Download: ZIP Archive All 22 models in each of three formats: NEC, ANT & MAA

^* Leading Half-Angle. Ref. Definition

At Lambda/2 above Sommerfeld-Norton ground

Models over SN ground are not horizontal. Each is tilted toward ground at an angle. The angle varies in accordance with length of boom. Said angle afords each element its optimal lambda/2 height above ground. (thus to obtain a unified pattern across all frequencies of operation). Models verified by Average Gain tests performed at 0th, 5th, and 10th-file intervals.

Z_in near 200 Ohms. Gain 10.26 dBi @ 27° Elev

Z_in between 200 and 300 Ohms. Gain 11.07 dBi @ 27° Elev

Z_in near 300 Ohms Gain 11.92 dBi @ 26° Elev

The above seems to suggest that Z_in is dependent on boom length. It is not. Alternately, one might suppose that Z_in maybe varies according to whether the saw teeth are sharp versus squat. Neither conjecture is supported, however, in comparison against these same three antennas when in Free Space (see further down). I am inclined to propose that Z_in varies instead a result of increased horizontality. This being due to increasingly shallower tilt angles needed for each element's lambda/2 height above ground.

  Boom Length = 10 m   10.26 dBi Azimuth Gain @ 27° Elevation; AGT = 1.00

  Boom Length = 11 m   10.24 dBi Azimuth Gain @ 27° Elevation

  Boom Length = 12 m   10.42 dBi Azimuth Gain @ 27° Elevation

  Boom Length = 13 m   10.67 dBi Azimuth Gain @ 27° Elevation

  Boom Length = 14 m   10.91 dBi Azimuth Gain @ 27° Elevation

  Boom Length = 15 m   11.07 dBi Azimuth Gain @ 27° Elevation; AGT = 0.98

  Boom Length = 16 m   11.34 dBi Azimuth Gain @ 26° Elevation

  Boom Length = 17 m   11.44 dBi Azimuth Gain @ 26° Elevation

  Boom Length = 18 m   11.67 dBi Azimuth Gain @ 26° Elevation

  Boom Length = 19 m   11.84 dBi Azimuth Gain @ 26° Elevation

  Boom Length = 20 m   11.92 dBi Azimuth Gain @ 26° Elevation; AGT = 0.96

In Free Space

Unlikely anyone could erect a 10m-long LPSA at free-space equivalent height above ground. As a contrast, however, it clearly reveals the effect of proximity to ground. AGT likewise performed at 5th-file intervals.

Gain 7.99 dBi

Gain 8.34 dBi

Gain 9.16 dBi

As seen above, Z_in remains between 200 and 300 Ohms on all three boom lengths. Z_in is therefor also not strongly related to a saw-tooth array's LHA^* measure.

^*Leading Half-Angle: a more visual stand-in for the somtimes cryptic sigma (relative spacing constant). Ref: LHA Definition & Examples

  Boom Length = 10 m   7.99 dBi Azimuth Gain @ 0° Elevation; AGT = 1.0

  Boom Length = 11 m   7.89 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 12 m   7.95 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 13 m   8.10 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 14 m   8.25 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 15 m   8.34 dBi Azimuth Gain @ 0° Elevation; AGT = 0.98

  Boom Length = 16 m   8.58 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 17 m   8.67 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 18 m   8.89 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 19 m   9.08 dBi Azimuth Gain @ 0° Elevation

  Boom Length = 20 m   9.16 dBi Azimuth Gain @ 0° Elevation; AGT = 0.96