# WAOK-AM Uses Method of Moments Proof

The antenna system employs Kintronic Labs VSU-1 voltage sampling units.

The characteristic impedance of the sample lines was also measured using the following mathematical expression:
Zo = √( √(R1² + X1²) x √ (R2² + X2²))

With R1 + X1 equal to the measured impedance at the +45 degree offset frequency and R2 + X2 equal to the measured impedance at the -45 degree offset frequency.

The results of the sample line characteristic impedance measurements are shown in Table 2.

 Tower +45 Degree Offset Frequency (kHz) +45 Degree Measured Impedance (ohms) -45 Degree Offset Frequency (kHz) -45 Degree Measured Impedance (ohms) Calculated Characteristic Impedance (ohms) 1 1118.889 7.54 - j48.15 1367.531 9.18 + j48.11 48.86 2 1120.2696 7.51 - j47.81 1369.2184 9.01 + j47.65 48.44 3 1120.0725 7.45 - j47.53 1368.9775 9.00 + j47.47 48.21 4 1119.5559 7.49 - j47.91 1368.3641 9.10 + j47.92 48.66 Table 2. Sample line characteristic impedances

In addition Crain conducted measurements of the sample line electrical lengths that were derived by measuring the open circuited 90 degree and 450 degree resonance frequencies. The results are shown in Table 3.

 Tower Sampling Line Open-Circuited 90 Degree Resonance (kHz) Sampling Line Open-Circuited 450 Degree Resonance (kHz) Sampling Line Calculated Electrical Length 1380kHz (degrees) Measured Impedance with VSU Connected 1380kHz (ohms) 1 246.105 1243.21 499.513 5.48 - j33.08 2 246.400 1244.744 498.898 5.42 - j33.13 3 246.725 1244.525 498.986 5.41 - j33.14 4 246.312 1243.951 499.216 5.44 - j31.91 Table 3. Resonance and measured impedances

The measured characteristic impedance and electrical length variations relative to the average are less than ±1 percent and ±0.5 degrees. Based on these measured data, Crain returned to his office to develop a computer model for this complex array using MININEC Broadcast Professional V14.5 software. Each of the four towers were modeled using 19 segments with the tower 1 and 2 segments having a uniform length and radius and towers 3 and 4 being modeled in a graduated diameter stacked cylinder approach where the bottom segments had longer segments and larger radii with uniform segment and radius at the top fixed cross section portion of the tower. The tower spacings and orientations from the FCC database were identical in the self-impedance and DA models. The segment lengths and radii were varied while maintaining the tower height as constant to converge on modeled tower impedances with the other towers shorted that matched the corresponding measured impedances. A table of the final modeled segments for towers 3 and 4 are shown in Table 4.

 Tower Segment Physical Height (degrees) Model Height (degrees) Model Percent of Height Model Radius (meters) Percent Equivalent Radius 3-1 22.72 24.5376 108.0 3.3479 100 3-2 22.719 24.5365 107.999 2.8398 100 3-3 17.671 19.0847 108.0 2.3881 100 3-4 15.146 16.3577 108.0 2.0212 100 3-5 12.622 13.6317 108.0 1.7106 100 3-6 10.098 10.9058 107.999 1.4566 100 3-7 10.097 10.905 108.002 1.2308 100 3-8 7.573 8.179 108.002 1.0332 100 3-9 7.574 8.18 108.001 0.8638 100 3-10 7.573 8.178 107.989 0.6944 100 3-11 7.573 8.179 108.002 0.525 100 3-12 5.049 5.453 108.002 0.3839 100 3-13 to 3-19 4.6979 5.0737 107.999 0.3274 100 3 Overall 179.3 193.644 108.0 . 100 Table 4. Modeled segments for towers 3 and 4

- continued on page 4

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