Effects of Conductor Diameter on Dipole Resonance
Magnitudes of Voltage and Current variation along a half-wave Dipole
The Figure 1 below shows the magnitudes of voltage and current variation along a half-wave dipole. The dipole has opposite voltages on either side of center, reaching a maximum at each end. The current at each end of a dipole is zero. If the dipole is one-half wavelength long, the current is maximum at the center. This is the distribution of voltage and current. The distribution of current, in particular, determines how a dipole radiates.
1. Antenna Length and Conductor Thickness
The physical length of a resonant 1/2-λ (half-wavelength) dipole antenna is not exactly equal to the half-wavelength of a radio wave of that frequency in free space. The length depends on the thickness of the conductor in relation to the wavelength.
- Thicker Conductors: A thicker conductor shortens the resonant length of the antenna. The relationship between thickness and antenna length is captured by a multiplying factor, K.
- Thinner Conductors: Thinner conductors result in a slightly longer resonant length.
2. Resonant Lengths for #12 AWG Copper Wire
The Table below provides resonant lengths for dipoles in free space made from #12 AWG bare copper wire:
3. K Factor and Antenna Shortening
The K Factor is a multiplying factor that accounts for the shortening of the antenna due to the conductor’s thickness.
- Larger Diameter Conductors: Increase the ratio of stored electric to magnetic field energy, thus lowering the resonant frequency.
- End Effect: Shortening also occurs due to capacitance added by insulators supporting the wire antenna, further affecting the electrical length.
4. K Factor Equation
The K factor for a given dipole is calculated as follows:
To calculate the K factor for a given dipole, use this simplified method:
Multiply the free-space half-wavelength by K to get the resonant length. From the graph in Figure 2 or using the equation, this ratio gives K = 0.974. Therefore, the resonant length of the half-wavelength dipole is:
0.974 × 68.3 = 66 feet 7 inches.
5. Graph Interpretation of K Factor
The Figure 2 below presents a graph of K as a function of the half-wavelength-to-diameter ratio. The K factor varies with the thickness of the conductor, as shown in the graph:
For most HF wire antennas, half-wavelength-to-diameter ratios range between 2,500 and 25,000, with values of K between 0.97 and 0.98.
6. Resonant Length Formula
For a dipole of half-wavelength in free space, the resonant length is given by:
Resonant length (in feet) = K × 491.786 / Frequency (MHz)
Where:
- Frequency = frequency in MHz
- K = constant that accounts for conductor thickness (typically between 0.97 and 0.98 for most HF antennas).
7. Traditional 4-6-8 Formula
For wire dipoles at and below 10 MHz, the traditional "4-6-8" formula is:
Resonant length (in feet) = 468 / Frequency (MHz)
This formula assumes a resonant length 5% shorter than the free-space half-wavelength, which translates to K = 0.95. This approximation tends to underestimate the length by a small margin, as shown in the examples below.
8. Examples
- Example 1: A half-wave dipole for 7.2 MHz using #12 AWG wire has a resonant length of 66 feet 7 inches.
- Example 2: A half-wave dipole for 7.2 MHz using the traditional 4-6-8 formula gives 65 feet, which is slightly shorter.
- Example 3: A half-wave dipole for 50.1 MHz made of 1/2-inch diameter tubing has a resonant length of 9 feet 4.5 inches.
9. Antenna Above Ground
The above formulas apply to antennas in free space. For wire antennas at HF frequencies above ground, ground effects may alter the resonant length. In such cases, antenna modeling software can provide more accurate results.
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