## Analysis of an 80m Base-Loaded Mobile Antenna

#### Cecil Moore, W5DXP, Rev. 1.3, 12/09/2010

__Introduction__

#### The previous article, Degrees of Antenna Occupied
by a Loading Coil, showed how
The Hamwaves Inductance Calculator can be used to determine the parameters of an
80m mobile loading coil. Note that the accuracy of the inductance calculator is thought
to be ± 10%. The specifications for the 80m mobile loading coil are:

### 2 inches in diameter (50.8 mm), 100 Total Turns, 10 inches long (254 mm)

#18 wire (1.024 mm in diameter), Design Frequency = 3.5 MHz

#### The above graphic shows how the parameters were entered into the inductance calculator
and the following outputs were obtained.

### Axial Propagation Factor = 1.8118 radians/meter,
Characteristic Impedance: Z0 = 4747 ohms

Effective Series Inductance at 3.5 MHz = 98.6 uH,
Effective Series Reactance at 3.5 MHz = 2168 ohms

Effective Series AC Resistance at 3.5 MHz = 3.86 ohms,
Effective Unloaded Q of Coil at 3.5 MHz = 562

#### In the previous article, we converted the Axial Propagation Factor to degrees/inch
by multiplying by 1.4553 and then multiplied by the 10 inch length of the coil to
obtain the number of electrical degrees occupied by the loading coil at 3.5 MHz. We also
calculated the Velocity Factor of the coil at 4% of the speed of light in free space. Knowing
the VF of the 10 inch coil allowed us to calculate the RF propagation time through the coil.

### Degrees Occupied by the Loading Coil = 26.37 degrees

Velocity Factor (VF) of the Coil = 0.04

RF Propagation Time through the Coil = 21.2 ns

__3.5 MHz Base-Loaded Mobile Antenna Analysis__

#### EZNEC was used to estimate the length of the whip above the coil in order to resonate the
mobile antenna on 3.5 MHz. That length is 8.83 feet. So the 3.5 MHz base-loaded mobile
antenna consists of the specified 10 inch coil at the base with a whip length of 8.83 feet. It
is interesting to note that the reactance of the coil is j2168 ohms while EZNEC says the
impedance looking into the 8.83 foot whip is 0.4-j2187 ohms which is in reasonable agreement.

One wavelength at 3.5 MHz is 281 feet so we can estimate that the whip occupies 8.83/281 = 0.0314 wavelength. We can calculate that the loading coil occupies 26.37/360 = 0.0733 wavelength.

### The coil occupies 0.0733 wavelengths (26.4 degrees). The whip occupies 0.0314 wavelength
(11.3 degrees). That adds up to 0.1047 wavelengths = 37.7 degrees. We know that the resonant
mobile antenna is 0.25 wavelength (90 degrees) long. Where are the missing 52.3 degrees?

__Smith Chart Representation__

#### When we represent the mobile antenna on one half of a Smith Chart, the "missing" degrees are revealed. The 0.0733 wavelength occupied by the coil is plotted "toward the load" with the
low impedance feedpoint on the left. The 0.0314 wavelength occupied by the whip is plotted
"toward the source" with the high impedance open end on the right. We can now calculate where
those "missing" 52.3 degrees come from.
The impedance at the coil to whip junction can be obtained by reading the approximately -j0.5
value from the Smith Chart at the top of the coil and multiplying by the characteristic
impedance of the coil where Z0 = 4747 ohms.

### Approximate impedance at coil/whip junction = 4747(-j0.5) = -j2374 ohms ± 10%

#### Note: The total resistance looking into the whip, e.g. 15 ohms, is very small compared to -j2374 and can therefore be ignored for the purpose of finding the "missing" degrees in this mobile antenna. Note that the ~15 ohms of losses plus radiation resistance is less than one percent of -j2374. Of course, that 15 ohms is of primary importance for calculating efficiency.

Knowing the impedance at the coil/whip junction allows us to calculate the characteristic
impedance (Z0) of the whip. The normalized value from the Smith Chart on the whip end is
approximately -j5.0 which we know from the above calculation is -j2374 ohms. Therefore, the
Z0 of the whip at the coil/whip junction is 2374/5 = ~475 ohms which is a reasonable value.

We can now conclude that the "missing" 52.3 degrees is caused by the impedance discontinuity
at the coil/whip junction, i.e. where the Z0 changes from 4747 ohms to 475 ohms. The same
thing happens when two transmission lines are connected together if they have different
characteristic impedances, i.e. different Z0s.

__Conclusion__

In a base-loaded mobile antenna, there are three components that contribute to the 90
electrical degrees required for resonance.

1. The coil contributes a certain number of degrees, 26.4 degrees in the above example.

2. The impedance discontinuity at the coil to whip junction point contributes a certain number
of degrees, 52.3 degrees in the above example.

3. The whip contributes a certain number of degrees, 11.3 degrees in the above example.

#### Thus a base-loaded mobile antenna can be analyzed in the same manner that
Dual-Z0 Shortened Stubs can be analyzed.
The following dual-Z0 1/4WL stub analysis is virtually identical to the above base-loaded mobile
antenna analysis. Note that the phase shift at the impedance discontinuity is proportional to
the ratio Z0H/Z0L and 4747/475 = 500/50.

__Other Configurations__

#### There are two reasons why a center-loading coil must have more reactance, i.e. occupy
more degrees of the antenna. The primary reason is that the whip on a base-loaded antenna
is longer than the whip on a center-loaded antenna of the same length. A four foot whip
exhibits a lot more capacitive reactance than an 8 foot whip so more inductive reactance in the loading coil is required to neutralize that added capacitive reactance.

A secondary reason is that
having a bottom section under the loading coil (center-loading) complicates the analysis
by introducing an additional negative phase shift at the bottom section to coil impedance
discontinuity where the Z0 changes from the low Z0 value of the bottom section to the high Z0
value of the coil. Because the phase shift at the bottom of a center loading coil is negative,
the inductance of the coil must be increased to add to the degrees occupied by the coil, in
order to compensate for the number of degrees lost by the negative phase shift.

W5DXP will follow up on center-loaded mobile antennas with a short future article. The primary
concepts concerning degrees occupied by a loading coil are covered in this present article.