## Degrees of Antenna Occupied by a Loading Coil

#### Cecil Moore, W5DXP, Rev. 1.3, 11/18/2010

__Introduction__

#### The purpose of this article is to provide a procedure for determining the
number of degrees of antenna occupied by a loading coil. A later article
will explain how that value applies to inductively loaded mobile antennas.

Assume we have a vertical 1/4 wavelength resonant wire monopole over a ground plane.
The vertical monopole is obviously electrically 90 degrees long because one
wavelength is defined as 360 degrees. Let's say we want to reduce the length
of the monopole to 1/8 wavelength, i.e. one half of the original length, and
make the shortened monopole resonant on the same original resonant frequency.
We can obviously install a loading coil to perform that function. The shortened
wire monopole is physically 45 degrees long, i.e. missing 45 degrees, but resonance
requires the same 90 electrical degrees as before. The question arises:

### How many degrees of the "missing" 45 degrees of wire does the loading coil
replace?

#### There are two extreme opinions:

Opinion 1: When part of a wire antenna is replaced with an inductive loading
coil, the loading coil does not supply any of the missing degrees. (This is in
agreement with the lumped-circuit model.)

Opinion 2: When part of a wire antenna is replaced with an inductive loading
coil, the loading coil supplies 100% of the missing degrees. (Some seem to
think this is in agreement with the distributed network model.)

### Unfortunately, both of the above opinions are incorrect.

#### The loading
coil does replace some, but not all, of the degrees missing from the
shortened antenna. In a base loaded mobile antenna, the degrees of antenna occupied
by the loading coil is only one of three phase shifts.

#### This inductance calculator is one of the most sophisticated available
and corrects some problems with earlier inductance calculators. If
inductance, reactance, and RF resistance values are needed, this calculator
will provide them. The coil data is entered in metric (mm) but that is
a small problem since multiplying inches by 25.4 will yield millimeters.
There is one output parameter from this calculator that will allow us
to easily estimate the number of degrees occupied by the coil at any particular
frequency, i.e. the number of degrees of antenna replaced by the loading
coil.

__80m Loading Coil Example__

#### Let's assume the following loading coil with dimensions given in inches. Since
the calculator requires a metric input, we will convert inches to millimeters
by multiplying by 25.4. Here are the specifications on our loading coil to be
entered into the Hamwaves Inductance Calculator:

### 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

#### Entering these values into the calculator yields a lot of useful information
about inductance, reactance, and RF resistance, but the one we are most
interested in is the __Axial Propagation Factor__ in radians/meter.

The 1.8118 radians/meter *Axial Propagation Factor* is associated with
how fast the RF signal is traveling through the length of the coil. We can change
the radians/meter to degrees/inch by multiplying by
(57.296 degrees/radian)/(39.37 inches/meter) = 1.4553.

(1.8118 radians/meter)(1.4553) = 2.6367 degrees/inch

### Since the coil is 10 inches long, the number of degrees occupied by the coil
at 3.5 MHz is 10(2.6367) = 26.4 degrees.

#### We can also calculate the velocity factor (VF) of the coil. One wavelength at
3.5 MHz is 281 feet which is 360 degrees. So at the speed of light in free space,
26.4 degrees would be a length of (281 ft)(26.4/360) = 20.607 feet or 247.3 inches.

The coil is 10 inches long so if we divide 10"/247.3", we get the velocity factor of the coil.

### The VF value for the specified loading coil is 0.04

#### The Axial Propagation Factor in radians/meter is represented by the Greek letter 'Beta'
on the Hamwaves Inductance Calculator. Given Beta in rad/m and the frequency in MHz, the formula for the Velocity Factor is:

### VF = 4*freq/191*Beta = 4(3.5 MHz)/191(1.8118) = 0.04

#### i.e. the speed of light through the medium of the coil is
4% of the speed of light in free space. In free space, it takes light 0.8486 ns to
travel 10 inches. With VF = 0.04, we can calculate:

### The length of time it takes RF to travel the length of the coil is
0.8486/0.04 = 21.2 ns.

#### Such a coil is known in the literature as a "slow wave structure", i.e. from
"... a general class of structures that possess waves with a phase velocity
(along the axis) much less than the velocity of light ...".[1]
Finally, the Hamwaves Inductance Calculator tells us that:

### The characteristic impedance
(Z0) of the coil is 4747 ohms.

#### Given that we know the characteristic impedance, velocity factor, and number of degrees
occupied by the coil, we are in position to perform a complete analysis of an inductively
loaded mobile antenna. The next article will present that analysis. Note: A base-loaded
mobile antenna is also a dual-Z0 environment with the base-loading coil having a ballpark
Z0 of approximately ten times the Z0 of the whip. Some of the concepts presented in this
article may be more easily understood by referencing an earlier article.[2]

[1] __Fields and Waves in Communications Electronics__, 3rd edition; Ramo, Whinnery, & Van Duzer

[2] Dual-Z0 Shortened Stubs