Calculate Impedance of Radio Antenna

[Fischer777]

After googleing the formula to calculate impedance of a radio antenna, I came up with this formula Z = R + jX where X is reactance and R is resistance. I figured resistance could be measured with an ohmmeter. Then, it seems like there are two kinds of reactances, capacitive and inductive. The formula for capacitive reactance seems to be -1/2πfC, and the formula for inductive reactance appears to be 2πfL.

So my questions are: can I simply add the two reactances together? And how does this work if j is an imaginary number?

 

[vk6kro]

After googleing the formula to calculate impedance of a radio antenna, I came up with this formula Z = R + jX where X is reactance and R is resistance. I figured resistance could be measured with an ohmmeter. Then, it seems like there are two kinds of reactances, capacitive and inductive. The formula for capacitive reactance seems to be -1/2πfC, and the formula for inductive reactance appears to be 2πfL.

So my questions are: can I simply add the two reactances together? And how does this work if j is an imaginary number?

The resistive component only applies at the frequency of measurement. An ohm meter may give a reading but it is for DC conditions, not at the radio frequency of the antenna where it will probably be quite different.

The reactance will be either zero, inductive or capacitive, not more than one of these at the same time.

It is common, though, to introduce a component to cancel out an antenna'a reactance. You might place a capacitor in series with an antenna feed point to cancel out inductive reactance of the antenna.
In this case, as you suggested, the two reactances are added together and tend to cancel each other since one is positive and one is negative.

As long as both reactance are imaginary (ie they both have "j"s) then you can add their magnitudes (including their signs) if they are in series.
The phrase "imaginary" is a bit misleading as these reactances are very real and can be measured with suitable equipment.

 

[Fischer777]

The reactance will be either zero, inductive or capacitive, not more than one of these at the same time.

I'm not sure I understand. Wouldn't the antenna have a capacitance (being a wire) https://en.wikipedia.org/wiki/Capacitance#Capacitance_of_simple_systems , and by virtue of carrying an electrical current also have some amount of inductance? Or is one usually small enough to be negligible?

 

[NascentOxygen]

An antenna has both capacitance and inductance, as does the feeder wire that connects your transceiver to the antenna. The design goal is to arrange that these reactive components cancel at the frequency (or narrow band of frequencies) your transmitter operates at. Otherwise, power can be reflected back into your transmitter's final stage and burn it out.

You shouldn't try to calculate the impedance of an antenna. Antenna design is a black art, the secrets of which are understood only by a select few mortals.

 

[vk6kro]

An antenna usually has two terminals and power is fed into these two terminals for transmitting or signal is taken from these two terminals for receiving.

No matter what is happening in the antenna, there will be a resulting effect at these terminals and it will either be capacitive or inductive, but not both.

It may be surprising, but you don't need a capacitor to make capacitive reactance and you don't need an inductor for inductive reactance either.

When a radio signal, which is just high frequency AC, travels along a wire or a pipe some of it will radiate, but some of it will reach the end of the wire and reflect back to the feedpoint.
It takes time for this to happen, so, the reflected signal may not be in phase with the new signal coming into the terminal.
Whether it is in phase, leading or lagging the incoming wave will decide the impedance of the feed point of the antenna and the sign of the reactance.
It all depends on the length of the wires of the antenna compared with the wavelength of the radio frequency.

This will seem pretty strange if you haven't come across it before.

Think about this. You can have two wires parallel to each other but not touching and yet at a frequency which is has a quarter wavelength equal to the length of the wires, they will appear to be a short circuit to the signal fed in at one end.

A dipole antenna is a piece of wire about one half wavelength long split in the middle and fed there.
Here is some data for a dipole near resonant and at resonance.

below resonant:...66.92 - j 48.6 at 3.56 MHz or 82 ohms
almost resonant...72.18 - j 0.50 ..at 3.66 MHz or 72 ohms
above resonant ...82.2 + j 50.5. at 3.76 MHz or 96 ohms

Can you see that the reactance varies dramatically when resonance is reached, but the resistive component only varies slightly?

 

[Fischer777]

Okay, so here's my understanding so far (I have a feeling this probably incorrect): When the current travels through the antenna and reaches the end, some 'bounces' back along the antenna. This can be likened to the ripples created from a drop of water in a bucket bouncing off the walls and going back through the origin (in the opposite direction). Resistance of the antenna and reactance determine the 'speed' of that wave through the antenna. The speed of the wave determines how far out of phase the returning wave is, which in turn determines the impedance (so a wave that is in phase with the current input at the terminal has the least impedance).

I also have two more questions. First, how does one determine if the reactance is going to be capacitive or inductive? And second, what is the math behind taking the imaginary component and the resistance and turning it into the ohms of impedance?

 

[vk6kro]

The current that flows into the input of an antenna depends on the phase of the reflected wave.

If it has a voltage greater than the incoming signal, then current will actually flow out of the antenna into the signal source. The phase of this current will make the input look like an inductor or a capacitor.
Capacitors and inductors can do the same thing.

Predicting the phase of this reflection is done by working out the distance traveled and the time taken and comparing this to the period of the signal. The speed is quite close to the speed of light for wires or pipes in air.

The reflection is very much like the ripples in a pond or the resonance of a musical instrument like an organ pipe.

The ideal situation is where the reflected wave appears in phase with the input. Then the input power will be close to the power being radiated. Note that power is only dissipated in the resistive component of the antenna input and most of this should be lost in the radiation resistance of the antenna.

A resonant antenna should not appear reactive at all, although this is difficult to achieve in practice.

Resistance and reactance together determine the magnitude and phase of the impedance of a circuit.
This is by definition.

You can represent it easily enough by drawing a right angled triangle with resistance and reactance along the shorter sides and impedance along the hypotenuse.
You can then use trigonometry to work out the angle between the current (along the resistive side) and the voltage (along the hypotenuse).
You can also use Pythagoras's equation to work out the impedance.
Like this:

You can read about it here if you like:
http://en.wikipedia.org/wiki/Electrical_impedance