Working With Antenna Impedance
This is an elaboration of a post I made on the Rap 'N Tap Discussion group. The objective was to explain how a multi-wire flat top antenna adds capacitance to the antenna, while making it more inductive. This is an exercise in complex numbers. For example the expression 20 + j40 displays 20 ohms resistance and 40 ohms of inductive reactance. The complex number approach allows us to simply add series resistance, capacitance and inductance. The resistance remains constant when the frequency changes. However, the reactance of an inductor increases with frequency. The reactance of a capacitor decreases with frequency.
For purposes of illustration, assume an antenna with the following parameters for R, L and C in series for F=1650 kHz (our frequency of interest):
Rtotal = 20 ohms (total of radiation resistance and losses, including ground losses, since we are using a Marconi type antenna, i.e., worked against ground.)
L = 20 uh; the reactance is calculated as:
XL= 2*pi*F*L (where 2*pi =6.28, F is in hertz, and L in henries)
=+j207 ohms(inductance has a plus sign)
C = 400 pF; the reactance is calculated as:
Xc=1/(2*pi*F*C) (Where F is in hertz and C in farads)
=-j241 ohms (capacitance has a minus sign)
The value of this complex impedance is:
= 20 + j207 - j241 ohms
= 20 - j34 ohms
As mentioned earlier the use of complex numbers allows us to simply add the resistance and reactance. The sign of the reactive component -j34 is negative and indicates the antenna is capacitive. It is also shorter than the resonant length ( at resonance XL + XC = 0).
Now if you increase the capacitance of the antenna to ground with a flat top section,
where the series capacitance increases to 500 pF, here is what happens:
= -j193 ohms
The value of this new complex impedance is:
=20 + j207 - j193 ohms
=20 + j14 ohms
The sign of the reactive component is now positive, indicating the antenna is inductive, or long compared to resonance.
Use of the complex number notations also lets you figure what value of series reactance you would need to bring the antenna circuit into resonance. You could tune out the reactance and then match the remaining resistance to the input of the crystal set.
Referring to our last set of complex numbers: 20 + j14, you could bring the antenna
circuit into resonance with a reactance of -j14 ohms. The negative sign indicates a
capacitor is needed. Here is the value for 1650 kHz:
Solving for C, we get:
This is a huge capacitor and would probably be a combination of fixed and variable capacitors.
Now, referring to the first antenna example before we added more capacitance, we can see what value of reactance is required to bring the antenna into resonance. Since that antenna had an impedance of 20 - j34 ohms, we would need a series inductor to combine with the capacitance of -j34 ohms. This inductance is:
Solving for L we get: L=XL/(2*pi*F)
L=34/(6.28*1650 kHz) L=3.28 uH
These calculations were all done at the higher end of the broadcast band. When moving down to 530 kHz, the antennas we have been analyzing would have smaller resistance and higher negative reactance.
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