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I know that antennas can be capacitive or inductive depending on the relation of antenna length to the \$\lambda\$. I know that crystal radios use both antenna and capacitor for tuning and I don't understand why because for me it would seem the antenna should only be capacitive and you should only need inductor for tuning. Why is there a capacitor in crystal radios and why is it used for tuning?

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    \$\begingroup\$ Just like a pendulum has two modes of energy storage ( movement, and height) to store energy, so does any other way to store energy, that is, to achieve a resonance. In electronics, the capacitor's electric field and the inductor's magnetic field are the two energy storage mechanisms usually used in radios. Your cellphone also uses quartz, with its own special two mechanisms, to precisely get the channels used. We can acid-etch the quartz more precisely to set a frequency, than we can build capacitors or inductors. \$\endgroup\$ – analogsystemsrf Nov 22 '18 at 23:44
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You are very astute. You can indeed construct a crystal set without a physical capacitor, relying instead on the capacitance of the antenna alone and tuning by varying the inductance. Of course, the value of inductance required will vary with the length of the antenna. Reckon on 10 pF per metre as a starting point. Perhaps a variable capacitor is popular because it can be arranged to dominate the antenna capacitance and make tuning less antenna length dependent. A wide range, reliable variable inductor can also present challenges. Go ahead and experiment.

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It's a transmission-line thing. The electric current propagates down the antenna and, approximately, reflects from the antenna's end back toward the transmitter. Upon reflection, the current (but not the voltage) reverses.

By the time the reflected wave reaches the transmitter again—importantly, this does not happen instantaneously—the reflected wave (with current reversed) superimposes itself upon the signal the transmitter is generating at that moment.

Are you handy with your phasors? The impedance is the ratio of the sum of the voltages to the sum of the currents, phasor style. If the imaginary part of the ratio happens to be positive—and this depends on the wavelength compared to the antenna's length—then the antenna is inductive. Otherwise capacitive.

If you aren't sure that this makes sense, then here is a smaller idea to get you started: at the end of the antenna, the sum of the primary and reflected currents (but not voltages) must be zero because no current flows out the end of the antenna. This smaller idea does not explain the whole problem but it does serve to orient your mind so that, after you have slept on it, you might have a better frame to start to grasp the rest of the problem.

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For the AM broadcast band, few long-wire antenna builders erect a full quarter-wavelength wire (about 75 m) and settle for a shorter wire. A shorter antenna benefits from added inductance to compensate for its capacitive reactance.
Once you include all the resistive components (don't forget ground losses), this inductor-compensated antenna has very low Q . To effectively select one station from another, an LC resonator should have loaded Q of at least 100. So you often see an antenna loosely coupled to a high-Q LC resonator.

Why is "C" varied rather than "L" for tuning?
Compare prices for each. A variable capacitor maintains high Q over its very large capacitance range. Variable inductors seem mechanically complex - some air-core types are bulky. Both tuning methods have been employed - there was a time when car radios used ferrite-slug-tuned inductors for tuning. These were not crystal radios.

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