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Handheld radios receive AM in the Khz to Mhz band but for these frequencies the antenna length will be in 10s of meters . So how do the radios receive high frequency FM and relatively low frequency AM with a single antenna ?

Thank You

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    \$\begingroup\$ Broadcast radio transmissions use a lot of RF power to obviate the need for sophisticated antennas at the receivers. \$\endgroup\$
    – Chu
    Commented Nov 9, 2016 at 15:41

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how do the radios receive high frequency FM and relatively low frequency AM with a single antenna ?

They use separate antennas for FM (circa 100 MHz) and AM (circa 1 MHz). The FM antenna is usually close to a quarter wave monopole and the AM antenna is usually a coil of wire that only receives the magnetic part of the transmission.

A quarter wave monopole at 100 MHz will have a length of about 0.75 metres i.e. suitable for one of these: -

enter image description here

And a coil of wire needs no further introduction!

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    \$\begingroup\$ just another question , do radios use coherent or non coherent demodulation ? . Thanks a lot for the answer \$\endgroup\$
    – Aakusti
    Commented Nov 9, 2016 at 16:22
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    \$\begingroup\$ Radios that receive broadcast transmissions use non coherent demodulation because it is cheap and usually broadcast powers are massive enough to give good reception in what might be otherwise regarded as fringe areas. \$\endgroup\$
    – Andy aka
    Commented Nov 9, 2016 at 16:45
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but for these frequencies the antenna length will be in 10s of meters

That is only true for an electrical antenna.

But these radios use a magnetic antenna for the low frequency bands, like this one:

enter image description here

It's a ferrite core with a coil wound around it.

It picks up the magnetic part of the EM waves.

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