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Reciprocity theorem for antennas

Short answer

If emf V applied at antenna 1 terminals produces current I at shorted antenna 2 terminals, the same emf at antenna 2 produces the same current at antenna 1, so Z12=Z21. This holds in a linear, passive, isotropic medium. Radiation pattern, gain, directivity, effective aperture, and input impedance are identical for transmitting and receiving.

State the reciprocity theorem for an antenna system.

Reciprocity theorem: If an emf V applied at the terminals of antenna 1 produces a current I at the (short-circuited) terminals of antenna 2, then the same emf V applied at the terminals of antenna 2 produces the same current I at the terminals of antenna 1.

 Case 1:  V --> [Ant 1] ~~~~~~> [Ant 2] --> I21
 Case 2:  I12 <-- [Ant 1] <~~~~~~ [Ant 2] <-- V
          Reciprocity: I12 = I21, i.e. Z12 = Z21

In impedance form: V1/I2 = V2/I1, so the mutual impedances are equal, Z12 = Z21.

Conditions: the medium between the antennas must be linear, passive and isotropic (it fails in a magnetized plasma such as the ionosphere with the earth's magnetic field, or in ferrites).

Consequences for an antenna:

  • The radiation pattern is the same for transmitting and receiving.
  • Gain, directivity and effective aperture are the same in both modes.
  • Input impedance of the antenna is the same as its internal impedance when receiving.
  • Antenna patterns can be measured with the antenna under test used as a receiver.

✓ Answer checked by Claude Opus 5.5 (high effort)

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