--- title: "Reciprocity theorem for antennas" subject: "Propagation and Antenna" chapter: 1 key: "semester_6/propagation_and_antenna:reciprocity-theorem" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/reciprocity-theorem site: BCT Engineering --- # Reciprocity theorem for antennas Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals - Asked in: 2079 Chaitra, 3 marks; 2078 Chaitra, 2 marks; 2077 Chaitra, 2 marks; 2075 Bhadra, 2 marks; 2074 Bhadra, 2 marks; 2073 Magh, 2 marks; 2072 Magh, 2 marks (asked 7 times) - Video: [Reciprocity Theorem for Antennas with Its Proof](https://www.youtube.com/watch?v=4p_hr76Pybc) — Padmasri Naban - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/reciprocity-theorem - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1 ## Question State the reciprocity theorem for an antenna system. ## 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. ## Answer **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. ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which statement correctly expresses the reciprocity theorem for two antennas? A. An emf V at antenna 1 gives a current I at antenna 2; the same emf at antenna 2 gives the same current I at antenna 1 B. An emf V at antenna 1 gives a current I at antenna 2; the same emf at antenna 2 gives a current -I at antenna 1 C. The power radiated by antenna 1 equals the power received by antenna 2 D. The current at antenna 2 is always twice the current at antenna 1 Answer: A. Reciprocity says the transfer is symmetric: the same emf applied at either antenna produces the same current at the other, so I12 = I21. (Key point: Statement of reciprocity) 2. In impedance form, what does reciprocity give for the mutual impedances? A. Z12 = -Z21 B. Z12 = 1/Z21 C. Z12 = Z21 D. Z12 = Z21* Answer: C. Since V1/I2 = V2/I1, the mutual impedances are equal: Z12 = Z21. (Key point: Impedance form) 3. In which medium does the reciprocity theorem fail? A. Free space B. A linear, isotropic dielectric C. Air near the ground D. A magnetized plasma such as the ionosphere, or ferrites Answer: D. The theorem needs a linear, passive, isotropic medium; magnetized plasma and ferrites are non-reciprocal. (Key point: Conditions for validity) 4. Which is a consequence of reciprocity for an antenna? A. Transmitting and receiving radiation patterns are the same B. Transmitting gain is always higher than receiving gain C. Receiving pattern is the inverse of the transmitting pattern D. Input impedance differs between transmit and receive modes Answer: A. Pattern, gain, directivity and effective aperture are the same in both modes, so patterns can be measured with the antenna used as a receiver. 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