##  [Reciprocity Theorem](/reciprocity-theorem-0) 

 Definition

In linear, passive, time‑invariant and non‑magnetized electromagnetic media (reciprocal media), the transfer relation between two points or ports is symmetric: the response (field, voltage or scattering parameter) measured at point B due to a source at A equals the corresponding response at A when the same source is placed at B, under the same loading and excitation conditions.

 

 

 

 

 

 





## Principle

Principle

Source and observation points can be interchanged without changing the measured linear transfer in reciprocal systems; S21 = S12 and Green's functions are symmetric under the reciprocity conditions.

 

 

 

 

 





## Demonstration

Demonstration

Illustrative measurement → Two matched antennas in a reciprocal medium are connected to a network analyser to measure S21. Recognition → Replace transmitter and receiver positions and repeat measurement. Action → Compare S21 and S12 under the same matching and excitation. Consequence → Measured S21 ≈ S12 (within measurement error), allowing one direction's measurement to substitute for the other.

 

 

 

 

## Misapplication

Misapplication

Assuming reciprocity in systems containing magnetized materials (biased ferrites), active devices that inject power or time‑varying elements, or in nonlinear components; these break the symmetry and invalidate interchangeability assumptions.

 

 

 

 

 





## Consequence

Consequence

Reciprocity reduces measurement burden and simplifies antenna and network design by guaranteeing symmetric transmission between ports in passive linear systems; it constrains admissible network responses and simplifies modelling.

 

 

 

 

## Reversal

Reversal

Reciprocity fails for non‑reciprocal media or components (magneto‑optic materials, biased ferrites, circulators), for active/time‑varying circuits, or when system conditions (matching, loading, bias) differ between source placements.

 

 

 

 

 





## Boundary

Boundary

Applies when the system is linear, passive, time‑invariant, and reciprocal (no magnetic bias or gyrotropy); ports must be equivalently defined and matched. Does not apply to nonlinear, active, time‑variant, or magnetically biased systems.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Reciprocity versus intentional isolation — designers rely on reciprocity for symmetry and measurement, but practical devices often require nonreciprocity (isolation, routing), which must be produced by breaking reciprocity intentionally.

 

 

 

 

 





## Synthesis

Synthesis

Reciprocity is a structural symmetry of passive linear EM systems that both simplifies analysis and constrains design: achieving asymmetric behaviors (isolation, one‑way transfer) requires explicit nonreciprocal elements or time‑variance.