Reciprocity theorem

Statement

Reciprocity theorem: in a linear, bilateral, passive network with a single source, if a voltage source V placed in one branch produces a current I in another branch, then the same source V placed in the second branch produces the same current I in the first branch. Equivalently, the ratio V/I (transfer resistance) is the same when the source and the measuring point are interchanged.

 Case 1: source in branch 1       Case 2: source in branch 2
  (V) in branch 1 --> I in 2       (V) in branch 2 --> same I in 1

Conditions

• The network must be linear and bilateral (resistors, inductors, capacitors; no semiconductors).
• Only one source is active at a time.
• The polarity of the source and the direction of the measured current must be kept consistent in both cases.

Example

A 10 V source with 2 Ω in series feeds a 3 Ω shunt, and a 2 Ω load is connected beyond it.

• Case 1: source in the input branch. Total R = 2 + (3 ∥ 2) = 3.2 Ω, I = 3.125 A. Voltage across the 3 Ω ∥ 2 Ω part = 3.125 × 1.2 = 3.75 V, so the current in the output 2 Ω = 1.875 A.
• Case 2: the 10 V source is moved into the output branch (in series with the output 2 Ω), and its old place in the input branch is replaced by a short circuit. The input 2 Ω stays in the circuit as the measured branch. Total R = 2 + (3 ∥ 2) = 3.2 Ω, I = 3.125 A. Voltage across the 3 Ω ∥ 2 Ω part = 3.75 V, so the current in the input 2 Ω = 1.875 A.

Both currents are equal (1.875 A), so the theorem is verified. It does not apply to circuits with dependent sources or non-linear elements.