The Reciprocity Theorem
A beautiful symmetry of linear circuits: swap the source and the meter and the current stays exactly the same. Learn the statement, the linear–bilateral–single‑source conditions, transfer resistance, a step-by-step method, a fully solved example, why a diode breaks it, and how it powers antenna theory.
Complete Learning Path — Reciprocity Theorem
From the statement and the interchange demonstration, to transfer resistance, the conditions, the step-by-step method, a solved example, two-port symmetry, when it fails and applications
What is the Reciprocity Theorem?
The reciprocity theorem describes a striking symmetry of linear circuits. It states that in a linear, bilateral network with a single source, if a voltage source in one branch produces a certain current in another branch, then moving that same source to the second branch produces the same current in the first branch.
In short: swap the source and the ammeter, and the reading does not change. The excitation and the response can trade places freely.
Take a network with a single voltage source V at one port and an ideal ammeter at another, reading the response current I.
Excitation and response
The excitation is the driving source (here a voltage V); the response is what you measure (here a current I). Reciprocity is a statement about how these two are allowed to swap positions.
The Interchange — Source ↔ Meter
Now move the very same source to the branch where the ammeter was, and put the ammeter where the source used to be. The theorem promises the ammeter reads the identical current.
Why it feels surprising
The two circuits can look completely different — different branches carrying the source, different paths for the current — yet the measured current is exactly equal. That symmetry is the whole point of reciprocity.
Transfer Resistance — the Invariant
The quantity that stays constant is the ratio of source voltage to response current — the transfer resistance (transfer impedance in AC circuits).
The Three Conditions
Reciprocity is not universal — it holds only when three conditions are met.
Linear
Built from R, L and C so the response is proportional to the source — the same idea that underpins superposition.
Bilateral
Every element behaves the same in both directions. Resistors qualify; diodes do not.
Single source
Exactly one independent source drives the network while the response is measured.
The Step-by-Step Method
Verifying or applying reciprocity is a short, repeatable routine.
The procedure in words
- Check the conditions — linear, bilateral, single source.
- Solve the original circuit for the response current I at the output port.
- Interchange the source and the meter.
- Solve again for the new response current at the original source port.
- Compare — the currents are equal, and V/I is the same transfer resistance.
Solved Example
Take a T-network driven by V = 9 V with R1 = 1 Ω (series), R2 = 6 Ω (shunt) and R3 = 3 Ω (series to the output). An ideal ammeter measures the current at the far port.
Source at port 1
The ammeter branch is an ideal short, so R2 and R3 sit in parallel: R2 ∥ R3 = 6∥3 = 2 Ω. Source current Is = 9 / (R1 + 2) = 9 / 3 = 3 A. The mid-node voltage is Vm = 3 × 2 = 6 V, so:
Source at port 2 (interchanged)
Now R1 ∥ R2 = 1∥6 = 6/7 Ω; source current = 9 / (R3 + 6/7) = 9 / (27/7) = 2.333 A; mid-node voltage = 2.333 × 6/7 = 2 V, so the current in R1 is:
Reciprocity confirmed
Both ways give 2 A. The transfer resistance is Rtr = V/I = 9/2 = 4.5 Ω in either direction.
Reciprocity as a Two-Port Property
Treating the network as a two-port black box makes the symmetry precise: the transfer impedance measured from port 1 to port 2 equals that from port 2 to port 1.
This is why the impedance matrix of a reciprocal network is symmetric — a fact used constantly in filter, transmission-line and microwave design.
When Reciprocity Fails
The moment a network stops being bilateral, reciprocity breaks. The classic culprit is a diode, which conducts in only one direction.
Non-reciprocal elements
Diodes, transistors and other active/non-linear devices, plus dependent sources and components like ferrite isolators and circulators, all violate reciprocity. For these, swapping source and meter changes the result.
Applications
The most famous consequence of reciprocity is in antennas — but it appears throughout linear network theory.
Antennas
Transmit and receive patterns, gain and impedance are identical.
Two-port networks
Symmetric impedance/admittance matrices simplify analysis.
Filter design
Ladder and lattice filters exploit reciprocal symmetry.
Transmission lines
S-parameter symmetry (S12 = S21) for passive lines.
Sensitivity & testing
Measure a hard-to-reach response from an easier port.
Network theorems
A building block of linear two-port theory.
Advantages & Limitations
Advantages
- Lets you measure a response from the easier port.
- Guarantees symmetric two-port parameters (z12 = z21).
- Simplifies antenna, filter and transmission-line analysis.
- Provides a quick sanity check on circuit solutions.
Limitations
- Only for linear, bilateral networks.
- Requires a single source.
- Fails with diodes, transistors and dependent sources.
- Gives a relationship, not a simplified equivalent circuit.
Key Terms — Glossary
| Term | Meaning |
|---|---|
| Reciprocity theorem | Swapping source and response in a linear bilateral single-source network leaves the ratio V/I unchanged. |
| Excitation | The driving source (a voltage or current source). |
| Response | The measured quantity (usually a branch current). |
| Transfer resistance | The ratio V/I between two different branches; invariant under interchange. |
| Transfer impedance | The AC version of transfer resistance (z12 = z21). |
| Bilateral element | Behaves the same in both directions (R, L, C) — unlike a diode. |
| Linear network | Response proportional to source; built from linear elements. |
| Two-port network | A network with an input and output port; reciprocity makes its matrix symmetric. |
| Non-reciprocal | A network (with diodes, transistors, isolators…) where the swap changes the result. |
Frequently Asked Questions
Quick, expert answers to the questions people ask most about the reciprocity theorem.
What is the reciprocity theorem in simple words?
In a linear, bilateral network with a single source, if a voltage source in one branch produces a certain current in another branch, then moving that same source to the second branch produces the same current in the first. Swapping the source and the meter does not change the reading.
What is the statement of the reciprocity theorem?
In any linear bilateral network with a single independent source, the ratio of the response (current) to the excitation (voltage) is unchanged when excitation and response are interchanged: V₁/I₂ = V₂/I₁, so the transfer resistance is the same both ways.
What are the conditions for the reciprocity theorem?
The network must be linear (R, L, C), bilateral (same both directions) and driven by a single independent source. Circuits with diodes, transistors or dependent sources are not reciprocal.
What is transfer resistance?
It is the ratio of the source voltage in one branch to the current it produces in another, V/I. Reciprocity says this ratio is unchanged when the branches are swapped; in AC circuits it is called transfer impedance.
Why does reciprocity fail with diodes and transistors?
Because they are non-linear and non-bilateral. A diode conducts in only one direction, so interchanging the source and the meter changes the current. Reciprocity holds only for linear, bilateral networks.
Does the reciprocity theorem work with current sources?
Yes, in dual form. A current source applied at port 1 produces an open-circuit voltage at port 2; moving the same current source to port 2 produces the same voltage at port 1.
What is antenna reciprocity?
It is a direct application of the theorem: an antenna has the same radiation pattern, gain and impedance whether transmitting or receiving, and the coupling between two antennas is the same regardless of which one transmits.
How is reciprocity different from other network theorems?
Unlike Thevenin, Norton or superposition, reciprocity does not reduce a circuit to an equivalent. It states a symmetry: in a linear bilateral single-source network, excitation and response can be swapped without changing the ratio between them.
Conclusion & Key Takeaways
The reciprocity theorem captures a deep symmetry of linear circuits: excitation and response are interchangeable without changing the ratio between them.
Swap freely
Source ↔ meter, same current.
Linear + bilateral
R, L, C only.
Single source
One excitation at a time.
Rtr invariant
V/I is the same both ways.
No diodes
Non-bilateral breaks it.
Antennas
Transmit = receive.