Tellegen's Theorem

One of the most general results in circuit theory: in any lumped network that obeys Kirchhoff's laws, the branch powers always sum to zero — Σ vk ik = 0. It needs no linearity, depending only on the wiring. Learn the statement, its KCL/KVL basis, the power-conservation meaning, a clean proof, a solved example and the quasi-power form.

Complete Learning Path — Tellegen's Theorem

From the statement and its KCL/KVL basis, to power conservation, a proof, the step-by-step method, a solved example, the quasi-power form, sign convention, generality and applications

What is Tellegen's Theorem?
The KCL/KVL Basis
Conservation of Power
The Proof
Step-by-Step Method
Solved Example
Quasi-Power Form
Sign Convention
Generality
Applications
Key Terms
Related Topics & Searches
FAQs
Conclusion

What is Tellegen's Theorem?

Tellegen's theorem is one of the most powerful and general results in all of network theory. It states that for any lumped circuit, if you multiply each branch's voltage by its current and add up those products over every branch, the total is always exactly zero.

What makes it remarkable is what it does not require: no linearity, no particular element type, nothing about what the components actually are. It follows purely from Kirchhoff's laws — that is, from the topology (the wiring) alone.

Tellegen's theorem concept: for a lumped circuit the sum over all branches of branch voltage times branch current equals zero
Tellegen's theorem: for any circuit obeying KCL and KVL, Σ vk ik = 0 — the branch powers always balance.
Σvₖiₖ=0
Branch powers sum to 0
KCL + KVL
All it needs
any element
Linear or not
topology
Depends on wiring only
Tellegen's theorem statement: the sum from k equals 1 to b of v k times i k equals zero
The formal statement, summed over all b branches with associated reference directions.
A theorem about wiring

Two of the branch quantities — the voltages and the currents — need not even come from the same physical situation. As long as the voltages obey KVL and the currents obey KCL for the same graph, the sum of their products is zero. That surprising generality is the essence of Tellegen's theorem.

The Two Ingredients: KCL & KVL

Tellegen's theorem rests on exactly two facts, both from Kirchhoff: currents balance at every node (KCL) and voltages balance around every loop (KVL).

Tellegen's theorem rests on Kirchhoff's current law at each node and Kirchhoff's voltage law around each loop
KCL: currents into a node sum to zero. KVL: voltages around a loop sum to zero. Together they force Σvkik = 0.

KCL (currents)

At every node, the algebraic sum of branch currents is zero — charge is conserved.

KVL (voltages)

Around every closed loop, the algebraic sum of branch voltages is zero — voltages derive from node potentials.

Conservation of Power

The most intuitive reading of Tellegen's theorem is conservation of power: the power delivered by the sources equals the power absorbed by everything else, at every instant.

Tellegen's theorem as conservation of power: the power delivered by the source equals the power absorbed by the resistors so the branch powers sum to zero
Power delivered (source, −) balances power absorbed (elements, +): −36 + 18 + 6 + 12 = 0.

Using the passive sign convention, a branch that absorbs power has a positive v·i, while a source delivering power has a negative v·i. Tellegen guarantees these always cancel exactly:

Σ (power absorbed) + Σ (power delivered) = 0
No energy is created or lost inside the network — a completely general statement, valid instant by instant.

The Proof

The proof is short and elegant — it turns the branch sum into a sum over nodes, where KCL finishes the job.

Proof of Tellegen's theorem: write each branch voltage as a difference of node potentials, regroup by node, and apply KCL so the sum is zero
Write vk = ea − eb, regroup by node, then apply KCL — each node contributes zero.

Because KVL holds, every branch voltage can be written as the difference of two node potentials, vk = ea − eb. Substituting and collecting terms by node turns the branch sum into a sum over nodes, each multiplied by the total current leaving that node. By KCL that total is zero at every node, so the whole sum vanishes:

Σk vkik = Σnodes en · (Σ currents leaving n) = Σnodes en · 0 = 0

The Step-by-Step Method

To verify Tellegen's theorem on a circuit, follow five short steps.

Five steps to verify Tellegen's theorem: assign directions, solve the network, form each product, add them up, confirm the sum is zero
The five-step Tellegen verification procedure.

The procedure in words

  1. Assign reference directions — a consistent associated (passive) reference for each branch's v and i.
  2. Solve the network for every branch voltage vk and current ik.
  3. Form each product vk·ik (the branch power).
  4. Add them up over all branches.
  5. Confirm it is zero — delivered power balances absorbed power.

Solved Example

Take a source E = 12 V feeding R1 = 2 Ω to a node B, where R2 = 6 Ω and R3 = 3 Ω sit in parallel to the return.

Solved Tellegen example where the branch powers minus 36, plus 18, plus 6 and plus 12 watts sum to zero
The branch powers −36, +18, +6 and +12 W sum to exactly zero.

Solving: R2 ∥ R3 = 6∥3 = 2 Ω, so the source current is I = 12 / (2 + 2) = 3 A and the node-B voltage is VB = 3 × 2 = 6 V. The branch currents are then IR2 = 6/6 = 1 A and IR3 = 6/3 = 2 A.

Branchv (V)i (A)p = v·i (W)
Source E12−3−36 (delivers)
R163+18
R261+6
R362+12
Sum——0
Σ vkik = −36 + 18 + 6 + 12 = 0
Tellegen confirmed. (Check KCL at B: 3 A in = 1 A + 2 A out.)

The Quasi-Power (Difference) Form

The deepest version of Tellegen's theorem couples two different networks that share the same graph. The voltages of one and the currents of the other still sum to zero — even though they belong to different circuits.

Quasi-power form of Tellegen's theorem: two networks with the same graph, the sum of branch voltages of one times branch currents of the other is zero
Same graph, two circuits: Σ vk(1) îk(2) = 0 (and the reverse).
Σ vk(1) ik(2) = 0   and   Σ vk(2) ik(1) = 0
These “quasi-power” products aren't physical powers, yet they still vanish — the basis of the adjoint-network method for sensitivity analysis.

Sign Convention

To make the sum come out to zero, every branch must use a consistent associated (passive) reference for its voltage and current.

Associated passive sign convention for Tellegen's theorem: current entering the plus terminal means power absorbed, a source shows a negative product
Current into the + terminal → power absorbed (positive); a source delivering shows a negative product.
Be consistent

Pick the passive convention for every branch: reference the current as entering the positive voltage terminal. Then loads give v·i > 0, sources give v·i < 0, and the grand total is exactly zero.

Complete Generality

Unlike almost every other network theorem, Tellegen's theorem places no restriction on the elements. It is true for any lumped network whatsoever.

Tellegen's theorem holds for linear, non-linear, time-varying, active and passive networks alike
Linear, non-linear, time-varying, active or passive — all obey Σvkik = 0.
Why so general?

Because the proof uses only KCL and KVL — statements about connections, not components. Whatever sits in a branch (a diode, a transistor, a time-varying capacitor), as long as it connects through two terminals the theorem holds.

Applications

Tellegen's theorem is both a sanity check and a serious analysis tool.

Applications of Tellegen's theorem: power balance checks, proving theorems, sensitivity analysis, simulation checks, non-linear circuits and topology
From power-balance checks to sensitivity analysis — Tellegen's theorem in practice.
Power balance

Verify Σpower = 0 in any solved circuit.

Proving theorems

Underlies reciprocity and other results.

Sensitivity analysis

The adjoint-network method uses the quasi-power form.

Simulation checks

Validate SPICE / numerical solutions.

Non-linear circuits

Works where linearity-based methods fail.

Topology insight

Shows what depends on wiring alone.

Key Terms — Glossary

TermMeaning
Tellegen's theoremΣvkik = 0 over all branches of any network obeying KCL and KVL.
BranchA single two-terminal element (or path) in the network graph.
Branch powerThe product vk·ik for one branch.
KCLKirchhoff's current law: currents at a node sum to zero.
KVLKirchhoff's voltage law: voltages around a loop sum to zero.
Node potentialThe voltage of a node relative to a reference; vk = ea − eb.
Topology / graphHow the branches connect — the only thing Tellegen depends on.
Quasi-powerProducts of one network's voltages with another's currents (same graph); also sum to zero.
Passive sign conventionCurrent referenced into the + terminal, so v·i is power absorbed.

Frequently Asked Questions

Quick, expert answers to the questions people ask most about Tellegen's theorem.

What is Tellegen's theorem in simple words?

If you multiply the voltage across each branch by the current through it and add the results for every branch, the total is always zero. It holds for any network that obeys Kirchhoff's current and voltage laws.

What is the formula for Tellegen's theorem?

Σk=1b vk ik = 0, where b is the number of branches, vk is the voltage across branch k and ik the current through it, using associated reference directions.

What does Tellegen's theorem depend on?

Only on the network topology — how the branches connect. As long as the branch voltages satisfy KVL and the branch currents satisfy KCL, the theorem holds regardless of what the elements are.

Is Tellegen's theorem the same as conservation of power?

In effect, yes. Because all branch powers sum to zero, the power delivered by the sources exactly equals the power absorbed by the rest. It is a general statement of conservation of instantaneous power.

Does Tellegen's theorem require a linear circuit?

No. It is completely general — linear or non-linear, time-invariant or time-varying, active or passive — because it follows only from Kirchhoff's laws, not from element characteristics.

What is the quasi-power form of Tellegen's theorem?

It applies to two different networks with the same graph: the sum over branches of one network's voltages times the other's currents is also zero. This underpins sensitivity and adjoint-network methods.

What are the applications of Tellegen's theorem?

Power-balance checks, proving other theorems such as reciprocity, sensitivity analysis via the adjoint network, validating circuit-simulator results, and analysing non-linear and time-varying networks.

How is Tellegen's theorem different from other network theorems?

Most theorems (Thevenin, Norton, superposition) need a linear circuit and simplify it. Tellegen instead states a universal property — the branch powers sum to zero — that holds for any network obeying Kirchhoff's laws.

Conclusion & Key Takeaways

Tellegen's theorem is the most general power statement in circuit theory: for any lumped network obeying Kirchhoff's laws, the branch powers sum to exactly zero.

Σvkik = 0

Over every branch.

KCL + KVL

The only requirements.

Power balance

Delivered = absorbed.

Any element

Linear or non-linear.

Topology only

Depends on the wiring.

Quasi-power

Powers sensitivity analysis.

Continue Learning

Kirchhoff's Laws Reciprocity Theorem Superposition Theorem All Topics