Two parallel pancake paths can have equal turn counts and still carry unequal current. Their resistance, leakage relationships and mutual coupling differ with route and position. Temperature can change the imbalance further.
Define the actual branches, including series sections, leads and joints. Verify polarity and compatible induced voltage before analyzing loaded sharing. A branch with different induced voltage can create circulating current that is not explained by a resistance-only model.
Write the coupled branch equations
For a linear harmonic illustration, V1 = R1I1 + jω(L11I1 + L12I2) and V2 = R2I2 + jω(L21I1 + L22I2). Parallel connection imposes V1 = V2 and terminal current I = I1 + I2. Resistances are in ohms, inductances in henries, angular frequency in radians per second and currents as phasors in amperes.
The matrix entries require a consistent flux-linkage and current reference. Mutual terms cannot be discarded merely because the physical branches are separate. If a separate induced-voltage source is used, ensure it is not counted again in the inductance representation.
| Input | What to include | Why it affects sharing |
|---|---|---|
| Branch resistance | Conductor, joints and leads | Changes resistive voltage drop |
| Self and mutual inductance | Actual spatial paths | Changes coupled reactive drop |
| Induced voltage and polarity | Turn and connection relationships | Can drive circulation at low terminal load |
| Temperature state | Each branch’s material response | Resistance need not change equally |
| Harmonic current components | Frequency-specific impedance | Fundamental sharing may not represent all heating |
Compare thermal states
Solve the branch currents at a documented initial temperature. Then repeat using a plausible unequal-temperature state with justified material resistance data. A hotter branch may shed some current through increased resistance, but mutual impedance and the rest of the network govern the complete result. Keep connection, tapping and network assumptions explicit in the application assessment [1].
If thermal feedback is important, couple branch losses and temperatures iteratively or dynamically. A single-temperature calculation cannot establish that the eventual operating state remains balanced.
Keep circulating and load-sharing components distinguishable. A small terminal current does not guarantee small internal branch currents when induced voltages differ.
Verify the observable outputs
Use qualified branch-current measurements or accessible loss and temperature evidence where practical. Total terminal current alone cannot reveal its division. Equal total resistance from a terminal test also does not uniquely identify both branch resistances and mutual relationships. Assess the relevant measured quantities with a defined measurement model and propagated uncertainty [2].
Trace an overloaded pancake path through its joints, leads and coupled impedance before increasing conductor area. Validate the predicted branch currents in the relevant operating state, especially the parameters that control the imbalance.
Solve the two branch currents from their differential impedance
Assume equal induced voltages, reciprocal mutual coupling and a 50 Hz terminal current of 100 A RMS. Let R₁ = 0.10 Ω, R₂ = 0.20 Ω, L₁₁ = 0.40 mH, L₂₂ = 0.60 mH and L₁₂ = L₂₁ = 0.20 mH. Subtracting the branch equations gives I₁/I₂ = [R₂ + jω(L₂₂ − L₁₂)]/[R₁ + jω(L₁₁ − L₁₂)] = 2. The currents are therefore 66.67 and 33.33 A; their copper losses are 444.4 and 222.2 W. Equal current division would miss the higher branch duty. Obtain branch parameters at the relevant temperature before accepting conductor capability. Record rated quantities and test conditions on the general power-transformer basis of IEC 60076-1 [3].
References
- IEC, IEC 60076-8:1997, Power transformers – Part 8: Application guide, 1997, official public scope.
- JCGM, JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement, 2008.
- IEC, IEC 60076-1:2011, Power transformers – Part 1: General.