Fixed-linear transformer factor completion

Page status: guarded fixed-linear transformer completion with an executable anatomy witness; broader rating and control models remain open.

The leakage factor is only one part of a transformer model. A linear phasor model may also contain fixed ideal voltage transfers, a no-load excitation branch, and transformer-internal neutral grounding. These components can be assembled exactly, but only after their placement, coordinates, and control status have been declared.

This chapter completes the fixed-linear factor. Here complete means complete for the declared linear steady-state components. Saturation, frequency dependence, thermal state, protection, and adjustable controls are not thereby eliminated or approximated.

The figure is generated by experiments/render_transformer_anatomy.py. It shows the source factor and its typed interfaces; it is not a physical winding cross-section or a claim that the device decomposes into independent two-winding elements.

Serialization contract

For transformer $x$, winding $k$ at bus $i$ retains:

• its stable winding identity and position $k$;
• ordered terminal labels $\mathbf N_{xk}$ and labelled coil coordinates $\mathcal C_{xk}$;
• the connection incidence $\mathbf A_{xk}$;
• the source impedance-test reference and selected leakage-compilation reference;
• a declared voltage-transfer mode, coefficient convention, and decision identity when adjustable;
• the placement and coordinates of every excitation or grounding factor.

The executable JSON contract uses the versioned convention

\[V^{\mathrm{leak}}_{xkc} =a_{xkc}V^{\mathrm{coil}}_{xkc}, \qquad c\in\mathcal C_{xk}.\]

Thus $a_{xkc}$ is not an unqualified tap field whose direction must be guessed. An adapter may map a package's tap convention into $a_{xkc}$, but it must record that map. The running contract is stored separately from the canonical BMOPFTools fixture because its excitation and grounding values are illustrative, not nameplate data.

Power-dual voltage transfer

Let $\mathbf A_x$ be the aligned block connection matrix from TR-XFMR-003 and let

\[\mathbf T_x =\operatorname{blkdiag} \left( \operatorname{diag}(\mathbf a_{x1}),\ldots, \operatorname{diag}(\mathbf a_{xn_x}) \right).\]

The connected-coil and leakage-coordinate voltages are

\[\mathbf V_x^{\mathrm{coil}}=\mathbf A_x\mathbf U_x, \qquad \mathbf V_x^{\mathrm{leak}} =\mathbf T_x\mathbf A_x\mathbf U_x.\]

A voltage map does not determine the same current map. Complex-power preservation requires the conjugate-transpose dual:

\[\mathbf I_x^{\mathrm{w,leak}} =\mathbf T_x^{\mathrm H}\mathbf I_x^{\mathrm{leak}}.\]

Indeed,

\[(\mathbf V_x^{\mathrm{coil}})^{\mathrm H} \mathbf I_x^{\mathrm{w,leak}} =(\mathbf T_x\mathbf V_x^{\mathrm{coil}})^{\mathrm H} \mathbf I_x^{\mathrm{leak}}.\]

This distinction is invisible for a real unit coefficient. It is essential for an off-nominal magnitude or an explicitly declared phase-shifting transfer. Connection-induced vector-group displacement should remain in the real winding incidences—such as the declared delta roll—rather than being duplicated in $\mathbf T_x$.

With $\mathbf Y_x^{\mathrm{coil}}$ from the leakage assembly, define

\[\begin{aligned} \mathbf B_x&=\mathbf T_x\mathbf A_x,\\ \mathbf I_x^{\mathrm{leak}} &=\mathbf Y_x^{\mathrm{coil}}\mathbf B_x\mathbf U_x. \end{aligned}\]

The leakage contribution at transformer terminals is therefore

\[\mathbf Y_x^{\mathrm{series}} =\mathbf B_x^{\mathrm H} \mathbf Y_x^{\mathrm{coil}} \mathbf B_x.\]

A complex phase-shifting transfer need not produce a complex-symmetric nodal matrix. The correctness condition here is the declared voltage/current dual and its complex-power identity, not symmetry copied from the real-incidence special case.

Excitation and core loss

Let winding $k_0$ carry the explicitly placed excitation branch. The real selection-incidence matrix $\mathbf S_x$ maps terminal voltages to its aligned coil voltages, and $\mathbf Y_x^0$ is its labelled coil admittance:

\[\mathbf V_x^0=\mathbf S_x\mathbf U_x, \qquad \mathbf I_x^0=\mathbf Y_x^0\mathbf V_x^0.\]

The implementation accepts a full reciprocal matrix $\mathbf Y_x^0$ rather than assuming diagonal phases. Its Hermitian part must be positive semidefinite within tolerance, so the shunt cannot generate real power. A diagonal $G_x^0+\mathrm jB_x^0$ is the common core-loss and magnetising special case.

Placement is part of the contract. The executable example follows the BMOPFTools/OpenDSS convention of placing the branch across winding 2's coils, but the book-level compiler does not infer that placement from an array position.

Transformer-internal grounding

For each declared transformer-internal grounding branch $g$, let $y_g$ be its passive terminal-to-earth admittance and $\mathbf e_g$ select the qualified terminal $(x,k,N)$. Then

\[\mathbf Y_x^{\mathrm{ground}} =\sum_g y_g\mathbf e_g\mathbf e_g^{\mathsf T}.\]

This factor is distinct from a neutral conductor, a voltage reference, and an external bus grounding. The compiler rejects an object whose scope is external_bus: absorbing it would change asset ownership and could invalidate later grounding, protection, or topology decisions.

Completed fixed-linear factor

The terminal current is the sum of the three retained contributions,

\[\mathbf I_x =\mathbf B_x^{\mathrm H}\mathbf I_x^{\mathrm{leak}} +\mathbf S_x^{\mathsf T}\mathbf I_x^0 +\mathbf Y_x^{\mathrm{ground}}\mathbf U_x,\]

and hence

\[\boxed{ \mathbf Y_x^{\mathrm{complete}} =\mathbf B_x^{\mathrm H}\mathbf Y_x^{\mathrm{coil}}\mathbf B_x +\mathbf S_x^{\mathsf T}\mathbf Y_x^0\mathbf S_x +\mathbf Y_x^{\mathrm{ground}} }.\]

The target retains maps for leakage-coil current, winding-side leakage current, excitation current, and internal-ground current. A leakage-path coil limit is therefore enforced on

\[\left| [\mathbf T_x^{\mathrm H}\mathbf Y_x^{\mathrm{coil}} \mathbf T_x\mathbf A_x\mathbf U_x]_{xkc} \right| \leq \overline i^{\mathrm{leak}}_{xkc}.\]

If a source instead defines a rating on total terminal current or apparent power, that semantic must be declared separately and applied to the corresponding recovered sum. The compiler does not silently reinterpret a leakage-path rating.

Adjustable taps are a different target

For a tap decision $t_x$ or discrete position $d_x$, the relation is

\[\mathbf V_x^{\mathrm{leak}} =\mathbf T_x(t_x,d_x)\mathbf A_x\mathbf U_x, \qquad \mathbf I_x^{\mathrm{w,leak}} =\mathbf T_x(t_x,d_x)^{\mathrm H}\mathbf I_x^{\mathrm{leak}}.\]

This is a parameterized factor in the decision model, not one fixed admittance matrix. Replacing $t_x$ by its start value changes the feasible set and loses the tap decision. The static compiler therefore returns the structured guard adjustable_winding_transfer_requires_factorized_decision_model for both continuous and discrete modes. The serialization still retains the mode and decision identity so a later optimization compiler can implement the correct factor.

Executable evidence

The illustrative completion of the running WYE/WYE/DELTA transformer has 11 external terminals, nine leakage-coil coordinates, a three-coordinate excitation branch on winding 2, and one internal neutral-ground branch. It records:

• component-current recovery residual $9.38\times10^{-14}\ \mathrm A$;
• complex-power residual $1.13\times10^{-13}\ \mathrm{VA}$; and
• difference $2.96\times10^{-17}\ \mathrm S$ from the independently constructed BMOPFTools.nwinding_yprim after removing the separately added internal-ground branch, which that implementation does not stamp for the n-winding subtype.

The tests also use nonuniform real tap coefficients and a complex fixed phase transfer, reorder labelled transfer coordinates, and reject active shunts, missing grounding terminals, external bus grounding, and adjustable controls. The machine-readable result is certificate TR-XFMR-004.

Model boundary

TR-XFMR-004 is exact for the declared fixed linear phasor factor. It is not a claim about saturation, harmonics, frequency-dependent core behavior, thermal state, inrush, protection, mechanical tap dynamics, or uncertainty. Those are additional factors and states. Parameterized transformer tap decisions provides the continuous/discrete continuation that preserves feasible sets and decision identities rather than evaluating one fixed snapshot.