Transformer

2-Winding Transformer

The transformer is a T-equivalent: the winding resistance and leakage inductance are split evenly between the two windings, a magnetizing branch is connected from the midpoint of that split to ground, and an ideal transformer carries the ratio. The single line diagram is depicted in the figure below.

Writing the total leakage impedance referred to the primary as $z = R + j \omega L$, the two series segments are $z/2$ each and the magnetizing branch is a resistance in parallel with an inductance,

The model creates virtual nodes for the midpoint and for the ideal part, and two more when the winding resistance is included, so a transformer with resistive losses occupies five virtual nodes and one without occupies three.

Magnetizing parameters

The branch is sized from the two no-load quantities a nameplate carries under IEC 60076, both given per unit of the rated power $S_r$: the no-load current $i_0$ and the no-load loss $P_0$. With the nominal voltage of the winding the impedance is referred to, $V_p$,

The defaults are $i_0 = 0.01$ and $P_0 = 0.001$, and setMagnetizingBranch(i0, P0) overrides them. Two conditions are checked rather than assumed. The rated power must be positive, because it is the only scale the branch can be sized against; a transformer without one is built with no magnetizing branch and says so in the log. And $i_0$ must exceed $P_0$, because otherwise the square root above has no real value and the branch has no reactive part at all, which is not a transformer. That case raises rather than silently producing a purely resistive shunt.

This is why the rating is a required parameter rather than documentation, and the same is true of the nominal voltages: the branch is referred to one winding, and which one is decided by the reference winding.

The ideal part and its extra equation

The ideal transformer imposes two constraints at once: the voltages are in a fixed ratio and the powers on the two sides are equal, which makes the currents inversely proportional to the same ratio,

Neither is a current balance at a node, so neither can be written as an admittance. This is the same situation as an ideal voltage source described under sources: the system is extended with the branch current as an unknown, the constraint occupies the added row, and the added diagonal entry is zero. The complete matrix stamp is

The variable $j$ denotes the high voltage node while $k$ is the low voltage node, and $l$ indicates the inserted row and column. The transformer ratio is $T = V_{j} / V_{k}$. The asymmetry of the stamp, $-1$ against $\overline{T}$ in the added column and $1$ against $-T$ in the added row, is exactly the statement that voltage scales by $T$ while current scales by $1/T$ with opposite sign.

A complex $T$ adds a phase shift, which is how a delta-wye connection is represented without modelling the windings: the magnitude carries the tap ratio and the angle carries the vector group. The conjugate on the low voltage row is what makes the phase shift preserve power rather than create it. The three-phase representation in SP::Ph3 and DP::Ph3 carries this, and EMT::Ph3 cannot, because a phase shift is a rotation in the complex plane and the EMT state is real.

The direction of the ratio

The reference winding decides which side the impedance is referred to. Referring an impedance across an ideal transformer scales it by $T^2$, so the choice of side is a choice of reference, not an approximation, and the parameters have to be given consistently with it.

By default the reference is resolved from the nominal voltages, taking the higher one, and the ratio is then greater than one, from high voltage to low. Passing a WindingReference explicitly overrides that: Primary, Secondary or Tertiary name the winding, and Auto restores the resolution from nominal voltages. If both windings are nominally equal the resolution is ambiguous, the primary is assumed and the model warns; an explicit reference removes the ambiguity.

Three-winding transformers are not implemented, so Tertiary raises rather than silently modelling something else.

Power flow and the behaviour gate

For power flow the T is reduced to a two-port, which is a Kron reduction eliminating the midpoint node. With the half-leakage $z_h = z/2$ and $\Delta = z + z_h^2 y_m$,

At $y_m = 0$ this degenerates exactly to the series admittance $1/z$, so the reduction is the textbook pi-with-tap plus a correction, not a different model.

Whether $y_m$ is present at all is decided by the component behaviour. Under Behaviour::Simulation the magnetizing branch is left out, so a plain load flow matches the textbook pi-with-tap that a user comparing against hand calculations expects. Under Behaviour::Initialization and Behaviour::MNASimulation it is included, so that the power flow solution seeds a time-domain run of the same circuit and the admittance the power flow sees is the limit of the MNA conductance as the step grows.

Scaling conventions across the domains

The domains do not agree on what a voltage or a current means, and the transformer touches all of them, so the conventions are collected here.

Power flow node voltages are line-line RMS. The DP::Ph3 and EMT::Ph3 states are phase-peak envelopes instead, so seeding one from the other needs the factor RMS3PH_TO_PEAK1PH, which is $\sqrt{2/3}$. Omitting it initializes a state $\sqrt{3/2}$ too small and produces a startup transient after an otherwise correct power flow initialization, which reads as a model defect and is not one. SP::Ph3 keeps the line-line RMS convention in every phase, so each phase entry carries the whole-transformer quantity rather than a third of it.

The sources are not uniform either. EMT::Ph3::VoltageSource is the only source class that scales its own reference, so it takes line-line RMS directly, while every other source takes the working unit of its own domain. Driving a DP::Ph3 circuit with a raw line-line value therefore places the operating point $\sqrt{3/2}$ above what the initialization assumed, and the model rings.

The power formula follows from the same split. In DP::Ph3 and EMT::Ph3, where the quantities are phase-peak,

while in DP::Ph1, SP::Ph1 and SP::Ph3 it is $\mathrm{Re}{ V \overline{I} }$ from a single entry. Applying the second form to a phase-peak domain gives exactly two thirds of the right answer.

Verification

dpsim/examples/cxx/Circuits/DP_Ph1_Transformer_OpenShortCircuit.cpp runs three units through the open-circuit and short-circuit tests in all five domains. Every unit recovers its nameplate: the no-load current and loss from the open-circuit test, and the short-circuit voltage $u_k$ from the short-circuit test, each to within the discretization error of the step used.

One reading of the open-circuit test needs care in EMT. The magnetizing loop has no resistive path that damps a DC component, so an offset left by energization decays with a time constant of the order of $L_m$ over the winding resistance, which is minutes rather than cycles. An RMS taken over a window at one second still contains it. Extracting the fundamental with a one-cycle Fourier component rejects it and returns the nameplate value; the phasor domains do not show the effect at all, because they do not represent DC.