Magnetic Saturation · EMTP®

Magnetic Saturation in Synchronous Machine Modelling

The basic synchronous-machine equations assume a linear magnetic circuit — flux proportional to current, constant inductances. That is useful for deriving the model structure, but a real machine is not linear: at low flux the magnetising current and air-gap flux are nearly proportional, while at higher flux the iron saturates and more field current is needed for the same terminal voltage. For load rejection, terminal overvoltage, excitation response, synchronisation and large disturbances, that nonlinearity matters. In EMTP® studies saturation is added after the linear model, and — crucially — it is applied mainly to the mutual air-gap flux path, not to every leakage inductance. This guide explains the open-circuit characteristic behind it, the saturated and unsaturated reactances, and how the saturation curve is represented.

Reading time ≈ 35 min · OCC, air-gap line, SCR & the saturation curve

A machine model that is fine for a small-signal or lightly loaded condition can become inaccurate near or above rated voltage, during field forcing, or during abnormal transients. Saturation modifies the mutual air-gap inductances according to the operating flux level — the linear model provides the d- and q-axis structure, and the saturation model corrects the mutual path.

Abbreviations used on this page

EMTP®Electromagnetic Transients Program

OCCOpen-circuit characteristic

SCCShort-circuit characteristic

SCRShort-circuit ratio

puPer unit

RMSRoot mean square

\(K_s\)Saturation factor

\(L_{ad}, L_{aq}\)Saturated d- and q-axis mutual inductances

\(X_{du}, X_{ds}\)Unsaturated / saturated d-axis synchronous reactance

\(I_{fNL}, I_{fSC}\)Field current at rated no-load / for rated SC current

Key idea

Saturation is applied to the mutual air-gap path, not to the leakage inductances — the leakage flux is treated as independent of saturation.
The OCC is the master curve; its departure from the air-gap line measures saturation: the real machine needs more field current than the linear model predicts for the same voltage.
The saturation factor reduces the mutual inductance, \(L_{ad}=K_s L_{adu}\), with \(K_s \le 1\); round-rotor machines saturate on the resultant flux, salient-pole machines mainly on the d-axis.
Keep the OCC, air-gap line, field-current base and saturated/unsaturated reactance basis consistent — that is what makes the saturation model reliable.

Key terms used on this page

01Magnetic saturation

The nonlinear fall in incremental permeability as iron flux rises.

02Open-circuit characteristic

No-load terminal voltage vs field current at rated speed.

03Air-gap line

Straight-line extension of the unsaturated lower OCC; the ideal linear response.

04Short-circuit characteristic

Steady armature current vs field current with terminals shorted; nearly linear.

05Saturation factor \(K_s\)

The multiplier (\(\le 1\)) that reduces the mutual inductance under saturation.

06Resultant air-gap flux

\(\psi_{at}\): the combined d/q air-gap flux that drives saturation.

07Short-circuit ratio

\(SCR=I_{fNL}/I_{fSC}\); a strength indicator, \(\approx 1/X_{ds}\).

08Saturated / unsaturated reactance

\(X_{ds}\) (actual OCC) and \(X_{du}\) (air-gap line); \(X_{ds} < X_{du}\).

09Reciprocal / nonreciprocal base

Two field-current per-unit conventions; must match the software.

10Three-segment curve

Unsaturated, exponential and fully-saturated regions of the OCC.

Section 1

What magnetic saturation is

The iron in the stator and rotor has finite permeability. At low flux the relationship between magnetising current and air-gap flux is almost linear; at higher flux the iron saturates and a larger field-current increase is needed to produce the same terminal voltage or air-gap flux. This nonlinearity is magnetic saturation, and for many studies it cannot be ignored: it affects terminal and internal voltage, excitation current, reactive capability, short-circuit behaviour, load-rejection overvoltage, synchronisation and large-disturbance response. In EMTP® the key principle is that saturation is not applied to every inductance — it is applied mainly to the mutual air-gap flux path, because that is the part of the magnetic circuit passing through the iron most affected by saturation.

Section 2

Why saturation is neglected first

The basic equations neglect saturation for a practical reason: without it the machine is a set of linear coupled circuits, the flux-linkage equations can be written with inductance matrices, and superposition applies. That linear foundation gives the stator and rotor voltage equations, the d- and q-axis flux-linkage equations, the equivalent circuits, the transient and subtransient inductances, the open- and short-circuit time constants, the operational inductances and the parameter-conversion procedures. Saturation is then added as a nonlinear correction to the mutual path. This avoids starting from a fully nonlinear field problem while still capturing the most important effect of iron saturation.

Section 3

The open-circuit characteristic

The most important saturation curve is the open-circuit characteristic (OCC), measured at rated speed with the stator open-circuited: the field current is varied and the no-load terminal voltage recorded. With the stator open the stator current is zero, so the terminal voltage is mainly related to the d-axis air-gap flux produced by the field current:

\[ i_d = 0 \qquad i_q = 0 \qquad v_d = 0 \qquad \psi_q = 0 \] \[ v_t = v_q = \psi_d \qquad\qquad \psi_d = L_{ad}\,i_F \]

\(i_d,\ i_q\): d- and q-axis stator currents (both zero on open circuit)
\(v_d,\ v_q\): d- and q-axis stator voltages
\(\psi_d,\ \psi_q\): d- and q-axis stator flux linkages
\(v_t\): no-load terminal voltage (in the selected per-unit convention)
\(L_{ad}\): d-axis mutual air-gap inductance
\(i_F\): field current

These relations assume rated speed and per-unit normalisation, so that the per-unit terminal voltage equals the per-unit d-axis flux linkage; the exact voltage–flux equality depends on that speed and per-unit convention. The OCC therefore gives the relationship between field current and d-axis flux (or no-load voltage) — the essential curve for d-axis saturation. At low field current it is nearly a straight line; at higher field current it bends downward relative to that line as the iron saturates.

Section 4

The air-gap line

The air-gap line is the straight-line extension of the lower, unsaturated part of the OCC — the ideal field-current-to-voltage relationship if the iron did not saturate (constant effective permeability). The departure of the OCC from the air-gap line is a direct measure of saturation: at a given terminal voltage the actual field current required on the OCC is higher than on the air-gap line, and the difference is the extra magnetising effort the saturated iron demands.

Physical meaning

The saturated machine needs more field current than the unsaturated magnetic model predicts for the same terminal voltage.

Section 5

The short-circuit characteristic

The short-circuit characteristic (SCC) is measured at synchronous speed with the stator terminals shorted: the field current is varied and the steady armature current recorded. Unlike the OCC, the SCC is normally almost linear up to rated armature current, because under short circuit the terminal voltage is very low and the resultant air-gap flux is low, so the iron is not significantly saturated. The contrast is the useful part: the OCC is strongly affected by saturation at higher voltage, the SCC is nearly linear in the test range — and together they let the saturated and unsaturated synchronous impedances be estimated.

Section 6

Saturated and unsaturated reactance, and SCR

Neglecting armature resistance, the unsaturated d-axis synchronous reactance uses the air-gap line, while the saturated value uses the actual OCC at rated voltage:

\[ X_{du} = \frac{I_{fSC}}{I_{fNL(ag)}} \qquad\qquad X_{ds} = \frac{I_{fSC}}{I_{fNL}} \]

\(X_{du},\ X_{ds}\): unsaturated and saturated d-axis synchronous reactance (pu)
\(I_{fSC}\): field current for rated armature current under steady three-phase short circuit
\(I_{fNL(ag)}\): field current for rated no-load voltage on the air-gap line
\(I_{fNL}\): field current for rated no-load voltage on the actual OCC

Because saturation needs extra field current at rated voltage, \(I_{fNL} > I_{fNL(ag)}\), so the saturated reactance is lower than the unsaturated value (\(X_{ds} < X_{du}\)). Saturated and unsaturated parameters should not be mixed carelessly: using a saturated reactance with an unsaturated saturation curve (or vice-versa) breaks the intended voltage/field-current relationship.

\[ SCR = \frac{I_{fNL}}{I_{fSC}} \qquad\qquad SCR = \frac{1}{X_{ds}} \]

\(SCR\): short-circuit ratio (armature resistance neglected)
\(I_{fNL},\ I_{fSC}\): field currents at rated no-load voltage and for rated SC current

Different standards and manufacturer data sheets may express the short-circuit ratio in equivalent forms, so the test basis should be confirmed before it is used. A higher SCR means a stronger machine — less sensitive to load changes, needing smaller field-current changes to hold voltage, but usually larger and more expensive. A lower SCR is more compact and cheaper but more sensitive and needs a stronger excitation system. In EMTP® the SCR is a useful high-level check: if the model’s saturated synchronous reactance is inconsistent with the expected SCR, review the OCC/SCC interpretation or the base conversion.

Section 7

Which part of the machine saturates

Saturation is applied to the mutual air-gap inductances, not to every leakage inductance. The main assumptions: stator and rotor leakage fluxes are largely independent of saturation and their contribution to iron saturation is neglected; zero-sequence leakage flux is likewise treated as non-saturable; the air-gap mutual flux sets the saturation level; the sinusoidal air-gap field shape is taken as sufficiently valid; and the d- and q-axes stay uncoupled in the saturation representation. The d-axis mutual inductance \(L_{ad}\) is the one most directly modified; whether the q-axis mutual path is also saturated depends on the machine type and the saturation model. Saturating every inductance indiscriminately would distort the physical behaviour. So saturation is not a full finite-element problem — it is a modification of the stator–rotor mutual inductance according to the air-gap flux level:

\[ L_{ad} = K_s\,L_{adu} \qquad\qquad L_{aq} = K_s\,L_{aqu} \]

\(L_{ad},\ L_{aq}\): saturated d- and q-axis mutual inductances
\(L_{adu},\ L_{aqu}\): unsaturated d- and q-axis mutual inductances
\(K_s\): saturation factor (\(K_s \le 1\) when saturation reduces the mutual inductance)

Saturation depends on the model formulation

Different simulation tools apply saturation through different machine formulations. In a reactance-and-time-constant model, the input data are usually unsaturated reactances and time constants, and the software adjusts the effective saturated values during the simulation according to the operating flux level. In a circuit-model formulation, the model is built from stator leakage, field leakage, damper branches and mutual air-gap inductances; in that case saturation is normally applied by adjusting the mutual air-gap inductances, mainly \(L_{ad}\) and, where the model supports it, \(L_{aq}\), while the leakage inductances remain linear. The important point is consistency: the saturation function used during time-domain simulation must be compatible with the same field-current base, air-gap line, OCC and unsaturated parameter set used during initialisation.

Initialise and simulate on the same saturation convention

If the model is initialised using one saturation convention but simulated using another, the machine may appear numerically stable but give the wrong no-load voltage, field current, reactive-power output or voltage-recovery behaviour.

Section 8

The saturation factor

The saturation factor \(K_s\) represents the reduction of effective mutual inductance, obtained from the OCC by comparing the actual air-gap flux with the unsaturated flux the air-gap line would have produced:

\[ K_s = \frac{\psi_{at}}{\psi_{at} + \psi_I} \qquad\qquad \psi_I = 0 \;\Rightarrow\; K_s = 1 \]

\(K_s\): saturation factor
\(\psi_{at}\): actual resultant air-gap flux linkage
\(\psi_I\): saturation departure term (distance from the air-gap line)

When the machine is unsaturated \(\psi_I = 0\) and \(K_s = 1\); as saturation grows \(\psi_I\) increases and \(K_s\) falls below one, reducing the mutual inductance. In other words, saturation is represented by reducing the mutual air-gap inductance so the model needs more magnetising current to produce the same air-gap flux.

Section 9

Resultant air-gap flux for round-rotor machines

For solid-rotor / round-rotor machines the saturation relationship is usually based on the resultant air-gap flux, assuming the loaded saturation can be represented using the no-load OCC:

\[ \psi_{at} = \sqrt{\psi_{ad}^{2} + \psi_{aq}^{2}} \] \[ I = \sqrt{(i_d + i_F + i_D)^{2} + (i_q + i_{Q1} + i_{Q2})^{2}} \]

\(\psi_{at}\): resultant air-gap flux linkage
\(\psi_{ad},\ \psi_{aq}\): d- and q-axis mutual air-gap flux linkages
\(I\): magnitude of the resultant magnetising current
\(i_d, i_F, i_D\): d-axis stator, field and d-damper currents
\(i_q, i_{Q1}, i_{Q2}\): q-axis stator and damper currents

This lets one OCC-derived saturation curve serve the combined d- and q-axis air-gap flux.

Section 10

Saturation in salient-pole machines

In a salient-pole machine the magnetic structure differs greatly between axes: the d-axis flux passes through the pole body and is strongly affected by iron saturation, while the q-axis flux path is largely through air and far less affected. So saturation is often applied only to the d-axis mutual inductance, keeping the q-axis mutual inductance constant:

\[ L_{ad} = K_s\,L_{adu} \qquad\qquad L_{aq} = L_{aqu} \] \[ \psi_{at} = \psi_{ad} \qquad\qquad I = i_d + i_F + i_D \]

\(L_{adu},\ L_{aqu}\): unsaturated d- and q-axis mutual inductances
\(\psi_{at} = \psi_{ad}\): saturation based on d-axis air-gap flux only
\(I = i_d + i_F + i_D\): d-axis magnetising current

A round-rotor machine and a salient-pole machine should not automatically use the same saturation treatment.

Section 11

The three-segment saturation curve

A practical way to represent the saturation curve is to split it into three regions, reproducing the OCC shape with a small number of parameters:

\[ \text{unsaturated:}\quad \psi_{at} \le \psi_{T1} \;\Rightarrow\; \psi_I = 0,\; K_s = 1 \] \[ \text{nonlinear:}\quad \psi_{T1} < \psi_{at} \le \psi_{T2} \;\Rightarrow\; \psi_I = A_{sat}\,\exp\!\left[B_{sat}(\psi_{at} - \psi_{T1})\right] \] \[ \text{fully saturated:}\quad \psi_{at} > \psi_{T2} \;\Rightarrow\; \psi_I = \psi_{G2} + L_{ratio}(\psi_{at} - \psi_{T2}) - \psi_{at} \]

\(\psi_{at}\): resultant air-gap flux variable used by the curve
\(\psi_I\): saturation departure — the flux-linkage distance of the OCC from the air-gap line (same units as \(\psi_{at}\), as in the saturation-factor definition above)
\(\psi_{T1},\ \psi_{T2}\): boundaries of the unsaturated and fully-saturated regions
\(A_{sat},\ B_{sat}\): exponential saturation-curve constants
\(\psi_{G2}\): air-gap-line value at the second transition
\(L_{ratio}\): ratio of the air-gap-line slope to the fully-saturated incremental slope

Physically, \(\psi_I\) is how far the OCC departs from the air-gap line, expressed in flux-linkage terms. The exponential middle gives a smooth rise in saturation; the final linear segment avoids extending the exponential indefinitely and gives the high-saturation part a controlled slope. The curve is defined by \(\psi_{T1}, \psi_{T2}, \psi_{G2}, A_{sat}, B_{sat}, L_{ratio}\), all extracted from the OCC — if the OCC is unavailable, saturation modelling is uncertain and should be stated as an assumption. Check that the transition points introduce no numerical discontinuity large enough to affect the simulation: a small mathematical discontinuity may be negligible, but any visible jump in the curve should be corrected.

Section 12

Reciprocal and nonreciprocal field-current base

Field current in per unit needs care. In the reciprocal system, the per-unit field current for 1.0 pu terminal voltage on the air-gap line is related to the reciprocal of the unsaturated mutual reactance. In the nonreciprocal system, 1.0 pu field current is defined as the field current that produces 1.0 pu terminal voltage on the air-gap line. Many programs prefer the nonreciprocal system because it is easier to visualise: 1.0 pu field current is directly the field current for rated no-load voltage on the air-gap line. The distinction matters when importing OCC data, excitation data or manufacturer field-current values — the wrong base scales the saturation curve incorrectly. State the field-current base convention whenever saturation or excitation data is central to the study.

Confirm the base before importing a curve

Be clear whether the software uses the reciprocal or the nonreciprocal field-current base described above before importing OCC, excitation or manufacturer field-current values. Importing the same OCC on the wrong field-current base shifts the saturation curve and distorts the excitation response — even when the simulation still runs without error.

Section 13

How saturation affects EMTP® results

Saturation reduces the effective mutual inductance at high flux and raises the field current needed for a given terminal voltage. Its influence is best considered by output quantity:

No-load and load-rejection terminal voltage — the saturated machine holds a different voltage for the same field current.
Excitation current and automatic voltage regulator (AVR) response — field forcing and regulator action depend on the saturated characteristic.
Voltage recovery after fault clearing — the post-fault voltage rise depends on the saturated magnetising path.
Internal voltage and reactive power — the voltage behind the reactance, and the reactive capability, shift with saturation.
Apparent short-circuit response — saturation changes the effective impedance during severe transients.
Electromagnetic torque during large disturbances — torque is affected when the flux is driven high.
Simulated-versus-measured agreement — neglecting saturation can move the model away from test results.

For very-fast-front studies saturation is usually not controlling — terminal capacitance and high-frequency behaviour dominate — but for low-frequency and slow-front studies it can be important and should be considered carefully.

Section 14

When saturation should be included

Include saturation for load rejection, generator terminal overvoltage, excitation-system response, field forcing, synchronisation, inadvertent energisation, large reactive swings, severe close-in faults, voltage recovery, studies where field current is a key output, and operation near or above rated terminal voltage. It may be less important when the study stays close to nominal flux, when only the initial subtransient current is required, when the event is extremely fast and controlled by terminal capacitance, or when the data does not support a reliable saturation curve. Even when saturation is not included, the report should state that assumption.

Section 15

Practical workflow for saturation modelling

Obtain the machine’s open-circuit characteristic.
Identify the air-gap line from the lower linear part of the OCC.
Determine whether the data and software use the reciprocal or nonreciprocal field-current base.
Convert the OCC into the same per-unit system the EMTP® model uses.
Identify the unsaturated region and the onset of saturation.
Select the saturation representation (direct curve, piecewise-linear, exponential-segment, or software-specific format).
Define the required saturation parameters.
Check that the curve reproduces the rated no-load point.
Verify that the saturated and unsaturated synchronous reactances are consistent with the OCC and SCC.
Run sensitivity cases if the saturation data is uncertain or the result depends strongly on voltage level.

Section 16

Practical checks

The OCC source and test basis are known, and the rated speed / frequency basis of the curve is confirmed.
It is clear whether the OCC is in phase voltage, line voltage, RMS or per unit, and it is converted to the model’s per-unit system.
The air-gap line is correctly fitted to the linear lower part of the OCC; the saturated OCC needs more field current than the air-gap line at rated voltage.
The field-current base convention (reciprocal or nonreciprocal) matches the software; the saturation factor does not exceed one in the saturated region.
The saturated and unsaturated reactance basis is consistent: \(X_{du} > X_{ds}\), and the SCR is consistent with \(1/X_{ds}\).
The SCC is approximately linear up to rated armature current.
The saturation curve is smooth and monotonic, with no unrealistic discontinuity, and the q-axis treatment matches the machine type.
The saturated mutual inductances \(L_{ad}\) and \(L_{aq}\) remain physically meaningful (positive and of sensible magnitude).
The simulated no-load voltage matches the OCC, and the field current at rated no-load voltage is reasonable.
Load-rejection and voltage-recovery results are physically plausible; the model is tested with and without saturation where saturation materially affects the conclusion.
The saturation curve used for initialisation is the same curve — or an explicitly consistent curve — as the one used during the time-domain EMTP® simulation.
The mutual inductances \(L_{ad}\) and \(L_{aq}\) entered as unsaturated values are the ones the saturation function is intended to modify.

Section 17

Common modelling mistakes

Avoid these

Applying saturation to leakage inductances instead of the mutual air-gap path.
Mixing saturated and unsaturated reactances without checking their source.
Importing OCC data with the wrong field-current base (confusing reciprocal and nonreciprocal).
Confusing air-gap-line voltage with the actual OCC voltage.
Using the SCC or SCR without confirming the test basis.
Applying a round-rotor total-saturation model to a salient-pole machine without checking its validity.
Assuming the q-axis saturation is known when only d-axis / OCC data is available.
Using a saturation curve that is not smooth or monotonic.
Ignoring saturation in load-rejection, overvoltage or voltage-recovery studies.
Accepting EMTP® convergence without checking whether the saturation response is physically plausible.

Section 18

Where this fits in the series

This guide explains how the saturation data used by the EMTP® synchronous-machine model is represented. It follows the test-procedure and data-conversion pages: the tests provide the OCC, SCC and field-current information; the conversion builds the linear per-unit equivalent circuit, whose reactances connect to the short-circuit response; and this page explains how the nonlinear saturation correction is added to the mutual air-gap flux path. The next page goes beyond the dq0 model to high-frequency winding and mechanical shaft modelling.

Section 19

Suggested report wording

Model statement — for a study report

“The synchronous-machine saturation model was based on the available open-circuit characteristic and the associated air-gap line. The saturated and unsaturated reactance bases, the field-current base convention, the rated-speed basis and the OCC voltage/current units were checked before use. Saturation was applied to the mutual air-gap flux path rather than to the leakage inductances, with the selected treatment reflecting the machine type and the available d- and q-axis saturation data. The implemented curve was checked for smoothness, monotonic behaviour, correct no-load voltage and field-current reproduction, and plausible voltage-recovery and load-rejection response in the EMTP® study.”

Section 20

Main takeaway

Saturation makes the model behave like a real magnetic machine

Magnetic saturation is not a general reduction applied to every machine inductance. In a practical EMTP® synchronous-machine model it is a nonlinear correction to the mutual air-gap flux relation, normally derived from the OCC and the air-gap line. A reliable saturation model requires a clear field-current base, the correct saturated/unsaturated reactance basis, a suitable round-rotor or salient-pole treatment and a smooth curve. The message: keep the OCC, air-gap line, field-current base, saturated/unsaturated reactance basis and machine type consistent, and validate against no-load, voltage-recovery and large-disturbance behaviour.

References

The reference text for the saturation treatment and the standard machine-modelling and test-procedure works.

J. A. Martínez-Velasco, Ed., Power System Transients: Parameter Determination, Ch. 5 (Synchronous Machines). Boca Raton, FL, USA: CRC Press, 2010.
P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1994.
IEEE Std 115-2009, IEEE Guide for Test Procedures for Synchronous Machines. New York, NY, USA: IEEE.
IEEE Std 1110-2019 (Revision of IEEE Std 1110-2002), IEEE Guide for Synchronous Generator Modeling Practices and Parameter Verification with Applications in Power System Stability Analyses. Piscataway, NJ, USA: IEEE, 2020.
IEC 60034-4, Rotating Electrical Machines – Part 4: Methods for Determining Synchronous Machine Quantities from Tests. Geneva, Switzerland: International Electrotechnical Commission.

Part Eight Reading now

Magnetic Saturation

The open-circuit characteristic and air-gap line, saturated and unsaturated reactance, and the saturation factor applied to the mutual air-gap inductance.

Series progress 8 of 9

Part Nine

High-Frequency and Shaft Modelling

Beyond dq0: high-frequency winding models for steep-front surge stress, and multimass mechanical shaft models for torsional and subsynchronous-resonance studies.