Machine Representation · EMT Studies

Synchronous Machine Representation for Electromagnetic Transient Studies

A synchronous machine is neither purely electrical nor purely mechanical: its behaviour is the continuous interaction of the stator network, the rotor magnetic system, the mechanical shaft, the excitation system and the prime mover or load. The familiar “voltage behind a reactance” picture is fine for load-flow and fault-level thinking, but it is not enough for many electromagnetic-transient (EMT) studies. The first engineering decision is not which equation to use — it is what frequency range and physical phenomenon the machine model must represent. This guide sets out the CIGRE frequency-range guidelines for that choice, then develops the phase-domain machine equations that sit behind every detailed representation.

Reading time ≈ 36 min · Frequency-range model selection & derivation

In steady state a synchronous generator is often described as an internal voltage behind a reactance. That is useful for load-flow and short-circuit thinking, but during a disturbance the internal flux linkages cannot change instantaneously: currents are induced in the field and damper windings, the electromagnetic torque changes, the rotor accelerates or decelerates, the shaft may oscillate and saturation may alter the magnetic coupling. The terminal voltage and current are therefore the output of a coupled electromagnetic and mechanical process — and the representation that captures it must be chosen for the study at hand. A model acceptable for a low-frequency load-rejection study is not the model needed for a fast-front lightning study.

Abbreviations used on this page

EMTElectromagnetic transient

EMTP®Electromagnetic Transients Program

mmfMagnetomotive force

d / q axisDirect / quadrature rotor axes

SSRSubsynchronous resonance

AVRAutomatic voltage regulator

PSSPower system stabiliser

CIGREInternational Council on Large Electric Systems

\(\theta,\ \omega\)Electrical rotor angle and angular speed

Key idea

The model must follow the transient. Choose the representation from the frequency range and the physical output required — not from a single “universal” machine model.
More detail is not automatically more accurate. A detailed low-frequency d-q model can be wrong for a very-fast-front study if it omits the terminal capacitance that actually controls the result.
The machine is electromechanical because its inductances depend on rotor position; that same position dependence produces electromagnetic torque.
Better question than “how detailed is the model?” → does the chosen representation include the physical effects that control this specific transient?

Key terms used on this page

01Low-frequency transient

Slow events (stability, load rejection, SSR) where mechanical inertia, shaft and controls participate.

02Slow-front transient

Switching-range events; the machine is an AC source behind a transient (frequency-dependent) impedance.

03Fast-front transient

Lightning-range events; a linear per-phase terminal circuit matching the machine frequency response.

04Very-fast-front transient

Steep-front events; a capacitance-to-ground per phase often dominates the first response.

05Direct / quadrature axis

Rotor reference axes; d aligns with the field, q is 90 electrical degrees away.

06Permeance

Ease of magnetic flux flow through a path; its variation with rotor angle makes inductances position-dependent.

07Magnetomotive force (mmf)

The magnetising “drive” of a winding; resolves into d- and q-axis components.

08Leakage inductance

Flux that links mainly its own winding without crossing the main air gap; a constant term.

09Damper winding

Short-circuited rotor circuit reproducing subtransient behaviour and oscillation damping.

10Magnetic coenergy

The field energy function whose derivative with rotor angle gives the electromagnetic torque.

11Parameter determination

Converting test/manufacturer reactances and time constants into the program’s internal circuit parameters.

12Multi-mass shaft

Turbine, generator and exciter as separate inertias joined by stiffness — needed for torsional / SSR work.

Section 1

The model must follow the transient

A synchronous machine requires special care in EMT studies because a representation that is acceptable for one transient can be completely inappropriate for another. A generator model used for a slow load-rejection study is not the representation needed for a fast-front lightning study; a model for subsynchronous resonance cannot be reduced to a source behind a reactance; and a model for very-fast-front transients may need only the machine terminal capacitance to ground, because the internal electromechanical dynamics are far too slow to influence the first part of the event. The first engineering decision is therefore not which equation to use, but:

The first question

What frequency range and physical phenomenon must the machine model represent? Once that is clear, the required level of synchronous-machine detail becomes much easier to select.

Section 2

Why the required model depends on transient frequency

The behaviour of a synchronous machine depends strongly on the time scale of the event.

Low-frequency transients develop slowly enough for the electrical circuits, mechanical inertia, shaft system, voltage control and speed control to participate — transient stability, load rejection, generator tripping, synchronisation, inadvertent energisation and SSR. A detailed electrical and mechanical representation is normally required, and saturation may matter because terminal voltage and flux can move far from normal.
Slow-front transients are faster than ordinary control action but slow enough that the transient impedance and some frequency-dependent behaviour matter. The machine can often be an ideal AC source behind a suitable transient (frequency-dependent) impedance; controls are usually unimportant for the first part of the event.
Fast-front transients (lightning) are far too quick for the mechanical and control systems. The machine is a linear per-phase circuit matching the terminal frequency response — the focus is how the terminal behaves to a fast surge, not rotor angle or field dynamics.
Very-fast-front transients are dominated by local capacitances and insulation geometry; a capacitance-to-ground per phase is often the most appropriate terminal representation.

This philosophy prevents two opposite mistakes: using a detailed low-frequency model for a very-fast-front study (adding irrelevant physics while omitting the controlling terminal capacitances), and using a source-behind-reactance model for a low-frequency electromechanical study (missing torque, rotor speed, saturation and shaft dynamics). The model must follow the transient.

Section 3

The CIGRE modelling guidelines

CIGRE Working Group 33.02 summarised the representation of synchronous machines by frequency range. The table below is the practical starting point for model selection: note how the mechanical and control systems are crucial for low-frequency and slow-front work but negligible for higher-frequency transients, while terminal capacitance moves the other way.

Table 1 — CIGRE WG 33.02 guidelines for representing synchronous machines by transient frequency range (after CIGRE Brochure 39, 1990).
Aspect	Low-Frequency	Slow-Front	Fast-Front	Very-Fast-Front
Representation	Detailed mechanical & electrical, incl. saturation	Ideal AC source behind frequency-dependent transient impedance	Linear per-phase circuit matching terminal frequency response	Capacitance-to-ground per phase
Voltage control	Very important	Negligible	Negligible	Negligible
Speed control	Important	Negligible	Negligible	Negligible
Capacitance	Negligible	Important	Important	Very important
Frequency-dependent parameters	Important	Important	Negligible	Negligible

Reading across the table, the trend is consistent: as the transient gets faster, the controlling physics moves from the rotor circuits and their controls toward the terminal capacitance and connection geometry — so the electromechanical detail that dominates a stability study becomes irrelevant to a very-fast-front study, and vice versa.

The key message

A synchronous machine is not represented by one universal model in all EMT studies. Its representation must be selected according to the transient frequency range and the physical output required.

Model selection also depends on the required physical output

Frequency range is the first filter for choosing the synchronous-machine representation, but the required output is the second filter. If the study is concerned with rotor-angle behaviour, the model must reproduce the electromagnetic torque response, not only terminal voltage and current. In stability terms, the torque change may be viewed as a synchronising component, which tends to restore rotor angle, and a damping component, which tends to reduce speed oscillations. Insufficient synchronising torque can lead to aperiodic loss of synchronism, while insufficient damping torque can lead to growing oscillations. For this reason, angle-stability and oscillation studies may require field-circuit dynamics, damper circuits, excitation controls and saturation. Voltage-stability studies place more emphasis on reactive capability and field-current limits, while frequency-stability studies require governor, protection, limiter and speed-dependent generator behaviour.

Figure 1 — Selecting the machine representation from two filters — the event frequency range and the required physical output. Low-frequency studies may need torque, field, damper, excitation, saturation and governor behaviour, while fast-front studies may need terminal frequency response or capacitance rather than rotor dynamics.

Section 4

Physical structure of the synchronous machine

The machine has stator circuits and rotor circuits. The stator is the three-phase armature windings (a, b, c), connected to the network and carrying the terminal currents. The rotor carries the field winding — on the direct axis, producing the main rotor field in a wound-field machine — and the damper windings, which are short-circuited rotor circuits that reproduce the transient electromagnetic behaviour of the rotor.

It is not necessary to model every physical rotor bar. A limited set of equivalent circuits is used; a common representation is one field winding, one d-axis damper and two q-axis dampers. The direct axis (d) aligns with the rotor field; the quadrature axis (q) is 90 electrical degrees away. The rotor position is the electrical angle \(\theta\) (the angle by which the d-axis leads the magnetic axis of phase a), and the rotor electrical speed is \(\omega\). Electrical and mechanical angle coincide only in a two-pole machine; for \(p\) poles (\(p/2\) pole pairs) the electrical angle advances faster than the mechanical angle by the pole-pair ratio.

Section 5

Main assumptions behind the classical equations

The classical machine equations are derived under simplifying assumptions that make the model practical for system-level work: the air-gap magnetomotive force is sinusoidally distributed (space harmonics neglected); the effect of stator slots on rotor inductances is neglected, so saliency is attributed to the rotor; magnetic hysteresis is neglected; and magnetic saturation is initially neglected so that the flux-current relations can be written with inductance matrices and superposition applied. Saturation is added later as a nonlinear correction. These assumptions are not weaknesses — the aim is not to reproduce the finite-element field in every tooth and slot, but to represent the machine response that matters at the terminals and the shaft. They are, however, system-level assumptions: they may not be adequate for high-frequency winding-stress and detailed insulation studies, slot and harmonic effects, saturation-heavy operation or special machine designs — cases handled by the specialist pages later in the series.

Section 6

Stator and rotor flux-linkage equations

The electrical model begins with flux linkages: each winding has a flux linkage that depends on the currents in several windings. Collecting the stator currents \(\boldsymbol{i}_s=[\,i_a\ i_b\ i_c\,]^{T}\) and the rotor currents \(\boldsymbol{i}_r=[\,i_F\ i_D\ i_{Q1}\ i_{Q2}\,]^{T}\) (field, d-axis damper and two q-axis dampers), with corresponding flux-linkage vectors \(\boldsymbol{\psi}_s\) and \(\boldsymbol{\psi}_r\), the coupled relationship is:

\[ \begin{bmatrix}\boldsymbol{\psi}_s \\ \boldsymbol{\psi}_r\end{bmatrix} = \begin{bmatrix}\mathbf{L}_{ss} & \mathbf{L}_{sr} \\ \mathbf{L}_{rs} & \mathbf{L}_{rr}\end{bmatrix}\begin{bmatrix}\boldsymbol{i}_s \\ \boldsymbol{i}_r\end{bmatrix} \]

\(\boldsymbol{\psi}_s,\ \boldsymbol{\psi}_r\): stator and rotor flux-linkage vectors
\(\boldsymbol{i}_s,\ \boldsymbol{i}_r\): stator and rotor current vectors (\(\boldsymbol{i}_r=[\,i_F\ i_D\ i_{Q1}\ i_{Q2}\,]^{T}\): field, d-axis damper, two q-axis dampers)
\(\mathbf{L}_{ss}\): stator–stator inductance matrix
\(\mathbf{L}_{sr},\ \mathbf{L}_{rs}\): stator–rotor and rotor–stator inductance matrices (\(\mathbf{L}_{rs}=\mathbf{L}_{sr}^{T}\))
\(\mathbf{L}_{rr}\): rotor–rotor inductance matrix

This matrix equation is the foundation of the machine electromagnetic model: each winding flux is produced not only by its own current but by the currents in every coupled winding through mutual inductance.

Section 7

Physical meaning of the inductance matrices

\(\mathbf{L}_{ss}\) describes the magnetic coupling among the three stator phases (self- and mutual inductances). \(\mathbf{L}_{sr}\) describes how stator currents link the rotor circuits; its elements depend on rotor position because the alignment of the stator phase axes and the rotor magnetic axes changes as the rotor turns. \(\mathbf{L}_{rs}\) is its transpose under reciprocity. \(\mathbf{L}_{rr}\) describes coupling among the field and damper windings and, under the stated assumptions, is constant because the rotor circuits see a constant permeance in their own frame. The key point: not all inductances behave the same way. Some are constant; some vary with rotor position — and that position dependence is exactly what makes the machine an electromechanical energy converter.

Section 8

Why stator inductance varies with rotor position

Flux from a stator winding crosses the air gap, passes through the rotor iron and returns — and the permeance of that path changes with rotor position because the air gap is not magnetically uniform. This is obvious in salient-pole machines and present, less visibly, in round-rotor machines because the field-winding slots make the two axes differ. The permeance can be approximated as:

\[ P = P_0 + P_2 \cos 2\alpha \]

\(P\): permeance of the magnetic path
\(P_0\): average permeance
\(P_2\): second-harmonic permeance component
\(\alpha\): angular distance from the d-axis along the air gap

The \(\cos 2\alpha\) term appears because the north and south poles have equal magnetic effect, so the pattern repeats twice per electrical revolution — which is why second-harmonic terms appear in the stator inductances. In the physical phase frame the stator inductances are therefore not constants; they vary as the rotor turns. This is one reason the d-q transformation is so useful: it moves the model into a frame where the main magnetic relations become constant.

Section 9

Stator self- and mutual inductances

A stator phase self-inductance has two parts: a constant leakage term (flux that links mainly the same winding without crossing the main air gap) and an air-gap term that follows the permeance and so varies with rotor position. For phase a:

\[ l_{aa} = L_{aa0} + L_{aa2}\cos 2\theta \]

\(l_{aa}\): instantaneous self-inductance of phase a
\(L_{aa0}\): average (constant) part, including leakage
\(L_{aa2}\): second-harmonic, position-dependent part
\(\theta\): electrical rotor angle (phase-a axis to d-axis)

Phases b and c have the same form shifted by \(\mp120^{\circ}\). The mutual inductance between two stator phases similarly has a constant part plus a second-harmonic part and is negative under the usual convention. In short: the stator phase inductance changes as the rotor moves because the magnetic path the winding sees changes with rotor position — and the position-dependent mutual terms drive torque production and unbalanced transient behaviour.

Section 10

Decomposition of phase mmf into d and q axes

The physical reason for the d-q model appears when one stator phase is examined. Phase a, carrying current \(i_a\), produces a sinusoidally distributed mmf wave in the air gap. That wave resolves into a component along the d-axis and a component along the q-axis. If the phase-a axis is separated from the d-axis by \(\theta\):

\[ \text{peak }MMF_{ad} = N_a i_a \cos\theta \qquad\qquad \text{peak }MMF_{aq} = N_a i_a \sin\theta \]

\(N_a\): effective turns per phase
\(i_a\): phase-a current
\(\theta\): electrical rotor position

A stator phase current therefore produces part of its magnetic effect along the d-axis and part along the q-axis. Because the rotor field and damper circuits are naturally aligned with these axes, the machine equations become far clearer in the d-q frame — this decomposition is the physical basis of Park’s transformation.

Section 11

Stator-rotor and rotor-rotor coupling

The mutual inductance between a stator phase and a rotor circuit depends on their relative position. For a d-axis rotor winding (the field or a d-axis damper), the mutual inductance with phase a varies as \(\cos\theta\) — maximum when the phase-a axis aligns with the d-axis, zero at 90 degrees. For a q-axis rotor winding it varies as \(\sin\theta\). Phases b and c use the appropriate \(\pm120^{\circ}\) shifts. So \(\mathbf{L}_{sr}\) is position-dependent, and that position dependence is what permits energy conversion between the electrical and mechanical systems.

The rotor-rotor matrix \(\mathbf{L}_{rr}\), by contrast, is treated as constant: with stator-slot effects neglected, the rotor circuits see a constant permeance from the rotor frame. The d-axis rotor circuits couple to each other and the q-axis circuits couple to each other, while d–q rotor coupling is zero in the idealised model because the axes are orthogonal. Keeping \(\mathbf{L}_{rr}\) constant concentrates the position dependence in the stator and stator-rotor terms.

Section 12

Voltage equations and sign convention

The voltage equations relate winding voltages to resistive drops and the rate of change of flux linkage. Using generator convention (currents leaving the terminals, terminal voltages taken as drops in the current direction):

\[ \boldsymbol{v}_s = -\mathbf{R}_s\,\boldsymbol{i}_s - \frac{d\boldsymbol{\psi}_s}{dt} \] \[ \boldsymbol{v}_r = -\mathbf{R}_r\,\boldsymbol{i}_r - \frac{d\boldsymbol{\psi}_r}{dt} \]

\(\boldsymbol{v}_s,\ \boldsymbol{v}_r\): stator phase-voltage and rotor voltage vectors
\(\mathbf{R}_s,\ \mathbf{R}_r\): stator and rotor resistance matrices
\(\boldsymbol{i}_s,\ \boldsymbol{i}_r\): stator and rotor current vectors
\(\boldsymbol{\psi}_s,\ \boldsymbol{\psi}_r\): stator and rotor flux-linkage vectors

The field winding has an applied voltage; the damper windings are short-circuited, so their voltages are zero and their currents are induced by changing flux during transients. The convention must be kept consistent with the software and the current directions: a mixed convention can show the wrong power direction, torque direction or field-voltage interpretation. The same physical machine runs as a generator (currents leaving) or motor (currents entering) — check the steady state so the signs of active power, reactive power and torque match the intended operation.

Section 13

Flux linkage or current as state variables

The equations can be written in state-space form using either flux linkages or currents as the state variables. Using flux linkages is physically natural — they are the continuous states in an electromagnetic transient — and currents are recovered by inverting the inductance matrix. Using currents requires the time derivative of the inductance matrix, because some inductances depend on rotor position and the position changes with time:

\[ \frac{d}{dt}\big(\mathbf{L}(\theta)\,\boldsymbol{i}\big) \;\neq\; \mathbf{L}(\theta)\,\frac{d\boldsymbol{i}}{dt} \qquad\text{(missing term)}\quad \frac{\partial \mathbf{L}}{\partial \theta}\,\frac{d\theta}{dt}\,\boldsymbol{i} \]

\(\mathbf{L}(\theta)\): position-dependent inductance matrix
\(\dfrac{\partial \mathbf{L}}{\partial \theta}\dfrac{d\theta}{dt}\): speed-induced coupling term (electromechanical interaction)

The extra term represents speed-induced coupling and is part of the electromechanical nature of the machine. EMTP®-type programs handle these relationships internally; the engineering message is simply that a synchronous machine is not a fixed inductor — its electrical parameters depend on rotor position and speed.

Section 14

Electromagnetic torque from magnetic coenergy

The electromagnetic torque comes from the magnetic coenergy stored in the air gap: if the stored field energy changes with rotor position, a torque exists. For a \(p\)-pole machine:

\[ T_e = \frac{p}{2}\,\frac{\partial W'}{\partial \theta} \]

\(T_e\): electromagnetic torque
\(W'\): magnetic coenergy of the air-gap field (a quadratic form in the currents and inductances)
\(p\): number of poles (the factor \(p/2\) relates electrical and mechanical angle)
\(\theta\): electrical rotor angle

The torque depends on the derivatives of the inductance matrices with respect to rotor angle — the same position-dependent inductances that appear in the flux equations also produce torque, so the model is energy-consistent. Physically, the torque is the interaction of stator currents with the rotor field; during a disturbance the stator currents change quickly, rotor currents are induced and the torque can contain large oscillatory components — which is why close-up faults, out-of-phase switching and sudden load rejection can impose severe mechanical stress.

Section 15

Electrical and mechanical interaction

The synchronous machine is a feedback system, solved at every time step: a network disturbance changes stator voltages → stator currents change → flux linkages change → rotor currents are induced → the air-gap field changes → electromagnetic torque changes → rotor speed and angle change → the inductance matrix changes → the next electrical response changes. This is why the mechanical model cannot always be ignored: in low-frequency and slow-front events the mechanical system can significantly affect the electrical result, whereas in fast- and very-fast-front events the disturbance is over before the rotor moves appreciably, so the mechanical dynamics can be neglected.

Section 16

Saturation: why it is initially neglected

The basic derivation neglects saturation so the machine is a set of linear coupled circuits to which superposition applies; saturation is then added as a nonlinear correction, mainly on the mutual air-gap flux path. It matters most for load rejection, generator overvoltage, transformer energisation near a generator, voltage recovery and other large disturbances, and least for very-fast-front events dominated by terminal capacitance or cases that stay near nominal flux. When in doubt, run sensitivity cases with and without saturation. The saturation data must be consistent with the reactance set used; the open-circuit characteristic, the saturated/unsaturated basis and detailed saturation modelling are covered on the magnetic-saturation page.

Section 17

Damper windings and transient behaviour

Damper windings are short-circuited rotor circuits that reproduce transient electromagnetic effects. When the stator current changes suddenly the air-gap flux tries to change, and the rotor circuits oppose it with induced currents — especially important in the first cycles after a disturbance. The d-axis damper shapes the direct-axis subtransient response; the q-axis dampers shape the quadrature-axis response and damping. A limited number of equivalent circuits (commonly one d-axis and two q-axis dampers) is normally sufficient — this is not a literal model of every rotor bar but an equivalent designed to reproduce the observed transient behaviour. Dampers matter for initial short-circuit current and subtransient reactance, oscillation damping, unbalanced-fault response, motor starting, out-of-phase switching torque and subsynchronous interaction. Omitting or mis-representing them can produce unrealistic current peaks, torque peaks or damping. The detailed subtransient and transient short-circuit response these circuits produce is set out on the parameters and short-circuit response page.

Section 18

Parameter determination philosophy

A major part of machine modelling is parameter determination. The equations need fundamental parameters — resistances, self- and mutual inductances, saturation data, inertia, damping and shaft stiffness — but manufacturers and standards usually supply characteristic quantities from tests instead: armature resistance, synchronous / transient / subtransient reactances, open- and short-circuit time constants, zero- and negative-sequence reactances, the saturation curve, inertia and shaft data. The modelling task is to convert these into the fundamental parameters the program uses to build its internal circuits — not a clerical exercise, but something that directly shapes the equivalent circuit. A transient reactance and time constant may set an equivalent field or damper branch; a subtransient reactance may need an extra damper branch; a saturation curve modifies the mutual-flux relation; shaft inertia and stiffness set the mechanical natural frequencies. The quality of the machine model depends on the quality and consistency of this data. The detailed procedures for converting test and manufacturer data into the program’s internal parameters are on the data-conversion page.

Section 19

Why standards matter

Machine-parameter determination is heavily standardised (IEEE and IEC test procedures), because the same term can be misread if the test basis is unclear. Transient and subtransient reactances may be reported under specific test assumptions; time constants may be open-circuit or short-circuit values; d- and q-axis quantities may come from different tests; per-unit values may be on the machine base, not the system base. A good study should therefore state the source of the data, the standard or test basis where known, the machine base values, whether values are saturated or unsaturated, whether time constants are open- or short-circuit, whether both d- and q-axis quantities are available, and whether any missing data was estimated. This matters most for fault-current envelopes, protection studies, torque assessment and generator-performance verification. The IEEE/IEC test procedures behind these characteristic quantities are described on the test-procedures page.

Section 20

From phase-domain equations to the d-q frame

The full phase-domain equations show the physics, but their position-dependent inductance terms are inconvenient to solve directly. The d-q-0 transformation expresses the stator quantities in a frame rotating with the rotor, so the stator and rotor circuits align with the same magnetic axes: the d-axis carries the field and direct-axis damper behaviour, the q-axis carries the quadrature-axis damper behaviour, and the zero-sequence component is separated. The transformation does not remove the physics — it reorganises it into a form that is easier to solve and to interpret, giving direct access to the d- and q-axis currents and fluxes, field and damper currents, electromagnetic torque, rotor angle and rotor speed. These variables are far more meaningful than raw phase-inductance terms when interpreting a machine transient. The full transformation and its sign and scaling conventions are derived on the Park transformation and dq0 page; the resulting equations are placed on a per-unit base and turned into d- and q-axis equivalent circuits on the per-unit and equivalent-circuit page.

Section 21

Representation across the four frequency ranges

Low-frequency transients

Represent the machine in detail — stator and rotor circuits, saturation, mechanical inertia and controls — because here the rotor speed and angle change enough to influence the electrical result, and reducing the machine to a simple impedance can remove the physics that controls the answer. In load rejection, for example, the mechanical input power cannot fall instantly when the electrical load is lost; the rotor accelerates, the voltage may rise, the excitation responds and saturation may matter — a source behind a reactance cannot reproduce this.

Slow-front transients

Faster than electromechanical events but slower than lightning. Controls may often be neglected for the first part of the event, but the electrical representation still matters: an ideal AC source behind a suitable (frequency-dependent) transient impedance is a practical choice. Use it where the internal mechanical dynamics are not the focus but the machine still contributes as a finite source — e.g. switching studies where generator proximity affects overvoltage, current or recovery voltage. Do not apply it where rotor angle, field response or shaft torque is part of the required result.

Fast-front transients

The internal electromechanical dynamics are too slow to respond; the transient is governed by terminal impedance, winding capacitance, transformer connection, arresters, cables, busbars and local reflections. Represent the machine by a linear per-phase circuit matching the terminal frequency response — the question is how the terminal behaves to a fast surge, not what the rotor does. This is the relevant case when a lightning surge reaches a generator terminal or connected bus.

Very-fast-front transients

The first response is dominated by local capacitances and insulation geometry, so a capacitance to ground per phase is often the most appropriate representation. This is the clearest example that “more detailed” does not mean “more correct”: a detailed low-frequency model can be inappropriate here if it omits the terminal capacitances that dominate the high-frequency behaviour.

Section 22

Mechanical representation

The electrical model must be connected to a mechanical equation. The simplest is a single inertia, acceptable when only the overall rotor acceleration is needed and the shaft is not a concern. The more complete option is a multi-mass shaft, where turbine stages, generator rotor and exciter are separate inertias joined by shaft stiffness and damping — necessary for torsional studies. The electromagnetic torque acts on the generator rotor; the prime mover applies mechanical torque; their difference accelerates the rotor:

\[ J\,\frac{d\omega_m}{dt} = T_m - T_e \]

\(J\): moment of inertia
\(\omega_m\): mechanical speed
\(T_m\): applied mechanical torque
\(T_e\): electromagnetic torque

For a multi-mass model this becomes a matrix equation — each mass has its own inertia, speed and angle, with shaft stiffness setting the torque between adjacent masses. Mechanical modelling is crucial for subsynchronous resonance, shaft-torque assessment, out-of-phase synchronisation, generator tripping and mechanical-duty studies; it is far less important for high-frequency surge studies where the event ends before the rotor can move appreciably. The multi-mass shaft model and its torsional/SSR detail are covered on the high-frequency and shaft-modelling page.

Section 23

What to check before using the model

Before drawing conclusions, check the model in steady state:

Terminal voltage matches the intended operating point; active and reactive power match the load-flow.
Stator current is reasonable; rotor speed is synchronous; electromagnetic torque balances the mechanical torque.
Field current is physically plausible; the initial fluxes do not produce artificial transients.
A no-disturbance run remains stable — if the model drifts or oscillates before the event, the disturbance results cannot be trusted.

After the disturbance, ask whether the result is controlled by the machine model or by an external network component: in a lightning study the terminal capacitance may matter more than the rotor circuits, while in a load-rejection study excitation and saturation may matter more than capacitance.

Section 24

Common modelling mistakes

Avoid these

Using the same machine model for every transient — the required representation changes with frequency range.
Using a voltage source behind a reactance for a low-frequency electromechanical study (missing rotor acceleration, torque and control response).
Using a detailed low-frequency d-q model for a very-fast-front study and assuming it is automatically accurate (it may omit the controlling terminal capacitance).
Neglecting saturation in load-rejection or overvoltage studies.
Using a single-mass shaft for a torsional study.
Mixing generator and motor sign conventions.
Using per-unit data without checking the base values.
Treating manufacturer reactances and time constants as if they were already the internal inductances and resistances.
Ignoring the origin of the data — test-based, manufacturer-supplied and estimated values do not carry the same confidence.
Trusting a result only because the simulation converged — convergence does not prove physical correctness.

Section 25

Engineering workflow for EMT studies

Define the transient frequency range: low-frequency, slow-front, fast-front or very-fast-front.
Define the required output: voltage, current, torque, shaft stress, rotor speed, field current, recovery voltage, surge reflection or insulation stress.
Select the machine representation suitable for that range.
Collect the required machine data; confirm the sign convention and base quantities.
Build the model and initialise it from the correct operating point.
Run a no-disturbance simulation; then apply the disturbance.
Monitor both terminal and internal variables where relevant.
Perform sensitivity checks on the uncertain parameters.

Tailor the sensitivity checks to the range: saturation, inertia, damping, shaft model and excitation response for low-frequency work; transient impedance, source strength and initial operating point for slow-front; terminal capacitance, frequency-response representation, connected transformer/cable models and arresters for fast-front; capacitance-to-ground and local layout for very-fast-front.

Section 26

Main takeaway

Match the representation to the transient, not the other way round

The correct synchronous-machine representation is selected from the transient frequency range and the physical quantity of interest. For low-frequency events, electromechanical dynamics, saturation, controls and shaft behaviour may control the answer; for fast- and very-fast-front events, terminal impedance, capacitance and insulation geometry may matter more than rotor dynamics. A reliable EMT model is therefore not the most detailed model, but the model that includes the physical effects controlling the studied transient.

References

The reference text for the derivation, the CIGRE modelling guidelines, and the IEEE / IEC standards and classic works behind this page.

J. A. Martínez-Velasco, Ed., Power System Transients: Parameter Determination, Ch. 5 (Synchronous Machines). Boca Raton, FL, USA: CRC Press, 2010.
CIGRE Working Group 33.02, Guidelines for Representation of Network Elements when Calculating Transients, Technical Brochure 39. Paris, France: CIGRE, 1990.
IEEE Std 115-2009, IEEE Guide for Test Procedures for Synchronous Machines. New York, NY, USA: IEEE.
IEC 60034-4, Rotating Electrical Machines – Part 4: Methods for Determining Synchronous Machine Quantities from Tests. Geneva, Switzerland: International Electrotechnical Commission.
IEEE Std 1110-2002, IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. New York, NY, USA: IEEE.
P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1994.

Part Two Reading now

Representation in EMT Studies

Choosing the representation from two filters — the CIGRE frequency range and the required physical output — on the phase-domain flux-linkage foundation.

Series progress 2 of 9

Part Three

Park Transformation and dq0

How Park’s transformation maps the time-varying phase-domain equations into the rotating dq0 frame: direct, quadrature and zero-sequence components, speed voltage and torque.