Per-Unit & Equivalent Circuits

Per-Unit Representation and Equivalent Circuits of a Synchronous Machine

The synchronous-machine equations can be written in SI units, but that is rarely the most convenient form for power-system work. In practice the quantities are converted to per unit before they are used in transient calculations — not merely for presentation, but because a well-chosen per-unit system makes the parameters easy to compare, easy to validate and easy to convert into the internal circuits an EMTP®-type program builds. The subtlety for machines is that the stator and rotor circuits do not naturally share the same current, voltage and flux bases: the rotor bases must be chosen so that rotor currents produce equivalent air-gap flux when referred to the stator. This guide sets out those base choices, the per-unit equations they produce, and the d- and q-axis equivalent circuits that follow.

Reading time ≈ 35 min · Bases, per-unit equations & equivalent circuits

Per-unit scaling is especially useful for synchronous machines because, on the machine rating base, the impedances fall in a fairly narrow range — a fast engineering check. A reactance that looks odd in ohms may be perfectly normal in per unit, while a per-unit value far outside the expected range quickly flags a wrong voltage base, a wrong MVA base or a bad conversion. The aim here is a consistent set of equations in which the d-axis, q-axis, field and damper circuits can be represented by equivalent networks suitable for time-domain solution.

Abbreviations used on this page

puPer unit

EMTP®Electromagnetic Transients Program

MVAMegavolt-ampere (apparent-power base)

RMSRoot mean square

DCDirect current

d / qDirect / quadrature axes

F, D, Q1, Q2Field, d-damper, q-dampers

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

\(S_B, V_B, I_B, Z_B\)Base power, voltage, current, impedance

\(\omega_B\)Base electrical angular frequency

\(T_e\)Electromagnetic torque

Key idea

Stator bases come straight from the machine rating; rotor bases do not — they are chosen so a rotor base current makes the same air-gap flux as the corresponding stator base current.
This equal-air-gap-flux principle is what lets field and damper quantities be referred consistently to the stator and converted into circuit parameters.
On consistent bases the per-unit equations are clean — diagonal stator inductances, axis-separated rotor coupling, and the compact torque \(T_e=\psi_d i_q-\psi_q i_d\).
The per-unit equations are the d- and q-axis equivalent circuits — the same physics expressed as resistances, leakage and mutual inductances close to how EMTP® builds the model.

Key terms used on this page

01Per unit (pu)

A quantity divided by a chosen base value, giving a dimensionless number in a comparable range.

02Base quantity

The reference value (power, voltage, current, impedance…) used to normalise a physical quantity.

03Equal-flux principle

Rotor base currents chosen so they produce the same mutual air-gap flux as the stator base current.

04Effective turn ratio

A ratio used to refer rotor quantities to the stator magnetic base — not a literal constructional turns ratio.

05Mutual (magnetising) inductance

\(L_{ad}\), \(L_{aq}\): the air-gap flux shared by the stator and rotor circuits on each axis.

06Leakage inductance

Flux linking mainly one winding through air; a separate branch from the mutual flux.

07d-axis equivalent circuit

A network of armature, field and d-damper branches sharing the d-axis mutual inductance.

08q-axis equivalent circuit

A network of armature and q-damper branches sharing the q-axis mutual inductance (no field on q).

09Transient / subtransient reactance

Effective reactances seen at successive time scales, set by the leakage and damper branches.

10Speed voltage

The \(\omega\psi\) cross-coupling between d- and q-axis voltage equations from the rotating frame.

Section 1

Why per-unit for synchronous machines

The per-unit system is more than a convenience. It makes machine impedances comparable across ratings, gives an immediate sanity check on entered data, and — most importantly for EMTP® modelling — provides the consistent base on which manufacturer reactances and time constants can be converted into the internal circuit parameters the program needs. For machines, per unit must be applied carefully because the stator and rotor circuits do not naturally share the same physical current, voltage and flux bases. The stator quantities are based directly on the machine ratings; the rotor quantities are selected so that the rotor currents produce equivalent mutual air-gap flux when referred to the stator. That rotor-base choice is the key idea of this page.

Section 2

Stator base quantities

A per-unit value is the actual value divided by its base value — a reactance of \(0.2\) pu means the actual reactance is 20% of the chosen impedance base. The value only has meaning once the base MVA, voltage and frequency are known. The stator base quantities are taken from the rated values of the machine — rated apparent power and rated line-to-line RMS voltage — from which the base current and impedance follow:

\[ S_B = S_{\text{rated}} \qquad V_B = V_{\text{LL,rated}} \qquad I_B = \frac{S_B}{\sqrt{3}\,V_B} \qquad Z_B = \frac{V_B^{2}}{S_B} \]

\(S_B\): base three-phase apparent power (VA) = machine rating
\(V_B\): base line-to-line RMS voltage (V) = rated voltage
\(I_B\): base stator phase current (A)
\(Z_B\): base impedance (\(\Omega\))

Line-to-line vs phase voltage base

Unless stated otherwise, the machine voltage base is the rated line-to-line RMS voltage \(V_{LL,B}\); the phase-to-neutral base is \(V_{\phi,B}=V_{LL,B}/\sqrt{3}\). Mixing line-to-line and phase bases changes the calculated base impedance and current and can corrupt every derived per-unit parameter — so on this page \(V_B\) always means \(V_{LL,B}\).

The frequency, angular-frequency, inductance and flux-linkage bases complete the set:

\[ \omega_B = 2\pi f_B \qquad \omega_{mB} = \frac{2}{p}\,\omega_B \qquad L_B = \frac{Z_B}{\omega_B} \qquad \psi_B = \frac{V_B}{\omega_B} \qquad T_B = \frac{S_B}{\omega_{mB}} \]

\(f_B,\ \omega_B\): base frequency (Hz) = rated, and base electrical angular frequency (rad/s)
\(\omega_{mB}\): base mechanical angular speed (rad/s); with \(p\) = number of poles, the pole-pair count is \(p/2\), so \(\omega_{mB}=\omega_B/(p/2)\)
\(L_B\): base inductance (H)
\(\psi_B\): base flux linkage
\(T_B\): base torque (= base power / base mechanical speed)

These bases let stator voltage, current, impedance, inductance and flux linkage all be expressed in per unit.

Section 3

Why d-q currents need careful base treatment

The stator base current is straightforward, but the d- and q-axis currents are not physical phase currents — they are transformed quantities obtained from the phase currents through Park’s transformation. Depending on the transformation scaling, the d- and q-axis current bases may not be numerically identical to the phase-current base, which is why the transformation convention must be consistent throughout the model. In a power-invariant transformation the scaling preserves instantaneous power between phase and dq0 variables — convenient, but it means the dq0 variables must be interpreted in that same convention.

The rule

Do not mix dq0 quantities from one convention with per-unit bases from another. If the machine data, the software model and the report equations use different dq conventions, the result can look plausible yet carry incorrect scaling, signs or torque interpretation.

Section 4

Rotor base quantities — the equal-flux principle

Rotor windings do not naturally share the stator current, voltage and flux bases: the field is a DC rotor winding, the dampers are short-circuited equivalent rotor circuits, and both have different turns from the stator. Their quantities must instead be referred to the stator magnetic base, with the rotor bases chosen to preserve the same air-gap flux effect. So although the field and damper physical currents may be very different from the stator current, they can produce equivalent air-gap flux. The rotor base currents are therefore chosen on an equal mutual flux-linkage basis: a base current in a rotor winding should produce the same fundamental air-gap flux as the stator base current acting in the corresponding fictitious d- or q-axis stator winding. For the d-axis:

\[ \psi_{ad} = L_{ad}\,I_B = L_{dF}\,I_{FB} = L_{dD}\,I_{DB} \] \[ I_{FB} = \frac{L_{ad}}{L_{dF}}\,I_B \qquad\qquad I_{DB} = \frac{L_{ad}}{L_{dD}}\,I_B \]

\(\psi_{ad}\): d-axis mutual air-gap flux linkage
\(L_{ad}\): d-axis mutual (magnetising) armature inductance
\(L_{dF},\ L_{dD}\): d-axis stator-to-field and stator-to-d-damper mutual inductances
\(I_{FB},\ I_{DB}\): field and d-axis damper base currents

The rotor base currents are therefore not arbitrary — they are fixed by the requirement that rotor currents refer consistently to the stator magnetic reference.

The same principle sets the q-axis damper base currents:

\[ I_{Q1B} = \frac{L_{aq}}{L_{qQ1}}\,I_B \qquad\qquad I_{Q2B} = \frac{L_{aq}}{L_{qQ2}}\,I_B \]

\(L_{aq}\): q-axis mutual (magnetising) armature inductance
\(L_{qQ1},\ L_{qQ2}\): q-axis stator-to-damper mutual inductances
\(I_{Q1B},\ I_{Q2B}\): first and second q-axis damper base currents

The same logic then defines the rotor base flux linkages, voltages, impedances and inductances.

Section 5

Referring rotor quantities to the stator

Rather than carrying separate rotor bases explicitly, rotor quantities can be referred to the stator using effective turn ratios. These are not literal constructional turns ratios — they are effective ratios that refer rotor quantities to the stator magnetic base:

\[ N_F = \frac{L_{dF}}{L_{ad}} \quad N_D = \frac{L_{dD}}{L_{ad}} \quad N_{Q1} = \frac{L_{qQ1}}{L_{aq}} \quad N_{Q2} = \frac{L_{qQ2}}{L_{aq}} \] \[ I_{FB} = \frac{I_B}{N_F} \quad \psi_{FB} = N_F\,\psi_B \quad V_{FB} = N_F\,V_B \quad Z_{FB} = N_F^{2}\,Z_B \quad L_{FB} = N_F^{2}\,L_B \]

\(N_F,\ N_D,\ N_{Q1},\ N_{Q2}\): effective turn ratios for field, d-damper and q-dampers
\(I_{FB},\ \psi_{FB},\ V_{FB}\): field base current, flux linkage and voltage
\(Z_{FB},\ L_{FB}\): field base impedance and inductance (scale with \(N_F^{2}\))

The field is shown as the example; the \(D\), \(Q1\) and \(Q2\) circuits follow the identical pattern with their own ratios. This refers rotor circuits to a consistent per-unit system without losing the physical link between rotor currents and stator air-gap flux.

Section 6

Why rotor base selection matters

Rotor base selection is not a cosmetic detail — it sets the numerical values of rotor resistances, rotor inductances, field voltage, field current and damper parameters. If the rotor quantities are not correctly referred to the stator base, the model can give incorrect transient current, field current, subtransient response and electromagnetic torque. This matters all the more because manufacturer data usually comes as reactances and time constants, not physical rotor inductances and resistances, and the program must convert that characteristic data into internal circuit parameters — a step only a consistent per-unit base makes possible. A good check is that the final per-unit mutual inductances show the expected relationships; in a properly chosen rotor base, the per-unit stator–rotor mutual inductances become consistent and easy to compare.

Section 7

Per-unit stator voltage equations

On a consistent base the stator voltage equations take their per-unit form, with the rotating-frame speed-voltage coupling between the d- and q-axes:

\[ -v_d = R_a\,i_d + \frac{1}{\omega_B}\frac{d\psi_d}{dt} + \omega\,\psi_q \] \[ -v_q = R_a\,i_q + \frac{1}{\omega_B}\frac{d\psi_q}{dt} - \omega\,\psi_d \] \[ -v_0 = R_a\,i_0 + \frac{1}{\omega_B}\frac{d\psi_0}{dt} \]

\(v_d, v_q, v_0\): per-unit d-, q- and zero-sequence stator voltages
\(i_d, i_q, i_0\): per-unit stator currents
\(\psi_d, \psi_q, \psi_0\): per-unit stator flux linkages
\(R_a\): per-unit stator resistance
\(\omega\): per-unit electrical speed; \(\omega_B\) = base electrical angular frequency

The signs depend on the generator/motor and d-q axis conventions; the physical structure — speed-voltage coupling between the axes — is fixed. If time is also in per unit, the \(1/\omega_B\) factor drops out of the derivative terms, giving a cleaner form for numerical implementation. Some EMTP® inputs give time constants in seconds while the internal equations are normalised by \(\omega_B\) in per-unit time, so always state whether a time quantity is in seconds or per unit.

Section 8

Per-unit stator flux-linkage equations

The stator flux linkages are produced by the stator currents and the rotor currents on the same axis:

\[ \psi_d = L_d\,i_d + L_{ad}\,i_F + L_{ad}\,i_D \] \[ \psi_q = L_q\,i_q + L_{aq}\,i_{Q1} + L_{aq}\,i_{Q2} \] \[ \psi_0 = L_0\,i_0 \]

\(L_d, L_q, L_0\): d-, q- and zero-sequence stator inductances
\(L_{ad}, L_{aq}\): d- and q-axis mutual armature inductances
\(i_F, i_D\): field and d-axis damper currents
\(i_{Q1}, i_{Q2}\): q-axis damper currents

The axis separation is explicit: d-axis flux couples to the field and d-damper, q-axis flux to the q-dampers, and the zero-sequence flux stands alone.

Section 9

Per-unit rotor voltage equations

The field has an applied voltage; the short-circuited dampers have zero terminal voltage, so their currents are induced by changing flux:

\[ v_F = R_F\,i_F + \frac{1}{\omega_B}\frac{d\psi_F}{dt} \] \[ 0 = R_D\,i_D + \frac{1}{\omega_B}\frac{d\psi_D}{dt} \] \[ 0 = R_{Q1}\,i_{Q1} + \frac{1}{\omega_B}\frac{d\psi_{Q1}}{dt} \qquad 0 = R_{Q2}\,i_{Q2} + \frac{1}{\omega_B}\frac{d\psi_{Q2}}{dt} \]

\(v_F, R_F, i_F\): field voltage, resistance and current
\(R_D, R_{Q1}, R_{Q2}\): d- and q-axis damper resistances
\(\psi_F, \psi_D, \psi_{Q1}, \psi_{Q2}\): field and damper flux linkages

Section 10

Per-unit rotor flux-linkage equations

The rotor flux linkages show the magnetic coupling between stator and rotor circuits, and between the rotor circuits on each axis:

\[ \psi_F = L_{ad}\,i_d + L_{FF}\,i_F + L_{FD}\,i_D \] \[ \psi_D = L_{ad}\,i_d + L_{FD}\,i_F + L_{DD}\,i_D \] \[ \psi_{Q1} = L_{aq}\,i_q + L_{Q1Q1}\,i_{Q1} + L_{Q1Q2}\,i_{Q2} \] \[ \psi_{Q2} = L_{aq}\,i_q + L_{Q1Q2}\,i_{Q1} + L_{Q2Q2}\,i_{Q2} \]

\(L_{FF}, L_{DD}\): field and d-axis damper self-inductances
\(L_{FD}\): field-to-d-damper mutual inductance
\(L_{Q1Q1}, L_{Q2Q2}\): q-axis damper self-inductances
\(L_{Q1Q2}\): mutual inductance between the q-axis dampers

The field and d-axis damper are coupled through the d-axis mutual flux \(L_{ad}\); the q-axis dampers are coupled through the q-axis mutual flux \(L_{aq}\).

Section 11

Per-unit electromagnetic torque

In the per-unit dq formulation the air-gap torque becomes very compact — the \(p/2\) factor is absorbed into the per-unit definitions:

\[ T_e = \psi_d\,i_q - \psi_q\,i_d \]

\(T_e\): per-unit electromagnetic torque
\(\psi_d, \psi_q\): d- and q-axis flux linkages
\(i_d, i_q\): d- and q-axis stator currents

In this per-unit form the torque base \(T_B\) has been selected so that the pole-pair factor is already included in the normalisation — which is why no \(p/2\) appears here. Torque is not set by phase-current magnitude alone but by the interaction of d-axis flux, q-axis flux, d-axis current and q-axis current. During faults, switching or synchronisation errors these can change rapidly, producing large torque oscillations — the reason this model matters for shaft-duty studies.

Section 12

Equivalent circuits of the synchronous machine

The per-unit equations can be drawn as equivalent circuits. These are not separate assumptions — they are another way of writing the same flux-linkage and voltage equations, in terms of familiar elements: resistances, leakage inductances, mutual inductances and voltage sources. They give the equations physical meaning and provide a practical bridge between manufacturer data and the internal EMTP® representation. They are equivalent representations of the differential equations, not physical wiring diagrams of the machine; their purpose is to reproduce the correct flux-linkage, current, voltage and torque behaviour over the intended frequency range. Even without a drawing, the meaning is clear once each branch is described by its resistance, its leakage path and the shared mutual air-gap flux.

Section 13

The d-axis equivalent circuit

The d-axis equivalent circuit represents the stator d-axis winding, the field winding and the d-axis damper sharing one main air-gap flux. Branch by branch:

Stator d-axis branch — armature resistance \(R_a\) and stator (armature) leakage inductance \(L_l\), in series with the shared d-axis mutual air-gap inductance \(L_{ad}\); its voltage also carries a speed-voltage term from the q-axis flux.
Field branch — field resistance \(R_F\) and field leakage inductance \(L_{lF}\), coupled to the air gap through \(L_{ad}\), and driven by the applied field voltage \(v_F\).
d-axis damper branch — damper resistance \(R_D\) and damper leakage inductance \(L_{lD}\), coupled through \(L_{ad}\), with zero terminal voltage because the damper is short-circuited.
Shared mutual flux — \(L_{ad}\) is the d-axis mutual (magnetising) inductance linking armature, field and damper; each leakage inductance (\(L_l, L_{lF}, L_{lD}\)) carries flux that links only its own winding.
Optional inter-rotor leakage — a path linking the field and damper but not the armature; sometimes omitted for simplicity, but it can matter for short-pitched dampers or solid-rotor iron paths.

The speed-voltage term in the stator equation is essential, not artificial — it is the voltage induced in the stator by rotation.

The d-axis equivalent circuit can be shown in two related forms. One form keeps the ideal transformer that represents the field-to-armature turns-ratio relationship. The per-unit form usually removes this transformer by referring the field voltage and current to the armature-side base. The physical circuit is unchanged, but the selected field-current base determines how the actual field quantities are represented in the per-unit model.

IEEE Std 1110-2019 Figure 2: d-axis equivalent circuits with a single d-axis damper winding, shown with and without the ideal field-to-armature transformer; stator, field and damper branches share the d-axis mutual air-gap inductance. — **Figure 1 —** D-axis equivalent circuits with a single d-axis damper winding: (a) the form that retains the ideal field-to-armature transformer (turns ratio); (b) the form with the field quantities referred to the armature, so the transformer is absorbed into the per-unit system. Both show how the stator d-axis branch, field branch and damper branch couple through the d-axis mutual air-gap path, and why field quantities must be referred consistently.

Use this figure to support the explanation of field-current referral and rotor-base selection. The actual measured field voltage and current are not automatically the same as the referred per-unit field quantities used inside the equivalent circuit. This is why the field-current base and equal-air-gap-flux convention must be stated clearly.

In one sentence

The d-axis equivalent circuit shows how the stator d-axis current, field current and d-axis damper current share the main air-gap flux while keeping their own leakage paths and resistive losses.

Section 14

The q-axis equivalent circuit

The q-axis equivalent circuit has the same structure as the d-axis, but with no field winding in the conventional wound-field machine. Branch by branch:

Stator q-axis branch — armature resistance \(R_a\) and stator leakage inductance \(L_l\), in series with the shared q-axis mutual air-gap inductance \(L_{aq}\); its voltage carries a speed-voltage term from the d-axis flux.
q-axis damper branches — damper resistances \(R_{Q1}, R_{Q2}\) and leakage inductances \(L_{lQ1}, L_{lQ2}\), coupled through \(L_{aq}\), each with zero terminal voltage (short-circuited).
Shared mutual flux — \(L_{aq}\) is the q-axis mutual (magnetising) inductance linking the armature and the q-axis dampers; the leakage inductances (\(L_l, L_{lQ1}, L_{lQ2}\)) carry winding-only flux.
No field winding — the q-axis carries only damper circuits; in a round-rotor machine the solid rotor adds eddy-current paths represented by extra equivalent q-axis dampers.

These damper circuits set the q-axis subtransient behaviour and rotor damping; the q-axis speed-voltage term, driven by the d-axis flux, is a major contributor to internal voltage in steady-state generator operation.

The q-axis circuit is structurally different from the d-axis circuit because a conventional wound-field synchronous machine has no field winding on the quadrature axis. The q-axis is therefore represented by the stator q-axis branch and one or more equivalent damper-current paths. These branches reproduce q-axis transient and subtransient behaviour, damping and solid-rotor eddy-current effects where relevant.

IEEE Std 1110-2019 Figure 3: q-axis equivalent circuit with a single q-axis damper winding; a stator q-axis branch and one damper branch share the q-axis mutual air-gap inductance, with no field winding. — **Figure 2 —** Q-axis equivalent-circuit representation with a single q-axis damper winding. The q-axis has no field winding in the conventional wound-field machine; its dynamic behaviour is represented through the q-axis armature branch and equivalent damper-current path.

Use this figure to clarify that q-axis model order is controlled by the number of equivalent q-axis damper branches. A single q-axis damper gives a first-order q-axis operational response, while additional q-axis damper branches may be needed for more detailed subtransient or solid-rotor behaviour.

In one sentence

The q-axis equivalent circuit shows how the q-axis stator current interacts with damper-current paths to reproduce quadrature-axis transient and subtransient behaviour.

Section 15

Why equivalent circuits are useful

Equivalent circuits earn their place for three reasons. First, they give physical meaning to the matrices — the engineer can see how stator leakage, magnetising flux, field winding, damper winding and resistance connect. Second, they help convert manufacturer data into simulation parameters, because reactances and time constants can be read as the response of equivalent inductive and resistive branches. Third, they help choose model complexity: a study may not need every branch, and the required number of damper circuits and leakage paths depends on the machine type, the available data and the objective. The detailed conversion of manufacturer reactances and time constants into these branch parameters — including operational-inductance fitting and the Canay characteristic inductance — is covered on the data-conversion page.

Section 16

Model complexity and available data

Equivalent circuits can be built at different levels: a simple model with the main d- and q-axis reactances and limited damper representation; a more detailed model with transient and subtransient branches on both axes; or a still more detailed model with extra leakage paths, additional damper branches or parameters fitted to frequency-response data. The most complex model is not always best — it may demand data that is unavailable or unreliable, and guessed parameters create false confidence. The right model is chosen from the machine type and rotor construction, the transient frequency range, the study objective, the available data, the required outputs, user experience and numerical stability.

The common model notation can be read as Model N.M, where N is the number of equivalent rotor windings represented on the direct axis and M is the number represented on the quadrature axis. For example, Model 2.1 represents the direct axis using the field winding plus one d-axis damper winding, and the q-axis using one q-axis damper winding. Higher-order models add additional equivalent damper branches, but they should only be used when the available data supports the additional parameters.

For low-frequency electromechanical transients — the regime where machine dynamics matter most — the d- and q-axis equivalent circuits reproduce the flux linkages, currents and torque over the study time range; which transient frequency range needs which level of machine detail is the subject of the earlier frequency-range page.

Section 17

How the equivalent circuits relate to EMTP®

In EMTP® the user may enter machine data as rated quantities, reactances, time constants and saturation data; the program converts those into an internal representation based on equivalent circuits and differential equations. The equivalent circuits are therefore not only explanatory — they are close to how the numerical model is built. The d-axis circuit determines how the field winding, d-axis damper and stator d-axis current interact; the q-axis circuit determines how the q-axis dampers and stator q-axis current interact; the voltage equations determine how currents and fluxes evolve in time; and the torque equation connects the electrical model to the mechanical one. This is exactly why correct per-unit scaling and parameter conversion are essential: an error in base current, base voltage, leakage inductance or mutual inductance changes the equivalent circuit and therefore the simulated response. Correct scaling does not by itself guarantee correct parameters, so the engineer must still validate the response against the expected current, field-current and torque behaviour.

Section 18

Practical engineering checks

After entering machine data, confirm the following before trusting any result:

The rated voltage, power and frequency match the nameplate, and the stator base current follows from the rated MVA and line-to-line voltage.
The machine MVA base is distinguished from the system MVA base, and per-unit data is converted between them where needed.
The line-to-line versus phase voltage convention is applied consistently.
The speed base and the pole-pair relation (\(\omega_{mB}=\omega_B/(p/2)\)) are correct.
The stator and rotor bases are consistent — rotor quantities are referred to the stator on an equal-air-gap-flux basis.
The derived resistances and leakage inductances are positive and physically meaningful.
The d- and q-axis reactances sit in a reasonable per-unit range and follow the expected transient/subtransient ordering.
A no-disturbance run stays stable, and the initial operating point gives reasonable stator current, field current, torque and power.
After a disturbance, the current, field-current and torque responses are physically plausible.

These checks matter because a machine model can converge numerically while still being physically wrong.

Section 19

Common modelling mistakes

Avoid these

Using stator base quantities directly for rotor quantities without proper referral.
Mixing line-to-line and phase voltage bases.
Mixing peak, RMS and dq quantities without respecting the Park-transformation scaling.
Using manufacturer per-unit data without confirming the MVA base.
Treating reactances and time constants as final internal branch values.
Selecting an over-detailed equivalent circuit without the data to support it.
Omitting q-axis damper representation where torque or damping matters.
Trusting a converged simulation without checking physical consistency.

Section 20

Where this fits in the series

This guide follows the Park/dq0 page and converts the transformed equations into a consistent per-unit equivalent-circuit form. The next page, Parameters and Short-Circuit Response, explains the reactances and time constants that populate these circuits; the data-conversion and magnetic-saturation pages then explain how reactances, time constants, saturation data and test results are interpreted and converted into the branch parameters.

Section 21

Suggested report wording

Model statement — for a study report

“The synchronous machine was represented in per unit using the rated apparent power, rated line-to-line voltage and rated frequency as the principal stator base quantities. Rotor field and damper quantities were referred to the stator using an equal-air-gap-flux basis so that stator and rotor currents produced consistent magnetic effects in the d-axis and q-axis equivalent circuits. The resulting per-unit branch parameters were checked for base consistency, positive resistance and leakage values, suitable d-/q-axis representation and stable no-disturbance initialisation before the transient event was applied.”

Section 22

Main takeaway

Consistent bases and physical equivalent circuits make the model trustworthy

Per-unit scaling is not only a reporting convenience — it is the step that makes the stator, field and damper circuits comparable in one equivalent-circuit model. Stator bases come from the machine rating; rotor bases are chosen so that rotor base currents produce equivalent mutual air-gap flux when referred to the stator, and then the d- and q-axis equations become equivalent circuits. The message: a reliable EMTP® synchronous-machine model requires correct stator bases, correct rotor referral, consistent Park scaling, physically meaningful leakage and mutual branches, and validation before the disturbance is studied.

References

The reference text for the derivation and the standard works on per-unit machine modelling and equivalent circuits.

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.
P. M. Anderson and A. A. Fouad, Power System Control and Stability, 2nd ed. Piscataway, NJ, USA: IEEE Press / Wiley, 2003.
P. C. Krause, O. Wasynczuk and S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems. Piscataway, NJ, USA: IEEE Press / Wiley.
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.
IEEE Std 115-2009, IEEE Guide for Test Procedures for Synchronous Machines. New York, NY, USA: IEEE.

Part Four Reading now

Per-Unit and Equivalent Circuits

Stator and rotor base quantities, the equal-air-gap-flux rotor referral, and the d- and q-axis equivalent circuits that the converted parameters populate.

Series progress 4 of 9

Part Five

Parameters and Short-Circuit Response

The meaning of the characteristic reactances and time constants, and the subtransient, transient and steady-state periods of the short-circuit current with its DC offset.