# Synchronous Machine SI Fundamental

Model dynamics of three-phase round-rotor or salient-pole synchronous machine using fundamental parameters in SI units

**Libraries:**

Simscape /
Electrical /
Specialized Power Systems /
Electrical Machines

## Description

The Synchronous Machine SI Fundamental block models a synchronous machine in generator or motor mode using fundamental parameters in SI units. The operating mode is dictated by the sign of the mechanical power (positive for generator mode or negative for motor mode). The electrical part of the machine is represented by a sixth-order state-space model and the mechanical part is the same as in the Simplified Synchronous Machine block.

The model takes into account the dynamics of the stator, field, and damper windings. The
equivalent circuit of the model is represented in the rotor reference frame
(*qd* frame). Stator windings are connected in wye to an internal
neutral point. All rotor parameters and electrical quantities are viewed from the stator
and identified by primed variables. The subscripts are:

*d,q*—*d*- and*q*-axis quantity*R,s*— Rotor and stator quantity*l,m*— Leakage and magnetizing inductance*f,k*— Field and damper winding quantity

The electrical model of the machine is shown in these diagrams.

### Dynamic Model with Unequal Mutual Inductance

The conventional theory of synchronous machine modeling for stability analysis assumes that
the mutual inductances between the armature, damper, and field on direct-axis
windings are identical. Generally, damper windings are near the air gap, and as a
result the flux linking damper circuits are almost equal to the flux linking
armature. This hypothesis produces acceptable outcomes for a wide range of stability
studies, especially those on the side of the network. However, when it comes to
field current studies, there is considerable error. The equivalent circuit dynamic
model of a synchronous machine may include an additional inductance representing the
difference between field-damper and field-armature mutual inductances on the
*D*-axis [1]. This
inductance is typically called *Canay inductance*. Canay
inductance corresponds to the leakage flux,
*Φ _{C}*, in the following figure and is
interpreted as a corrective element in an equivalent model that may have a negative
value [2].

The IEEE^{®} standard 1110-2002 [3] presents the
direct and quadratic axes of the synchronous machine dynamic model as shown in the
diagrams.

The relevant equations are:

$$\begin{array}{l}{V}_{d}=-{i}_{d}{R}_{s}-\omega {\psi}_{q}+\frac{d{\psi}_{d}}{dt}\\ {V}_{q}=-{i}_{q}{R}_{s}+\omega {\psi}_{d}+\frac{d{\psi}_{q}}{dt}\\ {V}_{0}=-{i}_{0}{R}_{0}+\frac{d{\psi}_{0}}{dt}\\ {V}_{fd}=\frac{d{\psi}_{fd}}{dt}+{R}_{fd}{i}_{fd}\\ 0=\frac{d{\psi}_{kd}}{dt}+{R}_{kd}{i}_{kd}\\ 0=\frac{d{\psi}_{kq1}}{dt}+{R}_{kq1}{i}_{kq1}\\ 0=\frac{d{\psi}_{kq2}}{dt}+{R}_{kq2}{i}_{kq2}\\ \left[\begin{array}{c}{\psi}_{d}\\ {\psi}_{kd}\\ {\psi}_{fd}\end{array}\right]=\left[\begin{array}{ccc}L{}_{md}+{L}_{l}& {L}_{md}& {L}_{md}\\ {L}_{md}& {L}_{lkd}+\text{}{L}_{f}{}_{1d}+\text{}{L}_{md}& {L}_{f}{}_{1d}+\text{}{L}_{md}\\ {L}_{md}& {L}_{f}{}_{1d}+\text{}{L}_{md}& {L}_{lfd}+\text{}{L}_{f}{}_{1d}+\text{}{L}_{md}\end{array}\right]\text{\hspace{0.17em}}\left[\begin{array}{c}-{i}_{d}\\ {i}_{kd}\\ {i}_{fd}\end{array}\right]\\ \left[\begin{array}{c}{\psi}_{q}\\ \begin{array}{l}{\psi}_{kq1}\\ {\psi}_{kq2}\end{array}\end{array}\right]=\left[\begin{array}{ccc}{L}_{mq}+{L}_{l}& {L}_{mq}& {L}_{mq}\\ {L}_{mq}& {L}_{mq}+{L}_{kq1}& {L}_{mq}\\ {L}_{mq}& {L}_{mq}& {L}_{mq}+{L}_{kq2}\end{array}\right]\left[\begin{array}{l}\begin{array}{c}-{i}_{q}\\ {i}_{kq1}\end{array}\\ {i}_{kq2}\end{array}\right]\end{array}$$

### Base Values, Transformation Ratio, and Rotor Parameters Referred to the Stator

The Synchronous Machine SI Fundamental block allows you to specify the fundamental parameters of a synchronous machine. You enter field and damper winding parameters (resistances, leakage inductances, and mutual inductances) in SI (Ω, H). The RL parameters of the field and damper windings are not the actual field RL values of the machine but the RL values referred to the stator.

**Stator Base Values**

This table displays the stator base values.

$${V}_{sbase}={\frac{{V}_{n}\sqrt{2}}{\sqrt{3}}}_{}$$
| Base stator voltage = peak nominal line-to-neutral voltage (V) |

$${I}_{sbase}={\frac{{P}_{n}\sqrt{2}}{{V}_{n}\sqrt{3}}}_{}$$
| Base stator current (A) |

$${Z}_{sbase}={\frac{{V}_{sbase}}{{I}_{sbase}}}_{}=\frac{{V}_{n}{}^{2}}{{P}_{n}}$$
| Base stator impedance (Ω) |

$${\omega}_{base}=2\pi \text{\hspace{0.17em}}{f}_{n}$$
| Base angular frequency (rad/s) |

$${L}_{sbase}={\frac{{Z}_{sbase}}{{\omega}_{base}}}_{}$$
| Base stator inductance (H) |

where:

*P*is the three-phase nominal power (VA)._{n}*V*is the nominal line-to-line voltage (Vrms)._{n}*f*is the nominal frequency (Hz)._{n}*i*is the nominal field current producing nominal stator voltage at no load (A)._{fn}

The stator parameters to specify in the block are:

R_{s} | Stator resistance per phase (Ω) |

L_{l} | Stator leakage inductance (H) |

L_{md} | Direct-axis magnetizing inductance viewed from stator (H) |

L_{mq} | Quadrature-axis magnetizing inductance viewed from stator (H) |

**Stator/Field Transformation Ratio**

This figure shows one phase of the stator winding coupled with the field winding.

In the diagram:

*N*and_{s}*N*are the equivalent number of sinusoidally distributed turns of the stator winding and of the field winding, respectively._{f}*R*and_{s}*L*are stator resistance and leakage inductance._{l}*R*′ and_{f}*L*′ are field resistance and leakage inductance._{lfd}

When the three stator windings are energized with a three-phase positive-sequence voltage
and the field winding is open, the stator magnetizing inductance is
*L _{md}*. However, when only one
phase is energized and the field winding is open, the magnetizing inductance is
2/3

*L*, as shown in the figure.

_{md}At no load, when the field winding is rotating at nominal speed and bears the nominal
field DC current *i _{fn}*, the amplitude of
AC voltage (peak value) induced on one phase of the stator is

*V*.

_{sbase}The maximum mutual inductance between one stator winding and the field winding is obtained when the two windings are aligned and is given by:

$${L}_{sfd}=\frac{2}{3}{L}_{md}\frac{{N}_{f}}{{N}_{s}}=\frac{{V}_{sbase}}{{i}_{fn}\text{\hspace{0.17em}}{\omega}_{base}}$$

from which we deduce the transformation ratio:

$$\frac{{N}_{s}}{{N}_{f}}=\frac{2}{3}{L}_{md}\frac{{i}_{fn}\text{\hspace{0.17em}}{\omega}_{base}}{{V}_{sbase}}.$$

The transformation ratio can be also expressed as:

$$\frac{{N}_{s}}{{N}_{f}}=\frac{2}{3}\frac{{I}_{fbase}}{{I}_{sbase}},$$

where *I _{fbase}* is the base field
current. The table shows how to compute field base values.

$${I}_{fbase}={i}_{fn}\text{\hspace{0.17em}}{L}_{md\_pu}{}_{}$$
| Base field current (A) |

$${V}_{fbase}={\frac{{P}_{n}}{{I}_{fbase}}}_{}$$
| Base field voltage (V) |

$${Z}_{fbase}={\frac{{V}_{fbase}}{{I}_{fbase}}}_{}$$
| Base field impedance (Ω) |

$${L}_{fbase}={\frac{{Z}_{fbase}}{{\omega}_{base}}}_{}$$
| Base field inductance (H) |

The actual field parameters are:

*R*′ is field resistance (Ω)_{f}*L*′ is field leakage inductance (H)_{lfd}

**Field Voltage, Current, and RL Values Referred to the Stator**

You can specify field parameters for the field resistance and leakage inductance referred
to the stator (*R _{f}*,

*L*).

_{lfd}If the nominal field current *i _{fn}* is known, the
transformation ratio

*N*/

_{s}*N*is calculated using the same equation as for the stator/field transformation ratio.

_{f}According to Krause [4], the
field voltage and current referred to the stator
(*V _{f}*,

*I*) are deduced from actual

_{f}*V*′ ,

_{f}*I*′ values as follows:

_{f}$${V}_{f}=\frac{{N}_{s}}{{N}_{f}}{V}_{f}\text{'}$$

$${I}_{f}=\frac{2}{3}\frac{{N}_{f}}{{N}_{s}}{I}_{f}\text{'}$$

When the actual field resistance *R _{f}*′
and leakage inductances

*L*′ (seen from the rotor) are known, the values referred to the stator are:

_{lfd}$${R}_{f}=\frac{3}{2}{R}_{f}\text{'}{\left(\frac{{N}_{s}}{{N}_{f}}\right)}^{2}$$

$${L}_{lfd}=\frac{3}{2}{L}_{lfd}\text{'}{\left(\frac{{N}_{s}}{{N}_{f}}\right)}^{2}$$

When the nominal field current is not known, and if the pu values of field
resistance and leakage inductances are known
(*R _{f_pu}*,

*L*), the corresponding SI values referred to the stator are computed as follows:

_{lfd_pu}$${R}_{f}={R}_{f\_pu}\times {Z}_{sbase}$$

$${L}_{lfd}={L}_{lfd\_pu}\times {L}_{sbase}$$

The same conversions are used for RL parameters of damper windings.

**Field Voltage and Current**

When you specify the nominal field current, the signal at the **Vf**
input port corresponds to the actual field voltage, as in real life. The field
current returned by the measurement output also corresponds to the actual field
current *If*.

The nominal field voltage producing nominal stator voltage at no load is given by:

$${e}_{fn}={R}_{f}\text{'}\times {i}_{fn}$$

When you do not specify the nominal field current, the signal at the
**Vf** input port corresponds to the actual field voltage
referred to the stator. In this case, the nominal field voltage referred to the
stator producing nominal stator voltage at no load is:

$${e}_{fn}(stator\text{}side)=\frac{{R}_{f\_pu}}{{L}_{md\_pu}}{V}_{sbase}=\frac{{R}_{f}}{{L}_{md}\text{\hspace{0.17em}}{\omega}_{base}}{V}_{sbase}.$$

The field current returned by the measurement output is the field current referred to the stator. The nominal field current referred to the stator is:

$${i}_{fn}(stator\text{}side)=\frac{{I}_{sbase}}{{L}_{md\_pu}}=\frac{{I}_{sbase}}{{I}_{fbase}}{i}_{fn}.$$

## Examples

## Assumptions and Limitations

In discrete systems, when you set the **Discrete solver model**
parameter of a Synchronous Machine block to ```
Trapezoidal non
iterative
```

, you might have to connect a small parasitic resistive load
at the machine terminals to avoid numerical oscillations. Large sample times require
larger loads. The minimum resistive load is proportional to the sample time. As a rule
of thumb, remember that with a 25 μs time step on a 60 Hz system, the minimum load is
approximately 2.5% of the machine nominal power. For example, a 200 MVA synchronous
machine in a power system discretized with a 50 μs sample time requires approximately 5%
of resistive load or 10 MW. If the sample time is reduced to 20 μs, a resistive load of
4 MW should be sufficient.

However, if you set the **Discrete solver model** parameter of a
Synchronous Machine block to ```
Trapezoidal iterative (alg.
loop)
```

, you can use a negligible parasitic load (below 0.1% of nominal
power) while preserving numerical stability. This iterative model produces an algebraic
loop and results in slower simulation speed.

## Ports

### Input

### Output

### Conserving

## Parameters

## References

[1] Canay, I.M. "Causes of Discrepancies
on Calculation of Rotor Quantities and Exact Equivalent Diagrams of the Synchronous
Machine." *IEEE Transactions on Power Apparatus and Systems*. PAS-88, no. 7
(1969): 1114–1120.

[2] Moeini, A., et al. “Synchronous
Machine Stability model, an Update to IEEE Std 1110-2002 Data Translation Technique.” *IEEE standards panel sessions*. 2018.

[3] *IEEE Guide for Synchronous Generator Modeling Practices and Applications in
Power System Stability Analyses.*
IEEE Std 1110-2002 (Revision of IEEE Std 1110-1991 [2003]): 1–72.

[4] Krause, P.C. *Analysis of Electric Machinery*. Section 12.5. New York:
McGraw-Hill, 1986.

[5] Kundur, P. *Power System Stability and Control*. New York, McGraw-Hill,
1994.

## Extended Capabilities

## Version History

**Introduced before R2006a**