Two-Winding Transformer (Three-Phase)

R2026b

Three-phase linear nonideal wye- and delta-configurable two-winding transformer with saturation capability

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Two-Winding Transformer (Three-Phase) block

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Libraries:
Simscape / Electrical / Passive / Transformers

Description

The Two-Winding Transformer (Three-Phase) block represents a linear nonideal three-phase two-winding transformer that transfers electrical energy between two or more circuits through electromagnetic induction. The block includes linear winding leakage and linear core magnetization effects.

You can parameterize the block using equivalent-circuit impedances or positive-sequence and zero-sequence test data (since R2026b).

The configuration options for both the primary and secondary windings are:

• Wye with floating neutral — Star or T configuration with Floating Neutral (Three-Phase)

• Wye with neutral port — Star or T configuration with Neutral Port (Three-Phase)

• Wye with grounded neutral — Star or T configuration with Grounded Neutral (Three-Phase)

• Delta 1 o'clock — Mesh configuration with a lagging 30 degree phase shift relative to the voltage of a connected wye configuration

• Delta 11 o'clock — Mesh configuration with leading 30 degree phase shift relative to the voltage of a connected wye configuration

If you parameterize the block using circuit impedances, you can specify the core type as one of these options:

• Three-phase five-limb

• Three-phase three-limb

Although a three-limb core is typically less expensive, a five-limb core offers these advantages:

• Lower impedance for the zero-sequence component of current, that is between the line and neutral, in the case of an unbalanced load

• Greater heat dissipation

Equations

Three-Limb Core

This block is implemented in the magnetic domain using basic magnetic reluctances, windings, and eddy currents blocks.

It is important to determine the relation between the electrical domain parameters from the block mask and the magnetic domain parameters used in the model:

• n₁ is the number of the primary winding turns.

• n₂ is the number of the secondary winding turns.

• L_m is the shunt magnetizing inductance.

• L₀ is the zero-sequence inductance.

• L_p is the primary winding leakage inductance.

• L_s is the secondary winding leakage inductance.

• R_m is the shunt magnetizing resistance.

• R is the magnetizing reluctance between phases.

  $$R=\frac{n_1^2}{L_m}$$

• R₀ is the zero sequence reluctance.

  $$R_0=\frac{1}{3}\frac{n_1^2}{L_0-L_p}$$

• Rl₁ is the primary winding leakage reluctance.

  $$R{l}_1=\frac{n_1^2}{L_p}$$

• Rl₂ is the secondary winding leakage reluctance.

  $$R{l}_2=\frac{n_2^2}{L_s}$$

• L_eddy is the conductance of eddy current loop

  $$L_{eddy}=\frac{n_1^2}{R_m}$$

For two-winding transformers (three-phase), the coupling between different windings in each phase is identical.

Five-Limb Core

In the case of a five-limb transformer, the extra magnetic flux paths provided by the extra limbs can be represented by zero sequence reluctances, which are originally designed for magnetic paths through the air in the three-limb transformer.

In a five-limb model, the magnetic reluctances from the phases to the extra limbs are supposed to be equal to the magnetic reluctances between phases.

$$R=R_0$$

Therefore:

$$R=R_0=\frac{n_1^2}{L_m}$$

Test Data

When you set the Specify parameterization by parameter to Test data, the block must derive the equivalent-circuit parameters from the measured test data.

The test data specifies the electrical quantities that you obtain from standard transformer tests, such as no-load and short-circuit tests. From these values, the block computes the corresponding equivalent-circuit impedances that characterize the transformer behavior under both positive‑sequence and zero‑sequence conditions.

For the positive-sequence equivalent circuit, the block defines the relationships between the test data and the equivalent parameters by using these mathematical expressions:

$$\begin{array}{l}{X}_{l1}+X_{l2}=X_{121}\\ P_{1\text{core}}=P_1-I_{\text{ext}1}^2\cdot R_1\end{array}$$

where:

• X_l1 is the per-unit primary leakage reactance.

• X_l2 is the per-unit secondary leakage reactance.

• X₁₂₁ is the per-unit positive-sequence short-circuit reactance.

• P₁ is the per-unit positive-sequence no-load loss.

• I_ext1 is the per-unit positive-sequence no-load excitation current.

• R₁ is the per-unit primary winding resistance.

The block calculates the shunt magnetizing resistance, R_m, from the specified positive-sequence core resistive loss, P_1core:

$$R_m=\frac{1}{P_{1\text{core}}}.$$

The block then derives the shunt magnetizing reactance, X_m from the no-load excitation current using the calculated shunt magnetizing resistance and impedance relationship:

$$\begin{array}{l}\left|\frac{1}{R_m}+\frac{1}{j{X}_m}\right|=I_{\text{ext}1}\\ X_m=\frac{R_m}{\sqrt{I_{ext1}^2\cdot R_m^2-1}}.\end{array}$$

X_m and R_m defines the shunt magnetizing branch of the positive-sequence equivalent circuit.

For the zero-sequence equivalent circuit, the block calculates the open-circuit impedance Z_open and the short-circuit impedance Z_short using the zero-sequence no-load and short-circuit test data,

$$\begin{array}{l}{Z}_{\text{open}}=\frac{1}{\frac{1}{R_m}+\frac{1}{R_0}+\frac{1}{j{X}_m}+\frac{1}{j{X}_0}}+R_1+j{X}_{l1}\\ Z_{\text{short}}=\frac{1}{\frac{1}{R_m}+\frac{1}{R_0}+\frac{1}{j{X}_m}+\frac{1}{j{X}_0}+\frac{1}{R_2+j{X}_{l2}}}+R_1+j{X}_{l1}\end{array}$$

where:

• X₀ is the per-unit zero-sequence reactance.

• R₀ is the per-unit zero-sequence resistance.

• R₂ is the per-unit secondary winding resistance.

By separating the real and imaginary components of these impedances, the block calculates the zero-sequence resistance and reactance,

$$\begin{array}{l}{X}_{120}=\text{Imag}\left\{Z_{\text{short}}\right\}\\ I_{\text{ext}0}=\left|\frac{1}{Z_{\text{open}}}\right|\\ P_0=\text{Real}\left\{\frac{1}{Z_{\text{open}}}\right\}\end{array}$$

$$\begin{array}{l}\text{Real}\left\{Z_{\text{open}}\right\}=\frac{P_0}{I_{\text{ext0}}^2}\\ \text{Imag}\left\{Z_{\text{open}}\right\}=\frac{\sqrt{I_{\text{ext0}}^2-P_0^2}}{I_{\text{ext0}}^2}\end{array}$$

where:

• X₁₂₀ is the per-unit zero-sequence short-circuit reactance-

• I_ext0 is the per-unit zero-sequence no-load excitation current.

• P₀ is the per-unit zero-sequence no-load loss.

These quantities characterize the transformer behavior under zero‑sequence excitation and model the zero‑sequence losses and leakage effects.

Generate Derived Data Sheet

Since R2026b

You can generate a derived data sheet for the Two-Winding Transformer (Three-Phase) block that contains summary tables and characteristic plots similar to those that device manufacturers provide in their data sheets. A built-in MATLAB^® script calculates the block-level characteristics based on the parameter values in your model. Use derived data sheets to explore the effect of your parameter choices on device characteristics, help you select manufactured parts, or share your component-level design with others.

The derived data sheet for the Two-Winding Transformer (Three-Phase) block includes these plots:

• Open-circuit test results — Secondary voltage versus excitation current

• Short-circuit test results — Current versus phase voltage

• Load test results — Voltage regulation and efficiency versus load

• Iron loss, copper loss, and total loss versus load

To generate a derived data sheet:

1. Open the MATLAB script by clicking the Open live script button next to the Derived data sheet parameter in the Utilities section of the block dialog box.

2. Click the Generate Data Sheet button in the script.

For more information about derived data sheets, see Generate Derived Data Sheets.

Display Options

You can display the transformer per-unit base values in the MATLAB command window. To display the transformer values, in the Utilities section, click the Display button next to the Base values parameter.

Variables

To set the priority and initial target values for the block variables before simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. You can specify nominal values using different sources, including the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Examples

Three-Phase Asynchronous Drive with Sensor Control

Control and analyze the operation of an Asynchronous Machine (ASM) using sensored rotor field-oriented control. The model shows the main electrical circuit, with three additional subsystems containing the controls, measurements, and scopes. The Controls subsystem contains two controllers: one for the Grid-Side Converter (AC/DC) and one for the Machine-Side Converter (DC/AC). The Scopes subsystem contains two time scopes: one for the Grid-Side Converter and one for the ASM. When the model is executed, a Spectrum Analyzer opens and displays frequency data for the A-Phase Supply Current.

• Open Model

Three-Phase Asynchronous Drive with Sensorless Control

Control and analyze the operation of an Asynchronous Machine (ASM) using sensorless rotor field-oriented control. The model shows the main electrical circuit, with three additional subsystems containing the controls, measurements, and scopes. The Controls subsystem contains two controllers: one for the Grid-Side Converter (AC/DC) and one for the Machine-Side Converter (DC/AC). The Scopes subsystem contains two time scopes: one for the Grid-Side Converter and one for the ASM. When the model is executed, a Spectrum Analyzer opens and displays frequency data for the A-Phase Supply Current.

• Open Model

Three-Phase Bridge Cycloconverter

A three-phase bridge cycloconverter. The cycloconverter consists of 36 thyristors and has the capacity to lower the frequency of the input voltage. The Control subsystem implements the cycloconverter RMS voltage control. It also provides pulse generation for the firing of the thyristors. The Visualization subsystem contains scopes that allow you to see the simulation results. The simulation time, t, is 1 second. The load increases when Load1 switches on at t = 0.75 seconds.

• Open Model

Model Transmission System Using Delta Reference (Three-Phase) Block

Use a Delta Reference (Three-Phase) block to provide a reference point for delta winding in a simple model of a transmission system.

• Open Live Script

IEEE 9-Bus System

Model a 9-bus three-phase power system network. This example is based on the IEEE® benchmark test case. For more information, see "Power System Control and Stability" by P. M. Anderson and A. A. Fouad (IEEE Press, 2003).

• Open Model

Ports

Conserving

~1 — Primary winding voltage

electrical

Expandable three-phase electrical conserving port associated with the three-phase, [a1 b1 c1], voltage of winding 1.

n1 — Primary winding neutral point

electrical

Electrical conserving port associated with the primary winding neutral point.

Dependencies

To enable this port, set Winding 1 connection type to Wye with neutral port.

~2 — Secondary winding voltage

electrical

Expandable three-phase electrical conserving port associated with the three-phase, [a2 b2 c2], voltage of the secondary winding.

n2 — Secondary winding neutral point

electrical

Electrical conserving port associated with the secondary winding neutral point.

Dependencies

To enable this port, set Winding 2 connection type to Wye with neutral port.

Parameters

Main

Electrical connection — Whether to model composite or expanded three-phase ports

Composite three-phase ports (default) | Expanded three-phase ports

Since R2026b

Whether to model composite or expanded three-phase ports.

Composite three-phase ports represent three individual electrical conserving ports with a single block port. You can use composite three-phase ports to build models that correspond to single-line diagrams of three-phase electrical systems.

Expanded three-phase ports represent the individual phases of a three-phase system using three separate electrical conserving ports.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

Parameter:	port_option
Values:	"ee.enum.threePhasePort.composite" (default) | "ee.enum.threePhasePort.expanded"

Rated apparent power — Apparent power at rated capacity

100e6 (default) | positive scalar

Apparent power flowing through the transformer when operating at rated capacity. The value must be greater than 0.

Rated electrical frequency — Connected network electrical frequency

60 (default) | positive scalar

Rated or nominal frequency of the AC network to which the transformer is connected. The value must be greater than 0.

Winding 1 connection type — Primary winding configuration

Wye with floating neutral (default) | Wye with neutral port | Wye with grounded neutral | Delta 1 o'clock | Delta 11 o'clock

Primary winding type.

Primary rated voltage — RMS line voltage applied to the primary winding

4160 (default) | positive scalar

RMS line voltage applied to the primary winding under normal operating conditions. The value must be greater than 0.

Winding 2 connection type — Secondary winding configuration

Wye with floating neutral (default) | Wye with neutral port | Wye with grounded neutral | Delta 1 o'clock | Delta 11 o'clock

Secondary winding type.

Secondary rated voltage — RMS line voltage applied to the secondary winding

24e3 (default) | positive scalar

RMS line voltage applied to the secondary winding under normal operating conditions. The value must be greater than 0.

Specify parameterization by — Option to specify parameterization of transformer

Circuit impedances (default) | Test data

Since R2026b

Option to specify the parameterization of the transformer, specified as one of these options:

• Circuit impedances — Parameterize the transformer using the equivalent-circuit impedances.

• Test data — Parameterize the transformer using positive-sequence and zero-sequence test data.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances.

Core type — Number of limbs

Three-phase three-limb (default) | Three-phase five-limb

Number of limbs that comprise the magnetic circuit.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances. (since R2026b)

Impedances

Primary winding resistance (pu) — Primary winding power loss

0.01 (default) | positive scalar

Per-unit power loss in the primary winding.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances. (since R2026b)

Secondary winding resistance (pu) — Secondary winding power loss

0.01 (default) | positive scalar

Per-unit power loss in the secondary winding.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances. (since R2026b)

Leakage reactance — Leakage reactance modeling

Exclude (default) | Include

Whether to model magnetic flux losses.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances. (since R2026b)

Primary leakage reactance (pu) — Primary winding magnetic flux loss

0.001 (default) | positive scalar

Per-unit magnetic flux loss in the primary winding. The value must be greater than 0.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b) and Leakage reactance to Include.

Secondary leakage reactance (pu) — Secondary winding magnetic flux loss

0.001 (default) | positive scalar

Per-unit magnetic flux loss in the secondary winding. The value must be greater than 0.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b) and Leakage reactance to Include.

Magnetizing resistance — Magnetizing resistance modeling

Exclude (default) | Include

Whether to model the transformer core losses.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances. (since R2026b)

Shunt magnetizing resistance (pu) — Transformer core losses

500 (default) | positive scalar

Per-unit losses in the transformer core. The value must be greater than 0.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b) and Magnetizing resistance to Include.

Magnetizing reactance — Magnetizing reactance modeling

Exclude (default) | Include

Whether to model magnetic effects of the transformer core.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances. (since R2026b)

Magnetic saturation representation — Representation of the magnetic saturation

None (default) | Lookup table (phi versus i)

Choose if and how you want to represent the magnetic saturation.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b) and Magnetizing reactance to Include.

Current vector (pu), i — Per unit vector of currents

[0, .002, 1, 2] (default)

Per unit vector of currents. The first value must be 0. This parameter must be strictly ascending.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b), Magnetizing reactance to Include, and Magnetic saturation representation to Lookup table (phi versus i).

Magnetic flux vector (pu), phi — Per unit vector of magnetic flux

[0, 1.2, 1.5, 1.51] (default)

Per unit vector of magnetic flux. The first value must be 0. This parameter must be strictly ascending.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b), Magnetizing reactance to Include, and Magnetic saturation representation to Lookup table (phi versus i).

Shunt magnetizing reactance (pu) — Transformer core magnetic effects

500 (default) | positive scalar

Per-unit magnetic effects of the transformer core when operating in its linear region. The value must be greater than 0.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b), Magnetizing reactance to Include, and Magnetic saturation representation to None.

Zero sequence reactance (pu) — Zero sequence reactance

0.5 (default) | positive scalar

Per-unit zero sequence reactance. The value must be greater than or equal to the primary winding magnetic flux loss.

Dependencies

To enable this parameter, set Specify parameterization by to Circuit impedances (since R2026b) and Core type to Three-phase three-limb.

Test Data

Primary winding resistance (pu) — Primary winding power loss

0.01 (default) | positive scalar

Since R2026b

Per-unit power loss in the primary winding.

Dependencies

To enable this parameter, set Specify parameterization by to Test data.

Secondary winding resistance (pu) — Secondary winding power loss

0.01 (default) | positive scalar

Since R2026b

Per-unit power loss in the secondary winding.

Dependencies

To enable this parameter, set Specify parameterization by to Test data.

Positive-sequence short-circuit reactance (pu) — Per-unit positive-sequence short-circuit reactance

0.1 (default) | positive scalar

Since R2026b

Positive-sequence reactance, in per-unit, measured from the primary winding when the secondary winding is short-circuited.

Dependencies

To enable this parameter, set Specify parameterization by to Test data.

Positive-sequence no-load loss (pu) — Per-unit positive-sequence no-load loss

0.01 (default) | positive scalar

Since R2026b

No-load power loss of the transformer, in per-unit, when you apply the positive-sequence nominal voltage at the primary winding. This parameter includes the core loss and the primary winding loss.

Dependencies

To enable this parameter, set Specify parameterization by to Test data.

Positive-sequence no-load excitation current (pu) — Per-unit positive-sequence no-load excitation current

0.02 (default) | positive scalar

Since R2026b

No-load excitation current, in per-unit, when you apply the positive-sequence nominal voltage at the primary winding.

The value of this parameter must be greater than the value of the Positive-sequence no-load loss (pu) parameter.

Dependencies

To enable this parameter, set Specify parameterization by to Test data.

Zero-sequence test data — Option to include zero-sequence test data

Exclude (default) | Include

Since R2026b

Option to include zero-sequence test data.

Dependencies

To enable this parameter, set Specify parameterization by to Test data.

Zero-sequence short-circuit reactance (pu) — Per-unit zero-sequence short-circuit reactance

0.09 (default) | positive scalar

Since R2026b

Zero-sequence reactance, in per-unit, measured from the primary winding when the secondary winding is short-circuited.

Dependencies

To enable this parameter, set Specify parameterization by to Test data and Zero-sequence test data to Include.

Zero-sequence no-load loss (pu) — Per-unit zero-sequence no-load loss

0.15 (default) | positive scalar

Since R2026b

No-load power loss of the transformer, in per-unit, when you apply the zero-sequence nominal voltage at the primary winding.

Dependencies

To enable this parameter, set Specify parameterization by to Test data and Zero-sequence test data to Include.

Zero-sequence no-load excitation current (pu) — Per-unit zero-sequence no-load excitation current

3.5 (default) | positive scalar

Since R2026b

No-load excitation current, in per-unit, when you apply the zero-sequence nominal voltage at the primary winding.

The value of this parameter must be greater than the value of the Zero-sequence no-load loss (pu) parameter.

Dependencies

To enable this parameter, set Specify parameterization by to Test data and Zero-sequence test data to Include.

Magnetic saturation representation — Representation of the magnetic saturation

None (default) | Lookup table (phi versus i)

Since R2026b

Option to represent the magnetic saturation.

Dependencies

To enable this parameter, set Specify parameterization by to Test data.

Current vector (pu), i — Vector of per-unit currents

[0, .002, 1, 2] (default) | vector of nonnegative elements

Since R2026b

Vector of per-unit currents. The first value must be 0. This parameter value must be strictly ascending.

Dependencies

To enable this parameter, set Specify parameterization by to Test data and Magnetic saturation representation to Lookup table (phi versus i).

Magnetic flux vector (pu), phi — Vector of per-unit magnetic flux

[0, 1.2, 1.5, 1.51] (default) | vector of nonnegative elements

Since R2026b

Vector of per-unit magnetic flux. The first value must be 0. This parameter value must be strictly ascending.

Dependencies

To enable this parameter, set Specify parameterization by to Test data and Magnetic saturation representation to Lookup table (phi versus i).

Utilities

Derived data sheet — Open live script

button

Since R2026b

Click this button to open a MATLAB script that generates a derived data sheet.

For more information about derived data sheets, see Generate Derived Data Sheets.

Base values — Display transformer per-unit base values

button

Since R2026a

Display the transformer per-unit base values in the MATLAB Command Window.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2019a

R2026b: Generate derived data sheet

You can generate a derived data sheet for the Two-Winding Transformer (Three-Phase) block that contains summary tables and characteristic plots similar to those that device manufacturers provide in their data sheets. A built-in MATLAB script calculates the block-level characteristics based on the parameter values in your model. Use derived data sheets to explore the effect of your parameter choices on device characteristics, help you select manufactured parts, or share your component-level design with others.

To generate a derived data sheet:

1. Open the MATLAB script by clicking the Open live script button next to the Derived data sheet parameter in the Utilities section of the block dialog box.

2. Click the Generate Data Sheet button in the script.

R2026b: Parameterize block using positive-sequence and zero-sequence test data

You can now parameterize the transformer by using positive-sequence and zero-sequence test data.

The test data specifies the electrical quantities that you obtain from standard transformer tests, such as no-load and short-circuit tests. From these values, the block computes the corresponding equivalent-circuit impedances that characterize the transformer behavior under both positive‑sequence and zero‑sequence conditions.

Before R2026b, you can parameterize this block only by using equivalent-circuit impedances.

R2026b: The Modeling option parameter has changed

The Modeling option parameter has been replaced with the Electrical connection parameter. You can now also set this parameter value programmatically by using the set_param function.

If you created a model containing this block in an earlier release, the software now automatically sets the value of the Electrical connection parameter to the previous value of the Modeling option parameter.

R2026a: Context menu usability improvements

The display option of the Two-Winding Transformer (Three-Phase) per-unit base values has been moved from the context menu to the block dialog box or Property Inspector. To display the per-unit base values of the Two-Winding Transformer (Three-Phase) block, open the block dialog box or Property Inspector and, in the Utilities section, click the Display button next to the Base values parameter.

Before R2026a, you plot the per-unit base values by right-clicking the block and selecting Electrical > Display Base Values from the context menu.