# Linear Transformer

Implement two- or three-winding linear transformer

## Library

Simscape / Electrical / Specialized Power Systems / Power Grid Elements

• ## Description

The Linear Transformer block model shown consists of three coupled windings wound on the same core. The model takes into account the winding resistances (R1 R2 R3) and the leakage inductances (L1 L2 L3), as well as the magnetizing characteristics of the core, which is modeled by a linear (Rm Lm) branch.

### The Per Unit Conversion

In order to comply with industry, the block allows you to specify the resistance and inductance of the windings in per unit (pu). The values are based on the transformer rated power Pn, in VA, nominal frequency fn, in Hz, and nominal voltage Vn, in Vrms, of the corresponding winding. For each winding, the per unit resistance and inductance are defined as

`$R\left(pu\right)=\frac{R\left(\Omega \right)}{{R}_{base}}$`
`$L\left(pu\right)=\frac{L\left(H\right)}{{L}_{base}}$`

The base impedance, base resistance, base reactance, and base inductance used for each winding are

`${Z}_{base}={R}_{base}={X}_{base}=\frac{{\left(Vn\right)}^{2}}{Pn}$`
`${L}_{base}=\frac{{X}_{base}}{2\pi fn}$`

For the magnetization resistance Rm and inductance Lm, the pu values are based on the transformer rated power and on the nominal voltage of winding 1.

For example, the default parameters of winding 1 specified in the dialog box section give the following bases:

`${R}_{base}=\frac{{\left(735e3\right)}^{2}}{250e6}=2161\Omega$`
`${L}_{base}=\frac{2161}{2\pi 60}=5.732H$`

Suppose that the winding 1 parameters are R1 = 4.32 Ω and L1 = 0.4586 H; the corresponding values to be entered in the dialog box are

`${R}_{1}=\frac{4.32\Omega }{2161\Omega }=0.002pu$`
`${L}_{1}=\frac{0.4586H}{5.732H}=0.08pu$`

To specify a magnetizing current of 0.2% (resistive and inductive) based on nominal current, you must enter per unit values of 1/0.002 = 500 pu for the resistance and the inductance of the magnetizing branch. Using the base values calculated previously, these per unit values correspond to Rm = 1.08e6 ohms and Lm = 2866 henries.

### Modeling an Ideal Transformer

To implement an ideal transformer model, set the winding resistances and inductances to 0, and the magnetization resistance and inductance (Rm Lm) to `inf`.

## Parameters

Units

Specify the units used to enter the parameters of the Linear Transformer block. Select `pu` to use per unit. Select `SI` to use SI units. Changing the Units parameter from `pu` to `SI`, or from `SI` to `pu`, will automatically convert the parameters displayed in the mask of the block. The per unit conversion is based on the transformer rated power Pn in VA, nominal frequency fn in Hz, and nominal voltage Vn, in Vrms, of the windings. Default is `pu`.

Nominal power and frequency

The nominal power rating Pn in volt-amperes (VA) and frequency fn, in hertz (Hz), of the transformer. Note that the nominal parameters have no impact on the transformer model when the Units parameter is set to `SI`. Default is `[ 250e6 60 ]`.

Winding 1 parameters

The nominal voltage V, in volts RMS, resistance, in pu or ohms, and leakage inductance, in pu or henries. The pu values are based on the nominal power Pn and on V1. Set the winding resistances and inductances to 0 to implement an ideal winding. Default is ```[ 735e3 0.002 0.08 ]``` when the Units parameter is `pu` and `[7.35e+05 4.3218 0.45856]` when the Units parameter is `SI` .

Winding 2 parameters

The nominal voltage V2 in volts RMS, resistance, in pu or ohms, and leakage inductance, in pu or henries. The pu values are based on the nominal power Pn and on V2. Set the winding resistances and inductances to 0 to implement an ideal winding. Default is ```[ 315e3 0.002 0.08 ]``` when the Units parameter is `pu` and `[3.15e+05 0.7938 0.084225]` when the Units parameter is `SI`.

Three windings transformer

If selected, implements a linear transformer with three windings; otherwise, it implements a two-windings transformer. Default is selected.

Winding 3 parameters

The Winding 3 parameters parameter is not available if the Three windings transformer parameter is not selected.

The nominal voltage in volts RMS (Vrms), resistance, in pu or ohms, and leakage inductance in pu or henries. The pu values are based on the nominal power Pn and on V3. Set the winding resistances and inductances to 0 to implement an ideal winding. Default is `[ 315e3 0.002 0.08 ]` when the Units parameter is `pu` and `[3.15e+05 0.7938 0.084225]` when the Units parameter is `SI`.

Magnetization resistance and inductance

The resistance and inductance simulating the core active and reactive losses. When selected, the pu values are based on the nominal power Pn and on V1. For example, to specify 0.2% of active and reactive core losses, at nominal voltage, use Rm = 500 pu and Lm = 500 pu. Default is `[ 500 500 ]` when the Units parameter is `pu` and `[1.0805e+06 2866]` when the Units parameter is `SI`.

Rm must have a finite value when the inductance of winding 1 is greater that zero.

Measurements

Select `Winding voltages` to measure the voltage across the winding terminals of the Linear Transformer block.

Select `Winding currents` to measure the current flowing through the windings of the Linear Transformer block.

Select `Magnetization current` to measure the magnetization current of the Linear Transformer block.

Select `All voltages and currents` to measure the winding voltages and currents plus the magnetization current.

Place a Multimeter block in your model to display the selected measurements during the simulation.

Default is `None`.

In the Available Measurements list box of the Multimeter block, the measurements are identified by a label followed by the block name.

Measurement

Label

Winding voltages

`Uw1:`

Winding currents

`Iw1:`

Magnetization current

`Imag:`

## Limitations

Windings can be left floating (that is, not connected to the rest of the circuit). However, an internal resistor is automatically added between the floating winding and the main circuit. This internal connection does not affect voltage and current measurements.

Due to limitations inherent to graph theory and its application to electric network theory as implemented in Simscape™ Electrical™ Specialized Power Systems software, the following topologies are unsolvable:

• Loops containing only ideal transformer secondary windings (for example, delta-connected ideal secondary windings of three-phase transformer). To solve this topology issue, you can add a small impedance in series with the loop.

• Loops containing only ideal transformer secondary windings and ideal voltage sources. To solve this topology issue, you can add a small impedance in series with the loop.

• Loops containing only ideal transformer secondary windings and capacitors. To solve this topology issue, you can add a small impedance in series with the loop.

• All topologies where an ideal transformer primary has at least one of its nodes that is connected to elements consisting only of ideal transformer primary windings or current sources (for example, wye-connected three-phase primary windings with floating common point). To resolve this circuit topology, you connect a small resistance to problematic node.

## Examples

The `power_transformer` example shows a typical residential distribution transformer network feeding line-to-neutral and line-to-line loads.