# Mutual Inductance

Implement inductances with mutual coupling

Libraries:
Simscape / Electrical / Specialized Power Systems / Passives

## Description

The Mutual Inductance block can be used to model two- or three-windings inductances with equal mutual coupling, or to model a generalized multi-windings mutual inductance with balanced or unbalanced mutual coupling.

If you choose to model two- or three-windings inductances with equal mutual coupling, you specify the self-resistance and inductance of each winding plus the mutual resistance and inductance. The electrical model for this block in this case is given below:

If you choose to model a general mutual inductance, specify the number of self-windings (not just limited to 2 or 3 windings) plus the resistance and inductance matrices that define the mutual coupling relationship between the windings (balanced or not).

### Equations

The resistance and inductance matrices are defined as

$R=\left[\begin{array}{cc}R1& Rm\\ Rm& R2\end{array}\right]$

and

$L=\left[\begin{array}{cc}L1& Lm\\ Lm& L2\end{array}\right],$

where:

• R is the resistance.

• R1 is the self-resistance of resistor R1.

• R2 is the self-resistance of resistor R2.

• Rm is the mutual resistance, such that $Rm and $Rm.

• L is the inductance.

• L1 is the self-inductance of inductor L1.

• L2 is the self-inductance of inductor L2.

• Lm is the mutual inductance, such that $Lm\le \sqrt{L1\cdot L2}$

### Examples

The power_mutual example uses three coupled windings to inject a third harmonic voltage into a circuit fed at 60 Hz.

## Assumptions and Limitations

• If you choose to model two or three windings inductances with equal mutual coupling, the following restrictions apply:

 R1, R2, ..., RN ≠ RmL1, L2, ..., LN≠ Lm. (1)
• Negative values are allowed for the self- and mutual inductances as long as the self-inductances are different from the mutual inductance.

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

## Parameters

expand all

To edit block parameters interactively, use the Property Inspector. From the Simulink® Toolstrip, on the Simulation tab, in the Prepare gallery, select .

Select Two or Three windings with equal mutual terms to implement a three-phase mutual inductance with equal mutual coupling between the windings. This is the default.

Select Generalized mutual inductance to implement a multi windings mutual inductance with mutual coupling defined by an inductance and a resistance matrix.

#### Dependencies

Self-resistance and inductance in ohms (Ω) and henries (H) for winding 1.

#### Dependencies

To enable this parameter, set Type of mutual inductance to Two or three windings with equal mutual terms.

Self-resistance and inductance in ohms (Ω) and henries (H) for winding 2.

#### Dependencies

To enable this parameter, set Type of mutual inductance to Two or three windings with equal mutual terms.

If selected, implements three coupled windings; otherwise, it implements two coupled windings.

#### Dependencies

To enable this parameter, set Type of mutual inductance to Two or three windings with equal mutual terms.

Self-resistance and inductance in ohms (Ω) and henries (H) for winding 3.

#### Dependencies

To enable this parameter, set Type of mutual inductance to Two or three windings with equal mutual terms and select the Three windings Mutual Inductance parameter.

The mutual resistance and inductance between windings, in ohms (Ω) and henries (H). The mutual resistance and inductance corresponds to the magnetizing resistance and inductance on the standard transformer circuit diagram. If the mutual resistance and reactance are set to [0 0], the block implements three separate inductances with no mutual coupling.

The mutual inductance can be expressed as a relationship between two self inductances as

 Lm = k*sqrt(L1*L2), (2)

where k is the coupling coefficient (−1 ≤ k ≤ 1).

#### Dependencies

To enable this parameter, set Type of mutual inductance to Two or three windings with equal mutual terms.

Number of self inductances.

#### Dependencies

To enable this parameter, set Type of mutual inductance to Generalized mutual inductance.

Inductance matrix, in Henry, that define the mutual coupling relationship between the self windings. It must be an N-by-N symmetrical matrix.

#### Dependencies

To enable this parameter, set Type of mutual inductance to Generalized mutual inductance.

Resistance matrix, in ohms, that define the mutual coupling relationship between the self windings. It must be an N-by-N symmetrical matrix.

#### Dependencies

To enable this parameter, set Type of mutual inductance to Generalized mutual inductance.

Select Winding voltages to measure the voltage across the winding terminals.

Select Winding currents to measure the current flowing through the windings.

Select Winding voltages and currents to measure the winding voltages and currents.

Default is None.

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

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:

## Version History

Introduced before R2006a