# Mutual Inductance

Implement inductances with mutual coupling

## Library

Simscape / Electrical / Specialized Power Systems / Fundamental Blocks / Elements  ## 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 $Lm\le \sqrt{L1\cdot L2}$

## Parameters

### Two or Three Windings Inductances with Equal Mutual Coupling:

Type of mutual inductance

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.

Winding 1 self impedance

The self-resistance and inductance for winding 1, in ohms (Ω) and henries (H). Default is `[1.1 1.1e-03]`.

Winding 2 self impedance

The self-resistance and inductance for winding 2, in ohms (Ω) and henries (H). Default is `[ 1.1 1.1e-03]`.

Three windings Mutual Inductance

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

Winding 3 self impedance

The Winding 3 self impedance parameter is not available if the Three windings Mutual Inductance parameter is not selected. The self-resistance and inductance in ohms (Ω) and henries (H) for winding 3. Default is `[ 1.1 1.1e-03]`.

Mutual impedance

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. Default is `[1.0 1.0e-03]`.

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

Lm = k*sqrt(L1*L2),

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

Measurements

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.

### Generalized Mutual Inductance:

Type of mutual inductance

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

Number of windings

The number of self inductances. Default is `3`.

Inductance matrix L

The inductance matrix, in Henrys, that define the mutual coupling relationship between the self windings. It must be a N-by-N symmetrical matrix. Default is ```[1.0 0.9 0.9 ; 0.9 1.0 0.9; 0.9 0.9 1.0 ] * 1e-3```.

Resistance matrix R

The resistance matrix, in ohms, that define the mutual coupling relationship between the self windings. It must be a N-by-N symmetrical matrix. Default is ```[1.0 0.9 0.9 ; 0.9 1.0 0.9; 0.9 0.9 1.0 ] ```.

Measurements

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:`

## Limitations

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

R1, R2, ..., RNRmL1, L2, ..., LNLm.

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.

## Examples

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