Mutual Inductance
Implement inductances with mutual coupling
Library
Simscape / Electrical / Specialized Power Systems / Passives
Description
The Mutual Inductance block can be used to model two or threewindings inductances with equal mutual coupling, or to model a generalized multiwindings mutual inductance with balanced or unbalanced mutual coupling.
If you choose to model two or threewindings inductances with equal mutual coupling, you specify the selfresistance 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 selfwindings (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 selfresistance of resistor R1.
R2 is the selfresistance of resistor R2.
Rm is the mutual resistance, such that $$Rm<R1$$ and $$Rm<R2$$.
L is the inductance.
L1 is the selfinductance of inductor L1.
L2 is the selfinductance 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 threephase mutual inductance with equal mutual coupling between the windings. This is the default. Winding 1 self impedance
The selfresistance and inductance for winding 1, in ohms (Ω) and henries (H). Default is
[1.1 1.1e03]
. Winding 2 self impedance
The selfresistance and inductance for winding 2, in ohms (Ω) and henries (H). Default is
[ 1.1 1.1e03]
. 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 selfresistance and inductance in ohms (Ω) and henries (H) for winding 3. Default is
[ 1.1 1.1e03]
. 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 1.0e03]
.The mutual inductance can be expressed as a relationship between two self inductances as
L_{m} = k*sqrt(L_{1}*L_{2}),
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 NbyN symmetrical matrix. Default is
[1.0 0.9 0.9 ; 0.9 1.0 0.9; 0.9 0.9 1.0 ] * 1e3
. Resistance matrix R
The resistance matrix, in ohms, that define the mutual coupling relationship between the self windings. It must be a NbyN 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 

Winding currents 

Limitations
If you choose to model two or three windings inductances with equal mutual coupling, the following restrictions apply:
R_{1}, R_{2}, ..., R_{N} ≠ R_{m}L_{1}, L_{2}, ..., L_{N}≠ L_{m}.
Negative values are allowed for the self and mutual inductances as long as the selfinductances 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.