# Mutual Inductor

Mutual inductor model with nominal inductance optional tolerances for each winding, operating limits and faults

Libraries:
Simscape / Electrical / Passive / Transformers

## Description

The Mutual Inductor block lets you model a mutual inductor (two-winding transformer) with nominal inductance tolerances for each winding. The model includes the following effects:

You can turn these modeling options on and off independently of each other.

In the unfaulted state, the following equations describe the Mutual Inductor block behavior:

`${v}_{1}={L}_{1}\frac{d{i}_{L1}}{dt}+\text{​}M\frac{d{i}_{L2}}{dt}+{i}_{L1}{R}_{1}$`
`${v}_{2}={L}_{2}\frac{d{i}_{L2}}{dt}+\text{​}M\frac{d{i}_{L1}}{dt}+{i}_{L2}{R}_{2}$`
`$M=k\sqrt{{L}_{1}{L}_{2}}$`

where:

• v1 and v2 are voltages across the primary and secondary winding, respectively.

• L1 and L2 are inductances of the primary and secondary winding.

• R1 and R2 are series resistances of the primary and secondary winding.

• M is mutual inductance.

• k is coefficient of coupling. To reverse one of the winding directions, use a negative value.

• t is time.

A parallel conductance is placed across the + and – terminals of the primary and secondary windings, so that iL1 = i1G1v1, where G1 is the parallel conductance of the primary winding, and i1 is the terminal current into the primary. Similar definitions and equation apply to iL2.

### Tolerances

You can apply tolerances separately for each winding. Datasheets typically provide a tolerance percentage for a given inductor type. Therefore, this value is the same for both windings. The table shows how the block applies tolerances to the nominal inductance value and calculates inductance based on the selected tolerance application option for the winding, L1 tolerance application or L2 tolerance application.

OptionInductance Value

```None — use nominal value```

L

`Random tolerance`

Uniform distribution: L · (1 – tol + 2· tol· `rand`)

Gaussian distribution: L · (1 + tol · `randn` / nSigma)

```Apply maximum tolerance value```

L · (1 + tol )

```Apply minimum tolerance value```

L · (1 – tol )

In the table:

• L is nominal inductance for the primary or secondary winding, Inductance L1 or Inductance L2 parameter value.

• tol is fractional tolerance, Tolerance (%) /100.

• nSigma is the value you provide for the Number of standard deviations for quoted tolerance parameter.

• `rand` and `randn` are standard MATLAB® functions for generating uniform and normal distribution random numbers.

Note

If you choose the `Random tolerance` option and you are in "Fast Restart" mode, the random tolerance value is updated on every simulation if at least one between the fractional tolerance, tol, or the Number of standard deviations for quoted tolerance, nSigma, is set to Run-time and is defined with a variable (even if you do not modify that variable).

### Operating Limits

Inductors are typically rated with a particular saturation current, and possibly with a maximum allowable power dissipation. You can specify operating limits in terms of these values, to generate warnings or errors if the inductor is driven outside its specification.

When an operating limit is exceeded, the block can either generate a warning or stop the simulation with an error. For more information, see the Operating Limits parameters section.

### Faults

To model a fault in the Mutual Inductor block, in the Faults section, click the Add fault hyperlink in the parameter that corresponds to the specific fault that you want to model. When the Create Fault window opens, you use it to specify the fault properties. For more information about fault modeling, see Fault Behavior Modeling and Fault Triggering.

Instantaneous changes in inductor parameters are unphysical. Therefore, when the Mutual Inductor block enters the faulted state, short-circuit and open-circuit voltages transition to their faulted values over a period of time based on this formula:

`CurrentValue` = `FaultedValue` – (`FaultedValue``UnfaultedValue`) · `sech`(∆t / τ)

where:

• ∆t is time since the onset of the fault condition.

• τ is the value of the Fault transition time constant parameter.

For short-circuit faults, the conductance of the short-circuit path also changes according to the `sech`(∆t / τ) function from a small value (representing an open-circuit path) to a large value.

The Mutual Inductor block lets you select whether the faults occur in the primary or secondary winding. The block models the faulted winding as a faulted inductor. The unfaulted winding is coupled to the faulted winding. As a result, the actual equations involve a total of three coupled windings: two for the faulted winding and one for the unfaulted winding. The coupling between the primary and secondary windings is defined by the Coefficient of coupling parameter.

The block can trigger the start of fault transition:

• At a specific time.

• After voltage exceeds the maximum permissible value a certain number of times.

• When current exceeds the maximum permissible value for longer than a specific time interval.

If you want to trigger a fault at a specific time, in the Fault Inspector window, set Trigger type to `Timed`. If you want to determine whether a system fails and, if so, when it fails, in the Fault Inspector window, set Trigger type to `Behavioral`.

If you select the behavioral trigger, the component fails as soon as one of the trigger conditions is true.

Faultable inductors often require that you use the fixed-step local solver, especially if your model transitions to a faulted state that includes short circuits. For more information, see Making Optimal Solver Choices for Physical Simulation.

### 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.

Use nominal values to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources. One of these sources is the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

The Primary current and Secondary current variables let you specify a high-priority target for the initial inductor current in the respective winding at the start of simulation.

## Ports

### Conserving

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Electrical conserving port associated with the primary winding positive terminal.

Electrical conserving port associated with the primary winding negative terminal.

Electrical conserving port associated with the secondary winding positive terminal.

Electrical conserving port associated with the secondary winding negative terminal.

## Parameters

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### Main

The nominal inductance value in the primary winding. Inductance value must be greater than zero.

The nominal inductance value in the secondary winding. Inductance value must be greater than zero.

The coupling between the primary and secondary windings. This coefficient defines the mutual inductance. To reverse one of the winding directions, use a negative value.

The inductor tolerance as defined on the manufacturer datasheet. Datasheets typically provide a tolerance percentage for a given inductor type. Therefore, this value is the same for both windings.

Select how to apply tolerance during simulation to the primary winding:

• `None — use nominal value` — The block does not apply tolerance, it uses the nominal inductance value.

• `Random tolerance` — The block applies random offset to the inductance value, within the tolerance value limit. You can choose Uniform or Gaussian distribution for calculating the random number by using the Tolerance distribution parameter.

• `Apply maximum tolerance value` — The inductance is increased by the specified tolerance percent value.

• `Apply minimum tolerance value` — The inductance is decreased by the specified tolerance percent value.

Select the distribution type for random tolerance:

• `Uniform` — Uniform distribution

• `Gaussian` — Gaussian distribution

#### Dependencies

Enabled when the L1 tolerance application parameter is set to ```Random tolerance```.

Number of standard deviations for calculating the Gaussian random number.

#### Dependencies

Enabled when the L1 tolerance distribution parameter is set to `Gaussian`.

Select how to apply tolerance during simulation to the secondary winding:

• `None — use nominal value` — The block does not apply tolerance, it uses the nominal inductance value.

• `Random tolerance` — The block applies random offset to the inductance value, within the tolerance value limit. You can choose Uniform or Gaussian distribution for calculating the random number by using the Tolerance distribution parameter.

• `Apply maximum tolerance value` — The inductance is increased by the specified tolerance percent value.

• `Apply minimum tolerance value` — The inductance is decreased by the specified tolerance percent value.

Select the distribution type for random tolerance:

• `Uniform` — Uniform distribution

• `Gaussian` — Gaussian distribution

#### Dependencies

Enabled when the L2 tolerance application parameter is set to ```Random tolerance```.

Number of standard deviations for calculating the Gaussian random number.

#### Dependencies

Enabled when the L2 tolerance distribution parameter is set to `Gaussian`.

### Resistance

Equivalent series resistance of the primary and secondary winding, specified as a two-element vector. The first number corresponds to the primary winding, the second number to the secondary winding. For a faulted winding, the block allocates the resistance to each segment in proportion to the number of turns in that segment.

Parallel leakage path associated with the primary and secondary winding, specified as a two-element vector. The first number corresponds to the primary winding, the second number to the secondary winding. The parallel conductances are placed directly across the + and – terminals of the primary and secondary winding, respectively.

### Operating Limits

Select `Yes` to enable reporting when the operational limits are exceeded. The associated parameters in the Operating Limits section become visible to let you select the reporting method and specify the operating limits in terms of power and current.

Select what happens when an operating limit is exceeded:

• `Warn` — The block issues a warning.

• `Error` — Simulation stops with an error.

#### Dependencies

Enabled when the Enable operating limits parameter is set to `Yes`.

Inductor saturation current, as defined in the manufacturer datasheets. If the net current into the primary and secondary windings exceeds this value, the core material enters saturation, and the block reports an operating limits violation. That is, the block compares the limit against |i1 + i2|, where currents are defined as being positive when they are into the + nodes.

#### Dependencies

Enabled when the Enable operating limits parameter is set to `Yes`.

Maximum instantaneous (total) power dissipation in the resistance and conductance elements associated with the mutual inductor. If the total power (including both primary and secondary winding) exceeds this number, the block reports an operating limits violation.

#### Dependencies

Enabled when the Enable operating limits parameter is set to `Yes`.

### Faults

Option to model a fault in the Mutual Inductor block.

Select whether the faults occur in the primary or secondary winding.

#### Dependencies

To enable this parameter, add a fault to the Mutual Inductor block by clicking the Add fault hyperlink in the Mutual inductor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

In practice, faults are enabled by segmenting the faulted winding into two coupled subinductors, connected in a series. The inductance is proportional to the square of the number of turns in the respective segment, and the series resistance of each subinductor is proportional to the number of turns in each segment. The parallel conductance spans both segments.

This parameter indicates the percentage of turns that are assigned to the subinductor that is in contact with the – port of the faulted winding. The remaining turns are assigned to the other subinductor. The default value is `50`, which means that the overall inductance of the faulted winding is divided into two equal, coupled subinductors.

#### Dependencies

To enable this parameter, add a fault to the Mutual Inductor block by clicking the Add fault hyperlink in the Mutual inductor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Select whether the fault results in one of the subinductor segments being short-circuited:

• `No` — The fault does not produce a short circuit.

• `To negative terminal` — The fault short-circuits the subinductor that is in contact with the – port of the block.

• `To positive terminal` — The fault short-circuits the subinductor that is in contact with the + port of the block.

#### Dependencies

To enable this parameter, add a fault to the Mutual Inductor block by clicking the Add fault hyperlink in the Mutual inductor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Select whether to apply an open-circuit fault between the two subinductor segments. The default is `No`. Even with an open-circuit fault, the characteristics of the subinductors are still related because they are magnetically coupled even in the faulted state.

#### Dependencies

To enable this parameter, add a fault to the Mutual Inductor block by clicking the Add fault hyperlink in the Mutual inductor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Select whether, in case of fault, there is a path for current to flow towards the ground node:

• `No` — The fault does not result in a connection to ground.

• ```Negative terminal side of fault node``` — The side that is in contact with the – port of the block is connected to ground.

• ```Positive terminal side of fault node``` — The side that is in contact with the + port of the block is connected to ground.

If the Open-circuit at fault node parameter is set to `Yes`, you need to specify which side (negative or positive) is connected to ground. If there is no open circuit, the two options behave similarly. Physically, this corresponds to a breakdown in the insulation between the windings and the grounded core or chassis.

#### Dependencies

To enable this parameter, add a fault to the Mutual Inductor block by clicking the Add fault hyperlink in the Mutual inductor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

If there is a ground fault, this parameter represents conductance of the current path to ground. For example, if the path to ground is through the core material, then specify a small conductance value depending on the core material being used. For highly conductive core material or for chassis-shorts, specify a higher conductance value.

#### Dependencies

To enable this parameter, add a fault to the Mutual Inductor block by clicking the Add fault hyperlink in the Mutual inductor fault parameter and set Ground fault to ```Negative terminal side of fault node``` or ```Positive terminal side of fault node```..

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Time constant associated with the transition to the faulted state, as described in Faults.

#### Dependencies

To enable this parameter, add a fault to the Mutual Inductor block by clicking the Add fault hyperlink in the Mutual inductor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Simulation time at which the block enters the faulted state.

#### Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to `Timed`.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Define the voltage threshold to a fault transition. If the voltage value exceeds this threshold a certain number of times, specified by the Number of events to fail when exceeding voltage parameter value, then the block starts entering the fault state.

#### Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to `Behavioral`.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Because the physical mechanism underlying voltage-based failures depends on one or more partial discharge events occurring, this parameter allows you to set the number of voltage overshoots that the inductor can withstand before the fault transition begins. Note that the block does not check the time spent in the overvoltage condition, only the number of transitions.

#### Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to `Behavioral`.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Current threshold to a fault transition. If the current exceeds this value for longer than the Time to fail when exceeding current parameter value, then the block enters the faulted state.

#### Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to `Behavioral`.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Maximum length of time that the current can exceed the maximum permissible value without triggering the fault.

#### Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to `Behavioral`.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

## Version History

Introduced in R2017a

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