Mutual inductor model with nominal inductance tolerances for each winding, operating limits, and faults
Simscape / Electronics / Passive Devices
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:
v_{1} and v_{2} are voltages across the primary and secondary winding, respectively.
L_{1} and L_{2} are inductances of the primary and secondary winding.
R_{1} and R_{2} are series resistances of the primary and secondary winding.
M is mutual inductance.
k is coefficient of coupling.
t is time.
A parallel conductance is placed across the + and – terminals of the primary and secondary windings, so that i_{L1} = i_{1} – G_{1}v_{1}, where G_{1} is the parallel conductance of the primary winding, and i_{1} is the terminal current into the primary. Similar definitions and equation apply to i_{L2}.
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.
Option | Inductance Value |
---|---|
| L |
| Uniform distribution: L · (1
– tol + 2· tol· Gaussian
distribution: L · (1 + tol · |
| L · (1 + tol ) |
| 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.
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.
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 user-defined time constant associated with the fault transition.
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
You can enable or disable these trigger mechanisms separately, or use them together if more than one trigger mechanism is required in a simulation. When more than one mechanism is enabled, the first mechanism to trigger the fault transition takes precedence. In other words, a component fails no more than once per simulation.
You can also choose whether to issue an assertion when a fault occurs by using the Reporting when a fault occurs parameter. The assertion can take the form of a warning or an error. By default, the block does not issue an assertion.
Faultable inductors often require that you use the fixed-step local solver, rather than the variable-step solver. In particular, if you model transitions to a faulted state that include short circuits, MathWorks recommends that you use the fixed-step local solver. For more information, see Making Optimal Solver Choices for Physical Simulation (Simscape).
Use the Variables section of the block interface to set the priority and initial target values for the block variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables (Simscape).
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.