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## Nonlinear Inductor Characteristics

This example shows a comparison of nonlinear inductor behavior for different parameterizations. Starting with fundamental parameter values, the parameters for linear and nonlinear representations are derived. These parameters are then used in a Simscape™ model and the simulation outputs compared.

Specification of Parameters

Fundamental parameter values used as the basis for subsequent calculations:

• Permeability of free space,

• Relative permeability of core,

• Number of winding turns,

• Effective magnetic core length,

• Effective magnetic core cross-sectional area,

• Core saturation begins,

• Core fully saturated,

mu_0 = pi*4e-7;
mu_r = 3000;
Nw = 10;
le = 0.032;
Ae = 1.6e-5;
B_sat_begin = 0.75;
B_sat = 1.5;


Calculate Magnetic Flux Density and Magnetic Field Strength Data

Where:

• Magnetic flux density,

• Magnetic field strength,

Linear representation:

Nonlinear representation (including coefficient, a):

% Use linear representation to find value of H corresponding to B_sat_begin
H_sat_begin = B_sat_begin/(mu_0*mu_r);
% Rearrange nonlinear representation to calculate coefficient, a
a = atanh( B_sat_begin/B_sat )/H_sat_begin;

% Linear representation
H_linear = [-500 500];
B_linear = mu_0*mu_r*H_linear;

% Nonlinear representation
H_nonlinear = -5*H_sat_begin:H_sat_begin:5*H_sat_begin;
B_nonlinear = B_sat*tanh(a*H_nonlinear);


Display Magnetic Flux Density Versus Magnetic Field Strength

The linear and nonlinear representations can be overlaid.

figure,plot( H_linear, B_linear, H_nonlinear, B_nonlinear );
grid( 'on' );
title( 'Magnetic flux density, B, versus Magnetic field strength, H' );
xlabel( 'Magnetic field strength, H (A/m)' );
ylabel( 'Magnetic flux density, B (T)' );
legend( 'B=\mu_0 \mu_r H', 'B-H curve', 'Location', 'NorthWest' )


Calculate Magnetic Flux and Current Data

Where:

• Magnetic flux,

• Current,

Linear representation:

Nonlinear representation:

% Linear inductance
L_linear = mu_0*mu_r*Ae*Nw^2/le;

% Linear representation
I_linear = [-1.5 1.5];
phi_linear = I_linear.*L_linear/Nw;

% Nonlinear representation
I_nonlinear = le.*H_nonlinear./Nw;
phi_nonlinear = B_nonlinear.*Ae;


Display Magnetic Flux Versus Current

The linear and nonlinear representations can be overlaid.

figure, plot( I_linear, phi_linear, I_nonlinear, phi_nonlinear );
grid( 'on' );
title( 'Magnetic flux, \phi, versus current, I' );
xlabel( 'Current, I (A)' );
ylabel( 'Magnetic flux, \phi (Wb)' );
legend( '\phi=I L/N_w', '\phi-I curve', 'Location', 'NorthWest' );


Use Parameters in Simscape Model

The parameters calculated can now be used in a Simscape model. Once simulated, the model is set to create a Simscape logging variable, simlog.

modelName = 'elec_nonlinear_inductor';
open_system( modelName );
sim( modelName );


Conclusion

The state variable for all representations is magnetic flux, . Current, I, and magnetic flux, , can be obtained from the Simscape logging variable, simlog, for each representation. Overlaying the simulation results from the representations permits direct comparison.

figure, plot( ...
simlog.Linear_Inductor.inductor.i.series.values,...
simlog.Linear_Inductor.inductor.phi.series.values,...
simlog.B_vs_H.inductor.i.series.values,...
simlog.B_vs_H.inductor.phi.series.values,...
simlog.phi_vs_I.inductor.i.series.values,...
simlog.phi_vs_I.inductor.phi.series.values,...
'o'...
);
grid( 'on' );
title( 'Magnetic flux, \phi, versus current, I' );
xlabel( 'Current, I (A)' );
ylabel( 'Magnetic flux, \phi (Wb)' );
legend( 'Linear (single inductance)', 'B-H characteristic', '\phi-I characteristic', 'Location', 'NorthWest' );