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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, $\mu_0, \rm{H/m}$

  • Relative permeability of core, $\mu_r$

  • Number of winding turns, $N_w$

  • Effective magnetic core length, $l_e, \rm{m}$

  • Effective magnetic core cross-sectional area, $A_e, \rm{m^2}$

  • Core saturation begins, $B_{sat_{begin}}, \rm{T}$

  • Core fully saturated, $B_{sat}, \rm{T}$

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, $B, \rm{T}$

  • Magnetic field strength, $H, \rm{A/m}$

Linear representation:

  • $B = \mu_0 \mu_r H$

Nonlinear representation (including coefficient, a):

  • $B = B_{sat} \tanh(a.H)$

% 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, $\phi, \rm{Wb}$

  • Current, $I, \rm{A}$

Linear representation:

  • $L = \mu_0 \mu_r A_e N_w^2/l_e$

  • $\phi = I L/N_w$

Nonlinear representation:

  • $I = H l_e/N_w$

  • $\phi = B A_e$

% 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, $\phi$. Current, I, and magnetic flux, $\phi$, 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' );

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