Influence of Non-Linearity in Losses Estimation of Magnetic Components for DC-DC Converters

In this paper, the problem of estimating the core losses for inductive components is addressed. A novel methodology is applied to estimate the core losses of an inductor in a DC-DC converter in the time-domain. The methodology addresses both the non-linearity and dynamic behavior of the core magnetic material and the non-uniformity of the field distribution for the device geometry. The methodology is natively implemented using the LTSpice simulation environment and can be used to include an accurate behavioral model of the magnetic devices in a more complex lumped circuit. The methodology is compared against classic estimation techniques such as Steinmetz Equation and the improved Generalized Steinmetz Equation. The validation is performed on a practical DC-DC Buck converter, which was utilized to experimentally verify the results derived by a model suitable to estimate the inductor losses. Both simulation and experimental test confirm the accuracy of the proposed methodology. Thus, the proposed technique can be flexibly used both for direct core loss estimation and the realization of a subsystem able to simulate the realistic behavior of an inductor within a more complex lumped circuit.


Introduction
DC-DC power converters are widely used in many electrical and electronic applications. The diffusion of wide-bandgap semiconductors, characterized by fast switching transients is increasing the operating frequency of DC-DC converters allowing for higher power densities [1][2][3].
Magnetic components are the bulkiest components of power converters, and their design must be accurate to avoid excessive weights and volumes [4]. Much effort has been spent in investigating the inductor losses generated in its winding and on its core [5]. The latter can result in both non-linear and dynamic behavior, because of the saturation and magnetic hysteresis phenomena [6]. This behavior is, in general, due to the material [7]. Considering the device geometry (i.e., the magnetic core shape), additional complexity arises due to the non-uniform distribution of the magnetic induction field across the core section, and this issue has only been partially investigated [8]. Considering these factors during the design of power converters is very difficult. For this reason, some manufacturers aim at constructive solutions that make the behavior of the magnetic component as simple as possible. In addition, they are looking for solutions with uniform distribution of magnetic field [9].
By neglecting the non-uniform magnetic induction distribution and only referring to the component data sheets, the designers are guided to choose a non-optimal magnetic converter operated under different operating conditions. Conclusion and final remarks close the manuscript.

Steinmetz Equations and Improved Generalized Steinmetz Approach
Core loss estimation can be achieved through the direct application of the Steinmetz equation. This equation relates the losses to the frequency of the excitation and the intensity of the magnetic induction, and thanks to its simplicity, represents a good method to predict the core losses under sinusoidal excitation. The average power loss is given by where C m , α and β are the Steinmetz coefficients, f is the frequency of the excitation and B m is the RMS value of the core magnetic flux density [26][27][28]. The main limit of this formula is that it results in an accurate estimation only under sinusoidal induction [26,27]. This limitation makes this formula difficult to use in time-domain simulations of non-linear devices operated at distorted currents (and, thus, H-B fields). To solve this limitation, several models were proposed. One of the most promising one is that based on the improved Generalized Steinmetz Equation (iGSE) [29,30].
Here, the average core loss is computed as where B m is the peak-to-peak flux density and This methodology accounts for an arbitrarily time-varying magnetic field, and for this reason, it is suitable for inclusion in time-domain simulation of magnetic materials. However, iGSE assumes a uniform distribution of the magnetic induction inside the core. It can be noted that the iGSE technique, differently from the SE approach, allows for a time-domain estimation of the core losses and can be therefore implemented in Spice environment.

Time-Domain Core Loss with Non-Uniform Field (TDNU)
A more recent method utilized to obtain a Time-Domain Core Loss estimation under non-sinusoidal excitation was presented in [23,24]. This approach allows to estimate the core power losses by using a time-domain approach rather than one based on a frequency-domain analysis. As a result, non-linearities can be considered, and estimations can be performed even with non-sinusoidal waveforms. In addition, by estimating the instantaneous power loss p(t), it is possible to predict the core losses during both the transient and steady-state operation. In addition, this approach takes into account the non-uniform magnetic field distribution inside the magnetic core. The instantaneous core loss is derived as where and B eff represents an equivalent flux density which considers the shape and the geometry of the core and approximates the effects of the non-uniformity of the magnetic field inside the core. This method consists of matching the area formed by an equivalent elliptical loop with the original hysteresis loop starting from the standard core loss coefficients. In (5), the parameter cos ϑ is computed as where B DC represents the DC induction bias. To compute the effective magnetic flux density B eff , a parameter ∆, called "field factor", is defined. This parameter depends on the magnetic core geometry and material, and relates the effective flux density with the current through the inductor according to For a toroidal core, the field factor can be calculated as in [24] where µ= µ 0 µ r is the magnetic material permeability, N is the number of turns, and R o and R i are the outer and inner radius of the toroidal core. Note that if β = 1, the field factor simplifies as Under this condition, (7) becomes which fully describes the case with a uniform magnetic field distribution. To properly compute the power loss p(t) given by (4), the actual values of B m and B DC must be cyclically updated. Since the proposed method works in the time-domain, the wipe out rule method is used: when the derivate of the magnetic flux density dB eff /dt is zero, a maximum B max or a minimum B min value of the actual hysteresis loop is reached and, therefore, the values of B DC and B m are updated for a correct estimation of the power loss. This technique has an important improvement which is of fundamental importance to perform time-domain simulations: in iGSE technique the estimation of the core loss is based on the knowledge of the mean average value of the magnetic flux B(t), while the method described in this section uses the effective magnetic flux B eff which is estimated by using (7). Moreover, this approach can be used in a lumped elements circuit and, also, it allows the SPICE subsystem modeling the inductor to be integrated in a more complex circuit, as shown in Figure 1. A detailed description of the LTSpice circuit used to compute the time-domain core power loss can be found in [24].
In the next sections, starting from the measurements of a DC-DC Buck converter prototype, a comparison between the power core losses on the inductor using the Steinmetz Equation, the iGSE and the proposed approach is presented.  In the next sections, starting from the measurements of a DC-DC Buck converter prototype, a comparison between the power core losses on the inductor using the Steinmetz Equation, the iGSE and the proposed approach is presented.

The Case Study: A DC-DC Buck Converter
To practically evaluate the accuracy of the three different techniques, the core loss on the inductor of a DC-DC Buck converter was analyzed. The electrical circuit of the buck converter is shown in Figure 2. The KIT-CRD-3DD065P, Buck-Boost Evaluation Kit [31] was used to realize the experimental converter. The components are summarized in Table 1.

The Case Study: A DC-DC Buck Converter
To practically evaluate the accuracy of the three different techniques, the core loss on the inductor of a DC-DC Buck converter was analyzed. The electrical circuit of the buck converter is shown in Figure 2.  In the next sections, starting from the measurements of a DC-DC Buck converter prototype, a comparison between the power core losses on the inductor using the Steinmetz Equation, the iGSE and the proposed approach is presented.

The Case Study: A DC-DC Buck Converter
To practically evaluate the accuracy of the three different techniques, the core loss on the inductor of a DC-DC Buck converter was analyzed. The electrical circuit of the buck converter is shown in Figure 2. The KIT-CRD-3DD065P, Buck-Boost Evaluation Kit [31] was used to realize the experimental converter. The components are summarized in Table 1. The KIT-CRD-3DD065P, Buck-Boost Evaluation Kit [31] was used to realize the experimental converter. The components are summarized in Table 1.

Component
Description Value Different operating frequencies and duty cycles were used to understand the performance of each core loss estimation technique and perform a comparison of their results. Indeed, the proposed topology works with unidirectional behavior. This turns out in a simpler driving system but excludes the possibility to evaluate the magnetic losses of typical bidirectional topologies. However, from the point of view of the core losses, the resulting waveforms will still include both the CCM and DCM condition, resulting in a complete behavioral analysis of the phenomenon.

Inductor Characteristics
The inductor of Cree's KIT-CRD-3DD065P Buck-Boost Evaluation Kit [31] is based on a toroidal high temperature rated powdered core, which results in a CWS-1SN-12877 inductor; the core material is KoolMu [32]. The geometric characteristics of the core are summarized in Table 2 along with the winding number of turns. The anhysteretic curve of the magnetic core and the magnetic permeability are shown in Figure 3a,b, respectively.

Power MOSFETs Q1
C3M0060065K RDS(on) = 60 mΩ Body Diode D2 C3M0060065K VF = 4.8 V Output Capacitor C MAL205956479E3 47 µF Load Resistance RL HS100 1R J 10 Ω Different operating frequencies and duty cycles were used to understand the performance of each core loss estimation technique and perform a comparison of their results. Indeed, the proposed topology works with unidirectional behavior. This turns out in a simpler driving system but excludes the possibility to evaluate the magnetic losses of typical bidirectional topologies. However, from the point of view of the core losses, the resulting waveforms will still include both the CCM and DCM condition, resulting in a complete behavioral analysis of the phenomenon.

Inductor Characteristics
The inductor of Cree's KIT-CRD-3DD065P Buck-Boost Evaluation Kit [31] is based on a toroidal high temperature rated powdered core, which results in a CWS-1SN-12877 inductor; the core material is KoolMu [32]. The geometric characteristics of the core are summarized in Table 2 along with the winding number of turns. The anhysteretic curve of the magnetic core and the magnetic permeability are shown in Figure 3a,b, respectively. The core Steinmetz coefficient are summarized in Table 3. The core Steinmetz coefficient are summarized in Table 3. By using in (5) the parameter values shown in Table 2, a coefficient C αβ = 8.51 is obtained. The curve giving the magnetic core permeability µ shown in Figure 3 can be interpolated and expressed as a function of the magnetic field as follows where, n 1 = 1.7650 × 10 −17 , n 2 = −1.8125 × 10 −12 , n 3 = 2.551 × 10 −7 , n 4 = 6.379 × 10 −5 , d 1 = 1.4782 × 10 −11 , d 2 = 5.520 × 10 −7 , d 3 = 0.0017, and d 4 = 1. As shown in Figure 3b, where the black dotted trace represents the plot of (11) and the red trace shows the real magnetic permeability the interpolation given by (7) perfectly matches the data sheet plot of the permeability.

Core Loss Estimation Algorithms
The Steinmetz procedure used to compute the core loss is summarized in the block diagram shown in Figure 4. The magnetic field H is deduced from the number of turns and the average axis of the toroidal core. Then, the magnetic field density B(t) is computed from the BH curve, and its RMS value is used in (1) to estimate the power core loss.
By using in (5) the parameter values shown in Table 2, a coefficient Cαβ = 8.51 is obtained. The curve giving the magnetic core permeability µ shown in Figure 3 can be interpolated and expressed as a function of the magnetic field as follows where, n1 = 1.7650e-17, n2 = -1,8125e-12, n3 = 2.551e-7, n4 = 6.379e-5, d1 = 1.4782e-11, d2 = 5.520e-7, d3 = 0.0017, and d4 = 1. As shown in Figure 3b, where the black dotted trace represents the plot of (11) and the red trace shows the real magnetic permeability the interpolation given by (7) perfectly matches the data sheet plot of the permeability.

Core Loss Estimation Algorithms
The Steinmetz procedure used to compute the core loss is summarized in the block diagram shown in Figure 4. The magnetic field H is deduced from the number of turns and the average axis of the toroidal core. Then, the magnetic field density B(t) is computed from the BH curve, and its RMS value is used in (1) to estimate the power core loss. The core loss procedure used by the iGSE technique is shown in Figure 5. The magnetic flux density and its derivative are calculated starting from the current iL and the BH curve. The Steinmetz coefficients are used to calculate the coefficient ki according to (3) and, finally, the core loss density is computed [29]. The core loss procedure used by the iGSE technique is shown in Figure 5. The magnetic flux density and its derivative are calculated starting from the current i L and the BH curve. The Steinmetz coefficients are used to calculate the coefficient k i according to (3) and, finally, the core loss density is computed [29]. The procedure for the Time-Domain Core Loss density computed using the approach proposed in [23] and [24] is shown in Figure 6. As already discussed, the field factor is calculated from the Steinmetz coefficients and the core geometry. Then, the effective magnetic flux density is calculated from the current. Finally, the core loss density is computed by using (4). The procedure for the Time-Domain Core Loss density computed using the approach proposed in [23,24] is shown in Figure 6. As already discussed, the field factor is calculated from the Steinmetz coefficients and the core geometry. Then, the effective magnetic flux density is calculated from the current. Finally, the core loss density is computed by using (4).  The procedure for the Time-Domain Core Loss density computed using the approach proposed in [23] and [24] is shown in Figure 6. As already discussed, the field factor is calculated from the Steinmetz coefficients and the core geometry. Then, the effective magnetic flux density is calculated from the current. Finally, the core loss density is computed by using (4).

Lossy Magnetic Hysteresis Cycle Reconstruction
The induction B(t) computed by iGSE and TDNU is in phase with the inductor current I (t). According to Lenz law, the inductor voltage is in quadrature with the current, resulting in a null average power loss. This means that the B-H trajectory would present no hysteresis. This is in conflict with the actual losses that are estimated by the two methodologies. To resolve this conflict, an additional artificial current term , in phase with the inductor voltage, must be considered. This term can be determined assuming an equivalent R-L parallel circuit model such as the one shown in Figure 7.

Lossy Magnetic Hysteresis Cycle Reconstruction
The induction B(t) computed by iGSE and TDNU is in phase with the inductor current I (t). According to Lenz law, the inductor voltage v L (t) is in quadrature with the current, resulting in a null average power loss. This means that the B-H trajectory would present no hysteresis. This is in conflict with the actual losses that are estimated by the two methodologies. To resolve this conflict, an additional artificial current term i LOSS , in phase with the inductor voltage, must be considered. This term can be determined assuming an equivalent R-L parallel circuit model such as the one shown in Figure 7. The procedure for the Time-Domain Core Loss density computed using the approach proposed in [23] and [24] is shown in Figure 6. As already discussed, the field factor is calculated from the Steinmetz coefficients and the core geometry. Then, the effective magnetic flux density is calculated from the current. Finally, the core loss density is computed by using (4).

Lossy Magnetic Hysteresis Cycle Reconstruction
The induction B(t) computed by iGSE and TDNU is in phase with the inductor current I (t). According to Lenz law, the inductor voltage is in quadrature with the current, resulting in a null average power loss. This means that the B-H trajectory would present no hysteresis. This is in conflict with the actual losses that are estimated by the two methodologies. To resolve this conflict, an additional artificial current term , in phase with the inductor voltage, must be considered. This term can be determined assuming an equivalent R-L parallel circuit model such as the one shown in Figure 7.  In this circuit, the inductor is ideal, and the instantaneous power related to the core losses is absorbed by the resistive element. Expressing the quantities in a time-discrete domain, a loss current i LOSS [k] can be computed by the ratio between the computed losses p[k] and the instantaneous voltage across the inductor v L [k]. The latter can be computed from the numerical expression of the Lentz law. Although the magnetic field H is not measured in this setup, it is possible to assume that, together with the induction B, it should accommodate the instantaneous value of the losses.
where S t is the cross-section of the toroid. From the loss current, the loss-affected H field can is derived as where λ is the magnetic path length of the toroid core.

Measurements and Simulation Results
The proposed methodology is validated, both in experimental and simulated environment, through a series of different tests, aimed at assessing the consistency of the three techniques. In the first test, the inductor current used for the different loss estimation methodologies is acquired from an LTSpice simulation. In the second test, the inductor current is measured on the real DC-DC Buck converter. For both tests, the four operating conditions described in Table 4 are considered to explore different current waveforms of the inductor. These operating conditions allow to compare the core loss estimation under significative different operating conditions, considering both the CCM and DCM case. For each operating condition:

•
The RMS losses are computed with three methodologies (SE, iGSE, TDNU); • The instantaneous losses are computed with two methodologies (iGSE, TDNU); • For the experimental data, the lossy B-H curve is reconstructed with two methodologies (iGSE, TDNU). The DC-DC converter circuit model simulated by using LTSpice is shown in Figure 7. Simulations were performed as transient analysis with a minimum timestep of 10 ns to capture the high frequency non-linear dynamics of the switching components. For the same reason, parasitic inductances were added on the MOSFET, diode and output capacitor. The small timestep allowed a detailed reconstruction of the inductor current waveform, which is a critical aspect, because the current is directly related to B as shown in Figure 8, and the B is numerically differentiated, as shown in Figure 4, to compute the instantaneous power loss.  In Figures 9-12, the waveforms related to the Buck DC-DC converter operating in Case I, II, III and IV are shown, respectively. Each figure represents the inductor current, the instantaneous magnetic induction and the instantaneous losses. In Figures 9-12, the waveforms related to the Buck DC-DC converter operating in Case I, II, III and IV are shown, respectively. Each figure represents the inductor current, the instantaneous magnetic induction and the instantaneous losses. In Figures 9-12, the waveforms related to the Buck DC-DC converter operating in Case I, II, III and IV are shown, respectively. Each figure represents the inductor current, the instantaneous magnetic induction and the instantaneous losses.      Different measurements at different operating frequencies and duty cycles were performed and the test parameters are shown in Table 4. The converter input voltage was fixed to Vi = 30 V, to reproduce conditions analogous to those used in the simulation test. Figure 13 shows the experimental setup used to measure the inductor current. Different measurements at different operating frequencies and duty cycles were performed and the test parameters are shown in Table 4. The converter input voltage was fixed to V i = 30 V, to reproduce conditions analogous to those used in the simulation test. Figure 13 shows the experimental setup used to measure the inductor current. Different measurements at different operating frequencies and duty cycles were performed and the test parameters are shown in Table 4. The converter input voltage was fixed to Vi = 30 V, to reproduce conditions analogous to those used in the simulation test. Figure 13 shows the experimental setup used to measure the inductor current. In Figures 14-17, the waveforms relative to the Buck DC-DC converter operating in Case I, II, III and IV are shown, respectively. Each figure represents the inductor current, the instantaneous magnetic induction field, the instantaneous losses, and the reconstructed lossy B-H profiles. In Table 5, the average losses for the four cases are compared between the methodologies. All the measured current waveforms are very close to the results predicted by the numerical simulations. The greatest difference between the In Figures 14-17, the waveforms relative to the Buck DC-DC converter operating in Case I, II, III and IV are shown, respectively. Each figure represents the inductor current, the instantaneous magnetic induction field, the instantaneous losses, and the reconstructed lossy B-H profiles. In Table 5, the average losses for the four cases are compared between the methodologies. All the measured current waveforms are very close to the results predicted by the numerical simulations. The greatest difference between the simulated and measured data is for the Case I. This is mainly due to the highest operating frequency which increases the effect of the parasitic components. However, as shown in Table 5, the estimated core losses are consistent with those derived from the simulations. This leads to the conclusion that, even if the current waveforms might exhibit some differences, the computed core losses are not very sensible to these variations.
Energies 2021, 14, x FOR PEER REVIEW 13 of 17 simulated and measured data is for the Case I. This is mainly due to the highest operating frequency which increases the effect of the parasitic components. However, as shown in Table 5, the estimated core losses are consistent with those derived from the simulations. This leads to the conclusion that, even if the current waveforms might exhibit some differences, the computed core losses are not very sensible to these variations.        Table 6 shows the average losses determined by using the current computed through simulation for the considered cases and methodologies. Table 6 shows the core losses derived by using the experimentally measured currents. The comparison is discussed in the conclusion section.

Conclusions
In this paper, a novel methodology for core losses estimation was compared against two state-of-the-art approaches in the study of a DC-DC power converter. Core loss estimation in time-domain is difficult due to non-linear, dynamic and geometrical phenomena involving the magnetic material. Moreover, practical applications such as power converters usually involve non-sinusoidal excitations which further complicates the study. A unidirectional topology has been considered. The proposed methodology and the two comparison methodologies allowed the estimation of the core losses in the inductor of a DC-DC Buck converter by considering detailed magnetic behavior of the core. The obtained results showed that the TDNU methodology results in a core losses estimation comparable with the other methods, yet the estimations are usually slightly higher than   Table 6 shows the average losses determined by using the current computed through simulation for the considered cases and methodologies. Table 6 shows the core losses derived by using the experimentally measured currents. The comparison is discussed in the conclusion section.

Conclusions
In this paper, a novel methodology for core losses estimation was compared against two state-of-the-art approaches in the study of a DC-DC power converter. Core loss estimation in time-domain is difficult due to non-linear, dynamic and geometrical phenomena involving the magnetic material. Moreover, practical applications such as power converters usually involve non-sinusoidal excitations which further complicates the study. A unidirectional topology has been considered. The proposed methodology and the two comparison methodologies allowed the estimation of the core losses in the inductor of a DC-DC Buck converter by considering detailed magnetic behavior of the core. The obtained results showed that the TDNU methodology results in a core losses estimation comparable with the other methods, yet the estimations are usually slightly higher than others resulting from the compared methods, thanks to the ability of the TDNU to consider the field non-uniform distribution inside the core.
The proposed accurate core loss model leads to the possibility of including a real induction model inside the SPICE environment. This model is able to exhibit a consistent behavior considering core non-linearities. In fact, since the TDNU method is inherently a time-domain approach and is natively implemented in the form of a Spice circuit, it is a promising candidate to be included in larger circuit designs and can benefit from the optimized integration engines coupled with circuit simulation software.