Accurate estimation of high frequency magnetic core losses under non-sinusoidal excitations is critical to the design of solid-state transformers. This paper proposes a waveform transformation principle, termed the equal derivative transformation (EDT), for piecewise linear flux density waveforms. Under the same maximum flux density, an asymmetric triangular flux density wave can be transformed into a symmetric one with a different frequency in terms of the calculation of core losses, hence the introduction of the improved waveform coefficient Steinmetz equation (iWcSE). The magnetic characteristics of three different toroidal samples (nanocrystalline alloy FT-3KL, amorphous alloy 1K101, ferrite N87) under square excitation at different duty ratios and certain frequencies are measured. Core losses predicted by the model are in good agreement with the measurement.

## I. INTRODUCTION

Solid-state transformers (SSTs), or power electronic transformers (PET), are an emerging technology with a broad sphere of applications.^{1} Magnetic components play a vital role in solid-state transformers, and the accurate estimation of the core loss lie at the heart of the magnetic design, multi-objective optimization and equivalent circuit modeling of solid-state transformers. In a typical three-stage SST topology,^{2} the DC/DC stage usually adopts the dual-active-bridge (DAB) converter operated by the phase-shift pulsewidth modulation (PWM) scheme. Symmetric triangular and trapezoidal flux density waveforms are generated for the full-bridge configuration, while symmetric and asymmetric triangular flux density waveforms are generated for the half-bridge configuration.^{3} For the estimation of core losses under such flux density waveforms, there are generally four approaches: (1) the Fourier decomposition; (2) the loss separation method; (3) dynamic hysteresis models; (4) empirical formulas. The Fourier decomposition decomposes the non-sinusoidal flux density waveform into sinusoidal components, and the total loss is the superposition of the loss generated by each component found through table lookup.^{4} Since the core loss is not linearly related to the flux density, this method introduces considerable errors. The loss separation method^{5} separates the core loss into three components, i.e., the hysteresis loss, the eddy current loss and the excess loss. It is originally derived upon the homogeneous flux density assumption, thus even the aim of its extensions to high-frequency non-sinusoidal cases is the better accuracy and clearer physical quantification,^{6,7} the lack of definite separation of all core loss contributions and the more curve-fitting parameters introduced make it no more logically reasonable than other empirical methods. As to dynamic hysteresis models,^{8} they have tons of parameters to be extracted (despite the highest accuracy) and have not been widely integrated into commercial finite element analysis software due to the numerical instability they may introduce. Thus, by now, this method has limited practical use. At present, the method most widely used is the empirical formula, or a series of extensions to or improvements on the original Steinmetz equation (OSE),^{9} among which the modified Steinmetz equation (MSE),^{10} the generalized Steinmetz equation (GSE),^{11} the improved generalized Steinmetz equation (iGSE),^{12} the waveform coefficient Steinmetz equation (WcSE)^{13} and the improved Steinmetz equation (ISE)^{14} are the most well-known ones.

In this paper, we introduce an improved version of the WcSE method, denoted as iWcSE, tailored for the estimation of core losses when dealing with asymmetric triangular flux density waveforms. Magnetic characteristics are measured for three distinct toroidal samples (nanocrystalline alloy FT-3KL, amorphous alloy 1K101, ferrite N87) under square excitations with varying duty ratios and specific frequencies to validate the proposed method. Experimental results confirm the soundness and accuracy of iWcSE in scenarios beyond the conventional application of WcSE with symmetrical triangular flux density waveforms.

## II. REVIEW OF EXISTING METHODS AND IMPORTANT CONCEPT

The original Steinmetz equation takes on the form: *P* = *kf*^{α}*B*_{m}^{β}, where *k*, *α*, *β* are parameters found through curve fitting, whereas *f* and *B*_{m} are the frequency and peak amplitude of the sinusoidal flux density waveform, respectively. When extending the OSE to non-sinusoidal applications, the philosophy usually adopted is to change the parameter *k*, as done by the GSE^{11}/iGSE^{12} in which a hypothetical loss dissipation function consistent with the OSE in the sinusoidal case is assumed, or the parameters *k, α*, *β* altogether, as done by the ISE^{14} in which the changing of *α* and *β* with frequency^{15} is considered. Since the improved waveform coefficient Steinmetz equation (iWcSE) to be proposed here is deeply related to the MSE, the WcSE, and the composite waveform hypothesis (CWH), the two methods and the important concept are reviewed as below.

### A. Review of the MSE and the WcSE

#### 1. MSE

*dM*/

*dt*(domain wall movement speed) is independent of the excitation frequency

*f*

_{0}, and an equivalent frequency

*f*in the OSE so that the core loss within one period is changed from

*kf*

^{α−1}

*B*

_{m}

^{β}to

*kf*

_{eq}

^{α−1}

*B*

_{m}

^{β}.

The variation in the variable *f* of the OSE results in a corresponding alteration in *k*. Upon closer examination of Eq. (1), it can be deduced that, for any given non-sinusoidal flux density waveform, the core loss within a single period is equal to that of a sinusoidal waveform per period at a different frequency. This intriguing assumption, which has not been experimentally verified, motivates the completion of this paper.

#### 2. WcSE

In Ref. 13, the parameter *k* is augmented by a constant factor denoted as Farea, representing the ratio of the area beneath the non-sinusoidal flux density wave to that under a sinusoidal waveform with the same peak amplitude *B*_{m} and frequency *f*. This implies an underlying assumption that the core loss is directly proportional to the area enclosed by the flux density and the horizontal axis. While this assumption offers a reliable and efficient reference point for estimating core losses in symmetrical triangular flux density waveforms, it falls short when dealing with asymmetric triangular ones. Experimental evidence indicates that triangular flux density waveforms with identical areas exhibit different core losses, as demonstrated in Ref. 16.

### B. Review of the composite waveform hypothesis

In Ref. 17, the zero levels of the rectangular wave voltage are disregarded. Additionally, the rectangular wave voltage is partitioned into one positive and one negative pulse for each period. The core loss attributed to each pulse is half of the loss per period associated with the corresponding square wave, as illustrated in Fig. 1. This establishes a relationship between the core loss of symmetric trapezoidal flux density waveforms and that of symmetric triangular waveforms.

Furthermore, it is explicitly assumed in the CWH that the magnitude of the voltage pulses, as well as their duration, directly influences the loss. Implicitly, this assumption suggests that, in the case of flux density waveform derivatives, a greater absolute value for each derivative function corresponds to a greater contribution to the overall loss. The equal derivative transformation to be proposed in the following section takes this into consideration.

## III. THE IMPROVED WAVEFORM COEFFICIENT STEINMETZ EQUATION

### A. The equivalent flux density derivative

It is not hard to notice that the OSE is a specific version of a generalized loss dissipation function in the sinusoidal case, however, as the natural waveform of alternating currents, the sine wave is extremely special in that it is not only infinitely differentiable, but also has sinusoidal derivatives of all orders if phase shifts are ignored.

Suppose *B*(*t*) = *B*_{m} sin(*ωt*), then $B\u0307$(*t*) = 2*π*(*fB*_{m})sin(*ωt* + *π*/2).

It can be understood as that the influence of the flux density derivative on the loss is lumped in the parameter *γ*_{0}, while the impact of the frequency on the loss lumped in *α* − *γ*_{0}. Similarly, the effect of the flux density on the loss is lumped in *β* − *γ*_{0}.

*γ*and rewrites the OSE for non-sinusoidal flux density waveforms as follows:

*f*

_{0}is the excitation frequency,

*B*

_{m}is the peak amplitude of flux density, $B\u0307eq$ is the equivalent flux density derivative to be defined, and

*k'*,

*α'*,

*β'*,

*γ*are the parameters to be fitted by the non-sinusoidal loss data. It is important to emphasize that: (1) Different definitions of $B\u0307eq$ will lead to their different calculated values, which will result in different sets of parameters (

*k'*,

*α'*,

*β'*,

*γ*). (2) Under the same definition of $B\u0307eq$,

*α'*=

*α − γ*

_{0},

*β'*=

*β*−

*γ*

_{0}, and

*γ*=

*γ*

_{0}may not necessarily hold, i.e., the exponential constants characterizing the impact of excitation frequency, flux density, and its derivative on the core loss in non-sinusoidal flux density waveforms differ from those in the sinusoidal case.

Based on the CWH, a specific definition of $B\u0307eq$ is proposed as follows.

*w*for all values of its derivative in one period:

Figure 2 displays the asymmetric/symmetric triangular and symmetric trapezoidal flux density waveforms and the corresponding duty ratio *D* defined for each waveform. Figure 2(a) shows the asymmetric triangular waveform, Fig. 2(b) shows the symmetric triangular waveform, and Fig. 2(c) shows the symmetric trapezoidal waveform.

Table I summarizes the calculated results of $B\u0307eq$ for the three flux density waveforms in question.

Waveform . | $B\u0307eq$ . |
---|---|

Symmetric triangular waveform | 4fB_{m} |

Asymmetric triangular waveform | $fBmD1\u2212D$ |

Symmetric trapezoidal waveform | $4fBmD$ |

Waveform . | $B\u0307eq$ . |
---|---|

Symmetric triangular waveform | 4fB_{m} |

Asymmetric triangular waveform | $fBmD1\u2212D$ |

Symmetric trapezoidal waveform | $4fBmD$ |

When the frequency *f* and peak amplitude *B*_{m} are held constant, the calculated results of $B\u0307eq$ for both waveforms match that of the symmetric triangular waveform when the duty ratio of the asymmetric triangular waveform equals 0.5 or when the duty ratio of the symmetric trapezoidal waveform equals 1. This aligns with the physical scenario, therefore, the definition of $B\u0307eq$ is reasonable.

### B. The equivalent derivative transformation and the iWcSE for asymmetric triangular flux density waveform

With the equivalent flux density derivative defined, the equal derivative transformation (EDT) is proposed as follows: *for piecewise linear flux density waveforms under the same peak flux density B*_{m}*, those with the same equivalent flux density derivative* $B\u0307eq$ *have equal core loss within one* *period.*

Applying the EDT to the asymmetric triangular flux density waveform in Fig. 2(a), it can be shown that with the correct constant factor *c*, the asymmetric triangular flux density waveform can be transformed into a symmetric triangular flux density waveform as in Fig. 2(b) with the period changing from *T* to *T*/*c*. The two different flux density waveforms have equal core loss per period.

Table II summarizes the relationship between the asymmetric triangular flux density waveform and the symmetric one with the same peak amplitude at various duty ratios. For each duty ratio of the asymmetric triangular waveform, the constant factor *c* is calculated.

Duty ratio (D)
. | $B\u0307eq$ of the asymmetric triangular waveform . | $B\u0307eq$ of the symmetric triangular waveform . | Constant (c)
. |
---|---|---|---|

0.1/.9 | 100fB_{m}/9 | 4cfB_{m} | 25/9 |

0.2/.8 | 25fB_{m}/4 | 4cfB_{m} | 25/16 |

0.3/.7 | 100fB_{m}/21 | 4cfB_{m} | 25/21 |

0.4/.6 | 25fB_{m}/6 | 4cfB_{m} | 25/24 |

Duty ratio (D)
. | $B\u0307eq$ of the asymmetric triangular waveform . | $B\u0307eq$ of the symmetric triangular waveform . | Constant (c)
. |
---|---|---|---|

0.1/.9 | 100fB_{m}/9 | 4cfB_{m} | 25/9 |

0.2/.8 | 25fB_{m}/4 | 4cfB_{m} | 25/16 |

0.3/.7 | 100fB_{m}/21 | 4cfB_{m} | 25/21 |

0.4/.6 | 25fB_{m}/6 | 4cfB_{m} | 25/24 |

Based on the above reasoning, the improved waveform coefficient Steinmetz equation is derived as below.

*f*, duty ratio

*D*and peak amplitude

*B*

_{m}is denoted as

*P*, then the core loss within one period,

*P*/

*f*, is equal to the core loss per period,

*P*

_{1}/

*f*

_{1}, of the symmetric triangular flux density waveform with frequency

*f*

_{1}=

*f*/[4

*D*(1 −

*D*)] and peak amplitude

*B*

_{m}. This can be approximated as (

*π*/4)

*kf*

_{1}

^{α−1}

*B*

_{m}

^{β}by WcSE, where

*k*,

*α*,

*β*are the OSE parameters for the corresponding soft magnetic material at frequency

*f*and within a specific temperature range. Consequently, the core loss of the asymmetric triangular flux density waveform in question is:

Based on the ISE, a more accurate version of iWcSE can be attained by replacing the parameters, *α* and *β*, obtained at frequency *f*, with *α*_{1} and *β*_{1}, which are determined at frequency *f*_{1} = *f*/[4*D*(1 − *D*)].

## IV. EXPERIMENTAL VERIFICATION

Under square excitation, the magnetic characteristics of three toroidal samples (nanocrystalline alloy FT-3KL, ferrite N87, amorphous alloy 1K101) are measured using the classical B–H loop measurement method, i.e., with the secondary winding open-circuited, the loss dissipated by the equivalent inductor (i.e., the no-load loss of the transformer) is taken as the core loss.

The schematic diagram and parts of the corresponding experimental setup are shown in Fig. 3. Therein, to filter out the DC component, a capacitor *C* (10μF) is connected in series on the primary side of the transformer. The square wave voltage is induced on the secondary side by pairs of semiconductor switches (S1, S4) and (S2, S3), which alternately conduct with unequal conduction times within the duty ratio range [0.1, 0.2, …, 0.9], excluding 0.5, or with equal conduction times specifically at a duty ratio of 0.5. The measured peak amplitudes are [0.1, 0.2, …, 0.9] T, [0.04, 0.08, 0.12, 0.2] T, and [0.1, 0.2, …, 1.1] T for the nanocrystalline alloy FT-3KL, ferrite N87, and amorphous alloy 1K101, respectively. The measured frequencies for the duty ratio range [0.1, 0.2, 0.3, 0.4] are 20 kHz, 20 kHz, and 10 kHz, respectively. Specifically, at a duty ratio of 0.5, the measured frequencies range from [55.56, 31.25, 23.81, 20.83] kHz, [55.56, 31.25, 23.81, 20.83] kHz, and [27.778, 15.625, 11.905, 10.415] kHz for those same materials.

To mitigate temperature-induced errors, an electric fan is employed to cool the test sample. Additionally, approximate input voltages for the corresponding peak flux densities are predetermined through testing to expedite the experiment and minimize temperature rise. Furthermore, an infrared thermal imaging camera is used to monitor the sample’s surface temperature when the duty ratio is set to 0.1, which is known to result in the maximum core loss at a fixed frequency, allowing an assessment of temperature control effectiveness. To enhance the accuracy of the measurement, a digital oscilloscope of frequency bandwidth of 200 MHz and maximum sampling rate of 2.5 GS/s, current and voltage probes of frequency bandwidth of 100 MHz, commercial MOSFET modules CAS120M12BM2 and corresponding gate drivers are used, the top view of which are shown in Fig. 3.

Table III lists the parameters, and the maximum surface temperature at *D* = 0.1 of the three toroidal samples.

Material . | A_{e} (mm^{2})
. | l_{e} (mm)
. | ρ (g/cm^{3})
. | Maximum temperature (°C) . |
---|---|---|---|---|

FT-3KL | 91.9 | 103.8 | 7.30 | 37.3 |

N87 | 110.4 | 103.7 | 4.37 | 29 |

1K101 | 75 | 141.4 | 7.10 | 46.8 |

Material . | A_{e} (mm^{2})
. | l_{e} (mm)
. | ρ (g/cm^{3})
. | Maximum temperature (°C) . |
---|---|---|---|---|

FT-3KL | 91.9 | 103.8 | 7.30 | 37.3 |

N87 | 110.4 | 103.7 | 4.37 | 29 |

1K101 | 75 | 141.4 | 7.10 | 46.8 |

The ambient temperature is about 27°C. The core loss exhibits a consistent increase as the duty ratio decreases from 0.5 to 0.1 with frequency held fixed. Therefore, the loss results for the ferrite material, known for its sensitivity to temperature fluctuations,^{18} are considered reliable. Furthermore, the magnetic characteristics of both the nanocrystalline alloy and the amorphous alloy remain relatively unaffected within the temperature range of 27–50°C. Consequently, the core loss results for these two materials are also deemed reliable.

Figure 4 presents a comparison of the core loss per period, scaled by a factor of 20 000, for FT-3KL under different duty ratios (ranging from 0.1 to 0.4) at 20 kHz and for a duty ratio of exactly 0.5 with varying frequencies. In Fig. 5, the maximum relative error among the core losses for the three materials is displayed. As can be seen, the relative error is 14% at most for all three materials, demonstrating the acceptable accuracy of the method. From Fig. 5, the correlation between the core loss per period of the asymmetric triangular flux density waveform and that of the symmetric triangular flux density waveform at a different frequency is apparent.

Figure 6 illustrates a comparison of iWcSE with other methods when applied to asymmetric triangular flux density waveforms for nanocrystalline alloy FT-3KL. As can be seen, while there are variations in measurements across different platforms, a minor discrepancy in the coefficients fitted from sinusoidal measurements can lead to a relatively significant divergence for non-sinusoidal loss predicted by MSE. However, iWcSE closely aligns with the non-sinusoidal measurements, provided that the relationship between non-sinusoidal core loss and sinusoidal core loss as predicted by WcSE remains reliable.

## V. CONCLUSION

This paper introduces a new empirical method to estimate the core loss for asymmetric triangular flux density waveform. The accuracy of the iWcSE proposed [Eq. (6)] is contingent upon the accuracy of WcSE. Nonetheless, it serves as a valuable supplement to the existing improved versions of OSEs. Ensuring the precise estimation of core loss for an asymmetric triangular flux density waveform at *D* = 0.1 and 20 kHz, the ISE necessitates loss data up to 200 kHz, whereas iWcSE only requires data at 55.56 kHz. Furthermore, iWcSE has unveiled an intriguing relationship that could contribute to a deeper understanding of core loss. Although the discrepancies between the loss results predicted by iWcSE and the actual measurements indicate that the equivalency posited by iWcSE is not exact, it does approximate the values closely.

## ACKNOWLEDGMENTS

This work was supported in part by the National Natural Science Foundation of China, (No. 52130710, 51777055), the Funds for Creative Research Groups of Hebei Province, (No. E2020202142).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Yongjian Li**: Funding acquisition (lead); Investigation (equal); Methodology (equal); Writing – review & editing (lead). **Kangshuo Qin**: Investigation (equal); Methodology (equal); Writing – original draft (lead). **Shenghui Mu**: Writing – review & editing (equal). **Jiatong Yin**: Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.