Time Domain Analysis for Measuring Core Losses in Inductive Elements for Power Electronics: An Investigation Study

The article proposes a method to measure core losses in magnetic elements for medium-frequency power electronics circuits working in conditions close to those in many real industrial circuits: supplied with square-wave voltage and working in linear magnetic region. The article presents analytical considerations regarding the proposed method. The proposed method allows determining core losses in power inductor with only voltage and current measurement - no additional EMF-measuring winding is needed. The method is based on the inductor model, which allows determining losses in the core, while at the same time enables obtaining a time-domain response consistent with rectangular voltage excitation measurements. The method was tested using the real industrial inductor with a copper litz winding and low-loss core (e.g. nanocrystalline core) at various operating points.


I. INTRODUCTION
The need to precisely determine the value of losses, especially in magnetic elements, is the consequence of the tendency in the construction of power electronic systems to reduce dimensions, weight and cost, and to increase efficiency [1], [2], [3], [4], [5], [6].
The values of magnetic materials parameters given by manufacturers in datasheets usually refer to the conditions The associate editor coordinating the review of this manuscript and approving it for publication was Liu Hongchen . of sinusoidal voltage supply [7], [8].Meanwhile, in modern power electronics systems, the voltage applied to magnetic elements is often in the form of a square wave [7], [9], [10].
The most reliable results are obtained during measurements under conditions similar to those in which an element will operate [1], [4], [10], [12].To determine the losses in a magnetic element, one can use well-known methods, which are based on the measurement of electrical quantities (voltage and current) [13], [14] or calorimetric methods [15], [16], [17], [18].These methods allow the determination of total losses, but usually it is also important to divide them into the so-called core losses and copper losses.
In the case of inductors, the division can be made using the classic two-winding method or special system solutions.
The common practice is the two-winding method, which requires winding an additional winding on the core of the tested power inductor [8], [10], [19], [20], [21], [22], [23], [24].The additional winding allows the measurement of the induced electromotive force (EMF) and, consequently, the determination of the power dissipated in the core.In addition, the two-winding method is sensitive to phase discrepancy.Phase discrepancy is the difference in phase between the actual inductor current waveform and the measured waveform.This discrepancy usually occurs due to the parasitic inductance of the resistive shunt, the mismatch between the probes and the oscilloscope sampling resolution.The problem becomes especially noticeable at high frequency of waveforms.
Articles [26], [27] propose methods for canceling the reactive voltage of the tested inductor by connecting a capacitor or another inductor in series.These methods, however, require a very precise selection of the compensation capacitance or inductance value.This is not an easy task.Modifications of reactive voltage compensation methods were proposed in [2] and [28].In both cases, inductance compensation is implemented, which allows the methods to be used for excitation waveforms of any shape.
According to the method proposed in [25], an air inductor with the same winding is connected in series with the tested core inductor.Simultaneously, separately, calorimetric measurements of losses of both inductors are made and it is assumed that the difference in measured values determines the losses in the core of the tested power inductor As mentioned earlier, the additional winding for EMF measurement is necessary when measuring core losses with the two-winding method.Currently, with the huge popularity of power electronic systems, their construction usually uses the rich market offer for ready-made inductors, and these often do not allow the winding of the additional measuring winding.
Therefore, it is important to develop a method that allows determining the losses in the core and in the copper of the inductor working in the conditions similar to those in the target system, without the need to modify the inductor or the system in which it operates.
The measurement method presented in this article allows determining the losses in the core and in the copper of the inductor, without the need for changes in the power electronic system or in the inductor itself.The method uses only the measurement of the inductor voltage and the inductor current.It is necessary to use a current and voltage probe with a possibly high bandwidth and high accuracy in order to minimize the implementation error of the developed algorithm.
The method is applicable to cases where the voltage on the tested inductor is rectangular and is forced by an appropriate power electronics system, with the possibility of adjusting both the voltage amplitude and its frequency, while maintaining a constant duty cycle D = 50%.Such excitation corresponds to the operating conditions of the inductor relatively often encountered in practical power electronic systems.
The article has been divided into four parts.Chapter II contains theoretical analysis of the proposed method and basic assumption of power inductor model.Chapter III is devoted to the practical implementation of the method and contains results of core losses measurements.The article ends with a major summary and conclusions in chapter IV.

II. PRINCIPLE OF THE PROPOSED METHOD A. INDUCTOR MODEL ASSUMPTION
In most power electronic medium frequency converters (10 kHz -200 kHz), the voltage across the inductor is a square wave, resulting in the flow of current shape close to a triangle (when working in a linear region).The simplest inductor model allowing one to separate the losses in the inductor into winding losses (in copper) and losses in the core, and at the same time allowing a good representation of the current waveform in the time domain, is the model shown in Fig. 1.The model contains resistances representing the winding losses (R Win ) and the core losses (R Core ) with inductor inductance L. In addition, the model has parasitic interturn capacitance (C), which is important in the case of high frequencies and will cause the flow of significant current at a high rate of voltage change (dv/dt) at the inductor terminals.Such a model is not a universal inductor model and in many cases it will be necessary to use a more extended model for precise inductor modeling, e.g. using the Foster or Cauer models [30], [31], [32], [33], [34].In the next part of the study, the core losses were determined using the model with neglected interturn capacitance C [35] (which is permissible in the frequency range of interest [36], [37]).This model is shown in Fig. 2.
From the differential equations describing this model, in response to a fast change of voltage v(t), one can obtain expressions on the time waveforms of currents in the following form: and the measured inductor current: where: I o -the initial value of the current in the inductance.
The time constant τ is determined by the equation: where: R Core ||R Win means the parallel connection of R Core and R Win .
If the condition t ≪ τ is satisfied, then the expressions (1) and ( 2) can be expanded to the Maclaurin series around time t = 0.In practical systems this condition is fulfilled.Relations (1) and (2) will simplify current equation to the following form: If the condition t ≪ τ is met, than: Thus, equations ( 5) and ( 6) can be simplified to the form: and: In practically constructed power inductors, there is a design relationship R Core ≫ R Win .It results from the fact that the inductor's magnetic circuit must be made of a material with low losses (i.e.large R Core in the equivalent circuit), and the winding must have a relatively low resistance R Win to limit winding losses.Then the expression (9) simplifies to the form: Further analysis will be carried out with the assumption that the inductor is powered by a square wave voltage with frequency f s = 1/T s with T s ≪ τ , which is true in practice, and that the system is in a steady state, that is, for each halfperiod of the supply voltage, there is the equation: In the following half-periods, the supply voltage is respectively: Fig. 3 shows the simulation results for a 3-element inductor model (model based on Fig. 2).As can be seen, the current step I at the moment of switching the supply voltage is equal to twice the current i Core (t), the value of which can be considered constant during the supply voltage half-period.It results directly from the expression (10).The jump in the current comes only from the change in direction of the current in R Core -a jump in the current in the inductance L is not possible.

B. POWER LOSS CALCULATION
The total power of losses in the reactor is determined by the relationship: The core losses can be expressed by the following relationships: To determine this power, it can be assumed with a good approximation that the voltage is equal to EMF v(t) ≈ e(t).Consequently according to the current jump method (CJM) the core losses can be calculated: The power balance shows that:

III. PRACTICAL TESTS OF THE ALGORITHM A. VARIABLE VOLTAGE AND FREQUENCY CONVERTER
The power supply system for the inductor test is shown in Fig. 4. It consists of a buck converter and a two-level inverter in a bridge system.This power electronic system with suitable control algorithm enables the tested inductor to be supplied with rectangular voltage, with the amplitude regulated by the buck converter and the frequency resulting from the bridge control.The mismatch in the gate -source circuit of the inverter drivers is compensated by additional close-loop, which force DC bias value to 0 A (the duty cycle is 50% +/-0.5%).
An additional winding was wound on the core of the tested inductor for the sole purpose of verifying the obtained results.The waveforms of the voltage supplying an exemplary inductor, the inductor current and the back electromotive force in the additional winding obtained in this system are shown in Fig. 5.

B. ALGORITHM REALIZATION
The algorithm requires measuring inductor voltage and current waveforms.The algorithm is implemented as follows: -Zero crossing points and the period T s (Fig. 5) are determined only from the current waveform.
-The period T s is divided into two time windows (falling and rising current) and in each of them the approximation of  the current waveform is performed with the first-degree polynomial.Moreover, the dead time is determined (DT in Fig. 6), and the current waveform in the dead time is approximated by a line parallel to the time axis.
-At the moment of switching, based on the approximation of the current during the dead time and, respectively, the rising or falling current, the current jump I is determined (Fig. 6).
-The inductor current and voltage are measured, and their product for the period T s (or a finite number of complete periods) is integrated; on this basis the power of total losses P c is determined from the expression (13), -P core is determined from the equation ( 15), -P Win is determined from the equation ( 16).
The current waveform at switching (actual measurement data) and the current jump estimation are shown in Fig. 6.

C. THE LIMITATION OF THE PROPOSED METHOD
The proposed method is insensitive to the phase discrepancy error.It is possible because only the voltage V value (kept constant during the half period T) and the current jump I during the switching event are measured.The only limitation is to correctly determine the current jump: not only the value of I but also the time point, when the jump occurs.It is significant for the method to measure current with high bandwidth current sensor with low uncertainty.
Except for the errors that can be made with determining the values of voltage, current, and current jump, it is important to remember that not every inductor can be tested, as stated in Chapter II.It is necessary to have the appropriate τ time constant value so that the current waveform is linear with a visible step change.

D. EXPERIMENTAL RESULTS
With this method and using the circuit shown in Fig. 4, the power losses were measured for exemplary inductor with a FINEMET core.The selected mechanical and electrical parameters of the inductor are summarized in Table 1.
The measurement system uses a Tektronix measurement set: MDO3104 oscilloscope (with band 1 GHz and 10 million points record length with 5 GS/s sampling rate), TCP0030A current probe with 120 MHz band and TPP1000 passive probes with the 1 GHz band, which were used to measure the voltage and the back electromotive force.In the measurement system, the voltage is rectangular and as the result the current shape is triangular.As stated in [14], to obtain a loss measurement error of 0.1% with such waveforms, the bandwidth of the measurement system must be at least 5-7 times higher than the switching frequency of the signal.The operating switching frequency of the tested inductors is 50 kHz, so this condition is met.
The inductor has been tested for various values of operating parameters and selected results are presented in Figs.7 and  8.The figure shows the core losses of the inductive element, determined using two methods: 1.The proposed method, based on the equation ( 15)current jump method (CJM), 2. A method based on an additional winding, which allows measurement of EMF -Base Method (BM) [19], [20], [21], [22], [39], [40]: The most frequently emphasized problem in the measurements with the BM, which is also known as the two-winding method is the error in measuring the phase shift between the current and voltage.This so called phase discrepancy, mentioned in the introduction section of this paper, may cause significant error in measurement of the losses [2], [5], [36], [38], [39], [41], [42], [43], [44], [45], [46], [47], [48].To minimize this error, probes compensation must be applied.In the case of the Tektronix kit mentioned above, compensation is performed after connecting the probes.According to [6], the measurement results using the BM with compensated probes  are very similar to those obtained for the method in which the phase shift measurement error is of minor importance.Fig. 7 also shows the relative difference in the losses values calculated with both methods for the selected inductor.It was assumed that BM is the reference method.
Additional tests were carried out to verify the algorithm using the same inductor with different peak to peak current and constant frequency (Fig. 8).The results show that in specific range of algorithm application the calculated losses can be close to measured value.In general the method cannot be applied when the core losses are low (no visible current jump).The significant relative error for high loss is caused by low resolution and high range of current and voltage oscilloscope probes.The relative error were calculated by: δ = | P core−BM − P core−CJM | P core−BM •100% The defined value of power losses also provides information about inductor parameters (e.g. for model based on Fig. 2).The method allows calculating specific values of both resistances for precision inductor modeling.

IV. CONCLUSION
The presented CJM makes it possible to determine the losses in the inductor core in conditions similar to the operating conditions in real power electronic converters.It is important that the following condition is met: T s ≪ τ i.e. that the rectangular voltage excitation corresponds to a current close to a triangular one.In the case of relatively small core losses, an increase in the measurement error must be taken into account.It results from the difficulties in determining the moment of switching (current jump event) and the value of the current jump, which are a consequence of the phenomena occurring at the moment of switching (e.g.oscillations shown in Fig. 6).
The Table 2 summarizes all available methods for core losses calculation and measurement.In comparison to other methods, the proposed method in comparison to other methods needs only current (vs time) and voltage (only DC value) measurements (without EMF) to determine the core and winding losses (Table 2).

FIGURE 3 .
FIGURE 3. Inductor voltage and current vs time.The simulation was carried out for the model shown in Fig. 2 with exemplary parameters (shown in Fig. 2): f s = 50 kHz, L = 100 uH, R Win = 0.1 , R Core = 100 and bipolar voltage supply 100 V.

FIGURE 4 .
FIGURE 4. Power electronic circuit for generating a rectangular voltage signal for inductors testing -variable DC source and frequency adjuster (Buck converter with H inverter).

FIGURE 5 .
FIGURE 5. Measured inductor current, voltage and EMF.The inductor (L = 70 µH) operated with 12 A peak to peak current and at frequency f s = 50 kHz.

FIGURE 6 .
FIGURE 6. Measured current for L = 70 µH, peak to peak current I pp equals to 8 A, one period T s with visible current ''jump.''

FIGURE 7 . 8 FIGURE
FIGURE 7. Core losses P core vs frequency for tested power inductor.During the tests the peak to peak current was fixed to 8

TABLE 1 .
Mechanical and electrical parameters of the tested inductor.

TABLE 2 .
General comparison of loss measurement methods for power inductors.
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