Predicting Common-Mode Inductor Current Using a Time-Domain Hysteresis Model

Common-mode inductors are generally designed so that their operation is in the magnetically linear region of the core material. Very few works consider magnetic hysteresis in the core in terms of the common-mode current waveform. This work validates one such magnetic model and enhances it by including the common-mode current due to the inductor's parasitic capacitance. This work establishes the considerable impact of hysteresis and parasitic capacitance on the ability of a common-mode inductor to reduce the common-mode current. (Statement A: Approved for Release. Distribution is unlimited #2022-0227)


I. INTRODUCTION
T HE use of increasingly fast-switching semiconductor devices has increased power density but exacerbated common-mode (CM)-current-related issues including converter and nearby equipment malfunctions due to electromagnetic interference. CM mitigation often involves the use of passive components including CM inductors/chokes (CMIs). The focus of this letter is to predict the performance of CMIs prior to hardware implementation to support their selection or design.
Several works regarding CMIs are available. For instance, a design procedure for a dc-side toroid-shaped CMI is presented in [1]. The effect of both differential-mode (DM) current and CM volts-seconds across the inductor is considered in the inductor sizing. An analytical model for the impedance of a toroidal CMI over a wide frequency band is presented in [2]. A drawback in [2] is that the relative permeability is presumed to be constant at a certain frequency and, thus, is independent of current in the windings. An integrated CM and DM choke is proposed in [3]. This work is also based on the linear analysis.
A common denominator in [1], [2], and [3] is that the CMIs are assumed to be magnetically linear. Typically, the CM current through the inductor is much smaller than the DM current. Since the core flux is primarily due to the CM component, the core is large (to support windings that carry a high DM current) in comparison to the amount of current driving flux through the core. It is argued in [4] that under such circumstances, the inductor CM current waveform is strongly influenced by magnetic hysteresis. A Cauer network model to simulate nanocrystalline-based CMIs in inverter-based drive systems is developed in [5]. It is extended in [6] to compute hysteresis and eddy current loss using the inductor CM current as an input and generating the corresponding B-H trajectory to find core loss. However, typically the CM voltage, and not the CM current, is known a priori.
A model for a UR-core CMI that takes the inductor CM voltage as an input and predicts the inductor CM current is presented in [4]. This model involves the use of a time-domain hysteresis model set forth in [7] to conduct magnetic analysis. However, the work in [7] has never been validated for complex pulse width modulation (PWM) excitations such as are encountered in power converters and the work in [4] has never been experimentally validated at all. Furthermore, the work in [4] does not consider the high-frequency component of the inductor CM current associated with its parasitic capacitance.
This work addresses both deficits providing an experimentally validated approach to predicting the CM current in a situation where only the CM voltage is known. Furthermore, the importance of including hysteresis and capacitance is experimentally demonstrated. It will be shown in the results section that the error in predicting the rms CM current in a test system was reduced from 27% to 6%.
The rest of this article is organized as follows. In Section II, a representative test system is set forth. A hysteresis-based magnetic model of the CMI is detailed in Section III. The calculation of the inductor current component due to the CM capacitance of the inductor is also included in the same section. Test results that validate the model are presented in Section IV. Finally, Section V concludes this article.

II. TEST SYSTEM AND CM MITIGATION STRATEGY
A schematic of the test system considered is shown in Fig. 1. This is very similar to the system considered in [4,Fig. 15.2] except for an added CM inductor (CMI 1) on the dc side. The   ac side ends in a delta-connected resistive load bank of phase resistance R l and parasitic capacitance from each node-toground C pl . The local grounds are denoted "g s ," "g i ," and "g l " where "s," "i," and "l" denote source, inverter, and load, respectively. The reference node "0" is not connected to the rest of the system.
The power block assembly is depicted in Fig. 2. Therein, the node "s" represents the heat sink, which is left floating. The capacitances C 1 and C 2 represent the parasitic capacitances from the dc and ac rails to the heat sink, respectively. The resistor-capacitor (R 1 -C 3 -R g ) network is added to bias dc railsto-ground voltage centrally around the ground. The grounding resistor of resistance R g is very large to avoid undue power dissipation.
A CM equivalent circuit (CMEC) of the test system is shown in Fig. 3. Therein, the dc source is represented by its Thevenin equivalent circuit in the CM. The quantity v pbcm represents the CM voltage associated with the power block. The CM impedances of the CMIs are denoted Z cmi1 and Z cmi2 . The source ground-to-inverter ground CM impedance is denoted Z si and the load ground-to-inverter ground CM impedance is denoted Z li .
The idea is to create CM shorts on the ac and dc sides of the inverter to prevent any CM interaction with the source or the load, followed by the insertion of a CMI between the shorts to limit the CM current through the inverter.
To mitigate switching frequency components in the CM current, the decoupling capacitors on the dc side and the bypass capacitors on the ac side are selected so that they are CM shorts over the frequency range of interest. The resistance R g is large and creates an open circuit in the CM. Also, the CM impedance of the three-phase inductor is due to leakage and is, therefore, very small. Furthermore, the parasitic capacitances C 1 and C 2 are negligible.
This results in a simplified CMEC, as shown in Fig. 4. It is evident from the figure that the CMI can now be designed to reduce the inverter CM current i cm based solely on v pbcm . This approach eliminates the trial-and-error associated with the CMI selection and/or design by forgoing the need to measure the inductor/ inverter CM current after the fact. By virtue of this, it also prevents any potential failure of the semiconductor switches.
Given the specifications of a CMI, the CM voltage in time v pbcm can be used to predict i cm . The goal here is to predict this current for an operating point of the test system and see how it compares to the measured CM current.
There are two ways to predict i cm -1) using a model based on time-domain hysteresis in the core, and 2) using a model based on the anhysteretic curve of the core material (that is, neglecting hysteresis). This is discussed in the next section.

III. PREDICTING INDUCTOR CM CURRENT
The test system described in the last section utilized two CMIs. Since CMI1 is fundamentally a part of the dc source, our discussion will focus on CMI 2. A UR-core CMI with an interleaved winding arrangement is considered in this work. The details of this inductor are given in [4] and [8]. The magnetic equivalent circuit (MEC) of this inductor in the CM is provided in [4] and redrawn in Fig. 5. Since the inductor utilizes an interleaved winding arrangement, the flux due to DM current is negligible.
Applying KVL to the circuit of Fig. 1 yields where r is the resistance of each CMI winding, and λ u and λ l are the flux linkages associated with the upper and lower windings of the CMI, respectively. The symbol "p" is a Heaviside notation for time derivative. Using the definitions of the CM voltage and current and neglecting the resistive term because it is small yields (2) It is noted from Fig. 4 that with the appropriate selection of circuit components, v cm ≈ v pbcm . Using this relationship and invoking the relationship between flux linkage and flux yields v pbcm = pλ cm = 2NpΦ cm (3) where N is the number of turns associated with the upper/ lower winding on one core post. The next step toward determining the inductor CM current is to predict flux density along the magnetic path shown in Fig. 5. To this end, using the definition of flux, the flux time derivative of the density in the core post and base can be expressed where x = {'p', 'b'} for post and base, respectively, A stands for the core area of the cross section, and N is the number of turns associated with the upper/ lower winding on one core post. Next, the flux density B x can be approximated by integrating (4) with time over one fundamental time period and subtracting the dc offset. Following this, the MMF drop along the magnetic path is approximated. The mean field intensity may be calculated from the anhysteretic model as where µ B () is a function with the absolute anhysteretic permeability as a function of flux-density as described in [9] and M is a structure of material parameters. Alternately, the impact of hysteresis may be included by using the extended Jiles-Atherton (EJA) model [7] H x = f EJA (t, B x , pB x , M).
The corresponding MMF drops are written as where l is the length of the reluctance associated with the core post or base and x is as noted before in (4). The MMF drop in the air gap can be expressed where R g is the reluctance of the air gap. The anhysteretic and hysteretic inductor CM current waveforms i cm,an and i cm,h can be written as The current component given by (10) or (11) is associated with the magnetic path of the CMI. There is also a parallel parasiticcapacitance-based path [8]. This component can be modeled as where C cm is the CM capacitance of the inductor. The total inductor CM current can then be expressed as the sum of the magnetic component given by (10) or (11) and the capacitive component given by (12).
To summarize, given the CM voltage source v pbcm , the first step is to predict the flux density in different parts of the core using (4) and numerically integrating it with time. The next step is to predict the corresponding field intensity using (5) and (6), followed by determining the MMF drops along the magnetic path using (8) and (9). The inductor CM current is then determined by summing the two components given by (11) and (12). Now that the procedure to predict the currents has been set forth, these currents are predicted for a given v pbcm with the test system under operation. These results are presented in the next section.

IV. RESULTS
The parameters of the test system shown in Fig. 1 are as follows: C ac = 1.1 µF, C d = 0.1 µF, R 1 = 100 kΩ, R g = 1 MΩ, C 3 = 150 µF, and C 4 = 1760 µF. The resistance of each phase resistor in the load bank R l is approximately 6.5 Ω and the measured value of C pl = 0.53 nF.
The CMI2 used here is referred to as D10 in [8] and its specifications are mentioned in [8, Table I (Column 5), Table II (Column 5)]. The magnetic core used was MN60LL, the material parameters of which are listed in [7] and [9].
The system was run with the inverter employing a sinetriangle modulation scheme. The switching frequency of the power block was set at 17 kHz. Furthermore, v src = 436.5 V, f sw = 17 kHz, duty cycle amplitude d = 92%, and load lineto-line voltage v ll = v de = 230.06 V rms.
The voltages of points u2, l2, a, b, and c with respect to g i , and the currents i u and i l were measured. The measured inductor CM current and power block CM voltage were then approximated as follows: In (14), v xgi , x = {'a', 'b', 'c', 'u2', 'l2'} represents the voltage of the point x with respect to g i . The power block CM voltage for the mentioned operating point is shown in Fig. 6.
The inductor CM current is determined as described in the last section. The component of this current associated with the magnetic path (i.e., without capacitive effects) is compared with the measured CM current in Fig. 7. The measured CM current was band-limited to the frequency range of 100 Hz-200 kHz to focus on the noncapacitive effects. The comparison is shown in Fig. 7. Observe there is a low-frequency component (∼924 Hz) in the CM current over which the higher-frequency behavior is superimposed. It is evident from the figure that the current predicted using the hysteresis model closely matches the measured current, which is significantly underestimated by the anhysteretic model.
The measured and predicted total CM currents (including both magnetic and capacitive components) are depicted in Fig. 8. A time offset of 50 µs is introduced between the compared waveforms to distinguish the sharp peaks due to the CM capacitance. Note that the predicted CM capacitance of 57.1 pF was used to determine the capacitive component of the current as opposed to the measured capacitance of 54.6 pF. This way, the predicted result only relies on material and geometrical data.
The percentage error in predicting the inductor rms CM current for the anhysteretic and hysteretic models was seen to be 27.0% and 5.92%, respectively.

V. CONCLUSION
This work validates the use of an EJA-based CMI model for PWM excitation and, in particular, for predicting the CM current through a CMI. The EJA model has never previously been experimentally demonstrated for PWM waveforms. The proposed approach provides a much better estimate of the CM current than an anhysteretic model. This establishes the importance of hysteresis in CMI design. The impact of parasitic capacitance on the CM current was also considered and found significant.