Uniﬁed Models for Multiphase Coupled Inductors

—Circuit models for multiphase coupled inductors are summarized, compared, and uniﬁed. Multiwinding magnetic structures are classiﬁed into parallel-coupled structures and series-coupled structures. For parallel-coupled structures used for multiphase inductors, the relationships between a) inductance matrix models, b) extended cantilever models, c) magentic circuit models, d) multiwinding transformer models, e) gyrator-capacitor models, and f) inductance dual models are investigated and discussed. These models represent identical physical relationships in the multiphase coupled inductors, but emphasize different physical aspects and offer distinct design insights. The circuit duality between the series coupled structure and the parallel coupled structure is revealed. Design equations linking these models are compared and uniﬁed. The models and design equations are veriﬁed through theoretical derivation, SPICE simulation, and experimental measurements.


I. INTRODUCTION
M ULTIPHASE coupled inductors are widely used in many power electronics applications. Using multiphase coupled inductors in power converters can improve the efficiency, enhance the functionality, reduce the passive component size, avoid saturation, and improve the transient response [1]- [6]. Designing high performance power converters with multiphase coupled inductors need advanced models and tools.
There are two common ways of modeling multiphase coupled inductors: 1) Math-based models, which focus on the mathematical coupling relationships between windings. Mathbased models are derived based on the inductance matrix, reluctance matrix, or permeance matrix. Information about the core geometry, material properties, and winding structure is not explicitly included. Math-based models can be represented by a few convenient circuit models whose element values may not have explicit physical meanings, such as the classic T model, π model, star model, delta model, and extended cantilever model [7]- [11]. The element values of math-based models may be obtained by many different approaches, including finite-element modeling or experimental measurements. 2) Physics-based models represent the physical geometry of the magnetic structure more directly. Each portion of the magnetic structure is represented by a lumped circuit element, and many lumped circuit elements are combined into a complete circuit. The reluctance circuit model, the gyrator-capacitor model, the inductance dual model, and the modular multi-layer model Minjie Chen is with the Department of Electrical Engineering and Andlinger Center for Energy and the Environment at Princeton University, Princeton, NJ 08540, USA.
Charles R. Sullivan is with the Thayer School of Engineering of Dartmouth College, Hanover, NH 03755, USA.
This work was jointly supported by the National Science Foundation (Award #1847365) and the Princeton University SEAS Innovation Fund. Fig. 1. Two major categories of multiphase coupled inductor structures: 1) series coupled structure; and 2) parallel coupled structure [22]. and 2) ideal current equalizing transformer. With infinite core permeability and negligible leakage, a series coupled structure can be modeled as an ideal voltage equalizing transformer, and a parallel coupled structure can be modeled as an ideal current equalizing transformer. For the ideal current equalizing transformer, we show the core topology with dashed lines, and indicate winding polarity with arrows rather than dots, indicating the direction of current that is equal in the corresponding current equation, just as the dots indicate the direction of voltage that is equal in the corresponding voltage equation for the voltage equalizing transformer.
In their basic, linear form, all models discussed in this paper are equivalent. However, different models offer different design insights. Math-based models are well suited to theoretical analysis of power converters, especially if the magnetic components are already designed. The main advantage of physics-based models is the natural extension to capture core loss, saturation, and the details of the flux distribution in the core. Designers should choose an appropriate model based on the information that is needed in the design process. Often, moving from math-based models to physics-based models can give experienced circuit designers useful insight on circuit operation that is harder to get from math-based models. Fig. 1 shows two major categories of multiwinding magnetic structures: 1) Series Coupled Structures, in which the flux paths of multiple windings are configured in series; 2) Parallel Coupled Structures, in which the flux paths of multiple windings are configured in parallel [22]. Fig. 2 shows their circuit symbols. With zero leakage flux, the series coupled structure forces the magnetic flux Φ of all windings to be the same, leading to a bonded voltage relationship across windings. With infinite permeability, the parallel coupled structure forces the magneto-motive-force (MMF) of all windings to be the same, leading to a bonded current relationship across windings. The series-coupled structure functions as an ideal voltage equalizing transformer in which N 1 i 1 + N 2 i 2 + ... + N M i M = 0, and v1 N1 = v2 N2 = ... = v M N M ; the parallel coupled structure functions as an ideal current equalizing transformer in which N 1 i 1 = N 2 i 2 = ... = N M i M , and v1 N1 + v2 N2 + ... + v M N M = 0. The goal of this paper is to investigate theoretical frameworks that are most practical for modeling sophisticated coupled magnetic structures. The series coupled structure is well studied in the literature [23]- [25]. Its mathematical symbol, an ideal voltage equalizing transformer, is commonly supported by commercial SPICE simulation platforms. The parallel coupled structure is less commonly found in power electronics applications. Moreover, ideal multi-winding current equalizing transformers are not well supported in mainstream SPICE simulation platforms. A majority of this paper focuses on investigating models for parallel coupled structures, and shows how to use these models to analyze and simulate buck converters with multiphase coupled inductors.
The remainder of this paper is organized as follows: Section II introduces general models for arbitrary multiwinding structures. Section III simplifies these models for idealized multiwinding coupled inductors. Section IV presents models for a ladder core coupled inductors and discusses the relationship between this model and a ladder modelf for a layered multiwinding transformer. Section V unifies these models and develops calculations to convert parameters between them. Section VI applies the models to symmetric coupled inductors in multiphase buck converters. Section VII verifies the effectiveness of these models, and presents simulation and experimental results of a multiphase coupled inductor buck converter. Finally, Section VIII concludes this paper.

II. GENERALIZED MULTIWINDING STRUCTURES
The voltage and current of an arbitrary multiphase coupled inductor is described by an inductance matrix L: Here V and I are the voltage and current of the M windings. The inductance matrix is a symmetric matrix with positive and negative element values. It describes the mathematical coupling relationship between windings and is applicable to all multiwinding coupled magnetics. The element values of this matrix can be identified by finite-element modeling or 3. An example four-winding coupled inductor with a combination of series, parallel, and air-gap configurations.  experimental measurements. Fig. 3 shows an example fourwinding coupled inductor with a combination of series, parallel, and air-gap configurations. Fig. 4 shows the inductance matrix model of this example [7]- [9]. Here the self and mutual inductance values are the element values of the inductance matrix. Many SPICE simulation platforms, e.g., PSIM, LT-Spice and Simplis, support the use of the inductance matrix model. A similar model which decouples the mutual coupling relationships is the extended cantilver model [10] as shown in Fig. 5. In an extended cantilever model, all elements are non-coupled inductors and ideal transformers.
Both the inductance matrix model and the extended cantilever model are math-based models. One drawback of mathbased models is that the geometry and material property information is not explicitly shown in the model. Limited insight on magnetic structure design is offered. Physics-based models 6. Magnetic circuit model of the coupled inductor in Fig. 3.
. Fig. 7. Gyrator-capacitor model of the coupled inductor in Fig. 3.  can offer more direct insight on the relationship between the physical structure and the model, and are considered next. The magnetic circuit model (reluctance circuit model) [11]- [13] as shown in Fig. 6 is one of the most widely used physicsbased models. Each portion of the magnetic core is modeled as a reluctance. Each winding is modeled as an MMF source driving the reluctance circuit as a voltage source. The throughvariable is the magnetic flux Φ, and the across variable is the MMF, F.
One limitation of the magnetic circuit model is that it cannot be easily simulated in SPICE. In a SPICE model, the across and through variables need to be current and voltage, instead of MMF and flux. The current is linearly related to MMF (F), and the voltage to the derivative of flux (dΦ/dt). Added circuits are needed to implement the linear scaling and timederivative/integral relationships [14].
The gyrator-capacitor model differs from the magnetic circuit model by replacing the through variable Φ with its time-derivativeΦ = dΦ/dt, replacing the reluctances with capacitors whose capacitance equal to the permeance P = 1/R, and replacing the MMF sources with gyrators which convert current into voltage [15], [16]. Fig. 7 shows an example gyrator-capacitor model derived from the magnetic circuit model in Fig. 6. The two lumped circuit models share the same net structure but have different component values.
In the gyrator-capacitor model, the through variable has units of V, and the across variable has units of A. It is sometimes beneficial to apply topological duality [29]- [32] to the gyrator-capacitor model and create an inductance dual model [17]- [22], as illustrated in Fig. 8. In the inductance dual model, the elements representing the magnetic core sections are inductors with inductance values equal to the permeance values P = 1/R, in units of henry (H). The through variable is current, and the across variable is voltage. The terminals of the model are ideal transformers with turns ratios equal to the physical turns of the windings.
All models represent the same math and physics. All models have strengths and weaknesses. The main purpose of this paper is to unify the mathematical derivations and physical insights behind these models, and systematically show their similarities, connections, and differences. Fig. 9 shows an example symmetric multiwinding coupled inductor with M legs on the side and one leg in the center. This is an example of a parallel coupled multiwinding structure with a symmetric geometry. Each of the outer legs is encircled by an N -turn winding. High permeability magnetic materials are used, and a gap is used in the center leg to control the inductance and avoid saturation with balanced dc currents. Optional small gaps are sometimes used in the outer legs to avoid saturation with small imbalances in the dc currents. Fig. 10 shows the magnetic circuit model [12], [13] of this parallel structure. The model comprises M outer leg reluctances R L , and one center leg reluctance R C . Each outer leg is driven by an MMF source F = N i. Fig. 11 shows the gyrator-capacitor model [15], [16] of this structure, following the same changes in variables as with Figs. 6 and 7.

III. IDEALIZED MULTIPHASE COUPLED INDUCTORS
Finding the topological dual of the magnetic circuit or the gyrator-capacitor model results in the inductance dual model shown in Fig. 12. To capture the dc bias of the magnetic core in circuit simulations, the ideal transformers in the inductance dual model should operate in dc, i.e., the transformer current relationship should apply to dc as well as ac currents. However, this is insufficient to make the the model in Fig. 12 capture the dc bias level in each core path, because there is a loop of ideal inductors for which the dc current is undetermined. To solve this problem, the model in Fig. 13 includes a small, non-physical resistance in series with each inductor (r L ). For proper dc balancing, the resistance of each is proportional to, but orders of magnitude smaller than, the impedance of the inductor it is connected in series with, divided by the quality factor of the core material at this frequency, to ensure that the losses in this dc balancing resistor are negligible where ω S is the switching Fig. 9. A symmetric multiphase coupled inductor with many windings. This structure has M side legs and a center leg.
12. Inductance dual model of the structure in Fig. 9. 1/R C and 1/R L represent the inductive elements of the center leg and the side legs, respectively. In SPICE simulations with ideal transformers, i C and i L are linearly related to Φ C and Φ L in the center leg and the side legs, respectively.
frequency and Q is the quality factor of the core material at this frequency. With the flux, including dc components, represented by inductor currents, one can probe the currents i L (t) and i C (t), divide them by the corresponding reluctances R L and R C , and visualize the magnetic fluxes Φ L (t) and Φ C (t) in circuit simulations. Fig. 13 also shows one example way to N:1 N:1 N:1 N:1 13. Inductance dual model including saturable inductors and core-loss resistors. The series resistor r L and r C determines the dc current. The parallel resistors R L and R C capture the core loss. By probing the current in the inductance dual model and divide it with the reluctance value, the magnetic flux and saturation effects in each portion of the core can be visualized.
14. Inductance matrix model of the structure in Fig. 9. The element values of this model come from the inductance matrix.
implement core loss and saturation effects in the inductance dual model. Each portion of the magnetic core is implemented as a saturable inductor (L L or L C ). An additional resistor is then connected in parallel with the saturable inductor and resistor to capture the core loss of each portion of the magnetic core. Fig. 14 shows the inductance matrix model of a multiwinding coupled inductor. The core of this model is a M × M inductance matrix which represents the mathematical mapping relationships between windings. The inductance matrix model is interchangeable with a multiwinding transformer model, as well as with all the other models under discussion. There are many ways to implement a multiwinding transformer model. Fig. 15 shows one example implementation of the multiwinding transformer model using an ideal current equalizing transformer [6]. The magnetizing inductance of each winding is M M −1 L µ and the leakage inductance of each winding is L l . M L µ . The leakage inductance of each winding is L l . More discussion of multiwinding transformer models is provided in [33]- [39].
The inductance matrix model, the extended cantilever model, and the multiwinding transformer models are mathbased models. The magnetic circuit model, the gyratorcapacitor model, and the inductance dual model are physicsbased models. Math-based models describe the mathematical equations at the interface, from port-to-port. Physics-based models illustrate the physical behaviors of the magnetic core and the windings.

IV. LADDER CORE AND LAYERED WINDING STRUCTURES
In this section, we first consider a commonly used coupled inductor structure, the ladder core, and relax the assumption that the top and bottom plates have negligible reluctance to develop a more accurate model. Then we consider a conventional multi-winding transformer with layered windings, to illustrate the topological dual relationship between this structure and the ladder-core structure.
A. Ladder Core Structure Fig. 17 shows a parallel coupled structure implemented with a ladder core [40]- [45]. This ladder core structure is attractive in designing coupled inductors for multiphase buck converters. If the reluctance of the top and bottom bars of this structure are neglected, any of the models previously discussed apply to this structure. But unless they are very thick and/or have very high permeability, their reluctance may be significant, and the magnetic circuit model in

Rµ
, the vector of current in all windings (I) is a function of the vector of fluxes (Φ) in all legs and the permeance matrix (P): Taking the derivative of (3) results in the inductance matrix relationship connecting the winding current vector (I) and the winding voltage vector (V):

B. Layered Winding Structure
In this subsection, we consider a conventional multiwinding transformer with layered windings. These may be concentric wire-or foil-wound windings or stacked PCB windings, as shown in Fig. 21. Fig. 21 19. Gyrator-capacitor model of the ladder structure in Fig. 17. Labeled values 1/R are capacitance values.  Fig. 24 can be further extended to cover skin and proximity effects [23]- [25]. A duality is observed between the ladder core models and the layered winding models. The ladder core structure has MMF sources in parallel, and the layered winding structure has MMF sources in series. The ladder core structure requires 21. Multiwinding series coupled planar magnetic structure. Windings are series coupled with equalized voltage-per-turn and zero summed MMF.    windings to have similar MMFs, and the layered winding structure requires windings to have similar fluxes. The ladder core structure is usually used to couple windings to make their ac currents match more closely, and the layered winding structure is usually used to couple windings to achieve similar voltage per turn. The ladder core structure functions well for coupled inductors in multiphase buck converters, and the layered winding structure functions well for multiport dc-dc converters, such as multi-active-bridge (MAB) converters and energy routers [26]- [28].

V. UNIFYING MODELS FOR IDEALIZED STRUCTURES
This section unifies and compares simplified models for coupled inductors in which the top and bottom plate or bar reluctance is neglected and the model is symmetric. The equations which describe the inductance matrix model in Fig. 14 is the inductance matrix L in (1). The multiwinding transformer model in Fig. 15 and Fig. 16 correspond to (1) with L S = L µ + L l and L M = − 1 M −1 L µ . The magnetic circuit model in Fig. 10 can be described by a reluctance matrix R based on R L and R C : The equation which describes the gyrator-capacitor model and the inductance dual model is the time derivative of (5): vx. (6) Note that L = N 2 R −1 . As a result: In a highly coupled structure, R L R C , the relationships between L S , L M , L µ , L l , R L and R C become: To design a multiphase coupled inductor, it is important to be able to estimate the dc flux density in each portion of the core. In the inductance dual model, the current that goes through L C = 1/R C , i C , is related to Φ C . The current in each of the legs, i L1 , i L2 , ..., i LM is related to the flux going through the corresponding side leg of the core, Φ 1 , Φ 2 , ..., Φ M : If the model in Fig. 13 is used, then Assuming the dc currents of all winding are all equal I 1 = I 2 = ... = I M = I DC , the dc flux densities in the side legs (Φ L,DC ) and center leg (Φ C,DC ) of the magnetic core, which determines the peak flux density when the system is working at full load, are functions of M , N , R L , R C , and I DC :

VI. MULTIPHASE COUPLED INDUCTOR BUCK CONVERTER
Coupled inductors can greatly enhance the performance of multiphase buck converters [34]- [36], [40]- [49]. By coupling the multiple inductors with a high permeability magnetic core, one can significantly reduce the current ripple in each of the phase in order to reduce the conduction loss in switches, windings, and printed circuit boards. Many methods have been developed to evaluate the performance of a coupled inductor design in multiphase buck converters. Based on the inductance matrix model, [47] predicts that the current ripple ratio perphase between the uncoupled and coupled cases with the same transient inductances in a two-phase buck converter is: where α = L M L S = − R C R L +R C , and D is the duty ratio. This ripple ratio is an important figure-of-merit (FOM) for evaluating coupled inductor design. A smaller ripple ratio is better. Substituting α into Eq. (20), the ripple ratio becomes a function of R L , R C , and duty ratio D: This equation was generalized in [43] for a M -phase coupled inductor buck converter with 0 < D < 1 M by using a multiwinding transformer model. We define ρ as the ratio between L µ and L l in the multiwinding transformer model as shown in Fig. 16: The ripple ratio is a function of M , D, R L and R C : Four design parameters were defined in [6] based on the inductance matrix model for a M -phase coupled inductor buck converter with an arbitrary duty ratio D by defining an index k to indicate the number of phases that are simultaneously energized. D and k are related by k M < D < k+1 M : 1) Overall steady-state inductance (L oss ): the multiphase coupled inductor results in the same total output peak-topeak ripple current amplitude as a single discrete inductor with inductance L oss . 2) Per-phase steady-state inductance (L pss ): each phase of the multiphase coupled inductor has the same peak-topeak steady-state current ripple as with individual discrete inductors with inductance L pss ; With these design parameters, the steady-state output current ripple, the output small-signal model, per-phase current ripple, and per-phase small-signal model of the multiphase coupled inductor, working in continuous-conduction-mode, can be rapidly estimated by applying them in the standard equations for a single-phase buck converter with the same duty ratio D and the same switching frequency f . These effective inductance values are also derived in Appendix I based on the inductance dual model. L pss , L ptr , L oss , and L otr expressed as functions of R L , R C , D, M , and k are: The per-phase transient inductance L ptr is a key parameter in the design process. It is equal to the leakage inductance L l in the transformer models, which show how it functions in the circuit. We define a coefficient δ such that L oss = L ptr /δ. δ also quantifies the combined output ripple reduction compared to using a single-phase converter with an inductor value L ptr . This factor is related to the use of multiphase interleaving [50]- [52], and is not affected by coupling. Note that a singlephase converter with an inductance value L ptr is a questionable choice for a reference point, given that such a converter would have worse transient response than the coupled-inductor design with the same L ptr The overall transient inductance L otr is simply L ptr scaled by 1/M ; effectively M discrete inductors with inductance L ptr = L l connected in parallel, as can be seen from the transformer models. Table I summaries the equations for calculating these design parameters based on the inductance matrix model, inductance Here β = R C R L . This ratio is equal to the per-phase current ripple ratio ∆icp ∆inoncp . Fig. 25 plots this FOM for a range of D, M , and β. TheFOM is always between zero and one. A smaller FOM is better. If β → +∞, R C R L , the inductors are strongly coupled, the benefits of coupling increase. If β → 0, R L R C , the inductors are weakly coupled, and the benefits of coupling decrease.
In practice, β can be increased by reducing R L , by using high permeability core material, reducing length of the legs, and increasing area of the legs. R C is then adjusted to maintain the selected L ptr to meet the transient requirements while maintain a small ripple. Tradeoffs exist between core loss, saturation margin, energy storage requirements, and transient response. In an optimal design, the core loss, winding loss, efficiency, power density, and transient and steady-state performance are highly correlated and need to be jointly optimized for a given design specification.
A good strategy to design high performance coupled inductors for multiphase buck converter is: 1) Selected a magnetic structure with R L R C ; 2) Choose an appropriate per-phase transient inductance (L ptr = L l ) based on the tradeoff between transient

References
[47]- [49] [41]- [45] R C /R L = =0.01 R C /R L = =0.1 to minimize R L and adjust R C to maintain the selected L ptr , optimize the loss, and ensure enough margin to avoid saturation under balanced excitation. 5) Evaluate the flux under the expected worst-case mismatch between phase currents. If this leads to saturation, add small gaps in the outer legs as necessary to accommodate the mismatch. There is always parasitic inductance adding to L l of the coupled inductor. When the targeted value of L ptr = L l is small, the parasitic inductance outside the transformer may provide a significant fraction of the necessary leakage inductance, thus providing an opportunity to reduce the required inductance and thus reduce the energy storage required in the magnetic structure and reduce its size. In some cases, the parasitic inductance may exceed the targeted value of L l , and careful layout to reduce parasitic inductance may be needed.
The absolute value of the current ripple per-phase impacts the loss in the windings and switches.The maximum current ripple is useful for evaluating the performance of the system if D spans across a wide range. When D = 0.5, M = 1, k = 0, the effective inductance is L pss = N 2 R C +R L , leading to the maximum inductor current ripple per-phase: The per-phase current ripple ∆i phase is: The normalized per-phase current ripple (∆ I   Fig. 25 directly shows how much benefit is provided at a given operating point, whereas Fig. 26 is useful for considering a range of operating points with different duty ratios, and assessing the ripple and ripple reduction at the worst-case point over that range.

VII. MODEL VERIFICATION
A four-phase coupled inductor design was selected to verify the models discussed in this paper. This design is not optimized for a particular application. Fig. 27 shows the geometry of a SIEMENS P1814 B65561-A0400 core with N26 material. The structure has four side legs and one center leg.
The reluctances R L and R L were calculated based on simple approximations. These values, denoted as R * L and R * C , are 492,070 H −1 and 1,860,700 H −1 , respectively. Fig. 29 shows the measured inductance of the prototype following the parameter extraction method described in Appendix II using a HP/Agilent 4395A impedance analyzer. With N = 1, the measured inductance with four windings connected in parallel L otr was 25.7 nH. The measured self inductance L S was 1.54 µH. The extracted R L was 496,100 H −1 , and R C was         multiwinding transformer model, L l is 133 nH and L µ is 1.43 µH. The input voltage is 3 V, the average output current is 10 A, the output capacitance is 200 mF, the output resistance is 0.05 Ω, the switching frequency is 125 kHz, and the duty ratio D = 1/6. The extracted L l and L µ were used to calculated L oss , L pss , L otr , L ptr as listed in Table II.  For an un-coupled four-phase buck converter with 133 nH inductance per-phase, the current ripple per-phase is expected to be 25.1 A. The expected rms current is 9.59 A. With the prototype four-phase coupled inductor, the transient inductance per-phase is maintained at 133 nH, but the steady-state phase current ripple is reduced by coupling, to an expected value of 3.98 A, 15.8% of that of the uncoupled case. The expected rms current is 2.70 A, 28.1% of that of the uncoupled case. Fig. 34 shows the simulated steady-state per-phase current of the three converters. The per-phase current ripple of the converter with four small inductors was 25.1 A, and the perphase current ripple of the buck converter with the coupled inductor was 3.98 A. The simulation results match with the calculated results. Fig. 35 shows the measured waveforms of the experimental prototype. As expected, the measured perphase peak-to-peak current ripple is about 4 A.
To demonstrate the capability of the inductance dual model to include magnetic saturation, the design was modified by changing the number of turns on one of the windings to two, while leaving the others windings with one turn. This results in an asymmetric structure that saturates early with equal dc currents in the different legs. Fig. 36 shows a circuit   simulation model (implemented in Powersim) with saturable inductor models. A non-linear current-flux-linkage relationship was implemented in the saturable inductor model in circuit simulations. The unbalanced MMF leads to unbalanced current and flux distribution, and may cause saturation when the flux mismatch is significant. Fig. 37 shows the measured waveforms of the modified design with 10 A as the overall output current. Strong asymmetry was observed in phase current, as predicted by the simulation results in Fig. 38.
The saturation flux density B sat of the SIEMENS N26 material is about 390 mT at 25°C. The saturation flux Φ sat = B sat A C of the side leg was 4.388 µWb (A C = 11.25 mm 2 ). R L was 496,100 H −1 . The threshold current for the saturable inductor is about 2.17 A. Fig. 39 shows the measured current waveforms when winding #1 with two turns is saturated (i 1 ). The total output current is about 20 A. The average current per phase is about 5 A. The per-phase inductance of this winding drops significantly as the inductor current (i 1 ) approaches the saturation limit (5 A), which matches well with the simulated results in Fig. 40. Fig. 41 shows the simulated current of L L and L C which reflects the side and center leg flux (i L = Φ L R L ). The magnetic core was saturated when i L approaches 2.17 A (Φ sat R L ). The non-linear saturation effects are captured in SPICE simulations.

VIII. CONCLUSIONS
This paper unifies the lumped circuit models for multiphase coupled inductors. Models including inductance matrix models, multiwinding transformer models, extended cantilever models, magnetic circuit models, gyrator-capacitor models, and inductance dual models are summarized and compared. These models represent identical mathematical relationships in the multiphase coupled inductors, but reveal different physical fundamentals and distinct design insights. Inductance matrix models, multiwinding transformer models, and extended cantilever models are math-based models. Magnetic circuit models, gyrator-capacitor models, and inductance dual models are physics-based models. The inductance dual model is particularly useful amongst these models because it is directly built on elements in the magnetic circuit model and offers convenience in circuit simulations. Core loss and saturation effects in each portion of the core can be captured. These models and design equations are verified through theoretical derivation, SPICE simulations, and experimental measurements.

IX. ACKNOWLEDGEMENTS
This work was supported by the the National Science Foundation under Award #1847365. The authors would like to thank Dr. Yenan Chen of Princeton University for collecting the experimental data of this work during COVID-19.
APPENDIX I: PERFORMANCE PARAMETERS Several performance parameters are derived here using the inductance dual model. The results are equivalent to the expressions for the same parameters in [6], derived based on the inductance matrix model. The effective overall steady-state inductance can be used to predict the effective output current ripple of the multiphase buck converter as if the multiphase coupled inductor is one single discrete inductor (Fig. 42). In a M phase coupled inductor buck converter with V IN as the input voltage, an arbitrary duty ratio k M < D < k+1 M , and V OU T = DV IN as the output voltage, the derivative of the overall output current is the summation of the derivative of the current of all phases: for an M phase coupled buck converter with k M < D < k+1 M . Here L ptr = N 2 R L +M R C is the equivalent inductance of the side leg of the coupled inductor as if the center leg is evenly divided into M pieces. It is also the per-phase transient inductance as will be derived. Appendix II introduced a method to measure L ptr . δ = (k+1−DM )(DM −k) (1−D)DM is the normalized ripple amplitude of an uncoupled interleaved M -phase buck converter with duty ratio D (Fig. 43) [50]- [52].
For an M -phase coupled inductor buck converter operating in continous conduction mode with duty ratio D, switching period T , and output voltage V OU T , the peak-to-peak ripple of the overall output current is a simple function of L oss : B. Effective Per-Phase Steady-State Inductance (L pss ) If k M < D < k+1 M , during the 0 < t < DT period, for each phase, there are k + 1 numbers of (D − k M )T sub-periods, in which k + 1 phases have V IN − V OU T across them, and M − k − 1 phases have −V OU T across them. The per-phase current ramps up. The current ripple in each winding during the k + 1 numbers of (D − k M )T sub-periods is: During the k numbers of ( k+1 M − D)T subperiods, k phases have V IN −V OU T across them, and M −k phases have −V OU T across them. The per-phase current ramps down. The current ripple in each phase during the k numbers of ( k+1 M − D)T subperiod is: The current ripple in each phase is the summation of all the ramp up and ramp down sub-periods: ∆i ss phase = ∆i up phase + ∆i down phase = T VOUT N 2 (− We define L pss as the effective inductance of each winding in steady-state ∆i ss phase = V OU T Lpss (1 − D)T : For an M -phase coupled inductor buck converter operating in continuous conduction mode with duty ratio D, switching period T , and output voltage V OU T , the peak-to-peak current ripple of the per-phase winding current is: Note that there are M ripple cycles within a switching period T . A perturbation ∆D results in a perturbation of ( the effective overall transient inductance of a M phase coupled inductor, regardless of the duty ratio D, is: L otr is effectively connecting M discrete inductors, each of reluctance value R L + M R C in parallel.

D. Effective Per-Phase Transient Inductance (L ptr )
Since the current equally distribute among the M parallel windings, with a purtubation of ∆D in the duty ratio, the perturbation of the current in each winding is 1 M of the overall current perturbation. As a result: The effective per-phase transient inductance of a M phase coupled inductor, regardless of the duty ratio D, is: APPENDIX II: MODEL PARAMETER EXTRACTION This Appendix introduces a method to extract parameters for the inductance dual model from impedance measurements. By connecting an N -turn winding to one of the side legs and measuring the inductance, we can measure the self inductance (L S ) of one winding: By connecting all windings in parallel, we can measure the overall transient inductance (L otr ) of the coupled inductor: R L and R C can be found from the measured L S , L otr , and the known values of N and M :