https://doi.org/10.6113/JPE.2019.19.2.344
ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718
Coupled Inductor Design Method for 2-Phase Interleaved Boost Converters
Dong Liang† and Hwi-Beom Shin*
†,*Department of Electrical Engineering, Gyeongsang National University, Jinju, Korea
Abstract
To achieve high efficiency and reliability, multiphase interleaved converters with coupled inductors have been widely applied. In this paper, a coupled inductor design method for 2-phase interleaved boost converters is presented. A new area product equation is derived to select the proper core size. The wire size, number of turns and air gap length are also determined by using the proposed coupled inductor design method. Finally, the validity of the proposed coupled inductor design method is confirmed by simulation and experimental results obtained from a design example.
Key words: 2-phase interleaved boost converter, Coupled inductor
Manuscript received Jul. 9, 2018; accepted Dec. 3, 2018
Recommended for publication by Associate Editor Honnyong Cha.
†Corresponding Author: liangd-628@163.com Tel: +82-55-772-1716, Fax: +82-55-772-1719, Gyeongsang Nat’l Univ.
*Department of Electrical Eng., Gyeongsang National University, Korea
Ⅰ. INTRODUCTION
The interleaved boost converter is widely applied in electric vehicle applications, power factor correction converters and photovoltaic arrays [1]-[3]. However, applying the interleaved technique requires additional inductors according to the number of phases. Since multiple discrete inductors make up a significant percentage of the volume and increase the complexity of converters, the coupled inductor has been proposed instead of multiple discrete inductors [4]-[7]. By integrating discrete inductors into one coupled inductor, the volume, price and number of inductors are further reduced.
As mentioned in [4], with inversely coupled inductors, the efficiency of a converter can be improved by 2 % under full loads and by 10 % under light loads when compared with non-coupled inductors. The coupled inductor can improve both the steady-state and dynamic performances of VRMs. In [8], a generalized steady-state analysis of multiphase interleaved boost converters with coupled inductors was addressed. The coupled inductors can improve both the performances of the input and the inductor ripple current.
Unfortunately, although coupled inductors can improve the performances of interleaved converters, the design methods of the coupled inductors are not mentioned in [4] and [8]. In addition, the core size is mentioned without giving a selection method, which leads to other designers not knowing how to start the design.
A coupled inductor design method was mentioned in [9]. The core selection was facilitated by calculating the core area product (AP) required by the application, and relating this calculation to the APs of the available cores. However, this coupled inductor design method is not suitable for the coupled inductor used in interleaved boost converters. Fig. 1 shows two existing coupled inductor structures of 2-phase interleaved boost converters. These structures were first presented by P. L. Wong in [4]. The winding structures of the coupled inductors are symmetrical. Therefore, the number of turns of phase 1 and phase 2 are the same. Considering the fabrication of EE or EI cores, the air gaps in the three legs are set to be the same. This is the simplest structure which means the cores do not need to be milled, and the two core parts can rely on separation to prevent them from saturating. The structure of the coupled inductor with two windings presented in [9] for a multiple-output buck derived regulator is shown in Fig. 2. The two windings are wound in the center leg, which is different from the coupled inductor structure of the 2-phase interleaved boost converter shown in Fig. 1. The inductor voltages of the coupled inductor shown in Fig. 2 are in phase, which means the maximum flux of the core is the sum of the maximum fluxes during windings 1 and 2. However, for interleaved converters, the inductor voltages are phase shifted. In addition, for the coupled inductor shown in Fig. 2, one core window area contains two windings. Thus, the currents flowing through both windings are taken into account when designing the inductor. However, for the coupled inductor used in 2-phase interleaved boost converters, there is only one winding in one core window area, which means that only one phase current needs to be considered in the design. The reasons described above lead to different AP equations for the coupled inductor of the 2-phase interleaved boost converter shown in Fig. 1 and the coupled inductor of the multiple-output buck derived regulator shown in Fig. 2.
(a) |
(b) |
Fig. 2. Structure of the coupled inductor presented in [9].
Therefore, in this paper, with the coupled inductor structures shown in Fig. 1, a detailed coupled inductor design method is presented for 2-phase interleaved boost converters operated in the continuous conduction mode (CCM). A new AP equation is also derived for selecting the core size. The wire size, number of turns and air gap length are also determined by using the proposed coupled inductor design method. Finally, the validity of proposed coupled inductor design method is confirmed by simulation and experimental results.
Ⅱ. AREA PRODUCT OF THE COUPLED INDUCTORS FOR 2-PHASE INTERLEAVED BOOST CONVERTERS
The area product is the magnetic cross-sectional area times the window area. The AP method is a good strategy for selecting the core size when designing magnetic components. Since the energy handling capability of a core is related to its area product, the core selection is facilitated by calculating the core area product required by the application and relating this calculation to the APs of the available cores. The smallest available core can be selected from catalog data, where the area product exceeds the calculated value and the inductance is adjusted by the air gap length [10]. For the coupled inductors used in 2-phase interleaved boost converters, the AP equation can be derived as follows. The international system of units (SI) is used in the follow equations. However, the dimensions of the AP equation are later changed from meters to centimeters.
When the maximum allowed dc bias current is exceeded, the inductor saturates and the inductor peak current becomes extremely large, which results in a drop in efficiency and anomalous behavior. For 2-phase interleaved boost converters, the area product of the coupled inductors is calculated at the maximum inductor dc current. The case where the inductor dc current is maximum is regarded as the worst case. It occurs at the minimum input voltage, maximum output power and maximum duty cycle of the MOSFET. The maximum inductor dc current can be written as:
where Pomax is the maximum output power, Vo is the output voltage, η is the estimation efficiency of the converter, and Dmax is the maximum duty cycle of the MOSFET, which can be expressed as:
where Vgmin is the minimum input voltage.
For one core window area, the ampere-turns of one phase is equal to the current density times the conductor area of one phase, which in the worst case can be expressed as:
where Jmax is the maximum current density, Wa is one of the core window areas, Ku is the core window utilization factor, and ILrmswt is the inductor rms current in the worst case, which can be derived as:
where fs is the switching frequency, and Leq is the equivalent inductance, which was analyzed in [4] and can be summarized as:
where Ls and M are the self and mutual inductances, respectively. ρ is called coupling parameter, ρ = 1 means direct coupling, and ρ = -1 means inverse coupling.
(a) |
(b) |
In order to keep the core from saturation, the maximum flux density of the core under the worst case should be considered. For the coupled inductor structures shown in Fig. 1, the maximum flux density of the core is the maximum flux density of the outer leg (the derivation is shown in the appendix) and it can be expressed as:
where Aeo is the cross-sectional area of the outer leg, and ϕdc is the dc flux of the outer leg under the worst case, which can be obtained from the magnetic circuits shown in Fig. 3 as:
where N is the actual number of turns of one phase, and Ro is the reluctance of the outer leg, which can be expressed as:
where Rc is the reluctance of the center leg, which can be expressed as:
where lg is the air gaps in the center and outer legs, Ae is the cross-sectional area of the center leg, μ0 is the permeability of air, ∆ϕ is the peak to peak flux of the outer leg under the worst case, which can be derived from Faraday’s law as:
Using the above equations, the maximum flux density under the worst case can be rewritten as:
As mentioned in [4], the self and mutual inductances can be given as:
Substituting (12) and (13) into (11) gives:
Solving the above equation for N yields:
Substituting (15) into (3) gives:
Since Ae ≈ 2Aeo, the above equation can be rewritten as:
Finally, the above equation is solved for the area product dimension in centimeters as:
Ⅲ. PROPOSED COUPLED INDUCTOR DESIGN METHOD FOR 2-PHASE INTERLEAVED BOOST CONVERTERS
A flowchart of the procedure for the proposed coupled inductor design method for 2-phase interleaved boost converters is presented in Fig. 4.
Fig. 4. Flowchart of the coupled inductor design method procedure for 2-phase interleaved boost converters.
Step-1: Specifications given
The starting point of the procedure is the specifications of the converter system.
Step-2: Determine the self and mutual inductances
The input ripple current analyzed in [8] can be expressed as:
where Vg is the input voltage and D is the duty cycle of the MOSFET.
Since Ae ≈ 2Aeo, substituting this into (8) and (9) gives:
Substituting (20) into (12) and (13) gives:
Then under the worst case, the self and mutual inductances can be derived from (19) and (21) as:
where ∆igmax is the maximum input ripple current, which can be expressed as:
where %ripple is the input current ripple at the minimum input voltage and a full load, given in the specifications. In addition, Igmax is the maximum input dc current, which can be written as:
When Dmaxis 0.5, the input current ripple is 0 for any self-inductances or mutual inductances. The inductance matrix cannot be determined.
Step-3: Determine the wire size
The wire area Aw can be determined as:
With the calculated Aw, a proper wire size can be selected from the wire table.
Step-4: Select the initial core size
The initial core size is selected by using the AP method. In addition, (18) shows the derived AP equation as:
Step-5: Calculate the number of turns
The minimum number of turns is expressed in (15) as:
The actual number of turns N is the next integer value greater than Nmin.
Step-6: Check the window
Check the conductor area with N wires to make sure it fits the area available in the core window, which is shown as follows:
If not, the next larger core size should be selected.
Step-7: Calculate the air gap length
The air gap length can be obtained by solving equations (8), (12) and (20) as:
Ⅳ. DESIGN EXAMPLE
A design example is shown as follows to demonstrate the proposed coupled inductor design method.
Step-1: Specifications given
The specifications are given in Table I.
Step-2: Determine the self and mutual inductances
From (2), the maximum duty cycle of the MOSFET is:
Then the maximum input dc current can be calculated from (25) as:
Next, the maximum input ripple current is calculated from (24):
Finally, the self and mutual inductances can be obtained from (22) and (23) as:
Step-3: Determine the wire size
In the worst case, with the calculated self and mutual inductances, Leq can be given from (5) as:
which means the inductor rms current in the worst case is (from (4)):
Jmax is selected as 600 A / cm2. Then the wire size can be determined from (26) as:
From the wire table, a 24 AWG wire size is selected with
Aw = 0.0025 cm2.
Step-4: Select the initial core size
The maximum inductor dc current can be calculated from (1) as:
The core material is a ferrite PC40. Thus, Bmax is set as 0.3 T. In addition, Ku is selected as 0.3. Then the area product can be given from (27) as:
An EI25 core is selected with AP = 0.339 cm4, Aeo = 0.203 cm2 and Wa = 0.772 cm2.
Step-5: Calculate the number of turns
The minimum number of turns is given from (28) as:
Therefore, the actual number of turns is 68.
Step-6: Check the window
The calculated actual number of turns and the selected wire size lead to:
In addition:
Check the window with (29), 0.17 cm2 < 0.23 cm2, to make sure the EI25 core is a proper size.
Step-7: Calculate the air gap length
The air gap length can be calculated from (30) as:
Ⅴ. SIMULATION AND EXPERIMENTAL RESULTS
Fig. 5 shows the simulation model for a 2-phase interleaved boost converter with a coupled inductor structure in PSIM. The input voltage is 18 V and the output power is 48 W. The simulated inductor current with a gate signal is plotted in Fig. 6(a), and the simulated input current is illustrated in Fig. 6(b). From Fig. 6(b), the simulated input ripple current is about 0.13 A, which has a small difference from the designed value 0.137 A. This is due to the fact that the efficiency of the converter in the simulation is 100 %. The simulated flux densities of the left and right outer legs are shown in Fig. 7, the maximum flux density is about 0.3 T , which satisfies the design very well.
Fig. 5. Simulation model of a 2-phase interleaved boost converter with a coupled inductor structure in PSIM.
(a) |
(b) |
Fig. 7. Simulated flux densities of the left and right outer legs.
A prototype of a coupled inductor with 68 turns per phase is shown in Fig. 8. By adjusting the air gap length to 0.3 mm, the self and mutual inductances measured with a precision LCR meter are 393 μH and 106 μH, respectively. When compared with the calculated inductances in step-2, the self-inductance has an error of 6 μH and the mutual inductance has an error of 27 μH. This is due to the fact that the air gap length of the coupled inductor during measurements and calculations is different. In addition, the leakage fluxes of the windings leaking into the air are neglected during calculations.
Fig. 8. Prototype of a coupled inductor.
Experimental waveforms of the input and inductor currents are shown in Fig. 9 with a 48 W output power and an 18 V input voltage. The inductor current trend with a gate voltage shown in Fig. 9(a) is similar to that shown in Fig. 6(a). The input and inductor waveforms illustrated in Fig. 9(b) coincide with those sketched in Fig. 6(b). From Fig. 9(b) it can be seen that the measured input ripple current is about 0.14 A, which matches the designed input ripple current in step-2. The measured efficiency of the converter and the inductor temperature with an 18 V input voltage are plotted in Figs. 10 and 11, respectively. Under a full load, the measured efficiency is about 96.3 % and the measured inductor temperature is about 50 °C at room temperature without fan cooling. These measurements also satisfy the design.
(a) |
(b) |
Fig. 10. Measured efficiency of a converter with an 18 V input voltage.
Fig. 11. Measured inductor temperature with an 18 V input voltage.
A prototype of a coupled inductor with desired self and mutual inductances are obtained by using the proposed coupled inductor design method. The desired input current ripple is also obtained from experimental results obtained with the designed prototype. Therefore, the validity of the proposed coupled inductor design method has been confirmed.
(a) |
(b) |
(a) |
(b) |
Ⅵ. CONCLUSIONS
In this paper, a coupled inductor design method for 2-phase interleaved boost converters has been proposed. To select a proper core size, an area product equation is newly derived. At the same time, by using the proposed coupled inductor design method, the wire size, number of turns and air gap length can be determined too. Finally, a design example indicates that the proposed coupled inductor design method is valid.
Appendix
Maximum flux density of the core
With the flux directions shown in Fig. 3, flux waveforms of the center and outer legs when 0 < D < 0.5 with inverse and direct coupling are shown in Fig. 12. For the inverse coupling, Fig. 3(a) shows that the flux of the center leg is the sum of the fluxes of the two outer legs. Therefore, during the DTs period, the peak to peak flux of the center leg can be shown as:
where ∆ϕo is the peak to peak flux of the outer leg. For the direct coupling, Fig. 3(b) shows that the flux of the center leg is the difference between the fluxes of the two outer legs. Therefore, during the DTs period, the peak to peak flux of the center leg can be given as:
Fig. 13 shows flux waveforms of the center and outer legs when 0.5 < D < 1 with inverse and direct coupling. For the inverse coupling, during the (1 – D) Ts period, the peak to peak flux of the center leg can be shown as:
For the direct coupling, during the (1 – D) Ts period, the peak to peak flux of the center leg can be given as:
Hence, the maximum flux of the center leg can be expressed as:
where ϕodc is the dc flux of the outer leg. The maximum flux of the outer leg for inverse coupling and direct coupling can be written as:
From (31)-(36) it can be seen that, whether it is inversely coupled or directly coupled, the maximum flux of the center leg is always less than twice the maximum flux of the outer leg in all of the ranges of the duty cycle. This means that:
The maximum flux densities of the center leg Bcmax and the outer leg Bomax are presented as:
Since Ae ≈ 2Aeo, substituting this into (37)-(39) gives:
This means that the maximum flux density of the core Bmax is the maximum flux density of the outer leg.
Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01058167).
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Dong Liang was born in China, in 1990. He received his B.S. degree from the Qilu University of Technology, Jinan, China, in 2012; and his M.S. degree in Electrical Engineering from Gyeongsang National University, Jinju, Korea, in 2016, where he is presently working toward his Ph.D. degree. His current research interests include magnetic components design and DC-DC converters.
Hwi-Beom Shin was born in Korea, in 1958. He received his B.S. degree from Seoul National University, Seoul, Korea, in 1982; and his M.S. and Ph.D. degrees in Electrical Engineering from the Korea Advanced Institute of Science and Technology (KAIST), Seoul, Korea, in 1985 and 1992, respectively. From 1990 to 1992, he was with Hyundai Electronics Industries Co., Ltd. as a Chief Engineer. Since 1993, he has been with the Department of Electrical Engineering, Gyeongsang National University, Jinju, Korea, where he is presently working as a Professor. His current research interests include power electronics and control, electric vehicles, and industrial drives. Dr. Shin is a Member of the IEEE Power Electronics, IEEE Industry Applications, and IEEE Industrial Electronics societies.