1 Unbalanced Three-Phase LLC Resonant Converters: Analysis and Trigonometric Current Balancing Sayed Abbas Arshadi, Student Member, IEEE, Martin Ordonez, Member, IEEE, Wilson Eberle, Member, IEEE, Mohammad Ali Saket, Student Member, IEEE, Marian Craciun, Member, IEEE, Chris Botting, Member, IEEE, Abstract— Three-phase LLC resonant converters can handle very high power levels beyond the capabilities of half-bridge and full-bridge LLC topologies. Among other characteristics, three- phase LLC structures reduce output current ripple (small output filter), enable parallel power processing (low peak current), and provide good thermal distribution. However, all these key advan- tages can be severely compromised due to passive components tolerances, leading to undesired current balance issues in three- phase LLC resonant converters. Tolerances in resonant tank passive components are inevitable and lead to unequal peak currents between phases, uneven temperature distribution, and large output current ripple. This paper investigates imbalances in three-phase LLC converters and proposes a novel Trigonometric Current Balancing (TCB) technique using phasor analysis. In this strategy the required input voltage phase-angles are calculated to achieve balanced phase currents, even under severe unbalanced conditions. In some cases, the output filter current ripple is reduced to less than half. The methodology is verified with a 3 kW experimental prototype, which validates the analytical framework and effectiveness of TCB. Index Terms - Three-Phase LLC, Current Sharing, Trigonometric Current Balancing, Interleaved Resonant Converter I. I NTRODUCTION During the past decade, LLC resonant converters have gained popularity in many applications, especially where there are strict requirements on efficiency, power density, and Electromagnetic Com- patibility (EMC). The popularity is due to the unique features of the LLC resonant converter, which include Zero Voltage Switching (ZVS) for the primary switches and Zero Current Switching (ZCS) for the secondary side rectifier diodes. Soft switching allows higher frequency operation, which reduces the size of passive components and increases the power density [1]-[8]. Half-bridge and full-bridge LLC resonant converters have been studied thoroughly for low and medium power levels in the literature [9],[10]. For medium to high power levels, three-phase structures have been investigated for several converter topologies. Three-phase structures have several advantages over half-bridge and full-bridge structures at high power levels. Three phase structures can achieve better loss distribution, easier thermal management, much lower input and output current ripple, and smaller size for active and passive components and heat sinks [11]. Due to these features, the power level can be increased to several kilowatts, where the half-bridge and full-bridge structures cannot be used effectively. Many three-phase PWM converters that operate with fixed fre- quency have been proposed in the literature [12]-[16]. However, there S. A. Arshadi, M. Ordonez, W. Eberle, and M. A. Saket are with the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, V6T 1Z4, Canada (e-mail: arshadi@ece.ubc.ca; mordonez@ieee.org; wilson.eberle@ubc.ca; alisaket@ece.ubc.ca). M. Craciun and C. Botting, are with Delta-Q Technologies Corpo- ration, Burnaby, BC V5G 3H3 Canada (e-mail: mcraciun@delta-q.com; cbotting@delta-q.com). has been limited work presented on three-phase resonant converters that employ variable frequency. In [17]-[19], the principle of operation and the performance of three-phase series-parallel and three-phase LCC resonant converters were discussed. Three-phase LLC resonant converters were studied in [20], [21], and [22]. In [23], a control strategy for a three-phase LLC resonant converter, fed by a PFC stage was proposed for an Electric Vehicle (EV) battery charger application. In that paper, the output voltage regulation was realized using the PFC stage, and the three-phase LLC resonant converter operated at a constant frequency. Three-phase LLC resonant converters with integrated magnetics were presented in [24] and [25]. In all the aforementioned work, balanced current operation was assumed and no analysis was done on the behavior for unbalanced operating conditions. Interesting work on LLC has been discussed using SiC and balancing transformer [26], and a star-connected transformers with a floating star point [27]. The balancing issues were identified and a compensation approach was preliminary presented based on a basic PID controller. The development of an in-depth theoretical analysis of the converter under unbalanced conditions remains as a new research opportunity, which is addressed in the technique proposed in the following sections of this paper. The conventional approach for avoiding unbalanced currents in a multi-phase resonant converter (collection of single-phase converters) is to control the phases separately. Unfortunately, such a method cannot be applied on a three-phase resonant converter because, with different switching frequencies for the phases, interleaved modulation of the phases cannot be achieved. The idea of using three separate PFC stages to balance the circuit has been presented in [28] and [29]. In that work, PFC rectifiers (3 units) have been employed to regulate the DC input voltages of each LLC phase. While balancing can be achieved, using 3 separate PFCs or 3 pre-regulators can be cost-prohibitive for some applications. The root cause of unbalanced operation is the tolerance in Cr , Lr , and Lm values of the LLC converter. Normally, 10% change in the passive component values can be expected, leading to detrimental unbalanced currents [30]-[32]. As shown in Fig. 1, unbalanced currents result in unwanted higher output current ripple, unequally distributed loss and consequently temperature in the circuit, higher voltage and current pick values, and higher conduction losses. Due to this unbalanced characteristics, there is a challenge in using three- phase LLC resonant converter effectively. While interesting work, the limited body of literature on three-phase LLC converters demands more analytical tools to model/explain their behavior and solve fundamental problems such as current balancing. In this paper, a detailed analysis of the current-sharing charac- teristics of the three-phase LLC resonant converter is developed to predict unbalanced operation. The theory shows how the tolerance in components affect the behavior of the converter leading to current balancing issues. In addition, a Trigonometric Current Balancing technique (TCB) is proposed to mitigate the unbalanced behavior of the circuit. In this strategy the required input voltage phase-angles are calculated to achieve balanced phase currents. The switching-angles of the inverter legs are updated (shifted) based on the calculated values. The analysis tool presented in this paper can predict the behavior of the converter, prevent worse case scenarios, and determine