State-Plane Trajectory-Based Duty Control of a Resonant Bidirectional DC/DC Converter with Balanced Capacitors Stress

Abstract

This paper presents the design, analysis, and control of a dual transformer-based bidirectional DC/DC resonant converter featuring balanced voltage stress across all the resonant capacitors. Compared to existing topologies, the proposed converter has a dual-rectifier structure on the secondary side, which allows operation over a wide load range with balanced voltage stress across all resonant components. The transformer stress is greatly reduced by employing two small transformers, thus greatly lowering thermal as well electrical stresses on the transformers’ windings. Furthermore, by operating the primary-side interleaved converter at a fixed 50% duty, input current ripples are significantly reduced. The proposed controller consists of a feedforward control part for effective system uncertainty compensation and a feedback control part for the convergence of system error dynamics. Notably, state-plane trajectory theory is employed to derive accurate feedforward compensation terms. Additionally, the effect of resonant elements’ parameter mismatch is analyzed in detail. The designed controller was implemented using the TI TMS320F28377D DSP on a 3.3 kW prototype hardware board. Detailed experimental investigations under tough, practical operating conditions corroborate an effective bidirectional power transfer operation with a balanced voltage stress distribution in each resonant element.

1. Introduction

Electric vehicles (EVs) are considered indispensable to meeting future decarbonization-related goals. In 2022, EVs accounted for 13% of all vehicles sold worldwide. Continuing this pace, EVs can become the main means of transportation as early as 2035 [1,2]. To support this technological and industrial revolution, EV charging systems have a major role to play in its wider adoption by offering fast, efficient, reliable, and safe charging operations. In particular, charging a battery EV (BEV) with an on-board charger (OBC) can be unidirectional or bidirectional, with the former being the most commonly used today. However, the trend is gradually shifting toward bidirectional OBCs where excess energy in a large EV battery (typically rated power > 50 kW) can be leveraged to support the grid (V2G: vehicle to grid), a load (V2L: vehicle to load), or home (V2H: vehicle to home), during either normal or emergency situations [3]. Note that operation over a wide voltage and power range is imperative in bidirectional OBCs because the battery voltage varies (60~100%) with the underlying change in its state of charge (SOC) [3,4].

Among others, the dual active bridge (DAB) [5] is a commonly adopted topology and consists of two active bridges, an isolation transformer, and a large leakage inductor. It should be noted that the bidirectional DAB topology features improved efficiency, large power handling capability, and the possibility of buck and boost operation [6].

Typically, power is transferred by controlling the phase shift (PS) between two full bridges (FBs) on the primary and secondary sides, whereas the pulse width (i.e., duty ratio) is usually held constant at 0.5 (i.e., half the switching period (T_sw)) [7,8]. The aforementioned approach is often referred to as single PS (SPS) modulation and presents useful advantages such as simple control, full range soft-switching operation (i.e., zero voltage switching (ZVS)), low peak currents, and less reactive power losses [8,9]. However, under input voltage variations, the SPS-based DAB topology suffers from a reduced soft-switching range, which eventually leads to a lower conversion efficiency [9]. To address the SPS issues, extended PS (EPS) modulation [10] has been introduced, which, in addition to controlling the phase shift, employs the duty ratio of primary-side FB as an additional control variable. Likewise, further control variables (e.g., phase shift control of primary as well as secondary-side FBs [11] and duty of the secondary-side FB [12]) add more degrees of freedom (DOF). Enabling more DOF helps to achieve additional optimization goals, such as minimizing losses due to reactive, conduction, and non-active currents and extending the overall ZVS range [11,12,13]. These control modifications help to leverage the benefits of the DAB topology; however, this comes at the cost of complex converter modeling, intricate control algorithm, heavy computational burden, and increased dependence on the converter parameters. In [14], an accurate model for triple-phase shift (TPS) is derived along with a global optimal optimization condition that helps minimize the current magnitudes for operation over the whole load range. Likewise, in [15] a hybrid control strategy is designed which employs TPS and asymmetric phase shift (PS) control. The controller achieves a lower magnitude of root mean square (rms) currents and an increased soft-switching range operation, but the optimization interval should be determined beforehand, which negatively affects the algorithm implementation cost.

Lately, several structural modifications [16,17,18,19] have been investigated to improve the bidirectional DAB operation (i.e., wide voltage range, lower cost, reduced thermal stresses, etc.). First, the integration of resonant elements on both sides (e.g., CLLC [16], CLLLC [17], etc.) helps to realize soft switching for the whole load range under bidirectional power flow. Nevertheless, large capacitor and small inductor values in secondary resonance are greatly influenced by the parasitic circuit parameters.

Next, in [18], the large isolation transformer has been disintegrated into two small transformers, which results in the added advantage of more effective thermal and voltage stress distribution, in addition to a wide voltage range operation. Similarly, a wide voltage and power range operation has been achieved in [19], whereas the cost of passive components is greatly reduced. Despite the advantages, the efficiency during the buck mode operation deteriorates under light load conditions. In order to improve the light load operation, a new asymmetric modulation method is presented [20], which enables stable buck-boost operation over a wide voltage range and reduced circulating currents. Although the topology presented depicts reasonable efficiency, the overall density and cost are greatly affected due to the increased number of devices. Then, in [21], a converter with a reduced number of components has been analyzed using a segmented control structure. Regardless of the full soft-switching range, circulating power losses are inevitable with this topology. In [22], a hybrid LLC resonant converter improves the efficiency at light load and low voltage by alternating between full-bridge and half-bridge structures. The integration of the leakage inductance in the transformer further improves the power density; however, the converter performance cannot be verified in backward operation.

In the case of bidirectional operation, the current drawn from the battery is desired to have minimal ripples to maintain its condition and extend the overall lifetime [23]. Unfortunately, most existing bidirectional DAB topologies [5,7,19,20,21,22,23,24] suffer from high battery current ripples during reverse power flow. In [25], an interleaved bidirectional DAB converter is investigated, which reduces the input current ripples to some extent; nevertheless, the converter suffers from reduced efficiency in the low-gain region. Additionally, due to the phase shift control, an unsymmetrical current is drawn, which inadvertently contributes to the increase in current ripples. It should be noted that unsymmetrical operation in a bidirectional DC/DC converter generates unequal voltage stress across all active and passive components. Consequently, unbalanced heat is distributed throughout the converter, which greatly compromises the reliability of the active and passive circuit components. Therefore, for bidirectional EV charging, it is desirable that the DC/DC converter has wide voltage regulation, minimal input current ripples, and balanced voltage stress across all components to ensure more efficient and reliable device utilization.

This article investigates the design, analysis, and control of a novel bidirectional DC/DC resonant converter. The main contributions can be summarized as follows:

• An in-depth design is presented of the novel bidirectional dual transformer-based resonant DC/DC converter featuring equivalent voltage stress distribution of the resonant capacitors. As a result, the designed approach enhances the efficiency of the DC/DC power conversion along with improved overall reliability with a lower active components count.
• A state-plane trajectory theory-based control law has been derived for proper control of the bidirectional DC/DC resonant converter. The proposed controller consists of two control terms, i.e., a feedforward term and a feedback term. Notably, a feedforward term is obtained using the state-plane trajectory theory for effectively mitigating the resonant converter nonlinearities.
• The introduction of a fixed 50% control duty cycle on the primary-side results in theoretically zero input current ripples. Consequently, the harmonics contents of the primary-side current are considerably diminished, providing significant benefits for applications involving battery charging.
• The proposed design and control strategy effectively address the variations in the leakage inductance of the two transformers. A comprehensive discussion is provided on the implications of these variations on the converter topology for different transformer connections. Furthermore, the designed control law is described, highlighting how it effectively compensates for these variations.

All theoretical findings have been validated through extensive experimentations on a laboratory-designed 3.3 kW converter prototype, whereas Texas Instruments (TI) digital signal processor (DSP) TMS320F28377D is employed for control algorithm implementation.

2. Topology Description and Detailed Operational Mode Analysis

2.1. Proposed Topology Description and Detailed Operational Mode Analysis

Figure 1 presents the circuit diagrams of conventional (Figure 1a) [24] and the proposed (Figure 1b) bidirectional DC/DC converters. The definitions of the notations used in this paper are as follows: L₁ and L₂ represent the interleaved inductances; S₁~S₄ are the primary-side switches; C_DC is the DC-link capacitor; T₁ and T₂ are the isolated transformers with turns ratio (n_1,2) N_s1/N_p1; (L_m1 and L_m2) and (L_lk1 and L_lk2) are the magnetizing and leakage inductances; L_r and C_r1~C_r4 are the resonant inductance and capacitances, respectively; S₅ and S₆ are the secondary-side switches; C_S is the secondary-side capacitor; (V_P, i_P), (V_S, i_S) are, respectively, primary and secondary-side voltages and currents; i_Lr1 and i_Lr2 are the resonant currents; and T_sw is the switching period.

The switches S₁~S₄ in the conventional converter (Figure 1a) operate with different control duties for output voltage/power regulation. As a result, this leads to higher ripples in the primary-side currents, less effective device utilization, and unbalanced voltage stress/heat distribution in S₁~S₁₀. Contrary to the conventional topology [24], the switches in leg 1 and leg 2 of the primary-side FB operate in a complementary manner. By employing a fixed 50% duty, a perfect interleaving between i_L1 and i_L2 can be achieved, which helps with realizing an almost ripple-free input current (i_P).

As can be observed from Figure 1a, the conventional bidirectional converter [24] employs more active power devices (i.e., 10 switches), which negatively affects the cost as well as the conversion efficiency. Compared to the traditional converter [24], the proposed structure (Figure 1b) considers lower active switches (i.e., only 6) while ensuring balanced voltage stress distribution across each resonant capacitor (C_r1~C_r4), all while having increased power processing capability. As a result, a balanced power flow can be ensured through each resonant tank, i.e., (L_r1, C_r1~2) and (L_r2, C_r3~4). The designed structure and feedforward control term help achieve equal voltage stress distribution across S₅, S₆, and C_r1~C_r4 and maintain the output voltage constant. The overall control law consists of the proposed feedforward duty and proportional-integral (PI) feedback control duty terms for the output voltage (V_S) regulation under both dynamic and steady-state conditions. Note that series resonance on the secondary side helps accomplish soft-switching operation (i.e., zero current switching (ZCS) and ZVS). To simplify the converter analysis, the following commonly used assumptions [26] are considered.

• The output capacitor (C_S) and clamp capacitor (C_DC) are significantly larger than the resonant capacitors (C_r1~C_r4).
• The transformers (T₁ and T₂) consist of magnetizing (L_m1, L_m2) and leakage inductances (L_lk1, L_lk2).
• A negligibly small dead time (T_dead) is considered.
• The resonant capacitance (C_r) consists of two identical capacitors (i.e., C_r = C_r1 + C_r2), whereas C_r1~C_r4 are identical.

2.2. Converter Analysis under Different Operation Modes

A fixed duty operation (i.e., 0.5 × T_sw) on the primary-side full bridge (FB) helps with achieving full interleaved operation with near-zero input current ripples. Here, the voltage-second balance in the two input inductors L₁ and L₂ (where L₁ = L₂) is given as

$$V_P\times \frac{1}{2}{T}_{sw}+\left(V_P-v_{Cdc}\right)\times \frac{1}{2}{T}_{sw}=0.$$

(1)

Rearranging (1) gives the DC-link voltage (v_Cdc) as

$$v_{Cdc}=2V_P(t).$$

(2)

Next, the input voltage (v_ab) for T₁ and T₂ can be modeled using v_Cdc as

$$v_{ab}=v_{Cdc}\times f_{ab}.$$

(3)

where f_ab = 1 for {S_1,4 = on and S_2,3 = off} and f_ab = −1 for {S_1,4 = off and S_2,3 = on}.

Figure 2 indicates the current paths of the proposed topology at different intervals, whereas Figure 3 depicts the important waveforms during Intervals 1~8, whereas Figure 4 presents the state-plane trajectory. To simplify circuit analysis, the converter is first analyzed with balanced, resonant inductance values (i.e., L_r1 = L_r2 = L_r1,2).

• Interval 1 [t₀–t₁]: During the first interval, S₁ and S₄ are switched on, and v_ab in (3) is applied across the primary windings of T₁ and T₂. At time t = t₀, both inductors (L_r1,2) are charged with resonant currents (i_Lr1 = i_Lr2 = i_Lr1,2). The state equations can be modeled as

$$L_{r1,2}\frac{d{i}_L{}_{r1,\hspace{0.17em}2}\left(t\right)}{dt}=-n{v}_{Cdc}-v_{Cr1,3}\left(t\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\because i_L{}_{r1,\hspace{0.17em}2}\left(t_0\right)=0$$

(4)

where v_Cr1,3 is the voltage across C_r1 and C_r3. The inductors currents i_Lr1,2 can be expressed as

$$\begin{array}{ll}{i}_L{}_{r1,\hspace{0.17em}2}\left(t\right)& =C_{r1,3}\frac{d{v}_{Cr1,3}}{dt}-C_{r2,4}\frac{d{v}_{Cr2,4}}{dt}\\ & =2C_r\frac{d{v}_{Cr1}}{dt}\end{array}$$

(5)

whereas v_Cr1(t₀) = V_S/2 + Δv_Cr1. Since V_S = v_Cr1 + v_Cr2 = v_Cr3 + v_Cr4, (5) can be rewritten as

$$i_L{}_{r1,\hspace{0.17em}2}\left(t\right)=C_{r1,3}\frac{d{v}_{Cr1,3}}{dt}-C_{r2,4}\frac{d\left(V_S-v_{Cr1,3}\left(t\right)\right)}{dt}.$$

(6)

Solving (4)–(6), i_Lr1,2(t) and v_Cr1,3 can be derived as

$$i_{Lr1,2}\left(t\right)=\left(\frac{r_{1,2}}{z_{r1,2}}\right)\sin \left(-{\omega }_{r1,2}\left(t-t_o\right)\right)$$

(7)

$$v_{Cr1,3}\left(t\right)=-n{v}_{Cdc}+\frac{V_S}{2}-r_{1,2}\cos \left({\omega }_{r1,2}\left(t-t_o\right)\right)$$

(8)

where r_1,2 (= nv_Cdc + V_S/2 + Δv_Cr1,2) are the radii of state-plane trajectory, z_r1,2 (= √[(L_r1,2)/(C_r1,3 + C_r2,4)] are the characteristic impedances, ω_r1,2 = 1/(√L_r1,2(C_r1,3 + C_r2,4)), and the closed path is centered at ((nV_P + V_S), 0). Interval 1 ends at t₁ (=t₀ + D_s1,2T_sw), whereas the trajectory moves from position x₁ (at t₀) to y₁ (at t₁) on the state-plane diagram (Figure 4).

• Interval 2 [t₁–t₂]: At t = t₁, S₅ turns off while T₁ and T₂ continue to output −nv_Cdc for the rest of the half cycle (i.e., 0.5 × T_sw). At the same time, i_Lr1,2 start flowing through the body diode of S₆. The energy stored in L_r1,2 is released during t = [t₁–t₂], and the trajectory moves from point y₁ (at t = t₁) to x₂ (at t = t₂). The state equations can be expressed as

$$\begin{array}{l}{L}_{r1,2}\frac{d{i}_L{}_{r1,2}\left(t\right)}{dt}=-v_{Cr1,3}\left(t\right)+V_S-n{v}_{Cdc}\\ i_L{}_{r1,2}\left(t\right)=2C_r\frac{d{v}_{Cr1,3}\left(t\right)}{dt}\end{array}$$

(9)

whereas the initial conditions are

$$i_{Lr1,2}\left(t_1\right)=\left(\frac{r_{1,2}}{z_{r1,2}}\right)\sin \left(-{\omega }_{r1,2}\left(t_1-t_o\right)\right)$$

(10)

$$v_{Cr1,3}\left(t_1\right)=-n{v}_{Cdc}+\frac{V_S}{2}-r_{1,2}\cos \left(-{\omega }_{r1,2}\left(t_1-t_o\right)\right).$$

(11)

Solving (8) and (9) gives the following equations:

$$i_L{}_{r3,4}\left(t\right)=\frac{r_{3,4}}{z_{r3,4}}\sin \left(\alpha -{\omega }_{r1,2}\left(t-t_1\right)\right)$$

(12)

$$v_C{}_{r1,3}\left(t\right)=n{v}_{Cdc}+r_{3,4}\cos \left(\alpha -{\omega }_{r1,2}\left(t-t_1\right)\right)$$

(13)

where

$$\begin{array}{l}{r}_{3,4}=-n{v}_{Cdc}+\frac{V_S}{2}+\Delta v_{Cr1,3}\\ \alpha =\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}\left(\left(\frac{r_{1,2}}{r_{3,4}}\right)\sin \left(-{\omega }_{r1,2}\left(t_1-t_o\right)\right)\right).\end{array}$$

(14)

Interval 2 ends (at t = t₂) with i_Lr1,2(t₂) = 0 and

$$v_{Cr1,3}\left(t_2\right)=\frac{V_S}{2}+\Delta v_{Cr1,3}.$$

(15)

• Interval 3 [t₂–t₃]: At t = t₂, i_Lr1,2 become zero, and the proposed converter switches into discontinuous conduction mode (DCM) (see Figure 3). Note that i_Lr1,2 turn zero just before the start of Interval 3, so the body diode of S₆ turns off with zero current switching (ZCS). A zero reverse recovery current flowing through each body diode helps significantly reduce the turn-off loss of the device. Note that during Interval 3, the trajectory stays at point x₂, the inductor currents (i_Lr1,2) remain zero, and the resonant capacitors maintain the voltage (v_Cr1,3) at their maximum value (i.e., V_S/2).
• Interval 4 [t₃–t₄]: At the end of the third interval, S₁ and S₄ (on the primary side) are turned off, where S₄ achieves ZVS turn-off as i_L2 was flowing at t₃. In this interval, the inductor current can be written as

$$i_{L1}\left(t\right)=i_{L1}\left(t_3\right)-\frac{V_P\left(t-t_3\right)}{L_1}.$$

(16)

Within this interval, the trajectory stays at x₂, whereas i_L2 decreases as follows:

$$i_{L2}\left(t\right)=i_{L2}\left(t_3\right)-\frac{\left[\left(v_{Cdc}-V_P\right)\left(t-t_3\right)\right]}{L_2}.$$

(17)

Next, Intervals 5~8 follow a similar pattern to 1~4 due to the symmetrical operation for the remaining duty cycle.

3. Feedforward State-Plane Trajectory-Based Control Derivation and Parameter Design Guidelines

This section provides a derivation of the proposed state-plane trajectory-based feedforward gain term (U_ff). Furthermore, the overall control law and its implementation, the effect of parameters mismatch, and parameters design guidelines are detailed later in this section.

3.1. Trajectory-Based Feedforward Term (U_ff) Derivation

The feedforward controller design of the proposed resonant converter can benefit from the state-plane trajectory theory, which is particularly adept at accurately modeling system nonlinearities. Note that in state-plane analysis, the analysis is performed on the trajectories or paths of the resonant converter states/variables (e.g., i_Lr, v_Cr, etc.) over time across different operating modes (Intervals 1~8). A detailed examination yields comprehensive insights into the resonant DC/DC converter behavior. Specifically, a feedforward term has been derived in this study that can effectively compensate for the system uncertainties. It is important to note that a resonant converter represents a complex, time-variant, and nonlinear power conversion system. Therefore, its behavior is precisely modeled through state-plane trajectory theory. The designed control law, based on state-plane trajectory theory, enables efficient and reliable control of the system.

The voltage ripples across the resonant capacitors Δv_Cr1,3 should first be calculated to derive the proposed feedforward term (i.e., U_ff). Since the resonant current is symmetrical in nature, the average inductor current is twice the secondary-side load current (i_S). First, i_S is estimated by calculating the charge balance of the output capacitor (C_S) during 0.5T_sw as

$$\begin{array}{c}{i}_S\left(t\right)=\left(\frac{1}{0.5T_{sw}}\right)[{\displaystyle \underset{t_0}{\overset{t_1}{\int }}\left(\frac{r_{1,2}}{2z_{r1,2}}\right)}\sin \left(-{\omega }_{r1,2}\left(\tau -t_0\right)\right)d\tau \\ \hspace{1em}-\hspace{0.17em}{\displaystyle \underset{t_1}{\overset{t_2}{\int }}\left(r_{3,4}/2z_{r1,2}\right)\sin \left(\alpha -{\omega }_{r1,2}\left(\tau -t_1\right)\right)}d\tau ].\end{array}$$

(18)

Solving the integral in (18) and then substituting the values of r_1,2 and r_3,4 gives

$$i_S(t)=\frac{4n{v}_{Cdc}\Delta v_{Cr1,3}}{T_{sw}{z}_{r1,3}{\omega }_{r1,3}{V}_S}.$$

(19)

After rearrangement, Δv_Cr1,3 is represented as

$$\Delta v_{Cr1,3}=\frac{P_{out}{T}_{sw}}{4n{v}_{Cdc}{C}_r}$$

(20)

where P_out denotes the output power.

In the state-plane trajectory (Figure 4), the circles with radii r_1,2 and r_3,4, respectively, have origins at (−nv_Cdc, 0) and ((V_S − nv_Cdc), 0). For both radii, the two circle equations are expressed as

$$\begin{array}{l}{\left(v_{Cr1,3}\left(t_1\right)-n{v}_{Cdc}\right)}^2+{\left(z_{r1,2}{i}_{Lr1,2}\left(t\right)\right)}^2=r_{1,2}^2\\ {\left(v_{Cr1,3}\left(t_1\right)-V_S+n{v}_{Cdc}\right)}^2+{\left(z_{r1,2}{i}_{Lr1,2}\left(t\right)\right)}^2=r_{3,4}^2.\end{array}$$

(21)

Here, the intersection point of the two circles is y₁ (as shown in Figure 4), thus rearranging and then equating (21) yields

$$\begin{array}{l}{\left(v_{Cr1,3}\left(t_1\right)-n{v}_{Cdc}\right)}^2+{\left(z_{r1,2}{i}_{Lr1,2}\left(t\right)\right)}^2-r_{1,2}^2=\\ {\left(v_{Cr1,3}\left(t_1\right)-V_S+n{v}_{Cdc}\right)}^2+{\left(z_{r1,2}{i}_{Lr1,2}\left(t\right)\right)}^2-r_{3,4}^2\end{array}$$

(22)

Note that the secondary-side switches remain turned on (i.e., U_ffT_sw) during Interval 1 [t₀–t₁]. The voltage across the resonant capacitors v_Cr1,3 at time instant t₁ can be written as

$$\begin{array}{ll}{v}_{Cr1,3}\left(t_1\right)& =-n{v}_{Cdc}+r_{1,2}\cos \left(-{\omega }_{r1,2}\left(t_1-t_0\right)\right)\\ & =-n{v}_{Cdc}+\left(\frac{V_S}{2}+\Delta v_{Cr1,3}+n{v}_{Cdc}\right)\times \cos \left(\frac{-2\pi D_{s1,2}}{F_{1,2}}\right)\end{array}$$

(23)

where F_1,2 is the ratio between ω_sw and ω_r1,2. Finally, substituting (22) in (23) gives the proposed feedforward controller gain (U_ff = [D_s1 + D_s2]) as follows:

$$\begin{array}{ll}{U}_{ff}& =D_{s1}+D_{s2}=2D_{s1,2}\\ & =\frac{2}{{\omega }_{r1,2}{T}_{sw}}\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left(\frac{1+G+\pi G{Q}_{1,2}/2F_{1,2}}{1+G+\pi G^2{Q}_{1,2}/2F_{1,2}}\right)\end{array}$$

(24)

where the voltage gain (G) and the quality factor (Q_1,2) are defined as

$$G=\frac{V_S}{2n{V}_P}=\frac{V_S}{n{v}_{Cdc}},\hspace{0.17em}\hspace{0.17em}{Q}_{1,2}=\frac{4}{{\omega }_{r1,2}{C}_r{R}_{Load}}=\frac{4{\omega }_{r1,2}{L}_{r1,2}}{R_{Load}}$$

(25)

Likewise, for the backward converter operation, the feedforward converter gain can be derived as follows:

$$U_{ff}=\frac{2}{{\omega }_{r1,2}{T}_{sw}}\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left(\frac{1-G^{-1}-G^{-2}H}{1-G^{-1}+G^{-1}H}\right)\hspace{0.17em}\hspace{0.17em}\because H=\frac{{\omega }_{r1,2}{Q}_{1,2}{T}_{sw}}{64n^2}$$

(26)

where G⁻¹ = 2nV_P/V_S.

3.2. Overall Control Law and Its Implementation Steps

The control law employed in this study is composed of two components, namely, a feedback term (U_fb) and a feedforward term (U_ff). First, a linear feedback controller, such as a proportional-integral (PI) controller, is utilized for converging the system errors to zero under dynamic operating conditions. Next, a precisely derived feedforward control term is introduced, which can effectively mitigate the nonlinearities inherent in the resonant converter. Particularly, the state-plane trajectory theory plays a crucial role in designing this feedforward control term. Given the highly complex and nonlinear dynamics characterizing the resonant converter, the state-plane trajectory theory provides an optimal method for modeling the system’s uncertainties.

The proposed feedforward control can be combined with linear controllers (e.g., proportional-integral (PI), linear quadratic regulator (LQR) [27], etc.) to improve the converter operation under dynamic operating conditions. Note that system uncertainties in both forward and backward operations are effectively mitigated by employing the proposed state-plane trajectory-based feedforward control term (i.e., U_ff). First, the PI current control (PICC) can be modeled as

$$U_{fb}=K_{PI}{e}_I+K_{II}{\displaystyle \underset{0}{\overset{t}{\int }}{e}_{I}d\tau }$$

(27)

where U_fb is the feedback control input; K_II and K_PI represent the integral and proportional current gains; and e_I is the current error (= i_P − i_{P_ref}). Then, the overall control law can be expressed as

$$U=U_{fb}+U_{ff}$$

(28)

Actually, the resonant converter depicts an extremely nonlinear system. Thus, the designed feedforward control can effectively handle the system uncertainties and disturbances, whereas the overall control law allows the output voltage to be regulated under dynamic operating conditions. Figure 5 presents the proposed control structure, while Figure 6 depicts the implementation flowchart. As can be observed, a fixed 50% duty on the primary-side interleaved converter helps with obtaining the ripple-free input current, whereas the secondary-side switches (S₅ and S₆) efficiently regulate the output voltage by employing the proposed control law (i.e., U (28)).

3.3. Converter Analysis under Parameters Mismatch

Parameters mismatch can inevitably occur (up to ±20%) during manufacturing inconsistencies. Notably, in dual transformer-based topologies, a slight variation between the two leakage inductances (L_r1 ≠ L_r2) can significantly affect the overall converter operation. That is why it is so critical to analyze the designed converter in terms of its operation and reliability prospects. For a more detailed analysis, two cases have been considered depending on the transformers’ connection on the primary side (i.e., parallel (Case I) or series (Case II)), as shown in Figure 7. First, a slight leakage inductance imbalance under Case I (see Figure 7) having a parallel primary-side connection can severely alter the voltage distribution across the resonant capacitors (C_r1~C_r4). Actually, a small mismatch in the leakage inductance causes an imbalance in the impedance values of the two resonant networks (i.e., z_r1 ≠ z_r2), which leads to an unequal distribution of the inductor currents (i_Lr1 ≠ i_Lr2). In particular, the parallel transformer connection (Case I) results in the flow of two independent and unequal primary-side currents, creating a significant difference in the i_Lr1 and i_Lr2 values. Subsequently, this results in an unbalanced and variable distribution of voltage stress across the resonant elements, thus reducing the reliability of the converter.

Contrarily, the proposed converter adopts a series winding connection on the primary side (Case II), which forces the converter to draw the same amount of current from the primary-side windings of both transformers. It can be observed that the imbalance in the two resonant tanks (i.e., z_r1 ≠ z_r2) still attempts to draw different resonant currents (Δi_Lr = i_Lr1 − i_Lr2 ≠ 0). However, Δi_Lr is almost negligible in Case II due to the series connection on the primary-side windings. Moreover, the proposed feedforward control terms in (24) (D_s1 and D_s2) compensate for the control duty (U) to effectively mitigate the system uncertainties. Thus, an almost equal voltage stress across the resonant capacitors can be ensured, which improves the reliability and overall lifetime of the proposed converter.

3.4. Key Components Design Guidelines

3.4.1. Interleaved Inductors (L_1,2) Design

The interleaved inductors (L_1,2) should be designed to ensure ZVS operation on the primary-side switches (i.e., S₁~S₄). To achieve ZVS, the current ripples (Δi_L) must be smaller than the inductor currents (i_L1,2) as follows:

$$\Delta i_L\le i_{L1,2},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\because \Delta i_L=\frac{V_P{T}_{sw}}{4L_{1,2}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{i}_{L1,2}=\frac{P_{out}}{2\eta V_P}$$

(29)

where η represents the converter efficiency. Using (29), the L_1,2 values can be selected by satisfying the following inequality

$$L_{1,2}\ge \frac{\left(\eta V_P^2\left(\mathit{max}\right)T_{sw}\right)}{\left(2P_{out}\right)}$$

(30)

3.4.2. Transformer Turns Ratio (n_1,2) Design

During the forward direction power flow, the proposed converter boosts the voltage, i.e., the secondary-side voltage (V_S) is always higher than the maximum primary-side voltage (V_P). Therefore, the converter gain (G) is always greater than 1. By rearranging (25), the turns ratio (n_1,2) can be derived as

$$n_{1,2}\le \frac{V_S}{2G_{\mathit{min}}{V}_P}$$

(31)

where G_min represents the minimum voltage gain.

3.4.3. Resonant Components (L_r1,2 and C_r1~4) Selection

During the converter operation, the voltages across all resonant capacitors (C_r1~4) remain positive (i.e., v_Cr1~4 > 0). Further, due to the voltage doubler structure, the condition max(Δv_Cr1~4) ≤ V_S/2 is always satisfied. Thus, C_r1~4 can be derived using (20) as

$$C_{r1~4}>\frac{P_{out}{T}_{sw}}{4n_{1,2}{v}_{Cdc}{C}_{r1~4}}\ge \frac{P_{out}^{\mathit{max}}{T}_{sw}}{8n_{1,2}{V}_P^{\mathit{min}}{V}_S}$$

(32)

To achieve the soft-switching operation (i.e., ZCS turn-off for the body diodes of S_5,6), the resonance frequency (ω_r) should be greater than ω_sw (i.e., 2πf_sw). Using the resonant frequency expression, L_r1,2 can be obtained as

$$L_{r1,2}\le \frac{1}{2{\omega }_r{C}_{r1~4}}\le \frac{1}{8{\pi }^2{f}_{sw}{C}_{r1~4}}.$$

(33)

4. Experimental Verification and Discussion

This section presents details on laboratory setup, evaluating conditions, and experimental results analysis. A comprehensive discussion is presented for each evaluating condition to confirm the viability of the proposed approach.

4.1. Experiment Prototype Details and Validation Conditions

The 3.3 kW converter board is developed to confirm the practicality of the proposed controller along with the overall converter performance. The TI TMS320F28377D DSP is considered for control law implementation, sensing, analog-to-digital (ADC) conversion, and pulse-width modulation (PWM) generation. Next, the designed converter is powered from a programmable DC power supply (IT6000C series), and the efficiency is recorded using the Yokogawa WT1800 power analyzer. Additionally, all waveforms have been displayed on Tektronix DPO 7254 digital phosphor oscilloscope. Figure 8 shows the complete laboratory setup, and

Table 1. Proposed converter specifications and components details.

Variable [Symbol]	Value [Unit]
Rated power [PRated]	3.3 [kW]
Secondary-side voltage [VS]	360~400 [V]
Primary-side voltage [VP]	150~220 [V]
Switching frequency [fsw]	50 [kHz]
Resonant capacitances [Cr1~4]	200 [nF]
Resonant inductances [Lr]	9.8 [uH]
Transformers [n1,2] (T1 [Lm1, Llk1]), (T2 [Lm2, Llk1])	17:24	(1.11 [mH], 1.63 [uH]) (1.12 [mH], 1.72 [uH])
Input inductances [L1, L2]	1 [mH]
Resonant frequency [fr1,2]	71 [kHz]
DC-link capacitance [CDC]	4.4 [uF]
MOSFET switches [S1~S6]	UJ3C065030K3S

lists the key converter parameters.

The practical feasibility of the proposed converter has been verified by the following tough operating scenarios:

• Scenario I: Forward direction load transient operation (i.e., 30% (962 W) → 80% (2672 W) → 30% (962 W)) with V_P (input) = 190 V for V_S (output) = 380 V.
• Scenario II: Load transient under backward operation (i.e., 15% (480 W) → 65% (2200 W) → 15% (480 W)) with V_S (input) = 170 V for V_P (output) = 400 V.
• Scenario III: Reliability, voltage stress, and efficiency analysis.

Moreover, the above-mentioned scenarios are also analyzed to corroborate the voltage balance of the resonant capacitors, the primary-side ripple-free current operation, and the soft-switching operation of the proposed converter.

4.2. Converter Analysis under Forward Direction Load Transient Operation (Scenario I)

Figure 9 presents the experimental results of the proposed converter under Scenario I, whereas the waveforms included are the gate-to-source voltages of S₁ (V_GS1) and S₅ (V_GS5), resonant current (i_Lr), S₅ current (i_S5), secondary-side voltage (V_S), primary-side current (i_P), and interleaved currents (i_L1 and i_L2), from top to bottom, respectively. The primary-side voltage (V_P) is fixed at 190 V, while the load varies as follows: 150 Ω (30%) → 54 Ω (80%) → 150 Ω (30%). For detailed analysis, the zoomed-in waveforms in the light and heavy load regions are presented separately in Figure 9. At the light load (962 W), the proposed control law accurately maintains the secondary-side terminal reference voltage (i.e., V_{S_ref} = 380 V). Mainly, the overall control law helps track the V_{S_ref} efficiently by properly adjusting the control term (U) that is applied on the two 180° phase-shifted secondary switches (S₅ and S₆). Note that the designed state-plane trajectory-based feedforward control term (U_ff) efficiently eliminates system nonlinearities while facilitating the task of the feedback controller (U_fb) to focus on achieving the desired dynamic and steady-state control performance.

Control of the secondary-side switches further helps with setting the primary-side control duty to 0.5. As a result, a perfect interleaved operation on the primary side results in a nearly ripple-free current (see Figure 9). For improved efficiency, the proposed converter realizes soft-switching operation on both primary and secondary-side switches. First, the S₁ body diode achieves ZVS operation because the magnetizing current (i_lm1,2) flows through it before the application of V_GS1. Note that the i_S1 has a sinusoidal shape due to resonance, which helps it to realize ZVS switching. In addition, the voltage stress across all primary-side switches (S₁~S₄) is equally distributed by applying 50% duty (0.5T_sw). Next, the resonant current (i_Lr) shows a linear increase in magnitude during the turn-on interval of S₅, while in the remaining half of the period, particularly in the high-power region (2672 W), the discharge current takes a more sinusoidal shape with a relatively longer discharge time. At this time, the body diode of S₆ (D_S6) turns on with soft switching. Similarly, when the resonance ends, the ZVS turn-off can be observed for D_S6. Finally, thanks to the symmetry of operation, the D_S5 also turns on and off with ZVS, which in turn, contributes to improved efficiency.

4.3. Performance Analysis under the Backward Operation with Primary-Side Load Transient Change (Scenario II)

Figure 10 presents the experiment results for the backward operation, in which a sudden load step is applied to evaluate the designed converter and its control performance and stability. The signals contained in Figure 10, from top to bottom, respectively, are the gating signals (V_GS1 and V_GS5), primary-side currents (i_L1, i_L2, and i_P), S₅ current (i_S5), and resonant current (i_Lr). Note that V_S is fixed at 400 V whereas the load, connected at the primary side, is switched as follows: 60 Ω (15%) → 13.25 Ω (65%) → 60 Ω (15%). As can be observed, the proposed converter accurately tracks the primary-side reference voltage (i.e., V_{P_Ref} = 170 V) in the low-power region due to the designed control law (U). For a thorough analysis, the zoomed regions under the light and heavy loading conditions are shown separately in Figure 10. First, under light load conditions (480 W), the resonant current (i_Lr) increases in magnitude without any reverse recovery current through the body diodes of S₅ and S₆. Then, a sudden increase in load (480 W → 2.2 kW) is applied; however, the proposed converter exhibits a stable behavior, maintaining V_P at 170 V with a slight undershoot during the transient interval. The enhanced control performance can be attributed to the designed control law, which helps with achieving reasonable dynamic control performance.

It is worth mentioning that during the heavy loading operation (2.2 kW), reactive current starts flowing through the body diodes of S₅ (i_SD5) and S₆ (i_SD6), which increases the losses, and thus decreases the overall conversion efficiency. The proposed state-plane trajectory-based control law solves this issue by adjusting the phase shift (θ_adj) of the final control input (U). Due to this phase compensation, the proposed circuit demonstrates an almost negligible reactive power loss under heavy loading (2.2 kW) (See Figure 10).

Notably, the designed controller tracks the V_{P_ref} value by applying a relatively higher control duty compared to that in the forward mode. This improves the conversion efficiency since a major part of the resonant current (i_Lr) flows through S₅ and S₆. Moreover, soft switching can be observed during the backward operation, which further improves the conversion efficiency. Additionally, the inherent nature of the proposed converter offers ripple-free primary-side current (i_P) due to the fixed 50% duty, allowing high-performance battery charging.

4.4. Reliability, Voltage Stress, and Efficiency Analysis (Scenario III)

During forward and reverse power flow, the designed resonant bidirectional converter features balanced voltage stress across the resonant capacitors (C_r1~4) as well as across the main control switches (S₅ and S₆). Figure 11 presents the switching signals (V_GS1, V_GS5) together with the voltages of resonant capacitors (i.e., v_Cr1~4). Note that the values of v_Cr1 and v_Cr3 are the same even under the presence of some variations in the leakage inductances, i.e., L_lk1 (1.63 μH) ≠ L_lk2 (1.719 μH). Likewise, v_Cr2 and v_Cr4 show a similar trend due to the symmetry of the S₅ and S₆ control duties. Notably, the designed control law compensates for voltage imbalance by updating the U_ff value. Furthermore, the application of the same control input (U) on S₅ and S₆ (with a phase shift of 180°) enables more efficient device utilization by evenly distributing the voltage stress between the two switches. Particularly, balancing the resonant capacitor voltages has great significance as it improves the overall reliability of the proposed converter. Finally, Figure 12 shows the converter efficiency curves recorded with a Yokogawa WT1800 power analyzer. The peak efficiencies of 96.5% and 97.3% are achieved during forward and backward modes, respectively. Note that during the backward mode, efficiency is slightly higher compared to the forward mode, as the majority of current flows through S₅ and S₆, lowering the power dissipation caused by conduction losses in the body diodes.

4.5. Comparative Analysis of the Proposed Converter with Existing Dual Transformer-Based Isolated DC/DC Converter Topologies

Table 2. Comparison of the proposed converter with existing dual transformer-based isolated topologies.

Parameter	[18]	[21]	[22]	[28]	[29]	Proposed
Components	6S, 4C	6S, 3C, 2D	4S, 3C, 4D	10S, 3C	4S, 3C, 4D	6S, 5C
Efficiency	95%	96.3%	98.5%	94%	95.3%	97.3%
Control Type	PI duty and phase shift control	PI and hysteresis control	Burst mode control	PI duty and phase shift control	PI phase shift control	State-plane trajectory-based control
Control Complexity	Medium	High	High	Medium	Simple	Medium
Circuit Complexity	Medium	Medium	High	High	Medium	Medium
Cost	Medium	Medium	Low	High	Medium	Medium
Rated Power	1 kW	1 kW	3.2 kW	300 W	1 kW	3.3 kW
Capacitor Voltage Balancing	Unbalanced	Unbalanced	Balanced	−	Unbalanced	Balanced
Power Flow	Bidirectional	Unidirectional	Unidirectional	Bidirectional	Unidirectional	Bidirectional

(S: switch, C: capacitor, D: diodes).

presents a comparative analysis between the proposed resonant converter and existing dual transformer-based isolated topologies. In [21], a segmented control is designed for a dual transformer-based linear DC/DC converter. While this converter achieves a wide load range operation and high efficiency, it suffers from unbalanced capacitor voltages due to unsymmetrical converter operation. In [22], a highly efficient charger is investigated, utilizing gallium nitride (GaN) semiconductor devices to reduce conduction losses. Although it improves conversion performance, the switching frequency varies significantly during converter operation, placing a heavy burden on the active devices and impacting the converter’s reliability negatively. Similarly, Ref. [28] extends the voltage operation range using a blocking capacitor on the primary side. However, the use of numerous active power components significantly affects the cost of the power converter. A dual transformer-based linear converter studied in [18] controls two variables: switches duty ratio and phase shift between the primary and secondary legs. Although it employs fewer active components, it suffers from low conversion efficiency under light load conditions due to increased switching and magnetic losses during turn-off. In [29], an LLC resonant converter with two resonant tanks is introduced to achieve a wide input voltage range. Pulse frequency modulation (PFM) is employed with a simple PI controller for load voltage regulation. Additionally, a hysteresis controller is included to smooth out oscillations during mode transitions. Despite its suitable efficiency, the use of hysteresis control can result in increased electromagnetic interference (EMI) noise due to variable switching frequency. Furthermore, a fully ripple-free input current cannot be achieved because of primary-side control. In contrast, the proposed converter achieves highly efficient bidirectional power conversion with a low count of active devices and balanced stress on resonant capacitors. This improves the reliability and lifetime of the DC/DC power converter.

5. Conclusions

This paper investigates the design and control of a bidirectional resonant DC/DC converter for BEV charging applications. First, the proposed dual-rectifier structure on the secondary side allows operation over a wide load range and more effective voltage stress distribution across the resonant capacitors. A fully ripple-free primary-side current is realized by achieving full interleaving between the two primary boosting legs. Then, for the proposed converter, a state-plane trajectory-based control law has been derived, which, besides eliminating the system nonlinearities, allows more efficient convergence of system errors to zero. Notably, the proposed converter and its derived controller naturally balance the voltages across the resonant capacitors, even in the case of an imbalance in the transformer leakage inductances. Finally, the proposed resonant converter achieves soft-switching operation (ZCS and ZVS), which further improves the overall converter efficiency. Detailed experiments are provided using a 3.3 kW laboratory prototype, whereas a Texas Instruments (TI) 320F283777D DSP is employed to implement the proposed controller.