Resonant Tank Design Considerations and Implementation of a LLC Resonant Converter with a Wide Battery Voltage Range

Abstract

This paper illustrates resonant tank design considerations and the implementation of a LLC resonant converter with a wide battery voltage range based on the fundamental harmonic approximation (FHA) analysis. Unlike the conventional design at zero load, the parameter K (the ratio of the transformer magnetizing inductor L_m to the resonant inductor L_r) of the LLC converter in this paper is designed with two charging points, (V_{o_min}, I_{o_max1}) and (V_{o_max}, I_{o_max2}), according to the battery charging strategy. A 2.9kW prototype with an output voltage range of 36V to 72V dc is built to verify the design. It achieves a peak efficiency of 96%.

I. INTRODUCTION

Since electric vehicles have developed so quickly, high capacity battery packs have been widely used. However, this highly demands high efficiency, low cost and compact smart chargers. Among the currently available charging solutions, the most common charger architecture consists of an ac-dc converter with power factor correction and an isolated dc-dc converter for galvanic isolation, as shown in Fig. 1 [1]-[3].

Fig. 1.Typical battery charging power architecture.

In Fig. 1, zero-voltage switching (ZVS) topologies are preferable for the isolated second stage to enhance efficiency. In particular, the multi resonance LLC topology has several advantages over other ZVS topologies. These advantages include: (a) the ability to operate with ZVS over a wide load range; (b) no diode reverse recovery losses through soft commutation; (c) low voltage stress of the output diodes; (d) only a capacitor as an output filter compared with the conventional LC filters; and (e) few EMI issues [3]. However, the LLC topology is difficult to analyze and to achieve an optimal design due to its multi resonance.

To simplify the analysis of the characteristics of LLC converters, the fundamental harmonic approximation (FHA) has been developed, where the voltages and currents are assumed to be sinusoidal waveforms, thereby permitting the traditional AC circuit analysis to be employed. As a result, the approximated DC gain in mathematical expressions can be easily derived. Some frequency-domain analysis methods take high-order elements into account to improve the accuracy of the FHA, such as the extended fundamental frequency analysis and Fourier series expansion [4], [5]. However, these become cumbersome in practical use. Other approaches like the state-plane or time-domain analysis are based on the converter's exact model to provide precise description of the circuit behavior. However, they are usually not very easy to interpret and can be difficult to use [6], [7]. Although the accuracy of the FHA will be degraded when the switching frequency deviates far from the resonant point, it still can meet the requirements of engineering. Moreover, it is simpler and more straightforward to calculate the dc gain and design the LLC converter. Therefore, it is adopted here to analyze wide output range applications.

Despite the aforementioned advantages, it is challenging to achieve an overall high efficiency over a wide battery voltage range [3]. In this paper, by combining the features of the LLC converter and the characteristics of the battery charging profile, an optimal design procedure is proposed to achieve an overall high efficiency.

Fig. 2 shows a simplified battery charging profile with five distinct operating modes: Pre-charge (Activate), Bulk, Absorption, Equalization, and Maintenance [2]. In the Pre-charge mode, the charger outputs a small current to activate the batteries. This current is about 20% of IMAX. In the Bulk mode, the charger charges the batteries with a constant current IMAX. It charges the batteries with a constant voltage VABS in the Absorption mode. When the current decreases to a preset value IMIN, the charger enters the Equalization mode with an overvoltage to equalize the cell voltages. After the batteries are fully charged, the charger only outputs a small current to offset the internal soft discharge in the Maintenance mode.

Fig. 2.Simplified lead-acid battery charging profile.

As shown in Fig. 3, the outer range of the lead acid battery V-I plane is constrained by the pre-charge, constant current (CC), constant power (CP), and constant voltage (CV) [2]. This indicates that the voltage range of a single lead-acid battery cell is generally 1.5 V~2.4 V, with a nominal voltage of 2.0 V at the maximum output current. Then it has significantly different design requirements for the resonant tank parameters when compared with those featuring a constant output voltage. The proposed resonant tank design procedure is based on the V-I plane in Fig. 3. It decides the design constraints for the full-bridge LLC converter, especially the resonant tank parameters K (the ratio of the transformer magnetizing inductor Lm to the resonant inductor Lr), and Qmax1 (the quality factor of the resonant tank at the minimum input voltage and the full load), which are vital in the LLC design.

Fig. 3.Desired lead-acid battery V-I plane on a 2900 W charger.

As indicated in Fig. 2 and Fig. 3, a small output current is required to activate the batteries in the pre-charge mode. This lasts for a short time during the whole charging process, such as 10min. Moreover, its output power, e.g. 36 V*10 A = 360 W, is much less than the nominal power 2900 W, and the increased heat dissipation can be easily handled. However, if the pre-charge mode is considered with the resonant tank design under the PFM control, the LLC converter has a wider output range, resulting in a smaller ratio K and Lm for the same frequency range. In addition, it will increase the primary conduction and turn-off losses, and deteriorate the efficiency in normal operation [8]-[10]. Therefore, the PFM&PWM control is adopted here [11], [12]. Furthermore, the charger will run into the PWM control with a small current and a low voltage output in the pre-charge mode, such as 10 A and 1.5 V/cell. The following sections will focus on the design of the LLC resonant tank parameters with a large output power under the PFM control.

 

II. RESONANT TANK DESIGN PROCEDURE

Because of its advantages over other ZVS topologies, the LLC topology has been used in many applications, such as adapters, induction heating, and fuel cells [13]-[15]. However, there are two major issues with the existing LLC design used in industry based on the FHA analysis: 1) the output voltage is considered constant (e.g. typical for telecom applications), which is not a valid assumption in battery charging, and 2) the ratio K of Lm to Lr is designed with a minimum normalized output voltage gain Mmin at zero load as Eq. (1) [9], which is not applicable to battery charging. Then, designing the resonant tank needs different steps for a battery charger. In the following work, the ratio K is designed with the two charging points (Vo_min, Io_max1) and (Vo_max, Io_max2) in Fig. 3. This is different from the design at zero load.

Where, Mmin = nVo_min/Vin_max with the transformer turn ratio n, and fr and fmax are the resonant frequency and maximum switching frequency.

Based on the FHA analysis, the nonlinear circuit of the full bridge LLC resonant converter in Fig. 1 can be transformed into the linear circuit in Fig. 4, where the ac resonant tank is excited by an effective sinusoidal input source that drives an effective resistive load Re as (2) [9]. This transformation allows for the use of traditional AC circuit analysis methods to study the circuit, and the normalized output gains of the LLC converter could be derived as (3) [9]. Fig. 5 illustrates the family of its typical characteristics.

Where, X is the normalized switching frequency fs/fr. Q is the resonant tank quality factor for different loads. Zo is the characteristic impedance of the LLC resonant tank.

Fig. 4.Two port model for LLC resonant converter based on FHA.

Fig. 5.Typical dc gain characteristics of the LLC converter.

For the LLC converter, as the switching frequency is varied closer to fr, the impedance of the resonant tank becomes smaller. This can reduce the circulating energy in the resonant tank and the conduction losses of the LLC converter [3]. In addition, the maximum efficiency can be achieved at fs = fr. Thus, the charger is designed to deliver the maximum output power at the unity gain point (fs = fr), which is marked as “Design Point fr” in Fig. 3. Furthermore, it is appropriate to operate the converter in Region 1 and Region 2 to maintain the primary switches’ ZVS operation for a wide DC gain range [9].

In order to avoid the primary switches working in the ZCS turn-off condition at (Vo_max, Io_max2) in Fig. 3, Qmax1 at this charging point can be designed at the blue dotted boundary between ZVS Region 2 and ZCS Region 3 in Fig. 5. In addition, the imaginary part of the resonant tank input impedance Zin(jω) in Fig. 4 is zero here. Then, Qmax1 has a relation with the inductor ratio K as (7) [9].

Where, Mmax=nVo_max/Vin_min.

If the maximum normalized switching frequency Xmax=fmax/fr at (Vo_min, Io_max1) is fixed, then its quality factor Qmax2 can be derived from (3) as:

As determined by the resonant tank parameters, Zo is same at the two charging points (Vo_min, Io_max1) and (Vo_max, Io_max2).

Then, the maximum ratio K of Lm to Lr can be obtained as (12) with (7) ~ (11).

Where:

After that, the LLC resonant tank parameters Lr, Cr and Lm can be determined. The following is the step-by-step design procedure, which is illustrated in Fig. 6.

Fig. 6.Flow chart of the resonant tank design.

Before it is designed, the initial parameters of the full bridge LLC converter should be defined, such as the current Io_max1 = IMAX = 50 A, Io_max2 =40 A, the input and output voltage range, the maximum output power and the maximum switching frequency. The input voltage seen by the LLC converter is determined by the front PFC output, whose variation is about ±20 V with a nominal value of 400 V. The output voltage is determined by the battery modules. Here, the charger is designed to charging two kinds of battery systems, 60 V systems with 30 cells and 48 V systems with 24 cells. Then, the output voltage of the charger varies from Vo_min = 36 V to Vo_max = 72 V, with a nominal value of Vo_nom = 58 V at the maximum output power. In this design, fr = 220 kHz is chosen.

A. Selecting the Transformer Turns Ratio, n

The transformer turns ratio should be selected at the resonant frequency, where the gain is unity. It can be calculated as:

Where, VF is the diode voltage drop of the output rectifier.

B. Maximum Switching Frequency, Xmax

The maximum switching frequency of the resonant converter is limited by the control circuit, the driver circuit and the reflected junction capacitance of the output rectifiers. Ref. [16] demonstrates that the normalized dc gain equation of the converter is modified with the rectifier junction capacitances. The circuit resonates with the parasitic capacitances when switching frequency is too high. As a result, the output voltage increases with the switching frequency, which deviates from the original design. Limiting the maximum switching frequency is an approach to prohibit this. In addition, the maximum switching frequency should be limited to 2fr to 2.5fr. Here, Xmax is chosen as:

C. Calculating the Ratio, K

The ratio K of Lm to Lr can be designed with the two charging points (Vo_min, Io_max1) and (Vo_max, Io_max2) in Fig.3. It can be calculated using (12).

D. Quality Factor Qmax1 and Characteristic Impedance Zo

The maximum quality factor Qmax1 at (Vo_max, Io_max2), i.e. (72 V, 40 A), can be calculated using (7). The characteristic impedance Zo of the resonant tank can be derived from (9) with Qmax1 and Re_max.

E. Calculating the Resonant Capacitor, Cr

The resonant capacitor value is given by:

If Cr is not an appropriate value for commercial product selection, it can have fine tuning while guaranteeing the two charging points (Vo_min, Io_max1) and (Vo_max, Io_max2) within the PFM control. Once the value of the resonant capacitor is determined, the resonant inductor value and the transformer magnetizing inductor value can be calculated using (19) and (20).

F. ZVS Requirements

In order to achieve ZVS at fmax with a duty cycle of 0.5, Lm must meet the inequality expressed of (21). If it is not satisfied, the dead time in (21) should be set again or the aforementioned design should be tuned.

Here, CHB is the equivalent parasitic capacitor in the middle point of the phase legs Q1 and Q2, and tdead is the dead-time.

 

III. PRACTICAL DESIGN CONSIDERATIONS

A. MOSFET and Output Diode Selection

When the resonant tank parameters are decided, the minimum switching frequency can be derived by (22) and (23). Then the resonant peak current through the MOSFET can be calculated approximately using (24). In order to decrease the losses, low RDS(ON) MOSFETs should be selected.

The output diode average current can be calculated by (25) with the same average output current.

Since the diodes at the secondary side operate with ZCS turn-off, the forward drop VF and junction capacitance Cj are among the main considering factors in diode selection. The available products include Schottky and Ultrafast diodes. Although Schottky diodes have a lower VF, they demonstrate a relatively higher Cj when compared with ultrafast diodes. As mentioned in section II, this will limit the maximum switching frequency. Therefore, ultrafast diodes are a better choice.

B. Resonant and Output Capacitors Selection

Assuming the resonant capacitor voltage waveform is sinusoidal, the RMS ac voltage value for the resonant capacitor is given by (26). Then, a Polypropylene film capacitor with a permissible voltage can be selected.

A very high ripple current on the secondary side needs to be handled by the output capacitors. This can be calculated approximately using (27). According to this value, the combination of a Polypropylene film capacitor and an Electrolytic capacitor is used.

C. Resonant Inductor and Transformer Design

The area product (AP) denotes the size of the magnetic components and is usually used for design. The AP of the transformer and inductor can be expressed as:

Where, IP and IS are the primary and secondary RMS currents. Δir and Δim are the peak-peak currents of the inductor Lr and the transformer magnetizing inductor Lm. ΔB, J and Ku are the flux density, the current density, and the window utility factor, respectively [17]. When magnetic cores are selected, the winding number NLr of Lr and the secondary winding number NTr_sec of the transformer Tr can be obtained from:

Where, AeLr and AeTr are the cross section areas of the inductor and transformer cores. Then, the resonant inductor and transformer can be designed.

D. Power Limit Restrictions and Control Design

The V-I plane provided in Fig. 3 illustrates the limitations on the output voltage, output power and output current, which are implemented by software. In order to easily use the PFM&PWM combined control, digital control has been adopted for the charger module. In addition, it can be easily paralleled by the upper controller to charge higher capacity battery packs.

From the analysis above, the converter is designed to work with the PFM control at the two charging points (Vo_min, Io_max1) and (Vo_max, Io_max2), which indicates that the batteries’ CC, CP, and CV charging modes are all within the PFM control range. Then, the boundary of the combined PFM&PWM control happens at the Pre-charge mode, which is just (48 V, 10 A) at fs=fmax with the design parameters in Table I. Fig. 7 is a simplified flowchart of the combined control in the pre-charge mode. At the boundary charging point, the hysteretic type control in Fig. 8 has been adopted to avoid control oscillation.

TABLE IDESIGN SPECIFICATIONS AND PARAMETERS

Fig. 7.Simplified flowchart of PFM&PWM combined control at the pre-charge mode.

Fig. 8.Diagram of the combined control with hysteresis.

When designing the controller, an exact expression for the control-to-output transfer function Gv(s) may be necessary. However, modeling the dynamic characteristics of the LLC converter is complex due to its multi resonance [13], [18]. Thanks to powerful simulation tools, they can be easily obtained from simulations, such as PSIM. Fig. 9 shows a LLC simulation model for ac analysis in PSIM. In this figure, Vdc and fs are the input voltage and output frequency of the VCO (voltage controlled oscillation), and vac is the small signal injected into the system for ac analysis. The conversion ratio fs/Vdc of the VCO is 136.4 kHz/V. Then, the switching frequency fs is 165.5 kHz at (72 V, 10 A) with Vdc =1.213V, while it is 162.3 kHz at (72 V, 40 A) with Vdc =1.190V. The dynamic characteristics of the converter at (72 V, 40 A) and (72 V, 10 A) in CV mode are shown in Fig. 10. As in Ref. [13], [18], the two-pole, one-zero voltage feedback Gc(s) in (32) is used to compensate the loop gain characteristics. When compensated with Kc=71.4, ωz=3.5×103 rad/s, and ωp=9.9×104 rad/s, the loop gain characteristics of the converter are shown in Fig. 11. It has enough stability at the cutoff frequency f = 3 kHz. The dynamic characteristic of the converter at other charging modes can be compensated with a similar procedure.

Fig. 9.LLC simultaion model for ac analysis in PSIM.

Fig. 10.Control to output transfer function at (72V, 40A), (72V, 10A) in CV charging mode.

Fig. 11.Loop gain characteristic after compensation at (72V, 40A), (72V, 10A) in CV charging mode.

 

IV. EXPERIMENTAL RESULTS

A prototype of the full-bridge LLC resonant converter, as shown in Fig. 12, was built to provide an experimental evaluation of the analytical work presented in this paper. The design criteria for the prototype are provided in Table I. When compared with the conventional design result K=1.2 by Eq. (1), a larger ratio K=4.1, as shown in Table I, has been obtained with the proposed design procedure. Table II gives the key components used in the prototype converter.

Fig. 12.Experimental prototype.

TABLE IIPOWER COMPONENT USED IN THE PROTOTYPE CONVERTER

Fig. 13 is a calculated loss comparison between the proposed design K=4.1 and the conventional design K=1.2 at a 58 V/2900 W output. This shows that the proposed design can cut down the losses of the primary switches’ and magnetic elements, which will improve the converter’s efficiency. This comparison has verified the effectiveness of the proposed solution. Efficiency curves of a converter with the proposed method are given in Fig. 14 for output voltages of 48 V, 53 V, 58 V, 65 V and 72 V. A peak efficiency of 96% has been achieved.

Fig. 13.Calculated losses comparison between the proposed design K=4.1 and the conventional design K=1.2 at 58V/ 2900W output.

Fig. 14.Measured Efficiency at Vin= 400V.

Fig. 15 show the experimental waveforms of the Q1 driver voltage (vGS1), voltage across Q1 (vDS1) and resonant tank current (ir) at Vin = 400 V. As shown in Fig. 15(a), the switching frequency is fs = 224.4 kHz at Vo = 58 V and Po = 580 W, which is close to the resonant frequency.

Fig. 15.Waveforms of Q1 driver voltage (vGS1), voltage across Q1 (vDS1), and resonant tank current (ir) at different load ((a) Vo = 58 V, Po = 580 W; (b) Vo = 58 V, Po = 2900 W; (c) Vo = 72 V, Po = 2900 W; (d) Vo = 48 V, Po = 480 W; (e) Vo = 36 V, Po = 1080 W).

The waveforms in Fig. 15(b) are given at Vo = 58 V and at the maximum output power Po = 2900 W. Fig. 15(c) presents the waveforms at the maximum output voltage and power, while operating at the minimum frequency. Fig. 15(d) provides the waveforms at Vo = 48 V and Po = 480 W with fs = fmax, which is the boundary charging point between the PFM control and the PWM control. With the same output voltage above resonant frequency fr, the switching frequency fs of the LLC converter decreases with increasing loads. As shown in Fig. 15(e), when the converter runs with fs = fmax at Vo_min = 36 V, its output current is 30 A and smaller than Io_max1 after fine tuning. Then, it can be inferred that fs at (Vo_min, Io_max1) is smaller than fmax, which fulfills the design requirement that the charging point should be within the PFM control range.

In Fig. 15, the primary switches can achieve the ZVS turn-on condition. Since (72 V, 40 A) at fmin and (36 V, 30 A) at fmax are the worst conditions for the LLC converter’s ZVS achievement within the PFM control, the charger can achieve ZVS turn-on during the whole range of the batteries’ CC, CP and CV charging modes.

 

V. CONCLUSION

For battery charging applications, this paper illustrates the resonant tank design considerations of a LLC resonant converter based on FHA analysis. By combining the features of the LLC converter and the characteristics of the battery charging profile, a simple design procedure is proposed to increase the inductance ratio K of Lm to Lr and to improve the efficiency of the charger.

When compared with the ratio of the conventional LLC design K=1.2 at the minimum normalized output voltage gain Mmin and zero load, the proposed design with two charging points, (Vo_min, Io_max1) and (Vo_max, Io_max2), can give a much larger ratio, K=4.1. As verified by a calculated losses comparison, the proposed design can cut down the primary conduction and turn-off losses.

A 2.9 kW prototype is built to verify the design. It converts 400 V from the Boost PFC to an output voltage range of 36 V to 72 V dc. Experimental results show that a prototype designed with the proposed method can realize the primary switches’ ZVS turn-on during the whole range of the batteries’ CC, CP and CV charging modes. In addition, a peak efficiency of 96% can be achieved by the proposed solution.