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 acdc converter with power factor correction and an isolated dcdc converter for galvanic isolation, as shown in
Fig. 1
[1]

[3]
.
Typical battery charging power architecture.
In
Fig. 1
, zerovoltage 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 frequencydomain analysis methods take highorder 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 stateplane or timedomain 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: Precharge (Activate), Bulk, Absorption, Equalization, and Maintenance
[2]
. In the Precharge mode, the charger outputs a small current to activate the batteries. This current is about 20% of I
_{MAX}
. In the Bulk mode, the charger charges the batteries with a constant current I
_{MAX}
. It charges the batteries with a constant voltage V
_{ABS}
in the Absorption mode. When the current decreases to a preset value I
_{MIN}
, 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.
Simplified leadacid battery charging profile.
As shown in
Fig. 3
, the outer range of the lead acid battery VI plane is constrained by the precharge, constant current (CC), constant power (CP), and constant voltage (CV)
[2]
. This indicates that the voltage range of a single leadacid 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 VI plane in
Fig. 3
. It decides the design constraints for the fullbridge LLC converter, especially the resonant tank parameters
K
(the ratio of the transformer magnetizing inductor
L_{m}
to the resonant inductor
L_{r}
), and
Q
_{max1}
(the quality factor of the resonant tank at the minimum input voltage and the full load), which are vital in the LLC design.
Desired leadacid battery VI 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 precharge 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 precharge 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
L_{m}
for the same frequency range. In addition, it will increase the primary conduction and turnoff 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 precharge 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
L_{m}
to
L_{r}
is designed with a minimum normalized output voltage gain
M_{min}
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 (
V_{o_min}
,
I
_{o_max1}
) and (
V_{o_max}
,
I
_{o_max2}
) in
Fig. 3
. This is different from the design at zero load.
Where,
M_{min}
=
nV_{o_min}
/
V_{in_max}
with the transformer turn ratio
n
, and
f_{r}
and
f_{max}
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
R_{e}
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
f_{s}/f_{r}
.
Q
is the resonant tank quality factor for different loads.
Z_{o}
is the characteristic impedance of the LLC resonant tank.
Two port model for LLC resonant converter based on FHA.
Typical dc gain characteristics of the LLC converter.
For the LLC converter, as the switching frequency is varied closer to
f_{r}
, 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
f_{s}
=
f_{r}
. Thus, the charger is designed to deliver the maximum output power at the unity gain point (
f_{s}
=
f_{r}
), which is marked as “Design Point
f_{r}
” 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 turnoff condition at (
V_{o_max}
,
I
_{o_max2}
) in
Fig. 3
,
Q
_{max1}
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
Z_{in}
(jω) in
Fig. 4
is zero here. Then,
Q
_{max1}
has a relation with the inductor ratio
K
as (7)
[9]
.
Where,
M_{max}
=
nV_{o_max}
/
V_{in_min}
.
If the maximum normalized switching frequency
X_{max}
=
f_{max}
/
f_{r}
at (
V_{o_min}
,
I
_{o_max1}
) is fixed, then its quality factor
Q
_{max2}
can be derived from (3) as:
As determined by the resonant tank parameters,
Z_{o}
is same at the two charging points (
V_{o_min}
,
I
_{o_max1}
) and (
V_{o_max}
,
I
_{o_max2}
).
Then, the maximum ratio
K
of
L_{m}
to
L_{r}
can be obtained as (12) with (7) ~ (11).
Where:
After that, the LLC resonant tank parameters
L_{r}
,
C_{r}
and
L_{m}
can be determined. The following is the stepbystep design procedure, which is illustrated in
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
I
_{o_max1}
=
I_{MAX}
= 50 A,
I
_{o_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
V_{o_min}
= 36 V to
V_{o_max}
= 72 V, with a nominal value of
V_{o_nom}
= 58 V at the maximum output power. In this design,
f_{r}
= 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,
V_{F}
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 2
f_{r}
to 2.5
f_{r}
. Here,
X_{max}
is chosen as:
 C. Calculating the Ratio, K
The ratio
K
of
L_{m}
to
L_{r}
can be designed with the two charging points (
V_{o_min}
,
I
_{o_max1}
) and (
V_{o_max}
,
I
_{o_max2}
) in
Fig.3
. It can be calculated using (12).
 D. Quality Factor Qmax1and Characteristic Impedance Zo
The maximum quality factor
Q
_{max1}
at (
V_{o_max}
,
I
_{o_max2}
), i.e. (72 V, 40 A), can be calculated using (7). The characteristic impedance
Z_{o}
of the resonant tank can be derived from (9) with
Q
_{max1}
and
R_{e_max}
.
 E. Calculating the Resonant Capacitor, Cr
The resonant capacitor value is given by:
If
C_{r}
is not an appropriate value for commercial product selection, it can have fine tuning while guaranteeing the two charging points (
V_{o_min}
,
I
_{o_max1}
) and (
V_{o_max}
,
I
_{o_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
f_{max}
with a duty cycle of 0.5,
L_{m}
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,
C_{HB}
is the equivalent parasitic capacitor in the middle point of the phase legs Q1 and Q2, and
t_{dead}
is the deadtime.
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 R
_{DS(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 turnoff, the forward drop V
_{F}
and junction capacitance C
_{j}
are among the main considering factors in diode selection. The available products include Schottky and Ultrafast diodes. Although Schottky diodes have a lower V
_{F}
, they demonstrate a relatively higher C
_{j}
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 (
A_{P}
) denotes the size of the magnetic components and is usually used for design. The
A_{P}
of the transformer and inductor can be expressed as:
Where,
I_{P}
and
I_{S}
are the primary and secondary RMS currents. Δ
i_{r}
and Δ
i_{m}
are the peakpeak currents of the inductor
L_{r}
and the transformer magnetizing inductor
L_{m}
. Δ
B
,
J
and
K_{u}
are the flux density, the current density, and the window utility factor, respectively
[17]
. When magnetic cores are selected, the winding number
N_{Lr}
of
L_{r}
and the secondary winding number
N_{Tr_sec}
of the transformer
T_{r}
can be obtained from:
Where,
Ae_{Lr}
and
Ae_{Tr}
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 VI 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 (
V_{o_min}
,
I_{o_max1}
) and (
V_{o_max}
,
I_{o_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 Precharge mode, which is just (48 V, 10 A) at
f_{s}
=
f_{max}
with the design parameters in
Table I
.
Fig. 7
is a simplified flowchart of the combined control in the precharge mode. At the boundary charging point, the hysteretic type control in
Fig. 8
has been adopted to avoid control oscillation.
DESIGN SPECIFICATIONS AND PARAMETERS
DESIGN SPECIFICATIONS AND PARAMETERS
Simplified flowchart of PFM&PWM combined control at the precharge mode.
Diagram of the combined control with hysteresis.
When designing the controller, an exact expression for the controltooutput transfer function
G_{v}
(
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,
V_{dc}
and
f_{s}
are the input voltage and output frequency of the VCO (voltage controlled oscillation), and
v_{ac}
is the small signal injected into the system for ac analysis. The conversion ratio fs/
V_{dc}
of the VCO is 136.4 kHz/V. Then, the switching frequency
f_{s}
is 165.5 kHz at (72 V, 10 A) with
V_{dc}
=1.213V, while it is 162.3 kHz at (72 V, 40 A) with
V_{dc}
=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 twopole, onezero voltage feedback
G_{c}
(
s
) in (32) is used to compensate the loop gain characteristics. When compensated with
K_{c}
=71.4,
ω_{z}
=3.5×10
^{3}
rad/s, and
ω_{p}
=9.9×10
^{4}
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.
LLC simultaion model for ac analysis in PSIM.
Control to output transfer function at (72V, 40A), (72V, 10A) in CV charging mode.
Loop gain characteristic after compensation at (72V, 40A), (72V, 10A) in CV charging mode.
IV. EXPERIMENTAL RESULTS
A prototype of the fullbridge 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.
Experimental prototype.
POWER COMPONENT USED IN THE PROTOTYPE CONVERTER
POWER 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.
Calculated losses comparison between the proposed design K=4.1 and the conventional design K=1.2 at 58V/ 2900W output.
Measured Efficiency at V_{in}= 400V.
Fig. 15
show the experimental waveforms of the Q1 driver voltage (
v
_{GS1}
), voltage across Q1 (
v
_{DS1}
) and resonant tank current (
i_{r}
) at
V_{in}
= 400 V. As shown in
Fig. 15
(a), the switching frequency is
f_{s}
= 224.4 kHz at
V_{o}
= 58 V and
P_{o}
= 580 W, which is close to the resonant frequency.
Waveforms of Q1 driver voltage (v_{GS1}), voltage across Q1 (v_{DS1}), and resonant tank current (i_{r}) at different load ((a) V_{o} = 58 V, P_{o} = 580 W; (b) V_{o} = 58 V, P_{o} = 2900 W; (c) V_{o} = 72 V, P_{o} = 2900 W; (d) V_{o} = 48 V, P_{o} = 480 W; (e) V_{o} = 36 V, P_{o} = 1080 W).
The waveforms in
Fig. 15
(b) are given at
V_{o}
= 58 V and at the maximum output power
P_{o}
= 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
V_{o}
= 48 V and
P_{o}
= 480 W with
f_{s}
=
f_{max}
, which is the boundary charging point between the PFM control and the PWM control. With the same output voltage above resonant frequency
f_{r}
, the switching frequency
f_{s}
of the LLC converter decreases with increasing loads. As shown in
Fig. 15
(e), when the converter runs with
f_{s}
=
f_{max}
at
V_{o_min}
= 36 V, its output current is 30 A and smaller than
I
_{o_max1}
after fine tuning. Then, it can be inferred that
f_{s}
at (
V_{o_min}
,
I
_{o_max1}
) is smaller than
f_{max}
, 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 turnon condition. Since (72 V, 40 A) at
f_{min}
and (36 V, 30 A) at
f_{max}
are the worst conditions for the LLC converter’s ZVS achievement within the PFM control, the charger can achieve ZVS turnon 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
L_{m}
to
L_{r}
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
M_{min}
and zero load, the proposed design with two charging points, (
V_{o_min}
,
I
_{o_max1}
) and (
V_{o_max}
,
I
_{o_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 turnoff 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 turnon 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.
Acknowledgements
This work was supported by the Natural Science Foundation of Jiangsu, China (BK2012794 BK20140812), the Industryacademic Joint Technological Innovations Fund Project of Jiangsu (BY201400312), and Jiangsu province university outstanding science and technology innovation team project.
BIO
Wenjin Sun was born in Jiangsu Province, China, in 1988. He received his B.S. degree in Electrical Engineering from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2011. He is presently working toward his Ph.D. degree in Electrical Engineering at NUAA. His current research interests include topologies and control methods for power converters, distributed power generation and spacecraft power systems.
Hongfei Wu was born in Hebei Province, China, in 1985. He received his B.S. and Ph.D. degrees in Electrical Engineering and Power Electronics and Power Drives from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2008 and 2013, respectively. From June 2012 to July 2012, he was a guest Ph.D. student at the Institute of Energy Technology, Aalborg University, Aalborg, Denmark. Since 2013, he has been with the Faculty of Electrical Engineering, NUAA. He is presently a Lecturer in the College of Automation Engineering, NUAA. He has authored or coauthored more than 90 peerreviewed papers published in journals and conference proceedings. He is the holder of more than 17 patents. His current research interests include power converters, distributed power generation and spacecraft power systems. Dr. Wu was awarded as an Outstanding Reviewer of the IEEE Transactions on Power Electronics (2013).
Haibing Hu received his B.S. degree from the Hunan University of Technology, Hunan, China, in 1995, and his M.S. and Ph.D. degrees in Electrical Engineering from Zhejiang University, Zhejiang, China, in 2003 and 2007, respectively. Since 2007, he has been with the Faculty of Electrical Engineering, Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China. He is presently a Professor in the College of Automation Engineering, NUAA. From 2009 to 2012, he was a Postdoctoral Research Fellow in the Department of Electrical Engineering, University of Central Florida, Orlando, FL, USA. He is the author or coauthor of more than 70 technical papers published in journals and conference proceedings. His current research interests include digital control in power electronics, multilevel inverters, digital control system integration for power electronics, and applying power electronics to distributed energy systems and power quality.
Yan Xing was born in Shandong Province, China, in 1964. She received her B.S. and M.S. degrees in Automation and Electrical Engineering from Tsinghua University, Beijing, China, in 1985 and 1988, respectively, and her Ph.D. degree in Electrical Engineering from Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2000. Since 1988, she has been with the Faculty of Electrical Engineering, NUAA, and is presently a Professor in the College of Automation Engineering, NUAA. She has authored more than 100 technical papers published in journals and conference proceedings. She has also published three books. Her current research interests include topologies and control methods for dcdc and dcac converters.
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