Design of a Resonant LLC Converter · simplified calculation methods such as the Fundamental...
Transcript of Design of a Resonant LLC Converter · simplified calculation methods such as the Fundamental...
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Abstract — Due to the its high power density and efficiency,
the resonant LLC converter topology has gained renewed
attention in recent years. However, given its complexity,
obtaining an accurate model that describes the converter’s
operation is a difficult and time consuming task. Therefore,
simplified calculation methods such as the Fundamental
Harmonic Approximation, FHA, are widely used by the scientific
community. The FHA assumes that, for an operating frequency
equal to the resonance frequency, the resonant tank filters the
higher order harmonics, allowing the use of only the fundamental
components of the electric variables. This is no longer valid for
frequencies different from the resonant frequency, which
introduces errors in the converter design. In this context, arises
the necessity to quantify the error associated to the FHA and
propose a model that, even though it’s simplified as well, obtains
more accurate results.
The design of the converter is done resorting to adequate
bibliography, and later studied with both models, while
simulations are made using the simulation program PSIM.
Finally, the theoretical results are compared with the simulation
results of the designed converter, with 500W and a resonant
frequency of approximately 400kHz, in order the evaluate the
accuracy of the two models.
Index Terms— Fundamental Harmonic Approximation, LLC
Resonant Converter, Power Density, Resonant Conversion,
Resonant Frequency.
I. INTRODUCTION
WITH the current growth of electronic and
telecommunications technologies, arises a necessity that
accompanies that growth, of miniaturizing the equipment,
while maintaining a high efficiency. The combination of these
traits results in a high power density equipment, which is the
desired goal.
However, most electronic applications operate with various
voltage levels that do not correspond to the voltage supplied
by the source, usually a battery or via the electric grid through
an AC-DC converter. This reveals the importance of a DC-DC
converter to adapt the power supply to the loads.
Being so, it is vital to design converters that transform these
voltage levels, maintaining a high efficiency and, at the same
time, reducing the volume to a minimum. This volume
reduction is achieved by increasing the operating frequency of
the converter, which in turn affects the size of the magnetic
components and capacitors. It’s possible to design hard
switching converters, for example, with 100 kHz and 100 or
200W output power, with an efficiency of approximately 85%
[1].
For higher frequencies, the efficiency of these converters
decreases rapidly, due to the increase of switching losses that
are proportional to the frequency, also increasing the
electromagnetic interference, or EMI. Such disadvantages can
be reduced with the insertion of resonant meshes composed by
inductors and capacitors that, by oscillating the voltages and
currents in the switching devices, allow for their zero voltage
switching, ZVS, or zero current switching, ZCS. There are
various types of converters that use the resonant switching
technique, but, in this paper, only the LLC series resonant
converter will be addressed.
Resonant converters are developed with the main objective
of maximizing the power density, by reducing the volume of
components, using a high operating frequency, and
minimizing the switching losses. Their operating principle
consists on energizing a LC circuit at a frequency close to its
resonant frequency in order to oscillate the voltage or current
in the devices, switching them at the zeros of their voltage or
current, leading to zero switching losses, approximately. The
use of resonant meshes has its disadvantages, such as:
- Increase in complexity of the circuit analysis;
- More difficult to control the converter;
- Inherent delay of the switching, as opposed to the hard
switching;
- Oversizing of components due to the increase of the
maximum values of voltages and currents for operating
conditions far from nominal.
However, the potential of the resonant converters to obtain
both high efficiency and high power density has led to an
intensive study in recent years, therefore, it is essential that the
methods used in the analysis and design of these converters
are as exact as possible or, if not so, that the error introduced
by them is quantified and accounted for.
In the context presented, the objectives of this paper are to
analyze one of the main methods used in resonant converters
studies - the Fundamental Harmonic Approximation, or FHA
– and the design of a resonant LLC converter. Specifically, the
objective is to verify the validity of the FHA for different
operating conditions. An analytical model is also proposed
which, hopefully, will produce more precise results than the
FHA.
To that end, partial objectives were established:
- Study the LLC converter through the FHA;
- Design the LLC converter based on specifications and
adequate bibliography;
- Propose an analytical model to study the converter;
- Study the LLC converter with the two models, and
Design of a Resonant LLC Converter
Nuno Oliveira, nº 75133, MEEC
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using a simulator, in order to evaluate both models;
- Compare the theoretical and simulation results, and
evaluate the errors of both models, in respect to the
simulation results.
II. RESONANT LLC CONVERTER
The circuit of the resonant LLC converter under study is
presented in Fig. 1, composed by the following energy
conversion stages:
- The half-bridge inverter, composed by the devices Q1
and Q2, that transforms the input voltage into a
square voltage ;
- The resonant tank, composed by the series inductor ,
the parallel inductor and the resonant capacitor ,
that, when excited by the square voltage at resonant
frequency, filters the high order harmonics and
generates an approximately sinusoidal voltage or
current;
- The rectifier, which consists in the diodes D1 and D2,
that converts the sinusoidal current into a constant
current;
- An output filter, the capacitor , to ensure that the
output voltage is approximately constant.
The output filter is a capacitor because the inductor in
series with the input of the rectifier makes it behave like a
current source which, in turn, forces the output of the rectifier
to act as a voltage source.
Additionally, a high frequency and high power transformer
with center-tap in the secondary is included, to ensure
galvanic insulation and to reduce the secondary voltage
through its turns ratio .
Fig. 1. Resonant LLC converter circuit.
In this point, the following topics will be addressed:
- A brief analysis of the LLC converter’s operation at
resonant frequency and in the continuous conduction
mode, or CCM, is made;
- A bibliographic review is made, focusing on the state-
of-the-art of the LLC converter, specifically, in the
application of the FHA in its study and development;
- The FHA is exposed, focusing on the approximations
made throughout its application;
- The analytical model is introduced.
A. Operation at resonant frequency
The typical waveforms of the LLC converter, for an
operating frequency equal to the resonant frequency and in
CCM operation are presented in Fig. 2, corresponding to the
voltages and currents marked in the circuit in Fig. 1. The CCM
is defined as the operation mode in which the current that goes
through the rectifier has no discontinuities.
Fig. 2. Typical waveforms of the LLC converter at resonant frequency.
The resonant frequency of the LLC converter, , is given
by
√
(1)
The resonant tank input voltage (Fig. 2 (a)) is generated
by the alternate switching of Q1 and Q2 (typically
MOSFETs), and consists in a square wave voltage with
amplitudes of and .
The resonant capacitor retains the average value of the
voltage , with a value of
, and the resulting voltage,
designated by (Fig. 2(b)) is applied at the parallel
inductor . This voltage creates a triangular current (Fig.
2 (c)), since the current in an inductor is given by
∫
(2)
The current in the resonant tank (Fig. 2 (d)), as long as
the frequency of the input voltage – which corresponds to
the operating frequency – is approximately equal to the
resonant frequency, is approximately sinusoidal. This happens
because of the filtering action of the resonant tank, at resonant
frequency, that filters the high order harmonics, remaining the
fundamental frequency, which is a sinusoid.
The voltages in the secondary of the transformer and
(Fig. 2 (e)) are square waveforms, with amplitudes
,
and is equal to but with a phase shift of .
Since they have the same waveform and amplitude, for
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simplicity, we refer only to the secondary voltage , which
we simply designate by . The diodes D1 and D2 rectify
these voltages, originating a nearly constant voltage, which is
then smoothed by the output capacitor . It’s this output
voltage, , that feeds the load .
The currents in diodes D1 and D2 (Fig. 2 (f)), which
correspond to the currents in the secondaries and ,
are rectified currents, equal but with a phase shift of . For the
same reasons described in the paragraph above, we refer only
to one of them, , which is designated by for
simplicity.
The current at the output of the rectifier (Fig. 2 (f)) has an
average value equal to the load current . The output
capacitor mitigates the ripple of this current, making it so that
only the DC component of the current, , feeds the load .
The LLC converter can achieve a high efficiency due to the
oscillation of the current in the resonant tank , the same
that passes through the MOSFETs Q1 and Q2. This oscillation
is obtained through the charging and discharging cycles of the
resonant elements and, combined with the phase shift
introduced by the resonant tank [2] – which is inductive at
resonant frequency – allows for ZVS of the MOSFETs, since
the current lags the input voltage .
B. State-of-the-art
The LLC converter has the potential to operate with high
frequency without a significant decrease of overall efficiency,
due to the ability to operate the switching devices with ZVS or
ZCS. Operating with high frequencies, the volume of the
magnetic components is reduced. Additionally, the non-
idealities of the transformer can be used as elements of the
resonant tank. Specifically, the leakage inductance can be used
as the series inductor and the magnetizing inductance can
be used as the parallel inductor , further decreasing the
overall volume of the converter.
As such, the LLC converter has been the subject of an
intensive study [3], driven by the global struggle to
miniaturize electronic equipment.
The non-linearity and multiresonant operation of the LLC
converter makes it difficult to accurately model it [4, 5],
meaning that obtaining an exact model is a very complex task
[5]. Knowing also that computer assisted modeling requires
solving transcendental equations [5, 6], the design of the LLC
converter – for many applications or experimental
development – is typically made based on approximated
models [7-12], with the FHA being the most commonly used.
In [7], two simplified models are used, based on the FHA,
to study the converter’s behavior, more specifically, to study
the ZVS operating regions. Based on the FHA results, various
expressions can be obtained to design different configurations
of the converter, however, the error introduced by the FHA
isn’t quantified.
In [8], the FHA is used to design an interleaved converter,
which operates with a fixed operating frequency equal to its
resonant frequency, therefore the error introduced by the FHA
is minimum.
In [9-12], the FHA is used to determine the voltage gain of
the LLC converter, in order to size the resonant elements,
switching devices, transformer and output filters. Once again,
the error is not determined.
In [13], the FHA and the extended fundamental harmonic
approximation, or eFHA, are introduced, and a detailed
analysis of the exactness of both models is conducted. The
analysis is done by obtaining of the relative errors of the
calculation of the maximum values of voltages and currents,
when compared to the exact time solution.
The FHA is based on the fact that, for operating frequencies
near the resonant frequency, the resonant tank filters the high
order harmonics, allowing for the consideration of only the
fundamental harmonic of the voltages and currents in the
resonant tank. However, for frequencies away from the
resonant frequency, this model introduces inevitable errors in
the converter’s design [6].
In this paper, similarly to what’s done in [13], the converter
is studied through the FHA. Then, an analytical model that is
proposed in [14] is developed and applied, that assumes only
that the current in the resonant tank is sinusoidal. The
maximum values of the most relevant voltages and currents in
the converter are determined, using both models. Finally, and
in order to evaluate the exactness of both models, the relative
errors are determined by comparing them with simulation
results, obtained using the simulation program PSIM.
This way, the two models are compared in detail, more
precisely, their accuracy in modeling the LLC converter, for a
range of frequencies below and above the resonant frequency.
C. Fundamental Harmonic Approximation
The FHA is used in many studies and applications of
resonant converters, among them, [7-12]. Using this
approximation, only the fundamental harmonics are accounted
for, which are sinusoidal waveforms. Therefore, a
considerable simplification of the circuit to analyze occurs,
due to the fact that a purely sinusoidal operation is considered.
Taking into account that the LLC converter presented in
Fig. 1 is operating at resonant frequency and in CCM mode,
the conditions allow for the application of the FHA. In the
application of this model several approximations are made,
which are explained throughout the study.
As mentioned before, the input tank voltage is a square
voltage waveform that assumes the values and , and the
resonant capacitor retains the average value (or DC
component).
The first approximation is to consider that, at resonant
frequency, the harmonics are all filtered with exception of the
fundamental, which is the only one considered. This
fundamental harmonic, , is given by
(3)
Where corresponds to the angular operating frequency,
(4)
And is the operating, or switching frequency.
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The voltage in the secondary, , is synchronous with the
resonant tank current , which means that, when the current
is positive, the voltage in the secondary is , when it’s
negative, the secondary voltage is . Mathematically,
{
(5)
Being so, the second approximation consists in considering
the secondary voltage to be a square waveform, and the third
approximation to consider only its fundamental harmonic,
which is
(6)
Where represents the phase shift between the
fundamental harmonics of the input voltage and the
secondary voltage [15]. Therefore, referring it to the
primary, the fundamental harmonic of the primary voltage
is obtained as
(7)
The current in the secondary, , is approximately
sinusoidal (according to the FHA) and so, at the output of the
rectifier, the current needs to have an average value equal to
the load current (since the output capacitor retains the
AC component). The current is a rectified sinusoid,
therefore, it is possible to demonstrate that, to have an average
value of , its maximum value has to be
(8)
And this is also the maximum value of the secondary
current :
(9)
The fourth and final approximation made with the FHA is
that, instead of considering the transformer, rectifier, filter and
load, transform this group into an equivalent resistance
that symbolizes the group’s behavior, since the operation is
purely sinusoidal [2, 15]. This resistance is determined
dividing the maximum values of the fundamental harmonics
of the voltage in the input and output of the rectifier [2]:
(10)
It’s important to note that, as pointed out in [15], if the
switching frequency is equal to the resonant frequency, the
impedance of the resonant inductance in series with the
resonant capacitor is null, meaning that the phase shift
is zero, and can be calculated. Mathematically:
(11)
Knowing that the load resistance is given by
(12)
The equivalent resistance is equal to
(13)
These approximations lead to a simplification of the circuit
under study, from the one presented in Fig. 1 to the one
presented in Fig. 3, where all the voltages and currents are
sinusoidal, simplifying the analysis significantly [15]. The
fundamental harmonic of the voltage in the secondary and
the secondary current are obtained dividing and
multiplying by the turns ratio of the transformer,
respectively.
Fig. 3. LLC converter circuit simplified by the FHA
Based on the circuit from Fig. 3, it’s possible to determine
some voltages and currents that are essential to the modeling
of the converter, the currents in the resonant tank and in
the secondary , the current in the parallel inductance
and the voltage drops in the series inductance and in the
resonant capacitor .
The current in the parallel inductance is given by
dividing the fundamental harmonic of the primary voltage
and the inductance’s reactance:
(14)
Knowing that the secondary current’s maximum value is
given by (9), can be determined as
(15)
Applying Kirchhoff’s current law in the node where the two
inductances intersect, the current in the resonant tank is
obtained, at resonant frequency:
(
)
(16)
Since the output capacitor retains the AC component of the
rectifier’s output current , the maximum value of the current
that flows through the output capacitor is
(
) (17)
The voltage drop in a capacitor is determined as
∫ (18)
Being so, and taking into account that the current that flows
in the resonant capacitor is , the voltage drop in the
capacitor is equal to
(
) (19)
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The average value of the voltage in the capacitor is
equal to the average value of the voltage , as can be seen
directly by applying Kirchhoff’s voltage law to the resonant
tank mesh, visible in Fig. 1. Therefore, the constant that
results from the integral is
(20)
Finally, the maximum value of the voltage drop in the
capacitor, , can be determined as
(
)
(21)
Analogously, knowing that that current flowing through the
resonant inductance is also , the voltage drop is
(
) (22)
Meaning that the maximum voltage drop at the resonant
inductor is
(
) (23)
Additionally, it’s important to determine the voltage gain of
the converter. Analyzing Fig. 3, it is possible to determine the
voltage across the equivalent resistance , through a voltage
divider, which translates to
(24)
From this, the voltage gain can be obtained:
(25)
Designating the voltage gain of a DC-DC converter
by
, and using the equations (it is possible to ascertain that
the two voltage gains are related through the following
equation
(26)
With phase shift being null at resonant frequency.
To generalize the study to any LLC converter with the same
topology, it is necessary to normalize the voltage gain given
by (25). Defining the quality factor as [15]
√
(27)
as the ratio between the two resonant inductors,
(28)
And as the normalized angular operating frequency
(29)
Where is the resonant angular frequency. After some
calculations, the normalized voltage gain is obtained, as a
function of the normalized frequency:
|
(
) (
)|
(30)
D. Approximated Analytical Model
In [14], a contactless battery charger is presented, in which
a series compensation in the primary side of the transformer is
used. In order to simplify the modeling of the system, a three
parameter equivalent model is used to describe the
transformer. A schematic of the proposed model is presented
in Fig. 4 (a).
Fig. 4. (a)Three parameter model of the transformer; (b) Coupled
inductors.
However, coherence between this model and the model of
two coupled inductors, presented in Fig. 4 (b), must be
guaranteed, which yields:
[
] [
]
[
] (31)
In which , and represent the self-inductances of
the primary and secondary and the mutual inductance,
respectively.
From (31) derive the equations
{
(32)
Through some manipulation of the equations (32) and
knowing that the mutual inductance is given by, as a
function of the magnetic coupling factor
√ (33)
The voltage can be determined by
√
(34)
Knowing also that the current corresponds to the current
in the resonant tank , analyzing the circuit in Fig. 4 (a), the
voltage is determined as
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(35)
Comparing equations (34) and (35, and taking into account
that the self-inductance of the primary is determined by
measuring the inductance in the primary with the secondary as
an open circuit,
(36)
It is possible to determine the parameters , and
that ensure coherence between the models:
{
√
(37)
The model proposed in [14] can be applied to the converter
in study, considering that the operating frequency is equal to
the resonant frequency. The model was developed knowing
that the current is approximately sinusoidal at resonant
frequency, and the only approximation made is to consider it
to be a pure sinusoid [14].
The resulting circuit is presented in Fig. 5 and the typical
waveforms are presented in Fig. 6. The model is only valid for
CCM operation, because, when the current in the secondary is
zero, the diodes turn off and the secondary voltage is
no longer a square voltage, with amplitude (output voltage),
like the one presented in Fig. 6.
Fig. 5. LLC converter circuit, with the three parameter model of the
transformer.
Fig. 6. Waveforms of the LLC converter. Adapted from [14].
In the application of this model, the leakage inductance
corresponds to the resonant inductor , the magnetizing
inductance corresponds to the parallel inductance and
the capacitor corresponds to the resonant capacitor .
Additionally, the fictitious turns ratio corresponds to the
turns ratio of the transformer.
The angles , and marked in Fig. 6 are defined as the
[14]:
- Angle : phase shift between the fundamental
harmonics of the inverter voltage and the current
in the resonant tank;
- Angle : phase shift between the fundamental
harmonics of the inverter voltage and the secondary
voltage ;
- Angle : phase difference between and :
(38)
The angle of the work [14] corresponds to the phase shift
introduced in the FHA study. Besides assuming that the
resonant tank current is a pure sinusoid, the following
conditions were also established in order to develop the model
[14]:
1) The converter operates at a frequency near the resonant
frequency;
2) The input power is equal to the output power:
(39)
3) The secondary current is null in the instant defined by
:
(
) (40)
4) The average value of the rectifier output current is equal
to the load current :
⟨| |⟩ ⟨ ⟩ (41)
5) The difference between the inverter voltage and the
secondary voltage referred to the primary is equal to
the voltage applied to the resonant circuit defined by the series
of and :
(42)
6) The converter operates in CCM for the current in the
secondary;
7) The output voltage is nearly constant.
Having established these conditions, the converter’s
operation near the resonant frequency can be modeled [14].
Switching the transistors Q1 and Q2 of the inverter
alternately, at a frequency near the resonant frequency , the
AC component of the inverter voltage is
(43)
As defined above, the current in the resonant tank is
approximately a sinus, described as
(44)
Where and have to be determined.
Knowing that the secondary’s voltage is a square voltage of
amplitude , referring it to the primary, the current in
can be determined, through the relation (2). The secondary
voltage is given by
(45)
Finally, the current in the secondary is obtained by
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subtracting the inverter current by the current in and
referring it to the secondary:
(46)
In order to conclude the modeling of the converter, some
parameters have to be calculated. Using the relations (39) to
(46), all the parameters can be determined, after some
mathematical manipulation:
(
)
(
)
(47)
Using the relations (47) in the equations (43) to (46), all the
electrical variables in the circuit can be determined [14]. The
voltages in the resonant capacitor and inductor are determined
through (18) and (22), resulting in
(48)
(49)
The voltage in the primary is obtained by referring the
secondary voltage to the primary:
(50)
The maximum value of the current in the output capacitor
is determined through equation (17). Additionally, the
voltage gain can be obtained by manipulating equations
in (46), resulting in
(51)
III. DESIGN OF THE LLC CONVERTER
In order to obtain results that can be analyzed, the LLC
converter has to be designed. First, some specifications need
to be established. The specifications were obtained by
reviewing the bibliography, and consist in:
- Input voltage ;
- Output voltage - Output power - Operating frequency range: ;
- Output voltage ripple ;
- Resonant frequency To facilitate the sizing of the converter, the three parameter
model of the transformer presented in Fig. 4 and, therefore, the
circuit under study is the one presented in Fig. 5.
In order to design the transformer, some parameters have to
be determined:
- Knowing that the voltage gain is unitary at
resonant frequency and given by (26), the turns ratio
of the transformer can be obtained:
(52)
- The load current is determined knowing that the
output power is equal to
(53)
- The load resistance , given by
(54)
- The maximum value of the current in the secondary
(55)
- The primary current’s maximum value, which equates
to
(56)
The transformer’s magnetizing inductance , which is a
parasitic element, acts as the parallel inductor of the resonant
tank. The magnetizing current , which is the current that
flows through this inductor, circulates in the primary and the
inverter’s devices, without entering the energy transfer [2].
Therefore, it contributes to the conduction losses in the
switching devices.
To initiate the transformer’s sizing, and to reduce the
conduction losses in the input switches, it is imposed that the
magnetizing current has a maximum value, for the entire
operating region, inferior to
(57)
Based on this condition and the relation (2), an expression
can be derived for calculating the magnetizing inductance that
satisfies condition (57), and for the minimum operating
frequency of 300kHz, the resulting inductance is
(58)
Afterwards, the transformer can be designed, following
these steps:
1) Choosing a core:
When choosing a core, the shape and material of the core
have to be studied and selected according to the application. In
this case, due to the high operating frequency, a relatively
small core can be selected, in order to reduce the overall
volume of the converter and the magnetic losses, which are
proportional to the volume. Due to this, and considering that
the ETD class of cores has a wide range of cores, the ETD
39/20/13 is selected.
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2) Number of turns to avoid magnetic saturation:
In order to avoid magnetic saturation, the maximum flux
density has to be inferior to the saturation flux density
of the material, for the entire operating frequency range.
That led, in this case, to a number of turns in the primary
of
(59)
And, consequently, a number of turns in the secondaries of
(60)
3) Determine magnetizing inductance and verify if a gap is
needed:
Knowing the number of turns and the core used, the
magnetizing inductance can be determined, to check if
condition (58) is met. The magnetizing inductance is
calculated by [16]
(61)
The resulting inductance is much higher than pretended,
which means that a gap must be added to the transformer,
because this high magnetizing inductance may lead to a
magnetizing current that is too low, possibly preventing the
ZVS of the inverter’s devices [15].
By adding a gap of approximately of 0,2mm, a
magnetizing inductance of 32 can be obtained.
4) Experimental implementation:
After obtaining the parameters of the desired transformer,
the transformer can be built. In this case, following the above
mentioned, an ETD 39/20/13 core transformer is built, with
the following characteristics:
{
(62)
In Fig. 7 the circuit of the transformer with all the parasitic
elements, more specifically, the leakage inductance and
resistance of the primary and , respectively, and the
leakage inductances and resistances of the secondaries,
and , and and , respectively.
Fig. 7: Circuit of a non-ideal transformer.
These elements are non-idealities that are essential for the
design of the converter, since the primary leakage inductance
can be used as the resonant inductor and the
magnetizing inductance can be used as the parallel
inductor . Measurements made resulted in the following
values for these parameters:
{
(63)
These result in a transformer that meets the condition (58)
and maintains a flux density below the saturation, reducing the
magnetic losses. Therefore, the resonant tank can be sized.
Implementing the primary leakage inductance as the
resonant inductor , the capacity of the resonant capacitor
that leads to the desired resonant frequency of 400kHz is
(64)
However, within the available capacities for capacitors, the
closest one is 0,56 which, using equation (1), leads to a
resonant frequency of
{
(65)
IV. RELATIVE ERRORS
With the designed LLC converter, it’s possible to obtain the
maximum values of the electric variables of the LLC
converter, by applying both models to the converter. The
electric variables that are relevant to this study are the ones
used in the design of the converter.
It is intended to determine the maximum values of these
variables for the designed LLC converter, for the specified
frequency range of 300kHz to 500kHz, in 5kHz intervals,
through both models. Then, the values are compared with
simulation results, obtained through the simulation program
PSIM. The electric variables in question are the voltages in the
resonant capacitor and inductor and
, and the voltages
in the primary and secondary and . The currents used
in the design are the magnetizing current , the current in
the resonant tank and the current in the secondary .
The maximum values, using the FHA, are obtained by
simulating, the equations (6), (7), (21) and (23) to determine
the mentioned voltages, and simulating the equations (14) to
(16) to determine the currents. The voltage gain is calculated
through equation (30).
Through the analytical model, the maximum values of the
voltages are determined simulating the equations (48) and (49)
and simulating the equations (44) and (46) to determine the
currents. The magnetizing current is obtained using the
equations (2) and (50). The voltage gain is calculated through
equation (51).
All the theoretical maximum values given by both models
are obtained using programs developed with the software
MATLAB.
Finally, the accuracy of both models is evaluated by
determining the relative error in the calculation of the
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maximum values, compared to the simulation results. The
Figures 8 to 13 display the relative errors of the FHA in
dashed lines and the analytical model in continuous lines. The
relative errors are presented in function of the operating
frequency and are determined for three loads: nominal load,
half the nominal load and double the nominal load , to
see if this change would impact the accuracy of the models.
Fig. 8. Relative errors of the FHA and the analytical model in
calculating the maximum values of the resonant tank current.
Fig. 9. Relative errors of the FHA and the analytical model in
calculating the maximum values of the magnetizing current.
Fig. 10. Relative errors of the FHA and the analytical model in
calculating the maximum values of the current in the secondary.
Fig. 11. Relative errors of the FHA and the analytical model in
calculating the maximum values of the voltage in the resonant
capacitor.
Fig. 12. Relative errors of the FHA and the analytical model in
calculating the maximum values of the voltage in the resonant
inductor.
Fig. 13. Relative errors of the FHA and the analytical model in
calculating the voltage gain.
1222
10
V. CONCLUSIONS
By analyzing the Figures 8 to 13, it is clear that the
analytical model is generally better in modeling the LLC
converter than the FHA, and for frequencies above the
resonant frequency, the model presents a relative error that is
always below the FHA’s error. However, for frequencies
below the resonant frequency, the FHA is better in
determining, for example, the maximum values of the currents
in the primary and secondary. It is also clear that the change in
the load doesn’t substantially affect the accuracy of the
models.
It is also clear that, for frequencies equal or near the
resonant frequency (around 416kHz), which is the frequency
for which both models are established, the models are quite
accurate, and provide a good basis for designing the converter.
While being accurate near the resonant frequency, as the
operating frequency shifts to frequencies far from the resonant
frequency, significant errors start to occur, which indicates
that these approximate models do not substitute the use of a
simulation tool when designing an LLC converter.
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