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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 This is no longer valid for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.

REFERENCES

[1] B. Borges, “Fonte de Alimentação Comutada de Topologia Forward

com Controlo em Modo de Corrente”, Relatório de Projecto, CEAUTL,

1986;

[2] Fairchild Semiconductor, “LED Application Design Guide Using Half-

Bridge LLC Resonant Converter for 160W Street Lighting”, 2012;

[3] W. Water, J. Lu, “A Systematic Approach of Resonant Tank Design for

LLC Converters Implemented in Solar Photovoltaic Energy Storage

Systems”, IEEE, APPEEC, 2015;

[4] J. F. Lazar, R. Martinelli, “Steady-State Analysis of the LLC Series

Resonant Converter”, APEC 2001, IEEE, vol. 2, p. 728-735, 2001.

[5] J. F. Lazar, R. Martinelli, “Analysis and Parameter Optimization of

Start-Up Process for LLC Resonant Converter”, IEEE, 2011;

[6] R. Yu, G. K. Y. Ho, B.M. H. Pong, B.W. K. Ling, and J. Lam,

“Computer aided design and optimization of high-efficiency LLC series

resonant converter,” IEEE, vol. 27, no. 7, pp. 3243–3256, Jul. 2012.

[7] C. Oeder, T. Duerbaum, “ZVS Investigation of LLC Converters Based

on FHA Assumptions”, IEEE, pp. 2643-2648, 2013;

[8] Z. Hu, et. al, “An Interleaved LLC Resonant Converter Operating at

Constant Switching Frequency”, IEEE Trans. On Power Electronics,

vol. 29, no. 6, 2014;

[9] M. H. Lokesha, S. G. Srivani, “LLC Resonant Converter Design and

Development”, IEEE India Conference, 2014;

[10] T. Zheng, H. Kim, “Research on the medical Nd: YAG laser power

system using LLC resonant converter”, IEEE AIM, pp. 1694-1698, 2015;

[11] K. Naraharisetti, “Design of Half Bridge LLC Resonant Converter using

Synchronous Rectifier”, IEEE, pp. 135-141, 2015;

[12] H. Park, et. al, “A Novel Adaptive Magnetizing Inductance Control

Scheme for High-Efficiency LLC Resonant Converter for PV

applications”, IEEE, 2015;

[13] C. Oeder, et. al, “A Comparison of Different Design Methods for the

Multiresonant LLC Converter with Capacitive Output Filter”, IEEE,

2010;

[14] H. Marques, B. Borges, “Contactless battery charger with high relative

separation distance and improved efficiency”, IEEE, 2011;

[15] H. Huang, “Designing an LLC Resonant Half-Bridge Power Converter”,

Texas Instruments, Power Supply Design Seminar, 2010;

[16] S. D. Umans, “Máquinas Elétricas”, McGraw Hill, 7ª Ed., 2014.