ST AN2450 Application note

AN2450

Application note

LLC resonant half-bridge converter design guideline

Introduction

The growing popularity of the LLC resonant converter in its half-bridge implementation (see Figure 1) is due to its high efficiency, low level of EMI emissions, and its ability to achieve high power density. Such features perfectly fit the power supply demand of many modern applications such as LCD and PDP TV or 80+ initiative compliant ATX silver box. One of the major difficulties that engineers are facing with this topology is the lack of information concerning the way the converter operates and, therefore, the way to design it in order to optimize its features.

The purpose of this application note is to provide a detailed quantitative analysis of the steady-state operation of the topology that can be easily translated into a design procedure.

Exact analysis of LLC resonant converters (see [1.] ) leads to a complex model that cannot be easily used to derive a handy design procedure. R. Steigerwald (see [2]) has described a simplified method, applicable to any resonant topology, based on the assumption that input- to-output power transfer is essentially due to the fundamental Fourier series components of currents and voltages.

This is what is commonly known as the "first harmonic approximation" (FHA) technique, which enables the analysis of resonant converters by means of classical complex ac-circuit analysis. This is the approach that has been used in this paper.

The same methodology has been used by Duerbaum (see [3] ) who has highlighted the peculiarities of this topology stemming from its multi-resonant nature. Although it provides an analysis useful to set up a design procedure, the quantitative aspect is not fully complete since some practical design constraints, especially those related to soft-switching, are not addressed. In (see [4] ) a design procedure that optimizes transformer's size is given but, again, many other significant aspects of the design are not considered.

The application note starts with a brief summary of the first harmonic approximation approach, giving its limitations and highlighting the aspects it cannot predict. Then, the LLC resonant converter is characterized as a two-port element, considering the input impedance, and the forward transfer characteristic. The analysis of the input impedance is useful to determine a necessary condition for Power MOSFETs' ZVS to occur and allows the designer to predict how conversion efficiency behaves when the load changes from the maximum to the minimum value. The forward transfer characteristic (see Figure 3) is of great importance to determine the input-to-output voltage conversion ratio and provides considerable insight into the converter's operation over the entire range of input voltage and output load. In particular, it provides a simple graphical means to find the condition for the converter to regulate the output voltage down to zero load, which is one of the main benefits of the topology as compared to the traditional series resonant converter.

October 2007

Rev 5

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www.st.com

Contents	AN2450

Contents

1	FHA circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. 4
2	Voltage gain and input impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	8
3	ZVS constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	14
4	Operation under overload and short-circuit condition . . . . . . . . . . . .	17
5	Magnetic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	18
6	Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	21
7	Design example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	23
8	Electrical test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	26
	8.1 Efficiency measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	26
	8.2 Resonant stage operating waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . .	27
9	Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	31
10	Revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	31

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AN2450	List of figures

List of figures

Figure 1. LLC resonant half-bridge converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Figure 2. FHA resonant circuit two port model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Figure 3. Conversion ratio of LLC resonant half-bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Figure 4. Shrinking effect of l value increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Figure 5. Normalized input impedance magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Figure 6. Capacitive and inductive regions in M - fn plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Figure 7. Circuit behavior at ZVS transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Figure 8. Voltage gain characteristics of the LLC resonant tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Figure 9. Transformer's physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Figure 10. Transformer's APR (all-primary-referred) model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Figure 11. Transformer construction: E-cores and slotted bobbin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Figure 12. LLC resonant half-bridge converter electrical schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Figure 13. Circuit efficiency versus output power at various input voltages. . . . . . . . . . . . . . . . . . . . . 27 Figure 14. Resonant circuit primary side waveforms at nominal dc input voltage and full load . . . . . . 28 Figure 15. Resonant circuit primary side waveforms at nominal dc input voltage and light load . . . . . 28 Figure 16. Resonant circuit primary side waveforms at nominal dc input voltage and no-load . . . . . . 29 Figure 17. Resonant circuit primary side waveforms at nominal dc input voltage and light load . . . . . 29 Figure 18. +200 V output diode voltage and current waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Figure 19. +75 V output diode voltage and current waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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FHA circuit model	AN2450

1 FHA circuit model

The FHA approach is based on the assumption that the power transfer from the source to the load through the resonant tank is almost completely associated to the fundamental harmonic of the Fourier expansion of the currents and voltages involved. This is consistent with the selective nature of resonant tank circuits.

Figure 1. LLC resonant half-bridge converter

Input source	Controlled	Resonant	Ideal	Uncontrolled Low-pass Load
	Switch Network	tank	transformer	rectifier	filter

Vdc

bridge-Half

Driver

Q1

Cr	Lr
	n:1	D1	Cout	Rout
	Irt

Lm	Vout

Q2

D2

The harmonics of the switching frequency are then neglected and the tank waveforms are assumed to be purely sinusoidal at the fundamental frequency: this approach gives quite accurate results for operating points at and above the resonance frequency of the resonant tank (in the continuous conduction mode), while it is less accurate, but still valid, at frequencies below the resonance (in the discontinuous conduction mode).

It is worth pointing out also that many details of circuit operation on a cycle-to-cycle time base will be lost. In particular, FHA provides only a necessary condition for MOSFETs' zerovoltage switching (ZVS) and does not address secondary rectifiers' natural ability to work always in zero-current switching (ZCS). A sufficient condition for Power MOSFETs' ZVS will be determined in Section 3: ZVS constraints still in the frame of FHA approach.

Let us consider the simple case of ideal components, both active and passive.

The two Power MOSFETs of the half-bridge in Figure 1 are driven on and off symmetrically with 50% duty cycle and no overlapping. Therefore the input voltage to the resonant tank vsq(t) is a square waveform of amplitude Vdc, with an average value of Vdc/2. In this case the capacitor Cr acts as both resonant and dc blocking capacitor. As a result, the alternate voltage across Cr is superimposed to a dc level equal to Vdc/2.

The input voltage waveform vsq(t) of the resonant tank in Figure 1 can be expressed in Fourier series:

Equation 1

vsq( t)	Vdc	2	∑
	= -------- + --Vdc
	2	π	n = 1, 3,

1	πfswt)
-- sin( n2
n
5...

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AN2450	FHA circuit model

whose fundamental component vi.FHA(t) (in phase with the original square waveform) is:

Equation 2

v		( t) =	2		sin( 2πf		t)
	iFHA.		--V	dc		sw
			π

where fsw is the switching frequency. The rms value Vi.FHA of the input voltage fundamental component is:

Equation 3

2

viFHA. = ---π---Vdc

As a consequence of the above mentioned assumptions, the resonant tank current irt(t) will be also sinusoidal, with a certain rms value Irt and a phase shift Φ with respect to the fundamental component of the input voltage:

Equation 4

irt( t) = 2Irt sin( 2πfswt – Φ)= 2Irt cosΦ • sin( 2πfswt)–2Irt sinΦ • cos( 2πfswt)

This current lags or leads the voltage, depending on whether inductive reactance or capacitive reactance dominates in the behavior of the resonant tank in the frequency region of interest. Irrespective of that, irt(t) can be obtained as the sum of two contributes, the first in phase with the voltage, the second with 90° phase-shift with respect to it.

The dc input current Ii.dc from the dc source can also be found as the average value, along a complete switching period, of the sinusoidal tank current flowing during the high side

MOSFET conduction time, when the dc input voltage is applied to the resonant tank:

Equation 5

Tsw

--------

2

Iidc. =	1	∫	irt( t)dt=	2
	---------			------Irt cosΦ
	Tsw	0		π

where Tsw is the time period at switching frequency.

The real power Pin, drawn from the dc input source (equal to the output power Pout in this ideal case) can now be calculated as both the product of the input dc voltage Vdc times the average input current Ii.dc and the product of the rms values of the voltage and current's first harmonic, times cosΦ :

Equation 6

Pin = VdcIidc.= ViFHA.Irt cosΦ

the two expressions are obviously equivalent.

The expression of the apparent power Papp and the reactive power Pr are respectively:

Equation 7

Papp = ViFHA.Irt

Pr = ViFHA.Irt sinΦ

Let us consider now the output rectifiers and filter part. In the real circuit, the rectifiers are driven by a quasi-sinusoidal current and the voltage reverses when this current becomes zero; therefore the voltage at the input of the rectifier block is an alternate square wave in phase with the rectifier current of amplitude Vout.

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FHA circuit model	AN2450

The expressions of the square wave output voltage vo.sq(t) is:

Equation 8

Vosq.( t) =	4	∑	1	πfswt – Ψ)
	π Vout		n sin( n2
	--		--

n = 1, 3, 5...

which has a fundamental component vo.FHA(t):

Equation 9

VoFHT.	( t) =	4	πfswt – Ψ)
		--Vout sin( 2
		π

whose rms amplitude is:

Equation 10

2 2 VoFHA. = -----π-----Vout

where Ψ is the phase shift with respect to the input voltage. The fundamental component of the rectifier current irect(t) will be:

Equation 11

irect( t)

where Irect is its rms value.

= 2Irect sin( 2 πfswt – Ψ)

Also in this case we can relate the average output current to the load Iout and also derive the ac current Ic.ac flowing into the filtering output capacitor:

Equation 12

Tsw

--------

2

Iout

=

2

∫

irect( t)

dt=

2 2

Pout

Vout

----

-----

----------Irect

= -----

-----

= -----

------

T

sw

0

π

V

out

R

out

Equation 13

Icac. = Irect2 – Iout2

where Pout is the output power associated to the output load resistance Rout.

Since vo.FHA(t) and irect(t) are in phase, the rectifier block presents an effective resistive load to the resonant tank circuit, Ro.ac, equal to the ratio of the instantaneous voltage and

current:

Equation 14

Roac.	=	voFHA.( t) VoFHA.				=	8	V2out	=	8	Rout
		---i--rect---------	(---t--)---	= ----	I--rect----------		π----2-	--P----out-------		π----2-

Thus, in the end, we have transformed the non linear circuit of Figure 1 into the linear circuit of Figure 2, where the ac resonant tank is excited by an effective sinusoidal input source and drives an effective resistive load. This transformation allows the use of complex acanalysis methods to study the circuit and, furthermore, to pass from ac to dc parameters (voltages and currents), since the relationships between them are well-defined and fixed (see equations Equation 3, Equation 5, Equation 6, Equation 10 and Equation 12 above).

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AN2450	FHA circuit model

Figure 2. FHA resonant circuit two port model

				H(jȦ)
	controlled						rectifier &
	switch						low-pass
	network						filter
dc input					n :1		dc output
Ii.dc	Irt		Cr	Lr		Irect	Iout
Vdc				Lm		Ro.ac	Rout	Vout
							Vo.FHA
	Vi.FHA
				ac resonant tank
	Zin (jȦ)

The ac resonant tank in the two-port model of Figure 2 can be defined by its forward transfer function H(s) and input impedance Zin(s):

Equation 15

	H( s)		=	VoFHA.	( s)	=	1n2Roac. | | sLm
				-V----iFHA.------------(---s---)-			-- --------------------------------------
							n	Zin( s)
Equation 16		ViFHA.( s) =
Zin	( s) =				----1----		+ sLr + n2Roac. | | sLm
			Irt( s)		sCr

For the discussion that follows it is convenient to define the effective resistive load reflected to the primary side of the transformer Rac:

Equation 17

Rac = n2Roac.

and the so-called "normalized voltage conversion ratio" or "voltage gain" M(fsw):

Equation 18

M( fsw)	= n		H( j2πfsw)				VoFHA.
							= n----------------
							ViFHA.

It can be demonstrated (by applying the relationships Equation 3, Equation 10 and Equation 18 to the circuit in Figure 2) that the input-to-output dc-dc voltage conversion ratio is equal to:

Equation 19

Vout	1
---------- =	------M( fsw)
Vdc	2n

In other words, the voltage conversion ratio is equal to one half the module of resonant tank's forward transfer function evaluated at the switching frequency.

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Voltage gain and input impedance	AN2450

2 Voltage gain and input impedance

Starting from Equation 18 we can obtain the expression of the voltage gain:

Equation 20

1

M( fn, λ, Q) = ----------------------------------------------------------------------------

1

λ

2

+ Q

2

f

1 2

+ λ – ------

– ---

f

2

n

fn

n

with the following parameter definitions:

resonance frequency: fr

1

= ----------------------

2π

LrCr

characteristic impedance:

Zo=

Lr

2πfrLr=

1

-----=

----------------

Cr

2

πfrCr

quality factor:

Zo

=

Zo

π2 Z0

Pout

Q = ---------

------------------=

-----------------------

R

ac

n

2

R

8

n

2

V

2

out

.

oac

Lr

inductance ratio: λ = ------

Lm

normalized frequency: fn

fsw

= -------

fr

Under no-load conditions, (i.e. Q = 0) the voltage gain assumes the following form:

Equation 21

MOL( fn, λ )	=	1		λ
			-1-----+-----λ-----–----
				fn2

Figure 3 shows a family of plots of the voltage gain versus normalized frequency. For different values of Q, with λ = 0.2, it is clearly visible that the LLC resonant converter presents a load-independent operating point at the resonance frequency fr (fn = 1), with unity gain, where all the curves are tangent (and the tangent line has a slope -2λ). Fortunately, this load-independent point occurs in the inductive region of the voltage gain characteristic, where the resonant tank current lags the input voltage square waveform (which is a necessary condition for ZVS behavior).

The regulation of the converter output voltage is achieved by changing the switching frequency of the square waveform at the input of the resonant tank: since the working region is in the inductive part of the voltage gain characteristic, the frequency control circuit that keeps the output voltage regulated acts by increasing the frequency in response to a decrease of the output power demand or to an increase of the input dc voltage. Considering this, the output voltage can be regulated against wide loads variations with a relatively narrow switching frequency change, if the converter is operated close to the loadindependent point. Looking at the curves in Figure 3, it is obvious that the wider the input dc

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AN2450	Voltage gain and input impedance

voltage range is, the wider the operating frequency range will be, in which case it is difficult to optimize the circuit. This is one of the main drawbacks common to all resonant topologies.

This is not the case, however, when there is a PFC pre-regulator in front of the LLC converter, even with a universal input mains voltage (85 Vac - 264 Vac). In this case, in fact, the input voltage of the resonant converter is a regulated high voltage bus of ~400 Vdc nominal, with narrow variations in normal operation, while the minimum and maximum operating voltages will depend, respectively, on the PFC pre-regulator hold-up capability during mains dips and on the threshold level of its over voltage protection circuit (about 1015% over the nominal value). Therefore, the resonant converter can be optimized to operate at the load-independent point when the input voltage is at nominal value, leaving to the stepup capability of the resonant tank (i.e. operation below resonance) the handling of the minimum input voltage during mains dips.

Figure 3. Conversion ratio of LLC resonant half-bridge

The red curve in Figure 3 represents the no-load voltage gain curve MOL; for normalized frequency going to infinity, it tends to an asymptotic value M∞:

Equation 22

	M∞ = MOL( fn	→∞, λ )=	1	λ
			1-----+-----

Moreover, a second resonance frequency fo can be found, which refers to the no-load condition or when the secondary side diodes are not conducting (i.e. the condition where the total primary inductance Lr + Lm resonates with the capacitor Cr); fois defined as:

Equation 23

fo =	1		λ
	2----π--------(--L----r---+-----L---m-----)--C-----r	=	fr 1-----+-----λ--

or in normalized form:

Equation 24

fno =	fo	λ
	---=	------------
	fr	1 + λ

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Voltage gain and input impedance	AN2450

At this frequency the no-load gain curve MOL tends to infinity.

By imposing that the minimum required gain Mmin (at max. input dc voltage) is greater than the asymptotic value M∞, it is possible to ensure that the converter can work down to no-load at a finite operating frequency (which will be the maximum operating frequency of the converter):

Equation 25

Mmin	=	2n	Vout	1
			------------------	> ------------
			Vdcmax.	1 +	λ

The maximum required gain Mmax (at min. input dc voltage) at max. output load (max. Pout), that is at max. Q, will define the min. operating frequency of the converter:

Equation 26

Vout

Mmax = 2n-----------------

Vdcmin.

Given the input voltage range (Vdc.min - Vdc.max), three types of operations are possible:

●always below resonance frequency (step-up operations)

●always above resonance frequency (step-down operations)

●across the resonance frequency (shown in Figure 3).

Looking at Figure 4, we can see that an increase of the inductance ratio value λ has the effect of shrinking the gain curves in the M - fn plane toward the resonance frequency fnr (which means the no-load resonance frequency fno increases) and contemporaneously reduces the asymptotic level M∞ of the no-load gain characteristic. At the same time the peak gain of each curve increases.

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