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 inputto-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 1/32
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
2/32
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
3/32
FHA circuit model AN2450
m
d
d
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
V
dc
Controlle
Switch Network
Q
Half-bridge
Driver
Q
Resonant
tank
1
C
L
r
r
I
rt
Ideal
transformer
n:1
Uncontrolle
rectifier
D
1
L
2
D
2
Low-pass
filter
C
out
Load
R
out
V
out
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
v
(t) is a square waveform of amplitude Vdc, with an average value of Vdc/2. In this case the
sq
capacitor C
voltage across C
The input voltage waveform v
acts as both resonant and dc blocking capacitor. As a result, the alternate
r
is superimposed to a dc level equal to Vdc/2.
r
(t) of the resonant tank in Figure 1 can be expressed in
sq
Fourier series:
Equation 1
V
vsqt()
4/32
---------
2
dc
2
-- -
V
dc
π
n 135.,,=
∑
1
-- -
n2πfswt()sin+=
n
..
AN2450 FHA circuit model
whose fundamental component v
(t) (in phase with the original square waveform) is:
i.FHA
Equation 2
2
t()
-- -
V
π
dc
2πfswt()sin=
of the input voltage fundamental
i.FHA
v
iFHA
.
where fsw is the switching frequency. The rms value V
component is:
Equation 3
2
v
iFHA
.
As a consequence of the above mentioned assumptions, the resonant tank current i
be also sinusoidal, with a certain rms value I
-------
=
V
dc
π
and a phase shift Φ with respect to the
rt
(t) will
rt
fundamental component of the input voltage:
Equation 4
irtt() 2I
2πfswt Φ–()sin 2I
rt
Φcos 2πfswt()2Irt–sin•Φ2πfswt()cos•sin==
rt
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, i
(t) can be obtained as the sum of two contributes, the first
rt
in phase with the voltage, the second with 90° phase-shift with respect to it.
The dc input current I
from the dc source can also be found as the average value, along a
i.dc
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
T
sw
---------
2
1
---------
where T
is the time period at switching frequency.
sw
The real power P
I
=
idc
.
, drawn from the dc input source (equal to the output power P
in
irtt() td
∫
T
sw
0
ideal case) can now be calculated as both the product of the input dc voltage V
average input current I
and the product of the rms values of the voltage and current's first
i.dc
-------
2
I
Φcos=
rt
π
in this
out
times the
dc
harmonic, times cosΦ :
Equation 6
P
inVdc
I
idc
.
V
.
iFHAIrt
Φcos==
the two expressions are obviously equivalent.
The expression of the apparent power P
and the reactive power Pr are respectively:
app
Equation 7
P
=
appViFHAIrt
.
PrV
iFHAIrt
.
Φsin=
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 V
out
.
5/32
FHA circuit model AN2450
4
The expressions of the square wave output voltage v
o.sq
(t) is:
Equation 8
4
t()
-- -
V
=
π
V
.
osq
which has a fundamental component v
out
n135.,,=
o.FHA
∑
(t):
1
-- -
n
..
n2πf
sw
t Ψ–()sin
Equation 9
t()
-- -
V
π
out
2(πfswt Ψ)–sin=
V
oFHT
.
whose rms amplitude is:
Equation 10
22
V
oFHA
.
---------- -
V
=
out
π
where Ψ is the phase shift with respect to the input voltage. The fundamental component of
the rectifier current
(t) will be:
irect
Equation 11
i
t() 2I
rect
where I
is its rms value.
rect
Also in this case we can relate the average output current to the load I
ac current I
flowing into the filtering output capacitor:
c.ac
rect
2(πfswt Ψ)–sin=
and also derive the
out
Equation 12
T
sw
---------
I
out
2
---------
T
sw
2
i
rect
∫
0
t() td
22
---------- -
π
I
rect
P
V
out
out
out
-----------==
R
out
---------- -==
V
Equation 13
2
2
I
–=
out
.
out
where P
Since v
is the output power associated to the output load resistance R
out
o.FHA
(t) and i
(t) are in phase, the rectifier block presents an effective resistive load
rect
to the resonant tank circuit, R
I
cac
, equal to the ratio of the instantaneous voltage and
o.ac
I
.
rect
current:
Equation 14
8
-----
π
V
------------- -
2
P
2
out
out
8
-----
====
R
out
2
π
t()
t()
V
oFHA
.
---------------- -
I
rect
v
.
oFHA
.
oac
---------------------- -
i
rect
R
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).
6/32
AN2450 FHA circuit model
k
2
Figure 2. FHA resonant circuit two port model
H(jȦ)
V
dc
dc input
I
i.dc
controlled
switch
networ
V
i.
FHA
n :1
I
rt
Cr L
L
r
m
I
rect
R
o.ac
rectifier &
low-pass
filter
V
o.
FHA
dc output
I
out
R
out
V
out
(jȦ)
Z
in
ac resonant tank
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 Z
in
(s):
Equation 15
Hs()
V
s()
.
oFHA
------------------------ -
V
s()
iFHA
.
n
1
-- -
--------------------------------------
==
n
||
R
.
oac
Z
sL
m
s()
in
Equation 16
Zins()
iFHA
-----------------------
s()
I
rt
1
--------- sLrn2R
++==
sC
r
oac
||
sL
.
m
V
s()
.
For the discussion that follows it is convenient to define the effective resistive load reflected
to the primary side of the transformer R
ac
:
Equation 17
R
n2R
=
ac
and the so-called "normalized voltage conversion ratio" or "voltage gain" M(f
oac
.
):
sw
Equation 18
V
.
oFHA
Mfsw()nHj2πfsw()n
==
---------------- -
V
.
iFHA
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
V
---------- -
V
out
dc
1
------ -
2n
Mfsw()=
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.
7/32
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
Mfnλ Q,,()
with the following parameter definitions:
---------------------------------------------------------------------------- -=
⎛⎞
1 λ
⎜⎟
⎝⎠
1
2
λ
------ -–+
2
f
n
⎛⎞
Q2f
+
⎝⎠
2
1
----–
n
f
n
resonance frequency:
f
r
characteristic impedance:
Z
o
---------
quality factor:
inductance ratio:
normalized frequency:
Q
R
ac
L
r
λ
------ -=
L
m
f
n
1
----------------------=
2π L
Z
o
Z
o
------------------ -
n2R
oac
.
f
sw
-------=
f
r
rCr
L
r
----- - 2πfrL
C
r
2
Z
π
0
-----
------
===
2
8
n
r
P
out
------------- -
2
V
out
1
---------------- -===
2πfrC
r
Under no-load conditions, (i.e. Q = 0) the voltage gain assumes the following form:
Equation 21
1
MOLfnλ,()
-----------------------------=
1 λ
λ
------ -–+
2
f
n
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 f
(fn = 1), with
r
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
8/32
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 V
the input voltage of the resonant converter is a regulated high voltage bus of ~400 V
- 264 Vac). In this case, in fact,
ac
dc
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 M
frequency going to infinity, it tends to an asymptotic value M
:
∞
; for normalized
OL
Equation 22
M∞MOLfn∞λ,→()
Moreover, a second resonance frequency f
can be found, which refers to the no-load
o
------------ -==
1 λ+
1
condition or when the secondary side diodes are not conducting (i.e. the condition where
the total primary inductance L
+ Lm resonates with the capacitor Cr); fois defined as:
r
Equation 23
-----------------------------------------
f
o
2π L
1
+()C
rLm
r
λ
------------ -==
f
r
1 λ+
or in normalized form:
Equation 24
f
o
----
f
no
f
λ
------------ -==
1 λ+
r
9/32
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 M
the asymptotic value M
, it is possible to ensure that the converter can work down to no-load
∞
(at max. input dc voltage) is greater than
min
at a finite operating frequency (which will be the maximum operating frequency of the
converter):
Equation 25
The maximum required gain M
V
out
M
min
(at min. input dc voltage) at max. output load (max. P
max
------------------ -
2n
V
.
dcmax
>=
1
------------ -
1 λ+
that is at max. Q, will define the min. operating frequency of the converter:
Equation 26
V
out
M
max
Given the input voltage range (V
● always below resonance frequency (step-up operations)
● always above resonance frequency (step-down operations)
● across the resonance frequency (shown in Figure 3).
dc.min
- V
----------------- -
2n
=
V
.
dcmin
), three types of operations are possible:
dc.max
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 - f
(which means the no-load resonance frequency f
reduces the asymptotic level M
of the no-load gain characteristic. At the same time the
∞
plane toward the resonance frequency fnr
n
increases) and contemporaneously
no
peak gain of each curve increases.
out
),
10/32
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