energies
Article
Small-Signal Modeling of Phase-Shifted Full-Bridge Converter
Considering the Delay Associated to the Leakage Inductance
Diego Ochoa 1, * , Antonio Lázaro 1, *, Pablo Zumel 1 , Marina Sanz 1 , Jorge Rodriguez de Frutos 2
and Andrés Barrado 1
1
2
*
Power Electronics System Group, Universidad Carlos III de Madrid, 28911 Leganes, Spain;
pzumel@ing.uc3m.es (P.Z.); cmsanz@ing.uc3m.es (M.S.); barrado@ing.uc3m.es (A.B.)
Power Smart Control S.L., 28919 Leganes, Spain; jorge.rodriguez@powersmartcontrol.com
Correspondence: dochoa@ing.uc3m.es (D.O.); alazaro@ing.uc3m.es (A.L.); Tel.: +34-605-646-713 (D.O.)
Abstract: This paper demonstrates that in the Phase-Shifted Full-Bridge (PSFB) buck-derived converter, there is a random delay associated with the blanking time produced by the leakage inductance.
This random delay predicts the additional phase drop that is present in the frequency response of the
open-loop audio-susceptibility transfer function when the converter shows a significant blanking
time. The existing models of the PSFB converter do not contemplate the delay and gain differences
associated to voltage drop produced in the leakage inductor of the transformer. The small-signal
model proposed in this paper is based on the combination of two types of analysis: the first analysis
consists of obtaining a small-signal model using the average modeling technique and the second analysis consists of studying the natural response of the power converter. The dynamic modeling of the
Phase-Shifted Full-Bridge converter, including the random delay, has been validated by simulations
and experimental test.
Citation: Ochoa, D.; Lázaro, A.;
Zumel, P.; Sanz, M.; Rodriguez de
Frutos, J.; Barrado, A. Small-Signal
Keywords: small-signal model; phase-shifted full-bridge converter; DC-DC power conversion;
zero-voltage-switching; injected-absorbed-current method
Modeling of Phase-Shifted
Full-Bridge Converter Considering
the Delay Associated to the Leakage
Inductance. Energies 2021, 14, 7280.
https://doi.org/10.3390/en14217280
Academic Editor: Nicu Bizon
Received: 30 August 2021
Accepted: 1 November 2021
Published: 3 November 2021
Publisher’s Note: MDPI stays neutral
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
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distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
1. Introduction
In recent years, the Phase-Shifted Full-Bridge (PSFB) converter has been widely used
in several applications such as renewable energy power systems [1,2], energy storage
systems for electric vehicles [3–5], railway applications [6], server power supply unit [7],
telecom power supplies or data centers [8,9], etc.
On the other hand, there is a strong tendency to increase the switching frequency
in order to achieve higher power density, and thus, reduce the volume of the switched
mode power supplies SMPS [10–13]. Increasing the switching frequency of the converter
leads to an increase the switching losses of the MOSFET. Furthermore, problems with
electromagnetic interference, EMI, can be generated. For this reason, it is common to opt
for topologies that could implement some of the soft-switching techniques, such as ZeroVoltage-Switching (ZVS). In the PSFB converter, the primary switches stress is reduced
when the ZVS condition is achieved. Additionally, the ZVS improves the reliability of the
converter [14,15].
The ZVS operation is achieved when the energy stored in the leakage inductance of
the transformer is able to discharge the output capacitor, Coss , before the MOSFET is turned
on [15]. Therefore, the ZVS avoids the overlap of the drain current and the drain-source
voltage in order to reduce switching losses. For this reason, the design of the leakage
inductance is very important in the PSFB converter. On the other hand, this leakage
inductance creates a blanking time interval, which depends on the operating point of the
converter. The blanking time is not a controllable quantity. Currently, many published
works can be found that deal with the design considerations to improve the static behavior
4.0/).
Energies 2021, 14, 7280. https://doi.org/10.3390/en14217280
https://www.mdpi.com/journal/energies
Energies 2021, 14, 7280
2 of 23
of the PSFB converter [15–19]. However, there is a limited number of references that
perform a detailed analysis of its dynamic behavior.
A small-signal model of the PSFB converter based on the buck converter approximation is presented in [15,20], which is a good estimation for cases where the blanking time
interval is small and the output filter inductance, L, is much higher than the transformer
leakage inductance referred to the secondary, n2 · Llk . The small-signal model of the PSFB
converter presented in [21–26] is based on the three-terminal PWM switch model [27]. Although the PWM switch modeling method is a simple way for modeling PWM converters,
the small-signal models of the PSFB converter obtained in [21–26] are a derivation of the
model presented in [15,20].
On the other hand, there are cases in which the leakage inductance is not able to store
the necessary energy to discharge the output capacitors of the MOSFETs, as occurs when
the converter is operating with light load. For this reason, in order to extend the range of the
ZVS, it is necessary to add an extra inductor in series to the transformer leakage inductor.
When the equivalent inductance (leakage inductance plus additional inductance) has a
considerable value with respect to the output filter inductance, a change in the output filter
inductor current slope occurs during the blanking time. Additionally, the blanking time
interval can be noticeably increased even with light load. On the other hand, the blanking
time also depends on the load current, so increasing the load current also increases the
blanking time interval. The small-signal models based on the approximation of the buck
converter could not be accurate, since there is an inherent delay associated to the blanking
time. When the blanking time interval increases, the delay starts to be noticed in certain
transfer functions, such as audio-susceptibility. This delay can be measured experimentally,
and it is discussed in detail in Section 3.
Another small-signal model of the PSFB converter is presented in [28]. This model is
based on a discrete-time modeling of the current waveforms. This approach is conceptually
very accurate, especially in the high frequency range, but it would become complex and
not very versatile model, since the number of state variables increases dramatically, when
external elements such as input filter or/and output post-filter are added to the converter.
Additionally, the explicit transfer functions of the converter are not shown.
Therefore, this paper proposes a new small-signal model of the PSFB converter, based
on an average model, which considers the additional switching stages due to the leakage
inductance. In addition, this model includes a delay term associated with the blanking
time produced by the leakage inductance.
The small-signal model proposed in this paper is based on the combination of two
types of analysis: the first analysis consists of obtaining an averaged model of the PSFB
converter. This averaged model is then linearized and perturbed in order to obtain the
characteristic coefficients of the injected-absorbed-current method (IAC) [29,30]. The IAC
method was chosen to obtain the proposed model of the PSFB converter, since this method
provides an analytical solution and allows the calculation of transfer functions, such as
audio-susceptibility, input impedance and output impedance in a more convenient way.
The characteristic coefficients allow to see the physical effects that each of the perturbations
(input voltage, output voltage and control quantity) produce on the currents of the input
and output ports of the converter. The second analysis consists of studying the natural
response of the power converter and this paper demonstrate that there is an inherent delay
due to the blanking time interval. This random delay predicts the additional phase drop
that is present in the frequency response of the open-loop audio-susceptibility transfer
function when the converter shows a significant blanking time. This random delay has not
been considered in previous works.
Since the proposed model is based on an averaging process, it only predicts accurately
the dynamic behavior up to half the switching frequency. However, in most cases, in order
to design the compensator, this model is accurate enough.
Energies 2021, 14, x FOR PEER REVIEW
Energies 2021, 14, 7280
3 of 2
The converter transfer functions and impedances have been validated by simulatio
3 of 23
and experimentally.
The original contribution of this paper is a new small-signal model of the PSFB con
verter. This model has two main advantages:
transfer functions
and impedances
been
validated
by simulation
- The
Theconverter
model predicts
more accurately
the gainhave
of the
transfer
function
because the ad
and experimentally.
ditional switching stage due to the leakage inductance is taken into account. This
The original contribution of this paper is a new small-signal model of the PSFB
especially important when the leakage inductance (referred to the secondary side o
converter. This model has two main advantages:
-
the transformer) is comparable to the output filter inductance.
The model predicts more accurately the gain of the transfer function because the
Additionally, this new model is capable of predicting the additional phase drop tha
additional switching stage due to the leakage inductance is taken into account. This is
is generated in the frequency response of the open-loop audio-susceptibility transfe
especially important when the leakage inductance (referred to the secondary side of
function
whenisthe
blankingtotime
is significant.
the
transformer)
comparable
the output
filter inductance.
Additionally,
this
new model
capable of
phase coefficients
drop that
The paper is
organized
asisfollows.
Inpredicting
Section 2,the
theadditional
characteristic
of th
generated
in the
frequency
of themodel
open-loop
audio-susceptibility
IACismethod
that
represent
theresponse
small-signal
of the
PSFB converter transfer
are obtained. I
function
when
the blanking
time
significant.
Section
3, the
natural
response
ofisthe
power converter is analyzed in order to complet
-
paper is
organized
as follows.
Section 2,by
thesimulation
characteristic
of the In Sec
the The
proposed
model.
In Section
4, a In
validation
hascoefficients
been performed.
IAC
method
that
represent
the
small-signal
model
of
the
PSFB
converter
are
obtained.
tion 5, experimental validation of the proposed small-signal model of the PSFBInconverte
Section 3, the natural response of the power converter is analyzed in order to complete the
is performed. Finally, conclusions are drawn in Section 6.
proposed model. In Section 4, a validation by simulation has been performed. In Section 5,
experimental validation of the proposed small-signal model of the PSFB converter is
2. Small-Signal
Based
the Average
Technique
performed.
Finally,Model
conclusions
areon
drawn
in SectionModeling
6.
The schematic of the power converter is shown in Figure 1. The converter is operatin
2.inSmall-Signal
Basedmode,
on the CCM,
Average
Modeling
Technique
continuous Model
conduction
and
the zero-voltage-switching,
ZVS, condition
The schematic
of the power
converter isof
shown
in Figure
1. The converter
achieved.
The steady-state
waveforms
the leakage
inductor
current,is𝚤̃operating
, output induc
in
continuous
conduction
mode,
CCM,
and
the
zero-voltage-switching,
condition
tor current referred to primary, 𝑛 ∙ 𝚤̃ , voltage at terminals A-B, 𝑣 ZVS,
, and
secondary vol
e
l Llk , output for th
isage,
achieved.
steady-state
of the leakage
inductor
current,
𝑣 , are The
shown
in Figurewaveforms
2. The blanking
time interval
has
been exaggerated
e
inductor current referred to primary, n·l L , voltage at terminals A-B, veAB , and secondary
analysis. The duty ratio, 𝑑, is determined by the phase shift, ∅, of the gating signals be
voltage, ves , are shown in Figure 2. The blanking time interval has been exaggerated for
tween the switches of the left leg (𝑄 − 𝑄 ) and the right leg (𝑄 − 𝑄 ). The blanking tim
this analysis. The duty ratio, d, is determined by the phase shift, ∅, of the gating signals
· , is caused
byleg
the(Q
leakage
inductance of the transformer. This interval rep
interval,the𝑑 switches
between
of the left
1 − Q3 ) and the right leg (Q4 − Q2 ). The blanking
Tsw
time
interval,
dl · 2 it, istakes
caused
theleakage
leakageinductance
inductance of
transformer.
This interval
resents
the time
forby
the
tothe
change
the direction
of the curren
represents
the time itthe
takes
for the leakage
inductance
to change the The
direction
of the current
flowing through
primary
winding
of the transformer.
secondary
voltage, 𝑣 ,
flowing
through
primarythis
winding
of the
transformer.
ves , isratio [15
maintained
at 0the
V during
interval;
therefore,
thereThe
is asecondary
reductionvoltage,
in the duty
maintained at 0 V during this interval; therefore, there is a reduction in the duty ratio [15].
The blanking time interval depends on the operating point of the converter. During th
The blanking time interval depends on the operating point of the converter. During the
Tsw𝑑) ·
interval
interval, the leakage inductance referred to the second
d)·−
d𝑑e · T2·sw interval
andand
(1 − (1
2 interval, the leakage inductance referred to the secondary
ary output
and output
inductance
in series.
However,
time interval,
and
inductance
are in are
series.
However,
during during
blankingblanking
time interval,
the two the tw
inductors
nono
longer
in series,
so there
is a change
in the output
current current
slope that
is that
inductorsare
are
longer
in series,
so there
is a change
in the output
slope
more
when
the leakage
inductance
referred to
the secondary
equal to or higher
morenoticeable
noticeable
when
the leakage
inductance
referred
to theis secondary
is equal to o
than
the than
output
inductance.
higher
the
output inductance.
Figure1.1.Schematic
Schematic
of the
PSFB
converter.
Figure
of the
PSFB
converter.
Energies 2021, 14, 7280
Energies 2021, 14, x FOR PEER REVIEW
4 of 23
4 of 23
𝟎
𝒕
𝟎
𝒕
𝟎
𝒕
𝟎
𝒕
𝒊𝟑
𝒊𝟐
𝒊𝟏
𝟎
𝒕
−𝒊𝟏
−𝒊𝟐
−𝒊𝟑
𝒗𝒊𝒏
𝒏 · 𝒗𝒊𝒏
𝟎
𝒕
𝟏
−𝒏 · 𝒗𝒊𝒏
𝟏
−𝒗𝒊𝒏
𝟐
Blanking time
𝟐
Figure2.
2. PSFB
PSFB steady-state
steady-state waveforms
waveforms(the
(the blanking
blankingtime
timeinterval
intervalhas
hasbeen
beenexaggerated
exaggeratedto
tohighlight
highlight
Figure
the effect).
effect).
the
The duty
duty ratio
ratio is
is defined
definedby
byEquation
Equation(1)
(1)[15]:
[15]:
The
(1)
𝑑 =
d=
dl 𝑑+ +
de 𝑑
(1)
Equations (2)–(4) are the characteristic values of the leakage and output inductor current Equations
waveform:(2)–(4) are the characteristic values of the leakage and output inductor
current waveform:
𝑣
𝑛 ∙ 𝑣 𝑑 𝑇
𝑖 =vin −n·vo d∙l T∙ sw
(2)
𝐿−
𝐿 · 2
2
i1 =
·
(2)
Llk
L
2 2
𝑛 ∙𝑣 − 𝑛 ∙𝑣
𝑇
(3)
𝑖 = n2 ·vin − n ·vo
∙ 𝑑Tsw
∙
+𝑖
𝐿 + 𝑛 ∙ 𝐿 ·de ·
2 i1
i3 =
+
(3)
2
2
L + n · Llk
𝑛 ∙𝑣
𝑇
𝑖 = 𝑖 − n ·vo
∙ (1 − 𝑑)
(4)
Tsw∙ 2
𝐿
+
𝑛
∙
𝐿
i2 = i3 −
·
1
−
d
(4)
(
)·
2
L + n2 · Llk
It can be inferred from Figure 2 that the averaged rectified voltage is given by Equation It
(5):can be inferred from Figure 2 that the averaged rectified voltage is given by
Equation (5):
2 Z Tsw
(5)
𝑣 =2
∙ 2 𝑣 (𝑡) ∙ 𝑑𝑡
vrec = 𝑇·
verec (t)·dt
(5)
Tsw 0
2
𝑣
= 𝑛 ∙ 𝑣 ∙ 𝑑 + 𝑛 ∙ (𝑖 − 𝑖 ) ∙ 𝐿2 ∙
(6)
vrec = n·vin ·de + n·(i1 − i2 )· Llk ·
(6)
𝑇
Tsw
By replacing
replacingEquations
Equations(1)–(4)
(1)–(4)into
intoEquation
Equation
rectified
voltage
as function
By
(6),(6),
thethe
rectified
voltage
as function
of vof
o,
𝑣
,
𝑣
,
𝑑
and
𝑑
is
obtained:
v , d and d is obtained:
in
l
𝑣
𝐿 ∙ 𝑣 ∙ 𝑛 ∙ 𝑑 − (𝐿 ∙ 𝑣 ∙ 𝑛 + 𝐿 ∙ 𝑣2 ∙ 𝑛 ) ∙ 𝑑 + 𝐿 ∙2𝑣 ∙ 𝑛
L·vin ·n·d − L·vin ·n + Llk ·vo ·n ·dl + Llk ·vo ·n
𝐿 ∙𝑛 + 𝐿
=
=
vrec
· n2
Llk
+L
The average output inductor current is now given by Equation (8):
(7)
(7)
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2021, 14,
14, xx FOR
FOR PEER
PEER REVIEW
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23
55 of
The average output inductor current is now given by Equation (8):
2 22
iL =
= · ∙∙
𝑖𝑖 =
Tsw𝑇𝑇 0
Z
Tsw
2
e
l𝚤̃L𝚤̃ ((𝑡)
t)·∙dt
(𝑡)
∙ 𝑑𝑡
𝑑𝑡
(8)
(8)
(8)
11
(9)
(𝑖 i−
(𝑖−−
+
= 1 ·[(∙∙ i (𝑖
−
− 𝑖 )) ∙ 𝑑 + 𝑖 + 𝑖
(9)
𝑑𝑑 (+
i L𝑖𝑖 ==
i2𝑖𝑖)·))d∙∙+
i2 (𝑖
−
(9)
3 )·𝑖dl ∙+𝑑 i2++𝑖 i3+] 𝑖
222·n∙∙ 𝑛𝑛 1
PuttingEquations
Equations (1)–(4)
(1)–(4) into
into Equation
Equation (9)
(9) and
and solving
solving for
for d𝑑𝑑l , ,,then
thenEquation
Equation(10)
(10)isis
is
Putting
Equations
(1)–(4)
into
Equation
then
Equation
(10)
Putting
obtained.
It
can
be
seen
from
Equation
(10)
that
the
blanking
time
interval
depends
on
the
obtained.
It
can
be
seen
from
Equation
(10)
that
the
blanking
time
interval
depends
on
the
obtained. It can be seen from Equation (10) that the blanking time interval depends on the
operatingpoint
pointconditions:
conditions:
operating
point
conditions:
operating
𝑑𝑑
Tsw · L· Llk · vin ·n2 · d2 − 2·d + vo ·n + 4·i L · L2 · Llk ·n + L· L2lk ·n3
(𝑑 −
(𝐿 ∙∙ 𝐿𝐿 ∙∙ 𝑛𝑛 +
− 22 ∙∙ 𝑑)
𝑑) +
+ 𝑣𝑣 ∙∙ 𝑛𝑛 +
+ 44 ∙∙ 𝑖𝑖 ∙∙ (𝐿
+ 𝐿𝐿 ∙∙ 𝐿𝐿 ∙∙ 𝑛𝑛 )) (10)
(10)
𝑇𝑇 ∙∙ 𝐿𝐿 ∙∙ 𝐿𝐿 ∙∙ 𝑣𝑣 ∙∙ 𝑛𝑛 ∙∙ (𝑑
dl =
(10)
=
=
Tsw
· L∙2𝑣·vin−−𝐿 L2lk∙ 𝑣·vo∙·𝑛n3 −
− 𝐿L∙· L
·∙v𝑣in ·n∙ 2𝑛 ++d𝑑· L∙·𝐿Llk
·vin∙ ·𝑣n2 ∙ 𝑛 )
lk
(𝐿
𝑇
∙
𝐿
∙
𝐿
𝑇 ∙ (𝐿 ∙ 𝑣 − 𝐿 ∙ 𝑣 ∙ 𝑛 − 𝐿 ∙ 𝐿 ∙ 𝑣 ∙ 𝑛 + 𝑑 ∙ 𝐿 ∙ 𝐿 ∙ 𝑣 ∙ 𝑛 )
On
On the
theother
otherhand,
hand, the
the waveform
waveform of
of the
theinput
inputcurrent
currentis
isshown
shownin
inFigure
Figure3,3,
3,so
soits
its
On
the
other
hand,
the
waveform
of
the
input
current
is
shown
in
Figure
so
its
average
value
is
given
by
Equation
(11):
average
value
is
given
by
Equation
(11):
average value is given by Equation (11):
2 22
𝑖 =
= · ∙∙
iin 𝑖=
Tsw𝑇𝑇 0
Z
Tsw
2
e
𝚤̃ ((𝑡)
(𝑡)
𝑑𝑡
l𝚤̃in
t)·dt
∙∙ 𝑑𝑡
11
1 11 ∙∙ (𝑖
=1 [i 𝑖𝑖 ++
+i 𝑖𝑖]·d∙∙ 𝑑
𝑑−
−
𝑖𝑖 d)) ∙∙ 𝑑𝑑
iin𝑖𝑖 ==
·(2i2(𝑖+ +
i+3 )·
−
3
2
l
22 1
22
(11)
(11)
(11)
(12)
(12)
(12)
̃𝒊𝒏
̃𝒊𝒏
𝒊𝒊𝟑𝟑
𝒊𝒊𝟏𝟏
𝟎
𝟎
𝒕𝒕
−𝒊𝟐𝟐
−𝒊
Figure3.
3.Input
Inputcurrent
currentwaveform.
waveform.
Figure
3.
Input
current
waveform.
Figure
Theaverage
averagemodel
modelof
ofthe
thePSFB
PSFBconverter
converterisis
isshown
shownin
inFigure
Figure4.4.
4.
The
average
model
of
the
PSFB
converter
shown
in
Figure
The
Figure 4.
4. Averaged
Averaged model
model of
of the
the PSFB
PSFB converter.
converter.
Figure
Figure 4. Averaged model of the PSFB converter.
Replacing Equation
Equation (10) into
into Equation (7),
(7), the average
average rectified voltage
voltage as function of
of
Replacing
Replacing
Equation (10)
(10) into Equation
Equation (7),the
the averagerectified
rectified voltageasasfunction
function
the
duty
ratio,
𝑑,
input
voltage,
𝑣
,
output
voltage,
𝑣
,
and
output
inductor
current,
𝑖
the
duty
ratio,
𝑑,
input
voltage,
𝑣
,
output
voltage,
𝑣
,
and
output
inductor
current,
𝑖
of the duty ratio, d, input voltage, vin , output voltage, vo , and output inductor current,,,
has been
been obtained.
obtained. After
After linearizing
linearizing and
and perturbing,
perturbing, then
then the
the small-signal
small-signal of
of the
the output
output
has
Energies 2021, 14, 7280
6 of 23
i L , has been obtained. After linearizing and perturbing, then the small-signal of the
output inductor voltage is obtained. The constant coefficients of Equation (14) are given in
Appendix A.
v̂ L = v̂rec − v̂o
(13)
v̂ L = Kvld ·dˆ + Kvlvi ·v̂in + Kvlvo ·v̂o + Kvlil ·lˆL
(14)
From the circuit of Figure 4, Equation (15) is obtained:
lˆL =
v̂ L
ZL (s)
(15)
Replacing Equation (15) into Equation (14), the output inductor current as function of
d,ˆ v̂in and v̂o is obtained:
lˆL =
Kvld
Kvlvi
Kvlvo
·dˆ +
·v̂in +
·v̂o
ZL (s) − Kvlil
ZL (s) − Kvlil
ZL (s) − Kvlil
(16)
Therefore, the characteristic coefficients of the output port are finally obtained:
Ao (s) =
Kvld
ZL (s) − Kvlil
Bo (s) = −
Co (s) =
Kvlvo
ZL (s) − Kvlil
Kvlvi
ZL (s) − Kvlil
(17)
(18)
(19)
Replacing Equations (2)–(4) and (10) into Equation (12), the average input current as
function of the duty ratio, d, input voltage, vin , output voltage, vo , and output inductor current, i L , has been obtained. After linearizing and perturbing, the characteristic coefficients
of the input port are obtained. The constant coefficients of Equations (20)–(22) are given in
Appendix A.
Ai (s) = Kiid + Kiidl ·Kdld + Kiidl ·Kdlil · Ao (s)
(20)
Bi (s) = −[Kiivo + Kiidl ·Kdlvo − Kiidl ·Kdlil · Bo (s)]
(21)
Ci (s) = Kiivi + Kiidl ·Kdlvi + Kiidl ·Kdlil ·Co (s)
(22)
3. Natural Response of a PSFB Converter
It is necessary to study the natural response of the PSFB converter in order to obtain
a complete small-signal model. In this section, the open-loop output inductor current response and the open-loop input current response due to a small input voltage perturbation,
small output voltage perturbation and small duty ratio perturbation is analyzed.
First, the output inductor current dynamic response to a small input voltage step
change is analyzed. The output voltage and duty ratio are assumed to remain constant.
If a small input voltage step occurs during the blanking time interval, then the output
inductor current does not change immediately since the output inductor current slope,
m3 , during this interval does not depend on the input voltage as shown in Figure 5a. The
term ∆i L represents the difference between the instantaneous output inductor current and
the operating point output inductor current. The variation of the output inductor current
occurs after the blanking time interval has elapsed, as shown in Figure 5b. Therefore, there
is a delay associated to the blanking time, since it depends on the moment in which the
input voltage perturbation is injected until the interval de · T2sw begins. If the input voltage
perturbation occurs in interval de · T2sw , the output inductor current varies immediately, since
the output inductor current slope, m1 , during this interval depends on the input voltage.
Therefore, there is no delay associated with this interval. Finally, if the input voltage
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77 of
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, there is no delay, since the rectiage perturbation
occurs
during
interval
−T𝑑)
sw ·
perturbation
occurs
during
the the
interval
(1 −(1d)·
2 , there is no delay, since the rectified
fied voltage
during
interval
depend
on the
input
voltage,
furthermore,
voltage
during
this this
interval
doesdoes
not not
depend
on the
input
voltage,
andand
furthermore,
if
a constant
duty
ratiois isconsidered,
considered,then
thenthis
thisinterval
intervaldoes
doesnot
notpresent
presentdynamics,
dynamics, and
and
aif constant
duty
ratio
during
this
interval
a delay
in the
something similar
similar happens
happensin
inthe
thebuck
buckconverter,
converter,since
since
during
this
interval
a delay
in
audio-susceptibility
transfer
function
is notisassociated.
Later,Later,
it willitbe
verified
by simuthe
audio-susceptibility
transfer
function
not associated.
will
be verified
by
sw
simulation
that(1
the−(𝑑)
1 −· d)· Tinterval
does
introduce
a delay.
does
notnot
introduce
a delay.
lation that the
2 interval
It takes time “ ” for
the current to react
Immediate variation
Figure 5.
5. Dynamic
Dynamicresponse
responsetotoan
aninput
inputvoltage
voltage
step
change.
Output
inductor
current,
(b) increFigure
step
change.
(a)(a)
Output
inductor
current,
(b) incremenmental
component
of
output
inductor
current,
(c)
input
current
and
(d)
primary
and
secondary
tal component of output inductor current, (c) input current and (d) primary and secondary voltage.
voltage.
The input current dynamic response to a small input voltage step change is analyzed.
The input
dynamic
response
to a the
small
input voltage
step change
analyzed.
If a small
inputcurrent
voltage
step occurs
during
blanking
time interval,
thenis the
input
If a small
input voltage
step occurs
thecurrent
blanking
time
interval,
the input
curcurrent
changes
immediately
since during
the input
slope
during
thisthen
interval
depends
rent
changes
immediately
since
the
input
current
slope
during
this
interval
depends
on
on the input voltage as shown in Figure 5c. If the input voltage perturbation occurs in the
Tsw
the
input
voltage
as
shown
in
Figure
5c.
If
the
input
voltage
perturbation
occurs
in
the
interval de · 2 , the input current varies immediately since the input current slope during
this
interval
on the
input voltage.
Therefore, there
noinput
delaycurrent
associated
with
this
interval
𝑑 ·depends
, the input
current
varies immediately
sinceisthe
slope
during
Tsw
interval.
If
the
input
voltage
perturbation
occurs
during
the
interval
1
−
d
,
there
is
(
)·
this interval depends on the input voltage. Therefore, there is no delay associated
with
2
no
delay,
since
it
has
the
same
effect
as
the
case
of
the
output
inductor
current.
this interval. If the input voltage perturbation occurs during the interval (1 − 𝑑) · ,
In order to check the effect that the input voltage perturbation has on the output
there is no delay, since it has the same effect as the case of the output inductor current.
inductor current, some simulations have been carried out using the PSIM software [31]. In
In order to check the effect that the input voltage perturbation has on the output inthese simulations, a periodic perturbation with fixed frequency is considered in order to
ductor current, some simulations have been carried out using the PSIM software [31]. In
observe how this delay that is present in the time domain affects the frequency response.
these simulations, a periodic perturbation with fixed frequency is considered in order to
In Figure 6, the schematic of the PSIM simulation is shown. The parameters of the first
observe how this delay that is present in the time domain affects the frequency response.
simulation are Vin = 45 V, Fsw = 100 kHz, Vo = 5 V, IL = 4.1 A, Llk = 3 µH, L = 36 µH,
6, D
the=schematic
of theofPSIM
simulation
is shown.
parameters
of the first
nIn=Figure
0.5 and
0.3. The ripple
the output
inductor
currentThe
is 200
kHz.
simulation are 𝑉 = 45 V, 𝐹 = 100 kHz, 𝑉 = 5 V, 𝐼 = 4.1 A, 𝐿 = 3 μH, 𝐿 = 36 μH,
𝑛 = 0.5 and 𝐷 = 0.3. The ripple of the output inductor current is 200 kHz.
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2021,
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PEER
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Figure 6. PSIM schematic of the PSFB converter in which a sinusoidal perturbation is injected into
the input voltage.
Figure 6.
of of
thethe
PSFB
converter
in which
a sinusoidal
perturbation
is injected
Figure
6.PSIM
PSIMschematic
schematic
PSFB
converter
in which
a sinusoidal
perturbation
is into
injected into
the
input
voltage.
the input
voltage. perturbation, 𝑣 , with a frequency of 80 kHz is injected into the input
A sinusoidal
voltage, as can be seen in Figure 6. The output inductor current is passed through a bandA sinusoidal perturbation, vsine , with a frequency of 80 kHz is injected into the input
sinusoidal
perturbation,
𝑣 is 80
, with
frequency
ofphase
80 kHz
is injected
the input
passAfilter
whose central
frequency
kHz,aand
thus, the
difference
thatinto
exists
voltage, as can be seen in Figure 6. The output inductor current is passed through a
betweenas
input
andoutput
the output
inductor
current
analyzed.
In thisa bandvoltage,
cansinusoidal
be seen inperturbation
Figure 6. The
inductor
current
is is
passed
through
bandpass filter whose central frequency is 80 kHz, and thus, the phase difference that exists
simulation,
the output
voltage
and theisduty
ratio are
constant,
so phase
the small
perturbation
pass
filter
whose
central
frequency
80
kHz,
and
thus,
the
difference
that exists
between input sinusoidal perturbation and the output inductor current is analyzed. In this
of
𝑣
and
𝑑
are
equal
to
0.
Therefore,
the
input
voltage
to
output
inductor
current
transbetween
input
sinusoidal
perturbation
and
theare
output
inductor
is analyzed. In this
simulation,
the output
voltage
and the duty
ratio
constant,
so thecurrent
small perturbation
fer function
is analyzed.
ˆ
simulation,
the
output
voltage
and
the
duty
ratio
are
constant,
so
the
small
perturbation
of v̂o and d are equal to 0. Therefore, the input voltage to output inductor current transfer
The
blanking
time
is
small
and
the
(1
−
𝑑)
·
interval
is
predominant,
as
can
be transfunction
is analyzed.
of
𝑣 and
𝑑 are equal to 0. Therefore, the input voltage to output inductor current
T
sw
seen
in Figure
7a.time
In this
test, the
pole,
and the phase
The
blanking
is small
andconverter
the (1 − dhas
interval
is predominant,
as difference
can be seenat
)· 2a single
fer
function
is analyzed.
high
frequencies
is test,
expected
to tend to
−90°.
The resulting
phase
difference
between
the
in
Figure
7a.
In
this
the
converter
has
a
single
pole,
and
the
phase
difference
at high
The blanking time is small and
the (1 − 𝑑) ·
interval is predominant,
as can be
input sinusoidal
perturbation
and
the◦ . output
inductor
current
is approximately
frequencies
is expected
to tend to
−90
The resulting
phase
difference
between the−90°,
inputas
seen
in seen
Figure
7a. In this
test,
the
converter
has
a single
pole,
and the
difference
at
◦ ,this
can be
in Figure
7b.
7c shows
the output
inductor
current.
From
test,
sinusoidal
perturbation
andFigure
the output
inductor
current
is approximately
−90phase
as can
beit
high
frequencies
isthat
expected
to tend
to𝑑)
−90°.
The
phase
difference
between
the
seen
Figure
7b. Figure
7c shows
the
current.
From athis
test,init the
can
be
can in
be
concluded
the
interval
(1output
−
· inductor
doesresulting
not
introduce
delay
freTsw the output inductor current is approximately −90°, as
input
sinusoidal
perturbation
and
1 − d)·predominant.
concluded
that the interval
a delay
the frequency
response
quency response
despite (being
In this test,
the indelay
introduced
by the
2 does not introduce
despite
being
predominant.
In
this
test,
the
delay
introduced
by
the
blanking
time
is
very
can
be
seen
in
Figure
7b.
Figure
7c
shows
the
output
inductor
current.
From
this
blanking time is very small because the blanking time interval is less than one-sixth of
the test, it
small
because
the
blanking
time
interval
is
less
than
one-sixth
of
the
switching
period.
can
be concluded
does not introduce a delay in the freswitching
period. that the interval (1 − 𝑑) ·
quency response despite being predominant. In this test, the delay introduced by the
blanking time is very small because the blanking time interval is less than one-sixth of the
switching period.
Figure7.7.Simulation
Simulationresults
resultsofofTest
Test1.1.(a)
(a)Primary
Primaryand
andrectified
rectifiedvoltage.
voltage.(b)
(b)Attenuated
Attenuatedsinusoidal
sinusoidal
Figure
perturbation
and
output
inductor
current
after
bandpass
filter.
(c)
Output
filter
inductor
current.
perturbation and output inductor current after bandpass filter. (c) Output filter inductor current.
Forthe
thesecond
secondtest,
test,the
theoutput
outputvoltage,
voltage,V𝑉
andthe
theduty
dutyratio,
ratio,D,
𝐷,have
havebeen
beenchanged
changed
For
o , ,and
to44VVand
and0.8,
0.8,respectively.
respectively.The
ThePSIM
PSIMschematic
schematicofofFigure
Figure66isisstill
stillvalid
validfor
forthis
thistest.
test.The
The
to
blankingtime
timeisisnoticeably
noticeablyincreased,
increased,as
ascan
canbe
beseen
seenininFigure
Figure8a.
8a.The
Theresulting
resultingphase
phase
blanking
Figure 7. Simulation results of Test 1. (a) Primary and rectified voltage. (b) Attenuated sinusoidal
difference between
the input
sinusoidal
perturbation
the(c)output
current
perturbation
and output
inductor
current after
bandpassand
filter.
Outputinductor
filter inductor
current.
For the second test, the output voltage, 𝑉 , and the duty ratio, 𝐷, have been changed
to 4 V and 0.8, respectively. The PSIM schematic of Figure 6 is still valid for this test. The
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9 of 23
difference between the input sinusoidal perturbation and the output inductor current is
is greater than −90◦ , as can be seen in Figure 8b. Figure 8c shows the output inductor
greater than −90°, as can be seen in Figure 8b. Figure 8c shows the output inductor current.
current. From this test, it can be concluded that the delay that appears in the time domain
From this test, it can be concluded that the delay that appears in the time domain also
also affects the frequency response. Having a considerable blanking time introduces an
affects the frequency response. Having a considerable blanking time introduces an addiadditional phase drop in the input voltage to the output inductor current transfer function.
tional phase drop in the input voltage to the output inductor current transfer function.
Figure8.8.Simulation
Simulationresults
resultsofofTest
Test2.2.(a)
(a)Primary
Primaryand
andrectified
rectifiedvoltage.
voltage.(b)
(b)Attenuated
Attenuatedsinusoidal
sinusoidal
Figure
perturbation and output inductor current after bandpass filter. (c) Output filter inductor current.
perturbation and output inductor current after bandpass filter. (c) Output filter inductor current.
Summarizing,the
thedelay
delayonly
onlyaffects
affectsthe
theoutput
outputinductor
inductorcurrent
currentininthe
thecase
caseofofsmall
small
Summarizing,
inputvoltage
voltageperturbation,
perturbation,
characteristic
coefficient
of the
output
is affected
input
soso
thethe
characteristic
coefficient
of the
output
portport
is affected
by
by delay
the delay
is given
by Equation
Expression
is valid.
still valid.
the
and and
is given
by Equation
(23).(23).
Expression
(22) (22)
is still
𝐾
𝐶 (𝑠) = 𝑒 ∙ ∙ Kvlvi
(23)
Co (s) = e−s·td · 𝑍 (𝑠)
(23)
−𝐾
ZL (s) − Kvlil
In order to quantify the value of 𝑡 in the frequency domain, the AC sweep tool of
In order
to quantify
the value
td in the frequency
domain,
the AC
toolvoltage
of the
the PSIM
software
has been
used.ofTherefore,
the frequency
response
ofsweep
the input
PSIM
software
has been
used.transfer
Therefore,
the frequency
input voltage
to
to output
inductor
current
function
has been response
obtained of
forthe
different
operating
output
inductor
current
transfer
function
has
been
obtained
for
different
operating
points,
points, OP. The simulation parameters are 𝐹 = 100 kHz , 𝑉 = 4 V , 𝐿 = 3 μH , 𝐿 =
OP.
The 𝑛simulation
kHz, of
Vo input
= 4 V,
Llk = and
3 µH,
L = currents
36 µH,
sw = 100values
36 μH,
= 0.5 andparameters
𝐷 = 0.689. are
The Fdifferent
voltages
output
nare
= shown
0.5 andinDTable
= 0.689.
The
different
values
of
input
voltages
and
output
currents
are
1.
shown in Table 1.
Table 1. Parameters to obtain the frequency response at different operating points.
Table 1. Parameters to obtain the frequency response at different operating points.
Operating Point
𝑽𝒊𝒏
𝑂𝑃
30 V
Vin
𝑰𝑳
21 𝐴
IL
𝑫𝒍
0.42
Dl
𝒕𝒅
td
𝑇𝑠𝑤
OP1
30 V
21 A
0.42
0.30· D2l · Tsw
2
𝑇𝑠𝑤Tsw
OP2
32 A 0.486
0.486
0.35
𝑂𝑃
40 V 40 V 32 𝐴
0.35 · 𝐷
· · Dl · 2
2
OP3
50 V
44 A
0.527
0.40𝑇𝑠𝑤
· Dl · Tsw
2
𝑂𝑃
50 V 60 V 44 𝐴
0.527
0.40 · 𝐷 ·
Tsw
OP4
55 A
0.554
0.42· D2l · 2
𝑇𝑠𝑤
𝑂𝑃
60 V
55 𝐴
0.554
0.42 · 𝐷 ·
2
The comparison of the frequency response obtained from the simulation and proposed
modelThe
is shown
in Figure
9a,b.
There isresponse
an excellent
agreement
between
the proposed
comparison
of the
frequency
obtained
from the
simulation
and promodel
the is
simulation
Table
1, it can
seen thatagreement
the value of
td is notthe
fixed,
posedand
model
shown inresult.
FigureIn9a,b.
There
is anbeexcellent
between
proTsw
which
is a randomresult.
delay.InHowever,
range
from
0 tothe
dl ·value
.
posedmeans
modelthat
andthere
the simulation
Table 1, its
it can
be is
seen
that
of
𝑡
is
2
In
order
to
design
a
control
loop
that
depends
on
the
audio-susceptibility
transfer
not fixed, which means that there is a random delay. However, its range is from 0 to 𝑑 ·
function,
the largest delay should be considered, which occurs when td is equal to the
.
blanking time. In this way, the stability of the control loop can be guaranteed.
Operating Point
0.30 · 𝐷 ·
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10 of 23
Figure 9. Comparison of the frequency response. (a) Operating point 1 and 3. (b) Operating point 2
and 3.
Figure
Comparison
the
frequency
response.
(a)
Operating
point
1 and
Operating
point
In9.9.
order
to design
afrequency
control
loop
that (a)
depends
on point
the
audio-susceptibility
transfer
Figure
Comparison
ofofthe
response.
Operating
1 and
3. 3.
(b)(b)
Operating
point
22
and
3.
function,
the
largest
delay
should
be
considered,
which
occurs
when
𝑡
is
equal
to
the
and 3.
blanking time. In this way, the stability of the control loop can be guaranteed.
In output
order
design current
acurrent
control
loop that
depends
the output
audio-susceptibility
transfer
The
dynamic
response
totoaon
The
outputtoinductor
inductor
dynamic
response
asmall
small
outputvoltage
voltagestep
stepchange
change
the
largest
delay
should
be
considered,
which
occurs
when
𝑡
is
equal
tothe
the
isisfunction,
analyzed.
If
a
small
output
voltage
step
occurs
in
any
of
the
three
intervals,
then
analyzed. If a small output voltage step occurs in any of the three intervals, then
the
blanking
time.
In
this
way,
the
stability
of
the
control
loop
can
be
guaranteed.
output
inductor
current
changes
immediately,
since
the
output
inductor
current
slope
output inductor
changes immediately, since the output inductor current slope aloutput
inductor
current
dynamic
response
to a10a,b.
small
output
voltage
step
always
depends
the
output
voltage,
as shown
in Figure
10a,b.
Therefore,
is
no
ways The
depends
onon
the
output
voltage,
as shown
in Figure
Therefore,
therethere
is nochange
delay
is
analyzed.
If
a
small
output
voltage
step
occurs
in
any
of
the
three
intervals,
then the
delay
associated
the output
associated
to thetooutput
port. port.
output inductor current changes immediately, since the output inductor current slope always depends on the output voltage, as shown in Figure 10a,b. Therefore, there is no delay
associated to the output port.
Immediate variation
Immediate variation
It takes time “ ” for the
current to react
It takes time “ ” for the
current to react
Figure 10. Dynamic response to an output voltage step change. (a) Output inductor current, (b)
Figure 10. Dynamic response to an output voltage step change. (a) Output inductor current, (b) increincremental component of output inductor current, (c) input current and (d) primary and secondary
mental component of output inductor current, (c) input current and (d) primary and secondary voltage.
voltage.
Figure 10. Dynamic response to an output voltage step change. (a) Output inductor current, (b)
incremental component of output inductor current, (c) input current and (d) primary and secondary
voltage.
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23
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The input current dynamic response to a small output voltage step change is anaThe input current dynamic response to a small output voltage step change is analyzed.
lyzed. If a small output voltage step occurs during the blanking time interval, then the
If a small output voltage step occurs during the blanking time interval, then the input
input current does not change immediately since the input current slope does not depend
current does not change immediately since the input current slope does not depend on
on the output voltage, as shown in Figure 10c. The variation of the input current occurs
the output voltage, as shown in Figure 10c. The variation of the input current occurs
after the
the blanking
blanking time
time interval
interval has
has elapsed.
elapsed. Therefore,
Therefore, there
there is
is aa delay
delay associated
associated to
to the
the
after
T
blanking time.
time. If
the input
input
blanking
If the
the output
output voltage
voltage perturbation
perturbation occurs
occursin
inthe
theinterval
interval 𝑑de ·· 2sw ,, the
current varies
varies immediately
immediately since
since the
the input
input current
current slope
slope during
during this
this interval
interval depends
depends on
on
output voltage.
voltage. Therefore,
Therefore, there
there is
is no
no delay
delay associated
associated with
with this
this interval.
interval. If
If the
the output
output
the output
voltage
the interval
interval (1
(1 − d𝑑))··T2sw ,,there
thereisisno
no delay,
delay,since
sinceitit has
has
voltage perturbation occurs during the
the
same
effect
as
the
case
of
the
output
inductor
current
response
when
an
input
voltage
the same effect as the case of the output inductor current response when an input voltage
step
step is
is applied.
applied.
Summarizing,
Summarizing, the
the delay
delay only
only affects
affects the
the input
input port
port in
in the
the case
case of
of small
small output
output voltage
voltage
perturbation,
so
the
characteristic
coefficient
of
the
input
port
is
affected
by
the
delay
perturbation, so the characteristic coefficient of the input port is affected by the delay and
and
is
is given
given by
by Equation
Equation (24).
(24). Expression
Expression (18)
(18) is
is still
still valid:
valid:
B𝐵i (s(𝑠)
e−s·td∙ ·[K
Kiidl
) ==−−𝑒
∙ iivo
𝐾 ++K𝐾
𝐾 −−
𝐾 ·K∙dlil
𝐾 · Bo∙ (𝐵s)](𝑠)
iidl · K∙dlvo
(24)
(24)
Finally, the
the output
output inductor
inductor current
current dynamic
dynamic response
response and
and input
input current
current dynamic
dynamic
Finally,
response
is
analyzed,
when
a
perturbation
is
introduced
into
the
duty
ratio,
∆𝑑.
For this
this
response is analyzed, when a perturbation is introduced into the duty ratio, ∆d. For
analysis, itit isisassumed
assumedthat
thatthe
theduty
dutyratio
ratioperturbation,
perturbation,∆d,
∆𝑑,
applied
interval
analysis,
is is
applied
in in
thethe
interval
de · T𝑑2sw ,·
, since
is a small-signal
perturbation
haseffect
an effect
onPWM
the PWM
since
it is it
a small-signal
perturbation
and and
has an
on the
reset.reset.
The perturbation of the duty ratio shown
shown in Figure
Figure 11
11 has
has been
been exaggerated
exaggerated in
in order
order
to clearly
clearly show
show the
the effect
effect that
that causes
causes on
on the
the waveforms
waveforms of
of the
the converter.
converter. The
The duty
duty ratio
ratio
veAB 𝑣voltage
perturbation
immediate
change
in the
as shown
in Figure
11a. Since
perturbationleads
leadstotoanan
immediate
change
in the
voltage
as shown
in Figure
11a.
Tsw
e
e
this
change
in
occurs
in
the
d
·
interval
then
the
secondary
voltage,
,
also
changes
v
v
s voltage, 𝑣 ,
AB in 𝑣
occurse in2 the 𝑑 ·
interval then the secondary
Since this change
e
e
l
l
as
shown
in
Figure
11b.
Therefore,
and
currents
change
instantaneously,
as can
be seen
L
in
also changes as shown in Figure 11b. Therefore, 𝚤̃ and 𝚤̃ currents change
instantanein
Figure
11c,d.
There
is
no
delay
in
the
input
and
output
port.
Expressions
(17)
and
ously, as can be seen in Figure 11c,d. There is no delay in the input and output port.(20)
Exare
still valid.
pressions
(17) and (20) are still valid.
Blanking time
Immediate
variation
Figure 11.
11. Dynamic
Dynamic response
response to
to aa duty
duty ratio
ratio step
step change.
change. (a)
at terminal
terminal A–B,
A–B, (b)
(b) secondary
secondary
Figure
(a) Voltage
Voltage at
voltage,
(c)
output
inductor
current
and
(d)
input
current.
voltage, (c) output inductor current and (d) input current.
In conclusion,
conclusion, from this
this analysis,
analysis, it
it has
has been
been shown
shown that
that the
the blanking
blanking time
time creates
creates aa
delay in the
the input
input current
current when
when there
there is
is aasmall
smalloutput
outputvoltage
voltageperturbation.
perturbation. This
This delay
delay
Likewise,the
theblanking
blanking time
time generates
generates aa delay
delay in
in the
the
directly affects the coefficient B𝐵i ((𝑠).
s). Likewise,
This
delay
dioutput inductor current
current when
whenthere
thereisisaasmall
smallinput
inputvoltage
voltageperturbation.
perturbation.
This
delay
rectly affects
the moment
moment
directly
affectsthe
thecoefficient
coefficient𝐶Co(𝑠).
Thedelay
delayisis random,
random, since
since it depends on the
(s).The
Energies 2021, 14, x FOR PEER REVIEW
12 of 23
Energies 2021, 14, 7280
12 of 23
begins. Its range is from 0
in which the perturbation is injected until the interval 𝑑 ·
Tsw
the perturbation
is injected
until thethat
interval
de · 2 the
begins.
range is from
0 to
· . The
final characteristic
coefficients
represent
new Its
small-signal
model
toin𝑑which
Tsw
d
·
.
The
final
characteristic
coefficients
that
represent
the
new
small-signal
model
of
the
l 2 PSFB converter have been collected in Table 2.
of the
PSFB converter have been collected in Table 2.
Table 2. Proposed characteristic coefficients of the PSFB converter.
Table 2. Proposed characteristic coefficients of the PSFB converter.
Output Characteristic Coefficients
Output Characteristic
𝐾 Coefficients
𝐾
𝐵 (𝑠) = −
𝐶 (𝑠) = 𝑒 ∙ −∙ s·td
Kvlvo
Kvlvi
(𝑠)
(𝑠)
Bo (s) =𝑍− Z (−
Co (s) = e 𝑍 · Z (−
𝐾
𝐾
L s )− Kvlil
L s )− Kvlil
Input
Characteristic
Coefficients
Input Characteristic Coefficients
𝐴 (𝑠) = 𝐾 + 𝐾 ∙ 𝐾 + 𝐾 ∙ 𝐾 ∙ 𝐴 (𝑠)
Kiid ++K𝐾
·Kdlil ·∙A𝐾o (s) ∙ 𝐵 (𝑠)
iidl · K∙dld
𝐵 (𝑠) = −𝑒Ai (∙s) ∙= 𝐾
𝐾 + Kiidl
−𝐾
−
s
·
t
d
=𝐾
−e + ·[𝐾Kiivo∙+
·K𝐾
· Bo (s)]
(s) =
dlvo −∙ K
dlil(𝑠)
𝐾Kiidl+
𝐾iidl ·K∙ 𝐶
𝐶Bi(𝑠)
𝐾
𝐴 (𝑠) =
vld
(𝑠)
Ao (s) 𝑍= Z (sK−
𝐾
L )− Kvlil
Ci (s) = Kiivi + Kiidl ·Kdlvi + Kiidl ·Kdlil ·Co (s)
4. Validation of the Small-Signal Model of the PSFB Converter by Simulation
4. Validation
of the
Small-Signal
Model
of the PSFB
Converter
by Simulation
The proposed
model
is validated
by simulation
using
the AC sweep
tool of the PSIM
software.
open-loop
audio-susceptibility
transfer using
function
since
it
TheThe
proposed
model
is validated by simulation
the has
AC been
sweepselected,
tool of the
PSIM
issoftware.
the transfer
thataudio-susceptibility
best represents thetransfer
effect offunction
the delay
by thesince
blankThefunction
open-loop
hasproduced
been selected,
it is
ing
The validation
by best
simulation
wasthe
developed
operating
points.
thetime.
transfer
function that
represents
effect offor
thethree
delaydifferent
produced
by the blanking
The
parameters
for each
operating point
are shownfor
in three
Tabledifferent
3. The parameter
𝑁 repretime.
The validation
by simulation
was developed
operating points.
The
sents
the peak
the modulator
The
schematic
is shown
parameters
forvalue
each of
operating
point arecarrier
shownsignal.
in Table
3. simulation
The parameter
Nr represents
the
inpeak
Figure
12. of the modulator carrier signal. The simulation schematic is shown in Figure 12.
value
Table
Table3.3.Parameters
Parameterstotoobtain
obtainthe
thefrequency
frequencyresponse
responseatatdifferent
differentpower
powerlevels.
levels.
Common
Parameters
for
Points
Common
Parameters
forthe
the33Operating
Operating Points
𝐿 = 36 μH
𝐿 = 10 μH
𝐹 = 100 kHz
L = 36 µH
Llk = 10 µH
Fsw = 100 kHz
𝐶 C ==
100
𝐷𝐶𝑅
=
10
mΩ
𝐸𝑆𝑅
=
180
mΩ
DCR = 10 mΩ
ESR = 180 mΩ
100μF
µF
fo
𝑛=
0.5
𝑁
=
249
n = 0.5
Nr = 249
Different
Power
Different
PowerLevels
Levels
𝑃 = 90 W
𝑃 = 280 W
𝑃 = 500 W
Po = 90 W
P = 280 W
P = 500 W
𝑉 =
100 V
𝑉 o= 150 V
𝑉 o= 150 V
Vin = 100 V
Vin = 150 V
Vin = 150 V
1414V V
𝑉 V= =
14.3
V
𝑉 V= =
14.85
V
𝑉 V= =
14.3 V
14.85 V
o
o
o
𝐷D
==
0.40.4
𝐷D
==
0.45
𝐷
=
0.65
0.45
D = 0.65
𝑅 Rload= =
0.733
𝑅 Rload==0.44
𝑅 Rload==2.2
2.2ΩΩ
0.733ΩΩ
0.44ΩΩ
Figure12.
12.PSIM
PSIM
schematicofofthe
thePSFB
PSFBconverter
converterused
usedfor
forvalidation
validationbybysimulation.
simulation.
Figure
schematic
Onthe
theother
otherhand,
hand,the
thesmall-signal
small-signalmodel
modelproposed
proposedininthis
thispaper
paperisisalso
alsocompared
compared
On
with
the
small-signal
models
presented
in
[20,28].
with the small-signal models presented in [20,28].
Thevalidation
validationfor
fora apower
powerlevel
levelofof9090WWhas
hasbeen
beenperformed.
performed.For
Forthis
thisoperating
operatingpoint
point
The
theblanking
blankingtime
timerepresents
representsapproximately
approximately 5%
5% of
of the
the duty
duty ratio.
the
ratio. Under
Under these
these conditions,
conditions,it
can
be
seen
in
Figure
13a,b
that
the
models
presented
in
[20,28]
apparently
match
with
the
it can be seen in Figure 13a,b that the models presented in [20,28] apparently match
with
frequency response obtained from PSIM software. In Figure 13c, a zoom of the phase in the
Energies 2021, 14, x FOR PEER REVIEW
Energies2021,
2021,
x FOR PEER REVIEW
Energies
14,14,
7280
13 of 23
13 ofof2323
13
the
the frequency
frequency response
response obtained
obtained from
from PSIM
PSIM software.
software. In
In Figure
Figure 13c,
13c, aa zoom
zoom of
of the
the phase
phase
high
frequency
region
is
shown.
From
Figure
13c,
it
can
be
seen
that
the
model
presented
in
the
high
frequency
region
is
shown.
From
Figure
13c,
it
can
be
seen
that
the
model
in the high frequency region is shown. From Figure 13c, it can be seen that the model
in
[20] is not accurate
in this frequency
range. However,
the proposed
model allows
to
presented
presented in
in [20]
[20] is
is not
not accurate
accurate in
in this
this frequency
frequency range.
range. However,
However, the
the proposed
proposed model
model
have
a
better
estimation
of
the
frequency
response
in
this
region.
allows
allows to
to have
have aa better
better estimation
estimation of
of the
the frequency
frequency response
response in
in this
this region.
region.
Figure
13.
Comparison
of
the
frequency
response
of
the
audio-susceptibility
Figure13.
13.Comparison
Comparisonof
ofthe
thefrequency
frequency response
response of
of the
the audio-susceptibility
audio-susceptibility transfer
transfer function
function
for
Figure
functionfor
foraa
power
level
of
90
W.
(a)
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[28].
(b)
power
level
of
90
W.
(a)
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[28].
(b)
a power level of 90 W. (a) Proposed small-signal model vs. small-signal model presented in [28].
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[20].
(c)
Phase
zoom.
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[20].
(c)
Phase
zoom.
(b) Proposed small-signal model vs. small-signal model presented in [20]. (c) Phase zoom.
The
validation
for
power
level
of
Thevalidation
validationfor
foraaapower
power level
level of
of 250
been performed.
performed.
From
Figure
14a,b,
it
The
250 W
W has
has been
performed.From
FromFigure
Figure14a,b,
14a,b,it
can
be
seen
that
the
models
presented
in
[20,28]
are
not
able
to
predict
the
phase
drop
itcan
canbe
beseen
seenthat
that the
the models
models presented
presented in [20,28] are not
not able
able to
topredict
predictthe
thephase
phasedrop
drop
produced
by
the
blanking
time.
However,
the
proposed
model
this
paper
matches
with
produced
producedby
bythe
theblanking
blankingtime.
time.However,
However,the
theproposed
proposedmodel
modelinin
inthis
thispaper
papermatches
matcheswith
with
the
frequency
response
obtained
from
PSIM
software.
The
delay
considered
in
the
the
response
obtained
from
PSIM
software.
The delay
considered
in the in
proposed
thefrequency
frequency
response
obtained
from
PSIM
software.
The delay
considered
the proprodl Tsw is
posed
model
·
.
For
this
operating
point,
the
blanking
time
represents
approxi·
model
is
.
For
this
operating
point,
the
blanking
time
represents
approximately
40%
posed model
· . For this operating point, the blanking time represents approxi2 2 is
mately
40%
of
of
the duty
mately
40%ratio.
of the
the duty
duty ratio.
ratio.
Figure
14.
Comparison
of
the
frequency
response
of
the
audio-susceptibility
for aa
Figure14.
14.Comparison
Comparisonof
ofthe
thefrequency
frequency response
response of
of the
the audio-susceptibility
audio-susceptibility transfer
transfer function
function
Figure
function for
for
power
level
of
280
W.
(a)
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[28].
(b)
power
level
of
280
W.
(a)
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[28].
(b)
a power level of 280 W. (a) Proposed small-signal model vs. small-signal model presented in [28].
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[20].
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[20].
(b) Proposed small-signal model vs. small-signal model presented in [20].
Finally,
the
proposed
model
is
for aa power
level of
500 W.
From
Finally,the
the
proposed
model
is validated
validated
power
W.Figure
From Figure
Figure
Finally,
proposed
model
is validated
for afor
power
level level
of 500ofW.500
From
15a,b,
15a,b,
it
can
be
seen
that
the
models
presented
in
[20,28]
do
not
predict
the
additional
can that
be seen
that thepresented
models presented
in not
[20,28]
do the
not additional
predict thephase
additional
it15a,b,
can beitseen
the models
in [20,28] do
predict
drop
phase
drop
transfer
function.
For
this
point,
the
phase
drop in
in the
the audio-susceptibility
audio-susceptibility
transfer
function.
For point,
this operating
operating
point,
the
in
the audio-susceptibility
transfer function.
For this
operating
the blanking
time
blanking
time
represents
approximately
60%
of
the
duty
ratio.
blanking time
represents 60%
approximately
represents
approximately
of the duty60%
ratio.of the duty ratio.
Energies
2021,
14,14,
7280FOR PEER REVIEW
Energies
2021,
Energies
2021,
14, xxFOR
PEER REVIEW
14
2323
14of
14
ofof 23
Figure15.
15.Comparison
Comparisonof
ofthe
thefrequency
frequencyresponse
responseof
ofthe
theaudio-susceptibility
audio-susceptibilitytransfer
transferfunction
function
for
Figure
15.
Comparison
Figure
response
of
the
audio-susceptibility
transfer
functionfor
foraa
power
level
of
500
W.
(a)
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[28].
(b)
small-signal
model
vs.vs.
small-signal
model
presented
in [28].
(b)
apower
powerlevel
levelofof500
500W.W.(a)(a)Proposed
Proposed
small-signal
model
small-signal
model
presented
in [28].
Proposed
small-signal
model
vs.
small-signal
model
presented
in
[20].
Proposed
small-signal
model
vs.vs.
small-signal
model
presented
in [20].
(b)
Proposed
small-signal
model
small-signal
model
presented
in [20].
Fromthis
thisvalidation,
validation,itititcan
canbe
beconcluded
concludedthat
thatifififthe
thepower
powerlevel
levelisisisincreased
increasedthen
thenthe
the
From
this
validation,
can
be
concluded
that
the
power
level
increased
then
the
From
phase
drop
becomes
more
noticeable.
This
occurs
because
the
blanking
time
is
proporphasedrop
dropbecomes
becomesmore
more
noticeable.
This
occurs
because
the blanking
is proporphase
noticeable.
This
occurs
because
the blanking
time time
is proportional
tional
tothe
the
loadcurrent,
current,
which
means
that
theload
loadcurrent
current
increased
then
theblankblankto
the load
current,
whichwhich
meansmeans
that ifthat
theifif
load
current
is increased
thenthen
the the
blanking
tional
to
load
the
isisincreased
ing
time
is
increased.
Therefore,
the
blanking
time
can
become
considerable
even
the
time
is
increased.
Therefore,
the
blanking
time
can
become
considerable
even
if
the
leakage
ing time is increased. Therefore, the blanking time can become considerable even ififthe
leakage
inductor
is
small.
On
the
other
hand,
it
has
been
observed
that
the
phase
drop
inductor
is small. On
the other
it has
been itobserved
the phase
drop
leakage inductor
is small.
On hand,
the other
hand,
has beenthat
observed
that
the could
phaseoccur
drop
couldoccur
occurbefore
before
one-half
ofthe
the
switchingfrequency.
frequency.
before
one-half
of the
switching
frequency.
could
one-half
of
switching
5.
ExperimentalValidation
Validationof
ofthe
theSmall-Signal
Small-SignalModel
Modelof
ofthe
thePSFB
PSFBConverter
Converter
5.5.Experimental
Experimental
Validation
of
the
Small-Signal
Model
of
the
PSFB
Converter
A
prototypeof
ofPhase-Shifted
Phase-ShiftedFull-Bridge
Full-Bridgeconverter
converterwas
wasbuilt
builtin
inorder
orderto
toexperimenexperimenAAprototype
prototype
of
Phase-Shifted
Full-Bridge
converter
was
built
in
order
to
experimentally
verify
the
proposed
small-signal
model
as
shown
in
Figure
16.
A
leakage
inductance
tallyverify
verifythe
theproposed
proposedsmall-signal
small-signalmodel
modelas
asshown
shownin
inFigure
Figure16.
16.AAleakage
leakageinductance
inductance
tally
that
generates
a
significant
blanking
time
has
been
selected
in
order
to
demonstrate
that
generates
a
significant
blanking
time
has
been
selected
in
order
to
demonstrate
that
that generates a significant blanking time has been selected in order to demonstratethat
that
there
is
an
excellent
agreement
between
the
proposed
small-signal
model
and
the
experithere
is
an
excellent
agreement
between
the
proposed
small-signal
model
and
the
experithere is an excellent agreement between the proposed small-signal model and the experimental
mentalresults.
results.The
Theconverter
converterisisiscontrolled
controlledby
byMicroZed
MicroZedboard
board[32]
[32]based
basedon
onaaa7Z020
7Z020Xilinx
Xilinx
mental
results.
The
converter
controlled
by
MicroZed
board
[32]
based
on
7Z020
Xilinx
Zynq
device
[33].
Zynq
device
[33].
Zynq device [33].
Figure16.
16.Experimental
Experimentalprototype
prototypeof
ofthe
thePSFB
PSFBconverter.
converter.
Figure
Figure
16.
Experimental
prototype
of
the
PSFB
converter.
For
Forthe
theexperimental
experimentalvalidation,
validation,two
twodifferent
differentoperating
operatingpoints
pointshave
havebeen
beenselected.
selected.
For
the
experimental
validation,
two
different
operating
points
have
been
selected.
Figure
17a,b
show
the
main
waveforms
for
a
power
level
of
8.5
W
and
70
W,
respectively.
Figure
17a,b
show
the
main
waveforms
for
a
power
level
of
8.5
W
and
70
W,
respectively.
Figure 17a,b show the main waveforms for a power level of 8.5 W and 70 W, respectively.
ItItItcan
a alarge
blanking
time.
canbe
beseen
seenthat
thataaalarge
largeL𝐿lk𝐿 provokes
provokes
alarge
large
blanking
time.
can
be
seen
that
large
provokes
blanking
time.
Energies 2021, 14, 7280
Energies 2021, 14, x FOR PEER REVIEW
15 of 23
15 of 23
Figure17.
17. Main
Main waveforms
waveforms of
8.58.5
W,W,
(b)
Figure
of the
the experimental
experimentalprototype
prototypeatatdifferent
differentpower
powerlevels:
levels:(a)(a)
70
W.
(b) 70 W.
AccordingtotoFigure
Figure1,1,the
themain
mainparameters
parametersfor
foreach
eachoperating
operatingpoint
pointare
arefound
foundinin
According
Table4.4.The
Thevalidation
validationshown
shownininthis
thissection
sectionconsists
consistsofofobtaining
obtainingthe
thefrequency
frequencyresponse
response
Table
themain
main
transfer
functions
of converter.
the converter.
The frequency
response
is obtained
by
ofofthe
transfer
functions
of the
The frequency
response
is obtained
by using
using
the
Venable
frequency
response
analyzer
model
3235.
The
open-loop
transfer
functhe Venable frequency response analyzer model 3235. The open-loop transfer functions are
tions are
chosen to
prevent
that from
any delay
from control
the digital
control
affecting
the measurechosen
to prevent
that
any delay
the digital
affecting
the
measurement.
ment.
Table 4. Main parameters for each operating point.
Table 4. Main parameters for each operating point.
Power Level of 8.5 W
Vin = 35 V
Llk𝑉= =
34 35
µHV
𝐿
= 34 μH
V𝑉
VV
in ==100
100
Llk = 34 µH
𝐿
= 34 μH
Power Level of 𝟖. 𝟓 𝐖
Fsw = 100 kHz
Vo = 3 V
Io = 2.8 A
D = 0.63
= 100 kHz C = 1 µF 𝑉 = 3 V n = 0.5 𝐼 = 2.8 A Nr =
𝐷 1= 0.63
L =𝐹36 µH
fo
𝐶 = 1 μF
𝐿 = 36 μH
𝑛 = 0.5
𝑁 =1
Power Level of 70 W
Power Level of 𝟕𝟎 𝐖
Fsw =𝐹100=kHz
100 kHz Vo = 12.4 V𝑉 = 12.4 VIo = 5.6 𝐼 = 5.6 D =𝐷0.7= 0.7
L = 36 µH
n = 0.5
𝐶 = 47 μF
𝐿 = 36 μH C f o = 47 µF
= 249
𝑛 = 0.5 Nr =𝑁249
5.1.
5.1.Control
ControltotoOutput
OutputVoltage
VoltageTransfer
TransferFunction
Function
The
output
voltage
transfer
function
is given
by Equation
(25). (25).
The term
Gm
Thecontrol
controltoto
output
voltage
transfer
function
is given
by Equation
The term
represents
the modulator
gain: gain:
the modulator
𝐺 represents
Gm𝐺· A∙o (𝐴s)·(𝑠)
Z ∙ 𝑍 (s) (𝑠)
𝐺 (s)(𝑠)
Gvvc
= = 1 + 𝐵 (𝑠)load
∙ 𝑍 (s) (𝑠)
1 + Bo (s)· Zload
(25)
(25)
where:
where:
(𝑠) = 𝑍
𝑍
∥𝑅
(26)
Zload (s) = ZC f o k Rload
(26)
The impedance of the output filter capacitor and output filter inductor consider the
The impedance of the output filter capacitor and output filter inductor consider the
parasitic
resistancesand
andare
aregiven
givenby
byEquations
Equations(27)
(27)and
and(28),
(28),respectively:
respectively:
parasitic resistances
1
(𝑠) = 𝐸𝑆𝑅 + 1
𝑠
∙
𝐶
ZC f o (s) = ESR +
𝑍
(27)
(27)
s·C f o
(28)
𝑍 (𝑠) = 𝐷𝐶𝑅 + 𝑠 ∙ 𝐿
ZL (s) = DCR + s· L
(28)
From Equation (25), it can be deduced that the control to output voltage transfer functionFrom
is notEquation
affected by
theit delay
thethe
blanking
since voltage
the characteristic
(25),
can beassociated
deduced to
that
controltime,
to output
transfer
function
is not𝐴affected
associated
blanking
time,
thein
characteristic
coefficients
(𝑠) andby𝐵the
(𝑠)delay
are not
affectedtobythe
the
delay, as
can since
be seen
Table 2. The
coefficients
Ao (scheme
s) and Biso (shown
s) are not
affected
measurement
in Figure
18.by the delay, as can be seen in Table 2. The
measurement scheme is shown in Figure 18.
Energies
2021,
x FOR PEER REVIEW
Energies
2021,
14,14,
7280
Energies 2021, 14, x FOR PEER REVIEW
16ofof2323
16
16 of 23
Venable model 3235
Venable model
ch2
ch1 3235
osc
+
- + ch2
ch1
osc
+
- +
𝑽𝒊𝒏
𝑽𝒊𝒏
PSFB
Converter
PSFB
Converter
+
_
_
+
𝑹𝒍𝒐𝒂𝒅
𝑹𝒍𝒐𝒂𝒅
𝑸𝟏 𝑸𝟐 𝑸𝟑 𝑸𝟒
𝑸𝟏 𝑸𝟐 𝑸𝟑 𝑸𝟒
M
M
perturbation
perturbation
+
+
+
∆𝒎
∆𝒎
+
MicroZed
MicroZed
ADC
ADC
Figure 18. Measurement block diagram of 𝐺 (𝑠).
Figure18.
18.Measurement
Measurementblock
blockdiagram
diagramof
ofG𝐺vvc ((𝑠).
Figure
s ).
The perturbation has been generated using the Venable oscillator. This perturbation
The
using
the
Venable
oscillator.
perturbation
is
Theperturbation
perturbation
hasbeen
beengenerated
generated
using
the
Venable
oscillator.
This
perturbation
is passed
through anhas
analog-digital
converter,
ADC,
with the
aim ofThis
injecting
the perturpassed
through
an
analog-digital
converter,
ADC,
with
the
aim
of
injecting
the
perturbation
is
passed
through
an
analog-digital
converter,
ADC,
with
the
aim
of
injecting
the
perturbation into the DPWM that is implemented in the MicroZed board. The ADC gain, modinto
theinto
DPWM
that
is implemented
the MicroZed
board.
The
ADC
modulator
bation
the
that
is implemented
in
theDPWM
MicroZed
board.
The
ADC
gain, modulator
gain
andDPWM
the total
delay
(ADCindelay
plus
delay)
has
beengain,
subtracted
from
gain
and
the
total
delay
(ADC
delay
plus
DPWM
delay)
has
been
subtracted
fromfrom
the
ulator
gain
and
the
total
delay
(ADC
delay
plus
DPWM
delay)
has
been
subtracted
the measurement in order to compensate these effects (post-processed).
measurement
in
order
to
compensate
these
effects
(post-processed).
the measurement
in
order
to
compensate
these
effects
(post-processed).
The magnitude and phase of the frequency response of the control to output voltage
The
frequency
of
the
totooutput
Themagnitude
magnitude
and
phase
ofthe
the
frequency
response
ofshown
thecontrol
control
output
voltage
transfer
function
forand
thephase
8.5 Wof
and
70
W powerresponse
levels are
in Figure
19a,b,voltage
respectransfer
function
for
the
8.5
W
and
70
W
power
levels
are
shown
in
Figure
19a,b,
respectively.
transfer
function
for
the
8.5
W
and
70
W
power
levels
are
shown
in
Figure
19a,b,
respectively. There is an excellent agreement between the proposed model and the experimental
There
is an excellent agreement between the proposed model and the experimental results.
tively.
results.There is an excellent agreement between the proposed model and the experimental
results.
Figure19.
19.Frequency
Frequencyresponse
responseofofcontrol
controltotooutput
outputvoltage
voltagetransfer
transferfunction.
function.(a)
(a)8.5
8.5W.
W.(b)
(b)7070W.W.
Figure
Figure 19. Frequency response of control to output voltage transfer function. (a) 8.5 W. (b) 70 W.
5.2.
5.2.Open-Loop
Open-LoopAudio-Susceptibility
Audio-SusceptibilityTransfer
TransferFunction
Function
5.2. Open-Loop
Audio-Susceptibility
Transfer
Function
The
open-loop
audio-susceptibility
transfer
The open-loop audio-susceptibility transferfunction
functionisisgiven
givenbybyEquation
Equation(29):
(29):
The open-loop audio-susceptibility transfer function is given by Equation (29):
Co (𝐶s)(𝑠)
Gvv𝐺(s)(𝑠)
==
(29)
𝐶 (𝑠)
(29)
𝐺 (𝑠) =Bo𝐵
(s(𝑠)
) ++Z 1 (11
(29)
𝑍 s) (𝑠)
load
𝐵 (𝑠) +
(𝑠)
𝑍
Energies 2021, 14, x FOR PEER REVIEW
Energies 2021, 14, 7280
17 of 23
17 of 23
From Equation (29), it can be deduced that the open-loop audio-susceptibility transFrom
Equation
(29), by
it can
deduced
that associated
the open-loop
audio-susceptibility
transfer function
is affected
theberandom
delay
to the
blanking time, since
the
fer
function
is
affected
by
the
random
delay
associated
to
the
blanking
time,
since
the
characteristic coefficient 𝐶 (𝑠) is affected by the delay, as can be seen in Table 2.
characteristic
coefficient
Cophase
by the delay,
as can
seen in Table
2.
(s) is affected
The magnitude
and
of the frequency
response
ofbe
open-loop
audio-susceptibilThe
magnitude
and
phase
of
the
frequency
response
of
open-loop
audio-susceptibility
ity transfer function for the 8.5 W and 70 W power levels are shown in Figure 20a,b, retransfer
function
foristhe
W and 70
W powerbetween
levels arethe
shown
in Figure
20a,b,
respectively.
spectively.
There
an8.5
excellent
agreement
proposed
model
and
the experiThere is an excellent agreement between the proposed model and the experimental results.
mental results. The delay considered in this measure is in · . However, the delay to
The delay considered in this measure is in d2l · T2sw . However, the delay to consider may be
consider may be different for other operating points due to its random nature, as mendifferent for other operating points due to its random nature, as mentioned in Section 3.
tioned in Section 3.
Figure20.
20. Frequency
Frequency response
transfer
functions:
(a) 8.5
Figure
responseofofopen-loop
open-loopaudio-susceptibility
audio-susceptibility
transfer
functions:
(a)W,
8.5(b)
W,70
W.
(b) 70 W.
5.3.
5.3.Open-Loop
Open-LoopInput
InputImpedance
Impedance
The
Theopen-loop
open-loopinput
inputimpedance
impedancetransfer
transferfunction
functionisisgiven
givenbybyEquation
Equation(30):
(30):
1 1
Bo 𝐵
++
(s)(𝑠)
Z 𝑍 (s) (𝑠)
load
Zin 𝑍
= =C (s)
(s)(𝑠)
i 𝐶 (𝑠)
+ C+i (𝐶s)·(𝑠)
Bo (∙ s𝐵) (𝑠)
− C−o (𝐶s)·(𝑠)
Bi (∙ s𝐵) (𝑠)
Zload
𝑍 (s) (𝑠)
(30)
(30)
From
FromEquation
Equation(30)
(30)it itcan
canbebededuced
deducedthat
thatthe
theopen-loop
open-loopinput
inputimpedance
impedanceisisaffected
affected
bybythe
) and
and
thedelay
delayassociated
associatedtotothe
theblanking
blankingtime,
time,since
sincethe
thecharacteristic
characteristiccoefficients
coefficientsC𝐶o (s(𝑠)
B𝐵
s
are
affected
by
the
delay,
as
can
be
seen
in
Table
2.
(
)
i (𝑠) are affected by the delay, as can be seen in Table 2.
For
Forthe
themeasurement
measurementofofthe
theinput
inputimpedance,
impedance,aahigh
highfrequency
frequencydecoupling
decouplingcapacitor,
capacitor,
C𝐶
, hasbeen
beenadded
addedand
andits
itsvalue
valueisisapproximately
approximately1 1µF.
𝜇𝐹.This
Thismeasurement
measurementisisnot
notaffected
affected
in , has
bybyany
the
effects
of of
digital
control,
since
thethe
current
source
amplifies
the perturbation
in
anyofof
the
effects
digital
control,
since
current
source
amplifies
the perturbation
aninanalog
way way
as shown
in Figure
21a. For
theFor
input
frequency
responseresponse
can be
an analog
as shown
in Figure
21a.
theimpedance
input impedance
frequency
emphasized
that the experimental
measurement
matches matches
with the with
theoretical
predictions,
can be emphasized
that the experimental
measurement
the theoretical
preasdictions,
shown in
Figure
21b.
as shown in Figure 21b.
5.4. Open-Loop Output Impedance
The terminated open-loop output impedance transfer function (considering the load)
is given by Equation (31):
Zo (s) =
Zload (s)
Zload (s)· Bo (s) + 1
(31)
Energies 2021, 14, 7280
18 of 23
From Equation (31), it can be deduced that the open-loop output impedance transfer
function
is not affected by the delay associated to the blanking time, since the characteristic
Energies 2021, 14, x FOR PEER REVIEW
18 of 23
coefficient Bo (s) is not affected by the delay, as can be seen in Table 2.
Figure 21. (a) Measurement block diagram. (b) Frequency response of the open-loop input impedance, 8.5 W.
5.4. Open-Loop Output Impedance
The terminated open-loop output impedance transfer function (considering the load)
is given by Equation (31):
𝑍 (𝑠) =
𝑍
(𝑠)
𝑍
(𝑠) ∙ 𝐵 (𝑠) + 1
(31)
Figure
21. (a)
Measurement
(b) Frequency
of the
open-loop
input impedFrom
Equation
(31), block
it block
candiagram.
bediagram.
deduced
thatFrequency
the response
open-loop
output
impedance
transfer
Figure
21.
(a)
Measurement
(b)
response
of the
open-loop
input
ance, 8.5 W.
function is8.5
not
impedance,
W.affected by the delay associated to the blanking time, since the characteris-
tic coefficient 𝐵 (𝑠) is not affected by the delay, as can be seen in Table 2.
5.4.The
Open-Loop
Output Impedance
phase
ofof
thethe
frequency
response
of the
open-loop
output
impedance
Themagnitude
magnitudeand
and
phase
frequency
response
of the
open-loop
output
impedThe
terminated
open-loop
output
impedance
transfer
function
(considering
the the
load)
transfer
function
is
shown
in
Figure
22.
There
is
an
excellent
agreement
between
proance transfer function is shown in Figure 22. There is an excellent agreement between
the
is given
by Equation
(31):
posed
model
and
the
experimental
result.
proposed model and the experimental result.
𝑍 (𝑠) =
𝑍
(𝑠)
𝑍
(𝑠) ∙ 𝐵 (𝑠) + 1
(31)
From Equation (31), it can be deduced that the open-loop output impedance transfer
function is not affected by the delay associated to the blanking time, since the characteristic coefficient 𝐵 (𝑠) is not affected by the delay, as can be seen in Table 2.
The magnitude and phase of the frequency response of the open-loop output impedance transfer function is shown in Figure 22. There is an excellent agreement between the
proposed model and the experimental result.
Figure22.
22.Frequency
Frequencyresponse
responseofofthe
theopen-loop
open-loopoutput
outputimpedance,
impedance,8.5
8.5W.
W.
Figure
6. Conclusions
A new small-signal model of the PSFB converter has been presented. This model
predicts more accurately the gain of the transfer function because the additional switching
stage due to the leakage inductance is considered. From the analysis of the natural response
of the PSFB converter, it has been shown that in this topology, the blanking time creates
Figure 22. Frequency response of the open-loop output impedance, 8.5 W.
Energies 2021, 14, 7280
19 of 23
a random delay in the input current response and the output inductor current response
when there is a small output voltage perturbation and small input voltage perturbation,
respectively. Therefore, the characteristic coefficients that are directly affected by this
delay are Bi (s) and Co (s). The delay is variable since it depends on the moment in which
the perturbation is injected until the interval de · T2sw begins. Its range is from 0 to dl · T2sw .
Therefore, in order to design a control loop that depends on audio-susceptibility transfer
function, the largest delay should be considered, which occurs when td is equal to the
blanking time. The proposed small-signal model of the PSFB converter is capable of
predicting the experimental phase drop that is generated in the audio-susceptibility transfer
function when the blanking time is significant.
Author Contributions: D.O. did the theoretical analysis, circuit implementation, experimental testing
and wrote the original draft paper. A.L. is the responsible for funding acquisition, investigation and
supervision; his contribution was related to the theoretical analysis, paper reviewing and editing. P.Z.
performed the investigation and his contribution was related to the theoretical analysis. M.S., J.R.d.F.
and A.B. reviewed and contributed with useful comments on the paper’s structure and main paper
contributions. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the European Regional Development Fund, the Ministry of
Science, Innovation and Universities and the State Research Agency, grant number DPI2017-84572C2-2-R.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: This work is partially supported by the European Regional Development Fund,
the Ministry of Science, Innovation and Universities and the State Research Agency through the
research projects “Modelling and control strategies for the stabilization of the interconnection of
power electronic converters”—CONEXPOT-2- (DPI2017-84572-C2-2-R) (AEI/FEDER, UE).
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
I AC
Gm
Gvv
Gvvc
Zo
Zin
A x , Bx , Cx
Zx
n
Llk
L
Dl · T2sw
td
Nr
Kx
V, I
ve, e
l
v, i
v̂, lˆ
Injected-absorbed-current method
Modulator transfer function
Audio-susceptibility transfer function
Control to output voltage transfer function
Output impedance transfer function
Input impedance transfer function
Characteristic coefficients of the injected-absorbed-current method
Impedances
Transformer turns ratio (number of turns of the secondary winding
divided by number of turns of the primary winding)
Leakage inductance of the transformer
Output filter inductance
Blanking time interval
Random delay
Peak value of the modulator carrier signal
Constant coefficients of the transfer function
Operating point of the currents and voltages
Switched currents and voltages
Averaged and characteristic value of the currents and voltage
Small-signal values of the currents and voltages
Energies 2021, 14, 7280
20 of 23
Appendix A
The full set of the equations of the small-signal model of the PSFB converter is given below:
Kvlvo
L2 ·Vin · D · Llk ·n2 + L K2
=−
· 2
Tsw ·( Llk ·n2 + L)
K1
K1 = L2 ·Vin − L2lk ·Vo ·n3 − L· Llk ·Vin ·n2 ·(1 − D )
K2 = Tsw ·Vin · D2 · L2lk ·n4 + L2 − L· Llk ·n2 + K4
K3 = Tsw ·Vin · D · L· Llk ·n2 − 2· L2lk ·n4
K4 = K3 + 4· IL · L· L2lk ·n3 + L3lk ·n5
Kvlil = −
4· L· Llk ·n2 ·( L·Vin + Llk ·Vo ·n)
Tsw ·K1
L·n·(K5 + K7 + K10 − K12 )
Tsw ·( Llk ·n2 + L)·K12
h
i
2
K5 = Tsw · D2 · Vin
·K6 + L4lk ·Vo2 ·n6
Kvlvi =
K6 = L3 · Llk ·n2 + L2 · L2lk ·n4
h
i
K7 = Tsw · L2 · L2lk ·Vo2 ·n2 + K8
K8 = D · K9 + Vo2 · L· L3lk ·n4 − L4lk ·n6
2
K9 = Vin
· L4 − L2 · L2lk ·n4
h
i
K10 = 4· IL ·Vo · L2 · L3lk ·n4 ·( D + 1) + K11
K12
K11 = D · L· L4lk ·n6 + L3 · L2lk ·n2
= 2· Tsw · D ·Vo ·Vin · L2 · L2lk ·n3 + L· L3lk ·n5
(A2)
(A3)
(A4)
(A5)
(A6)
(A7)
(A8)
(A9)
(A10)
(A11)
(A12)
(A13)
(A14)
(A15)
L·Vin ·n·(K13 + K15 + K17 + K19 )
Tsw ·( Llk ·n2 + L)·K12
(A16)
2
K13 = 2· Tsw · D ·Vo ·Vin ·K14 + Tsw · L4 ·Vin
(A17)
Kvld =
K15
(A1)
K14 = L· L3lk ·n5 − L2 · L2lk ·n3
h
i
= 4· IL · K16 + Vo · L2 · L3lk ·n4 + L· L4lk ·n6
K16 = Vin · L3 · L2lk ·n3 + L2 · L3lk ·n5
2 4
K17 = Tsw ·Vo ·Vin ·K18 − Tsw · L2 · L2lk ·Vin
·n
K18 = L2 · L2lk ·n3 − L3lk ·n5 · L· D2 + 2
K19 = Tsw ·Vo2 · L· L3lk ·n4 − n6 · L4lk − 2· D · L4lk
Kiivi = K20 · L· Dl + D · Llk ·n2 − Llk ·n2 · Dl
K20 =
Tsw ·( D − Dl )
4· L2lk ·n2 + 4· L· Llk
(A18)
(A19)
(A20)
(A21)
(A22)
(A23)
(A24)
(A25)
Energies 2021, 14, 7280
21 of 23
Kiivo =
Llk · Tsw ·n3 · Dl2 − D · Llk · Tsw ·n3 · Dl
+ K21
4· L· Llk ·n2
K22 − 2· Tsw ·n· Dl2 + Tsw ·n· Dl
4· Llk ·n2 + 4· L
K21 =
(A27)
K22 = 2· Tsw ·n· D · Dl − Tsw ·n· D2
4· L· Llk ·n· Llk ·n2 + L
Kdlil =
Tsw ·K1
(A28)
(A29)
Tsw · K23 + 2· D · L· Llk · Vin ·n2 − Vo ·n
4· L· Llk ·( Llk ·n2 + L)
K23 = Vin · Dl · L2 − 2· L· Llk ·n2 + K24
K24 = Vo · Dl · L· Llk ·n − L2lk ·n3
Kiid =
Tsw ·(K25 + K27 + L· Llk ·Vo ·n)
4· L· Llk ·( Llk ·n2 + L)
K25 = K26 + 2·Vo · Dl · L2lk ·n3 − L· Llk ·n
K26 = 2·Vin · Dl · L· Llk ·n2 − L2
K27 = D · L2 ·Vin − K28 + L· Llk ·Vo ·n
Kiidl =
K28 = L2lk ·Vo ·n3 + 2· L· Llk ·Vin ·n2
Kdld = 1 −
(A26)
K29 + K31
Tsw ·K12
(A30)
(A31)
(A32)
(A33)
(A34)
(A35)
(A36)
(A37)
(A38)
K29 = 4· IL ·Vin · L3 · L2lk ·n3 + L2 · L3lk ·n5 + K30
(A39)
K30 = Tsw · L4lk ·Vo2 ·n6
(A40)
K31 = Tsw · L
4
·Vin2
− Tsw · L
2
· L2lk ·K32
2 4
K32 = Vo ·Vin ·n3 + Vin
·n
Kdlvo =
(A42)
L· Llk ·n·(K33 + K35 )
Tsw ·K12
(A43)
K33 = Tsw ·Vin · D ·K34 + Tsw ·Vin · L2
(A44)
K34 = L· Llk ·n2 + D · L2lk ·n4 − 2· L2lk ·n4
(A45)
K35 = 4· IL ·K36 − Tsw ·Vin · L· Llk ·n
2
K36 = L· L2lk ·n3 + L3lk ·n5
L· Llk ·n·(K37 + K39 )
Tsw ·K12
= 4· IL · D ·K38 + 4· IL · L3 − L· L2lk ·n4
Kdlvi = −
K37
K39
(A41)
(A46)
(A47)
(A48)
(A49)
K38 = L2 · Llk ·n2 + L· L2lk ·n4 .
= Tsw ·Vo · D ·K40 + Tsw ·Vo · L2 − L· Llk ·n2
(A51)
K40 = L· Llk ·n2 − 2· L2lk ·n4 + D · L2lk ·n4
(A52)
(A50)
Energies 2021, 14, 7280
22 of 23
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