Engineering Science and Technology, an International Journal xxx (xxxx) xxx
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Engineering Science and Technology,
an International Journal
journal homepage: www.elsevier.com/locate/jestch
Full Length Article
Stability analysis of the main converter supplying a constant power load
in a multi-converter system considering various parasitic elements
Rashmi Patel, R. Chudamani ⇑
Electrical Engineering Department, Sardar Vallabhbhai National Institute of Technology, Surat 395007, India
a r t i c l e
i n f o
Article history:
Received 25 October 2019
Revised 15 February 2020
Accepted 18 March 2020
Available online xxxx
Keywords:
Constant Power Load (CPL)
Negative impedance characteristics
Parasitic elements
Stability
a b s t r a c t
Multi-converter power electronic system is nowadays popular in mobile vehicular applications. Different
converters behave as Constant Power Loads (CPL) in a multi-converter system. These offer a negative
impedance characteristic which results in a destabilizing effect on a converter driving them.
Researchers have analyzed that the Buck converter supplying a CPL is unstable, but stability can be
achieved by considering Constant Voltage Load (CVL) and parasitics of converter components in the system. In this paper, the effect of CVL in parallel with CPL and parasitic of switch, diode and inductor of
main converter is mathematically evaluated on system stability. Then capacitor internal resistance is considered for the further analysis and the generalized expression is derived with consideration of parasitics
of switch, diode, inductor, capacitor and CVL connected in the system and consistency of the generalized
expression is verified with the established cases in the literature. The effect of the equivalent series resistance of capacitor on stability is analysed and compared with the stability due to parasitic elements of
switch, diode and inductor. Further minimum and maximum range of parasitic of capacitor is derived
to obtain the desired stability limit for given operating condition.
Ó 2020 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Nowadays, multi-converter power electronic systems are
mainly used in mobile applications such as Electric Vehicles
(EVs), International Space Station (ISS), Hybrid Electric Vehicles
(HEVs), aircrafts and ships. In addition, they can also be employed
in various applications such as residential, portable electronics, DC
power distributed system etc.
Each converter performs its stand-alone operation in a multiconverter system as shown in Fig. 1. Whenever these converters
interact with each other the converter performance may degrade
or the system may become unstable. Several CPLs such as electric
motor, actuators and power electronic converters have to be controlled such that output power is maintained constant. Various
DC/DC multi-converter applications are reviewed in detail [1–5].
Various stability criteria for DC distributed power system have
been reviewed in [2]. This review mainly focuses on behavioral
modeling, effect of CPL in DC power system and associated challenges in the control of DC/DC power system. Stability analysis of
a telecom power supply loaded by CPL in parallel with standby bat⇑ Corresponding author.
E-mail address: rc@eed.svnit.ac.in (R. Chudamani).
teries has been done in [3] as a case study. The stability of DC/DC
power converter with CPL for electric vehicles is analysed in [4]
for different operating modes. The effect of instability in DC Microgrid system with CPL load is discussed in [5] and different compensating techniques are reviewed to stabilize DC Microgrid system.
A simple model of CPL is shown in Fig. 2. The tightly regulated
system which behaves like a CPL causes instability in the main converter. It is mainly due to negative impedance characteristic
offered by CPL at the output terminal of the main converter.
The negative impedance behaviour of CPL can be seen in Fig. 3.
The instantaneous value of impedance is positive ðV=I > 0Þ, but the
incremental impedance is always negative as dV=dI < 0. So the
current through the CPL decreases when the voltage across it
increases and vice versa. Negative impedance instability for
multi-converter system is explained elaborately in [6–8].
The researchers have attempted to make the main converter
stable by positioning the pole from right half of s-plane to left half
of s-plane. Mainly two approaches have been suggested to make
the system stable viz., (i) closed loop control techniques [8–22]
and (ii) open loop control techniques [6,23–26].
Basically the open loop systems loaded by CPL are unstable and
can be made stable by closing the loop. Different closed loop
control techniques are discussed to achieve the stability of a main
converter [8–22]. The shifting of unwanted pole in right half of
Peer review under responsibility of Karabuk University.
https://doi.org/10.1016/j.jestch.2020.03.007
2215-0986/Ó 2020 Karabuk University. Publishing services by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: R. Patel and R. Chudamani, Stability analysis of the main converter supplying a constant power load in a multi-converter system
considering various parasitic elements, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.03.007
2
R. Patel, R. Chudamani / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 1. Multi-converter power electronic system.
Fig. 2. A DC/DC converter with a tightly regulated load.
Fig. 3. Negative incremental impedance characteristic of a Constant Power Load.
s-plane to left half of s-plane using the pole placement technique is
discussed in [8–11] to make the system stable. Different quantities
are utilized as feedback like output current [12], voltage [13–16]
and power [17,27] of system to achieve the stability of a system.
In addition to the loop-cancellation technique [18], Feedback linearization technique [19], sliding mode control with feedback linearization technique [28] and sliding mode duty ratio control
technique [20–22].
It is claimed that the nonlinearities due to CPL is cancelled out
by input–output linearization technique proposed in [29] for a
boost converter feeding a combination of CPL and resistive load.
The stability of the main converter feeding a cascaded converter
in multi-converter system is obtained by Kharitonov rectangle
based programming technique in [30]. Instability caused due to
CPL in DC Microgrid is analyzed and attempted to make a system
stable by an active damping method in [31] and a droop mode
method in [32]. In [33], a new multi-objective modulated method
have been developed to control power electronics converter instability and nonlinearity behavior. In [34], authors had presented a
robust pulse-width modulation-based sliding mode controller for
DC/DC converter. The issue due to asynchronous sampling has
been investigated using alterable sampled-data terminal method
in [35].
The stability can also be improved without closing the loop but
by loading the converter with a constant voltage load (CVL). The
system stability for a buck converter supplying both CPL and CVL
has been analyzed and necessary and sufficient condition for stability of converter has been derived in [6]. For the stability analysis,
model is assumed to be ideal/lossless. The parasitic effects are
introduced in DC/DC converter for a model accuracy and an
improvement in stability of converter is observed [23,24]. In [23],
a state space averaged model has been derived with parasitic effect
for a converter loaded by CVL. The improvement in stability is analyzed mathematically for power electronic converter loaded by CPL
with parasitic effect in [24]. The issue of stability is addressed and
design criteria are suggested by deriving the control-to-output
transfer functions for different conduction modes (CCM and DCM).
Moreover, it is also observed that the parasitic elements play an
important role in shifting the poles from right half of s-plane to left
half of s-plane to make the system stable. In the available literature
the parasitic elements of switch, diode and inductor are considered
and their effect on stability is analysed but it is done without considering parasitic of capacitor. It is discussed in [25] that the consideration of capacitor parasitic resistance rC introduces zero in
left half of s-plane of the system. In [26], stability and dynamic
response analysis of DC-DC buck converter in CCM is carried out
considering rC by using transfer function approach for resistive
load. It is to be studied that the parasitic of capacitor varies subject
to use of composite material. Researchers have attempted to analyse equivalent series resistance of capacitor using different composite materials [36–38]. Effect of carbon based composite on
equivalent series resistance of capacitor is analysed in [36]. The
reduced graphene oxide–silica (rGOS) composite material is used
for capacitor by replacing Polyaniline (PANI). It is claimed that
rGOSP composite yielded a higher capacitance and lower ESR value
compared to that of its individual components [37]. In [38], the
capacitor ESR is reduced by embedding the mesh into the surface
of the adjacent electrode. It is attempted to reduce the interfacial
resistance between the electrode and the current collector to
reduce ESR. However in these papers, the effect of composite material in reducing the equivalent series resistance of the capacitor
and yielding a higher capacitance value has been presented. In this
paper the role played by the equivalent series resistance in improving the stability of the system has been studied and anaysed.
In this paper, an attempt has been made to study the effect of parasitics of switch, diode, inductor and capacitor on stability of the
main converter supplying CVL connected in parallel with CPL. The
generalized expression is derived to evaluate system stability and
its consistency has been verified with available literature. Furthermore, the improvement in system stability by considering the parasitics of different components of the buck converter is
mathematically analyzed and discussed. The improvement in stability range is compared considering parasitics of switch, diode, inductor and capacitor. Based on this analysis minimum to maximum
range of parasitic of capacitor is obtained considering different operating condition.
This paper is organized as follows. Section II briefs CPL and negative impedance instability in converter. Section III presents stability
criterion for DC/DC converter supplying a CPL in CCM with parasitic
elements of switch, diode, inductor and capacitor. Section IV
describes comparative analysis of a converter to improve stability
considering parasitic elements. Finally the significance of rC in stability improvement has been highlighted and it is concluded in section
V that out of the four parasitic elements considered, rC gives the maximum stability at any given operating conditions.
2. Stability analysis of buck converter supplying a CPL in CCM
The buck converter operating in continuous conduction mode
with switching period ‘T’ and duty cycle ‘D’ loaded by CPL ‘P’ is
Please cite this article as: R. Patel and R. Chudamani, Stability analysis of the main converter supplying a constant power load in a multi-converter system
considering various parasitic elements, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.03.007
R. Patel, R. Chudamani / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 5. A practical Buck converter supplying a CPL.
Fig. 4. Buck converter loaded by Constant power load.
presented in Fig. 4. The state-space averaging technique is applied
b
to obtain control-to-output transfer function V o ðsÞ for the given sysbd ðsÞ
tem [4,7,8,27,28].
The state space equations of buck converter supplying a CPL
when the switch is ON and diode is OFF are given by (1) and (2).
diL 1
¼ ½v in v C L
dt
dv C 1
P
iL ¼
C
dt
vC
ð1Þ
ð2Þ
Similarly, the state space equations of buck converter loaded by
CPL when the switch is OFF and diode is ON are given by (3) and
(4).
diL 1
¼ ½v C L
dt
dv C 1
P
iL ¼
C
dt
vC
ð3Þ
ð4Þ
The transformed input–output relation for a buck converter
loaded by CPL is derived using the state-space averaging technique
and it is given by,
v o ð sÞ
d ð sÞ
¼
s2
1
V in
LC P
CV 2o
1
s þ LC
3
ð5Þ
Similarly the state space equations for the OFF state of switch
are given by (8) and (9).
diL 1
¼ ½r D iL r L iL v C L
dt
dv C 1
P
iL ¼
C
dt
vC
ð9Þ
Using the state-space averaging method, the dynamic model of
the buck converter supplying a CPL with parasitic effect can be
written as,
diL 1 ¼ Dv in v C r L þ DrQ m þ ð1 DÞr D iL
L
dt
dv C 1
P
iL ¼
C
dt
vC
ð10Þ
ð11Þ
The output voltage of the main converter is describe in terms of
state variable as,
vo ¼ vC
ð12Þ
For studying the small-signal stability of converter, small disturbances in state variables, duty ratio and output voltage are considered. These small disturbances are assumed as follows:
iL ¼ IL þ i L
ð13Þ
vC ¼ VC þ vC
The poles of the transfer function in the above equation have
positive real parts (located in right half of the plane) thereby indicating instability of the buck converter loaded with CPL. This instability can be overcome with the insertion of parasitic elements. In
the next section the effect of parasitic elements on stability has
been studied and the scope for improving the stability limit by parasitic elements has been explored.
ð8Þ
d¼Dþd
ð14Þ
ð15Þ
vo ¼ Vo þ vo
ð16Þ
Here D and Vo are the average values of d and v o respectively, b
d
b o is the smallis the small signal variations of duty ratio and V
b in is assumed
signal variations of the output. The perturbation V
to be zero in input voltage to simplify the analysis, which gives
v in ¼ V in .
Substituting (13) to (16) in (10) and (11) gives the dynamic
model of the buck converter as,
3. Effect of parasitic elements on system
3.1. Effect of parasitic elements r Q m , r D and r L
In this subsection the effect of the parasitic elements r Q m , rD and
rL on the stability of system is presented, where r Q m , rD and rL are
parasitic resistances of the switch, diode and inductor respectively.
Fig. 5 shows a buck converter supplying a CPL with parasitic
elements.
For a practical buck converter supplying a CPL, the state-space
equations in the ON state of switch are given by (6) and (7).
diL 1 ¼ v in r Q m iL r L iL v C
L
dt
dv C 1
P
iL ¼
C
dt
vC
ð6Þ
ð7Þ
diL 1
DV in v o r L þ DrQ m þ ð1 DÞr D i L
¼
L
dt
"
#
dv C 1
P
IL þ i L ¼
C
dt
Vo þ vo
ð17Þ
ð18Þ
Since the average value of inductor current IL ¼ P=V O , substituting the same in (18) gives,
"
#
dv C 1 P iL þ 2 vo
¼
C
dt
Vo
ð19Þ
The output voltage VO of the main converter is equal to the voltage across capacitor V C (ignoring rC of capacitor) as given in (12).
The control-to-output transfer function of the system is obtained
as,
Please cite this article as: R. Patel and R. Chudamani, Stability analysis of the main converter supplying a constant power load in a multi-converter system
considering various parasitic elements, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.03.007
4
R. Patel, R. Chudamani / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
v o ðsÞ
d ð sÞ
¼
where,
s2
þ
P
l
L
is
r D r Q m
I
LC L
P
CV 2o
the
1
þ LC
V
in
1
P
s þ LC l LCV
2
ð20Þ
o
constant
l ¼ rL þ DrQ m þ ð1 DÞrD
power
supplied
to
CPL
and
The poles of the system described by (20) have negative real
parts (left half of the plane) indicating that the system is stable.
Thus it is concluded that introducing parasitic element makes
the system stable. Applying the Routh stability criteria to (20),
the stability limit is found to be,
(
)
lCV 2o V 2o
lCV 2o
Pmax ¼ min
; ½1 þ ðr D r Q ÞIL þ V in ¼
L
l
L
ð21Þ
Since the value of l is very small the first term in (21) decides
the stability limit of the system.
The simulation study is carried out using MATLABÒ/Simulink to
verify the effect of parasitic on stability region of buck converter
supplying a CPL. The parameters of buck converter and CPL considered for analysis are given in Table 1.
The stability region of the buck converter supplying a CPL with
variation in parasitic resistances rQ m , rD and r L considering one at a
time and keeping the remaining two constant is shown in Fig. 6.
Considering the operating point P max = PO = 100 W, the parasitic
resistances rL ¼ 3X, r D ¼ 5:6X and r Q m ¼ 6:5X are obtained from
Fig. 6. It is concluded from this analysis that required value of rL
is less compared to r D and r Q m to achieve maximum stabilityPmax = 100 W. Further it is clearly seen that for given value
(3X) of r Q m , r D and r L ; r L gives the maximum stability.
3.2. Buck converter supplying a CPL-CVL considering parasitic element
rQ m , r D andrL
In this subsection, the effect of CVL connected in parallel with
CPL along with parasitic elements on system stability is analysed.
The improvement in system stability for such system is mathematically analysed and the results are evaluated in terms of stability
limit. Fig. 7 shows a buck converter supplying a CPL in parallel with
CVL with parasitic resistancesrQ m , r D and rL of switch, diode and
inductor respectively.
The state space equations of the buck converter supplying a CPL
in parallel with CVL and considering the parasitic effect in the ON
state of switch are given by (22) and (23).
diL 1 ¼ v in rQ m iL rL iL v C
L
dt
ð22Þ
dv C 1
P vC
iL ¼
C
dt
vC R
ð23Þ
Fig. 6. Stability region of the converter for variation in parasitic element rQ , rD
and.rL
Fig. 7. Buck converter supplying a CPL in parallel with CVL considering parasitic
elements.
diL 1
¼ ½r D iL r L iL v C L
dt
ð24Þ
dv C 1
P
vC
iL ¼
C
dt
vC R
ð25Þ
Using the state-space averaging technique, the dynamic model
of the buck converter supplying a CPL in parallel with CVL and considering parasitic effect can be written as,
diL 1 ¼ DV in v C r L þ DrQ m þ ð1 DÞr D iL
L
dt
ð26Þ
dv C 1
P
vC
iL ¼
C
dt
vC R
ð27Þ
bin is assumed to be zero in input voltage to
The perturbation V
simplify the analysis, which gives v in ¼ V in . The dynamic model
of the buck converter supplying a CPL with parasitic effect is
derived by substituting (13) to (16) in (26) and (27). It is obtained
as,
Similarly the state space equations for the OFF state of switch
are given by (24) and (25).
diL 1
DV in v C r L þ DrQ m þ ð1 DÞr D i L
¼
L
dt
Table 1
Specification of Buck converter.
"
#
dv C 1 P vC
i L þ 2 vC ¼
C
dt
R
VC
Parameter
Value (Unit)
Input voltage (Vin )
Output Voltage (Vo )
Switching frequency (f s )
Coil Inductance (L)
Filter capacitor (C)
Output power of CPL (PCPL )
Output power of CVL (PCVL )
Total output powerðPo Þ
On-time drain-to-source resistance of the deviceðRDSON Þ
Diode on-state resistanceðRFD Þ
Coil resistance (RDCR )
24 V
12 V
20 kHz
504 lH,
100 lF
50 W
40 W
100 W
0.044 X
0.0675 X
1.9456 X
ð28Þ
ð29Þ
Since v o ¼ v c , the control-to-output transfer function of the system is written as,
v o ðsÞ
d ðsÞ
r r
¼
1 D Qm
IL þ 1 V
L
C
LC in
l
Pl
l
1
1
s2 s CVP 2 RC
L LCV
2 LC þ LCR
o
ð30Þ
o
The poles of the transfer function in (30) have negative real
parts (located in left half of plane). It indicates that the system is
stable. Applying R-H criteria to (30), the necessary and sufficient
condition for stability of this system is given by,
Please cite this article as: R. Patel and R. Chudamani, Stability analysis of the main converter supplying a constant power load in a multi-converter system
considering various parasitic elements, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.03.007
R. Patel, R. Chudamani / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
(
Pmax ¼ min
lCV 2o
L
þ
V2
V 2o V 2o 1 þ r D r Q m IL þ V in þ o
;
R l
R
)
ð31Þ
The value of l is very small, thus the first term in (31) gives the
stability limit of the system and it is given by (32).
Pmax ¼
lCV 2o
L
þ
r L þ Dr Q m þ ð1 DÞr D CV 2o V 2o
V 2o
¼
þ
R
L
R
ð32Þ
The simulation study is carried out to verify effect of parasitic
elements (for variation in rQ m , rD and rL ) on stability region of buck
converter supplying a CPL in parallel with CVL and results are presented in Fig. 8.
ApplyingPCPL = 50 W andPCVL = 40 W to the main converter of
capacityPCONV = 100 W, the steady state stability limit
Pmax > PCONV is obtained for rL ¼ 1:65X, rD ¼ 3:5X and rQ m ¼ 4X
as seen in Fig. 8. Thus required value of rL is small compared to values of rQ m and rD to achieve given stability limit for same value of
Vo .
Fig. 9 shows the realization of variable rQ m , rD and rL . It is realized by connecting switches Q Q , Q D and Q L across rQ m , rD and rL
respectively as shown in part (a), part (b) and part (c) of Fig. 10.
The effective value of parasitic element rQ meff , rDeff and rLeff are given
by,
r Q meff ¼ ð1 DQ ÞDr Q m
ð33Þ
r Deff ¼ ð1 DD Þð1 DÞr D
ð34Þ
r Leff ¼ ð1 DL Þr L
ð35Þ
where DQ , DD and DL are the duty ratios of the control switches Q Q ,
Q D and Q L respectively.
Since rQ meff and rDeff in (33) and (34) are dependent on the duty
ratio D of the main switch Q m , it adds nonlinearity in the system
and hence makes the control complex. The rLeff is independent of
D and makes the control of rLeff simpler. So instead of controlling
5
rQ meff and rDeff , it is preferred to control rLeff to achieve desired
stability limit of the main converter. Moreover to achieve a given
stability the value of rL required is much smaller than rQ m and
rD . This shows that varying the values of rQ m and rD to achieve a
given stability will only increase the conduction losses in the
switch.
3.3. Buck converter supplying a CPL-CVL considering parasitic element
rQ m , r D , rL and equivalent series resistance rC of the capacitor
It is discussed in previous section that a buck converter supplying a CPL and CVL considering parasitic elements is stable. It is
attempted to verify effect of parasitic resistance of capacitor rC
on system stability. Fig. 10 shows a buck converter supplying a
CPL in parallel with CVL considering parasitic resistancesr Q m , rD ,
rL and rC of switch, diode, inductor and capacitor respectively.
Due to resistive drop in rC , Vo –VC and hence the output equation is given by,
v o ¼ v C þ rC
P vo
iL vo R
ð36Þ
Considering small perturbation in output voltage and state
variables,
"
v o ¼ v C þ rC
iL þ
P
V 2o
vo vo
#
ð37Þ
R
By simplifying the above equation
vo ¼
1
rC
P
2 rC
Vo
þ
rC
R
iL þ
1
1
P
r
V 2o C
þ
rC
R
vC
ð38Þ
The state-space equations of the buck converter supplying a CPL
in parallel with CVL considering the parasitic effect in the ON state
of switch are given by (39) and (40).
diL 1 ¼ v in r Q m iL rL iL v o
L
dt
dv C 1
P vo
iL ¼
C
dt
vo R
ð39Þ
ð40Þ
Similarly the state space equations for the OFF state of switch
are given by (41) and (42).
diL 1
¼ ½r D iL r L iL v o L
dt
dv C 1
P vo
iL ¼
C
dt
vo R
ð41Þ
ð42Þ
By applying state space averaging technique
Fig. 8. Stability region of buck converter loaded with CPL-CVL considering parasitic
effect.
Fig. 9. Circuit diagram for variation in control of parasitic elements.
diL 1 ¼ DV in v o r L þ DrQ m þ ð1 DÞr D iL
L
dt
dv C 1
P vo
iL ¼
C
dt
vo R
ð43Þ
ð44Þ
Fig. 10. Buck converter supplying a CPL in parallel with CVL considering all
parasitic elements.
Please cite this article as: R. Patel and R. Chudamani, Stability analysis of the main converter supplying a constant power load in a multi-converter system
considering various parasitic elements, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.03.007
6
R. Patel, R. Chudamani / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Assuming zero perturbation in input voltage, the dynamic
model of the buck converter supplying a CPL with parasitic effect
is derived by substituting (13) to (16) in (43) and (44). It is given
by,
diL 1
DV in v o r L þ Dr Q m þ ð1 DÞr D i L
¼
L
dt
ð45Þ
"
#
dv C 1 P vo
iL þ 2 vo ¼
C
dt
R
Vo
ð46Þ
Substituting (38) in (45) and (46), the dynamic model of buck
converter can be obtained as follows:
9
2 8
=
diL 1 4 <
rC
r L þ DrQ m þ ð1 DÞr D þ
¼
iL
C
:
L
dt
1 Pr
þ rRC ;
V 2o
3
1
v C þ DV in 5
C
þ rRC
L 1 Pr
V2
Fig. 11. Stability region of buck converter loaded with CPL-CVL with parasitic rL and
rC.
ð47Þ
o
2
1R
3
dv c 1 4
1
vC5
iL þ ¼
PrC
rC
C 1 Pr2C þ rRC
dt
C 1 V2 þ R
Vo
P
V 2o
ð48Þ
o
From (47) and (48) the control-to-output transfer functions of
the system is obtained as,
v o ðsÞ
d ð sÞ
K ðs þ K 1 Þ
¼ 2
s þ K2s þ K3
n
where, K ¼
r D r Q m
L
ð49Þ
IL þ VLin
!
K2 ¼
K3 ¼
V2
o
1
Pr C r C
þR
V2
o
LC 1
n
P<
r D r Q m
L
2 R
1
L
rC
Pr C r C
þR
V2
o
PrC r C
þR
V2
o
C 1
l þ 1PrC þrC
rC
V2
o
R
1
Pr C r C
þR
V2
o
r C C 1
P 1
V2 R
o
, K1 ¼
1
l þ 1PrC þrC rC
1
L
!
o
Pr C r C
þR
V2
o
C 1
!0 P 1 1
2 R
@ Vo
A
PrC r C
þR
V2
o
o
2
2
2
2
IL þ VLin r C V 2o C þ VRo þ lVLo C þ rC lLRV o C þ rC VLo C
1 þ rCLlC
Case
Stability Equation
Only with CVL
P max ¼
Combination with Parasitics of
switch, diode and inductor
Combination with CVL and
Parasitics of switch, diode
and inductor
Only with parasitic of capacitor
P max ¼
Combination with PCVL , parasitic
of switch, diode, inductor and
capacitor
V 2o
R (A)
lCV 2o
L
¼
P max ¼ l L þ
CV 2o
2
ðrL þDrQ m þð1DÞrD ÞCV 2o
L
V 2o
R
¼
(B)
ðrL þDrQ m þð1DÞrD ÞCV 2o
L
2
þ VRo
2
P max ¼ rC VLo C þ V in rCLV o C
rD rQ
2
2
2
2
m I þV in r V 2 CþV o þlV o C þr C lV o C þr C V o C
L
C o
R
L
LR
L
L
L
P max ¼
r C lC
1þ
L
P 1
V2 R
o
,
C 1
Table 2
Equation of Pmax for different combinations with CPL Specification of Buck converter.
ð50Þ
The poles of the transfer function in (49) have negative real
parts (located in left half of s-plane) indicating a stable system.
Applying R-H criteria to (49), the maximum value of stability is
given by (50).
As it is discussed in previous section that control of r Leff is preferred over rQ meff and r Deff , the simulation study is carried out to verify effect of parasitic for variation in rL and rC on stability region of
buck converter supplying a CPL in parallel with CVL and results are
presented in Fig. 11.
ApplyingPCPL = 50 W andPCVL = 40 W to the main converter of
capacityPO = 100 W, the steady state stability limitPmax = 200 W
(double the converter capacity) is obtained for rC ¼ 0:2X. The
required value of rL should be very large to achieve the same stability. Thus it is concluded that system stability is maximum with
rC for the same value of rL and considering CVL load as constant.
all parasitic of the system is given by (50). The equation of Pmax
for different cases are derived from the generalized equation and
they are tabulated in Table 2. In the generalized equation (50) if
rC is made equal to zero it results into (A) and (B) for the given
combinations considered and it is consistent with the established
conditions derived in [24].
Fig. 12 shows variation in stability region by varying parasitic
rQ m , rD; rL and rC . It is also clear that rC gives maximum stability
compared to rQ m , rD and rL .
The converter must have a minimum stability range of PO where
PO is the maximum output power of the converter. The variation of
rC in its minimum and maximum value is derived by substituting
rQ m , rD and rL equal to zero in (50) and considering maximum load
(PCVLmax ¼ 50) and minimum load (P CVLmin ¼ 0) supplying by the
main converter. The derived equations of rC min and r C max are given
by,
4. Comparative analysis for improvement in stability
considering different parasitic
A detailed study of effect of parasitic on system stability is presented in section-3. A generalized expression for Pmax considering
Fig. 12. Stability region of buck converter loaded with CPL-CVL with parasitic rQ, rD,
rL, and rC..
Please cite this article as: R. Patel and R. Chudamani, Stability analysis of the main converter supplying a constant power load in a multi-converter system
considering various parasitic elements, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.03.007
R. Patel, R. Chudamani / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
r Cmim ¼
r Cmax ¼
ðP O PCVLmax Þ L
CV 2o ðV in þ 1Þ
ðPO PCVLmin Þ L
CV 2o ðV in þ 1Þ
ð51Þ
[11]
ð52Þ
Thus maximum stability limit of the main converter is obtained
by controlling rC in the range of rCmin to rCmax for given operating
conditions. In addition, introduction of rC will effectively reduce
the output of main converter and increase system internal instability. Keeping in mind these aspects the maximum value of rCmax is
chosen.
5. Conclusions
In this paper a multi-converter system is studied and analyzed
from the aspect of system stability. It is mathematically evaluated
that the buck converter supplying a CPL is unstable. The effect on
system stability is verified for individual and combined parasitic
elements in the system through mathematical analysis. The effect
on system stability is verified by considering the parasitic resistance of switch, diode and inductor individually. It is observed that
the consideration of parasitic of inductor (rL ) gives maximum stability compared to parasitic of switch (r Q m ) and diode (rD ). Afterwards parasitic of capacitor rC has been introduced in the system
and the generalized expression for the stability limit Pmax has been
derived. Further it is also verified that making rC ¼ 0 results into
equation (A) and (B) which are consistent with the established conditions derived in [24]. The effect of parasitic of capacitor rC on system stability is analyzed and it is concluded that rC gives maximum
stability compared to rQ m , rD and rL . In addition the range of rC min to
r C max is derived to achieve maximum stability limit equal to at least
output capacity of the main converter for given operating
condition.
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
Declaration of Competing Interest
[23]
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
[24]
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Please cite this article as: R. Patel and R. Chudamani, Stability analysis of the main converter supplying a constant power load in a multi-converter system
considering various parasitic elements, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.03.007