Pulse-width modulation is widely used to control electronic converters. One of the most topologies used for high  DC voltage/low DC voltage conversion is the Buck converter. It is obtained as a second order system with a LC filter  between the switching subsystem and the load. The use of a coil with an amorphous magnetic material core instead  of air core lets design converters with smaller size. If high switching frequencies are used for obtaining high quality  voltage output, the value of the auto inductance L is reduced throughout the time. Then, robust controllers are needed  if the accuracy of the converter response must not be affected by auto inductance and load variations. This paper  presents a robust controller for a Buck converter based on a state space feedback control system combined with an  additional virtual space variable which minimizes the effects of the inductance and load variations when a not-toohigh switching frequency is applied. The system exhibits a null steady-state average error response for the entire  range of parameter variations. Simulation results are presented

García García, Alfonso

Morales Cabrera, Rafael

Morón Fernández, Carlos

Somolinos Sánchez, José Andrés

English

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1
PWM Control of a Buck Converter with an Amorphous Core Coil 
 
José A. Somolinos1, Rafael Morales2, Carlos Morón3 and Alfonso García3 
 
1ETS de Ingenieros Navales. Universidad Politécnica de Madrid, Arco de la Victoria s/n 28040 Madrid, SPAIN 
2ETS de Ingenieros Industriales. Universidad de Castilla-La Mancha, Campus Universitario s/n 02071 Albacete, SPAIN 
3Grupo de Sensores y Actuadores. Avenida de Juan de Herrera s/n 28040 Madrid, SPAIN 
 
Pulse-width modulation is widely used to control electronic converters. One of the most topologies used for high 
DC voltage/low DC voltage conversion is the Buck converter. It is obtained as a second order system with a LC filter 
between the switching subsystem and the load. The use of a coil with an amorphous magnetic material core instead 
of air core lets design converters with smaller size. If high switching frequencies are used for obtaining high quality 
voltage output, the value of the auto inductance L is reduced throughout the time. Then, robust controllers are needed 
if the accuracy of the converter response must not be affected by auto inductance and load variations. This paper 
presents a robust controller for a Buck converter based on a state space feedback control system combined with an 
additional virtual space variable which minimizes the effects of the inductance and load variations when a not-too-
high switching frequency is applied. The system exhibits a null steady-state average error response for the entire 
range of parameter variations. Simulation results are presented. 
 
 
Index Terms—Buck Converters, Induction aging, PWM control  
 
I. INTRODUCTION 
UCK CONVERTERS are widely used for DC high-
voltage/DC low-voltage conversion when high efficiency 
is required. It consists on a switching system which is 
controlled by PWM (Pulse Width Modulation) and a L-C 
network which is coupled with the load [1]. Since the 
parameter values L, C, and load RL define the system 
dynamics, variations of any of them modify the load voltage 
which is the variable controlled by the converter. Variations 
over RL are defined by the relation between the output voltage 
and the current to be drawn from the converter. The capacity 
C is not expected to vary throughout the time and the magnetic 
induction L is affected by errors on the nominal value when an 
amorphous core coil is used. These magnetic induction 
properties could shift thoroughout the time due to multitude of 
causes. These variations may be caused by changes in the coil 
specifications and/or changes in the core magnetic material 
properties. These changes in the coil properties may happen 
for different reasons: variations in the packing factor of coils 
are usually caused by quickly changes in temperature 
(heating) that may be permanent or not. Short circuits between 
contiguous turns of the coil could appear because of any kind 
of loss of isolation. This may be caused by overheating or 
abrasion. Depending on the number of turn affected and the 
total turns, inductance variations could be significant. 
Additionally, in magnetic core coils, it is well known that 
the induction value decreases throughout the time because of 
changes in the core magnetic properties such as anisotropy 
direction or coercive field. These variations are produced by  
material heating or heating-cooling cycles that are similar to 
undesirables annealing treatments. Different authors have 
shown that the coercive field increases with annealing 
temperature and time due to the growth of the magnetic 
material grain size or the magnetic anisotropy direction 
dispersion [2]-[5]. At the same time, it is well known that the 
magnetic losses (and therefore the activation energy in 
magnetic cores) increase with the magnetic field frequency 
[6],[7]. Finally, if temperature and/or ambient conditions 
enables it, chemical changes in the core may happen [8],[9] 
which are clearly associated with heating. Thus, the higher 
frequency of the current applied to the induction is, the most 
significant aging parameter which affects the induction 
magnetic cores. This aging effect is translated into a time 
decreasing induction value.  
Traditional control systems for L-C circuits use a linear time 
invariant model with small parameter changes. They exhibit 
accurate responses when modulation frequencies are much 
higher than the frequency of the system. However, a non-
nominal design of the control system is required when not too 
high modulation frequency is chosen (in order to limit the 
heating of the switching elements) and the L-C network 
parameters cannot be considered as constant.  
This paper presents a new state-space control system which 
is robust against changes in the inductance L and large 
changes in the load RL (voltage/current). If a second order 
state-space feedback is used, a null steady-state average error 
is obtained only for changes in the inductance L. However, if 
large changes in RL are produced too, steady-state errors in the 
output voltage appears and the addition of a virtual state 
variable that combined with a state-space feedback (producing 
a third order system) allow to obtain accurate responses when 
variations in L and RL are produced in the system.  
B
Manuscript received January 1, 2008 (date on which paper was submitted
for review). Corresponding author: J.A. Somolinos (e-mail:
joseandres.somolinos@upm.es). 
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2
The paper is organized as follows: Section II describes the 
dynamic model. Section III develops the proposed control 
system. Section IV shows the steady-state behavior of the 
converter. Section V presents several simulation results and 
Section VI states some conclusions.  
II. DYNAMIC MODELING OF THE CONVERTER 
Figure 1 illustrates a typical Buck converter with a switch 
implemented as a N depletion mode MOSFET transistor.  
 
Fig 1. Buck Converter 
 
If current over the coil iL(t) and voltage of the capacitor 
uC(t) are used as state variables ( )TCL uix = , the model of 
the converter is: 
⎩⎨
⎧
⋅=
⋅+⋅=
)()(
)()()(
txCty
tuBtxAtx
C
CC&  (1)
With matrices AC, BC and CC respectively being: 
( )10,
0
1
,11
10 =⎟⎟⎠
⎞
⎜⎜⎝
⎛=⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−
−
= CC
L
C CLB
CRC
LA  (2)
Where u(t) denotes the system input, y(t) denotes the 
converter output y(t) = uC(t) = uR(t) and RL denotes the load. 
Nominal parameters values are L0 = 10 mH, C = 25 μF and   
RL0 = 10 Ω with input UIN = 100 V and a desired output of   
UR = 50 V. Under these nominal values the system exhibits a 
critically damped dynamics with a frequency ω0 = 2000 rad/s 
which is taken from: 
2
0
2
0
0 2
11
)CR(CL L⋅
== ω  (3)
If it is represented by its input-output relation, a transfer 
function between U and Y is obtained.  
 
CLsCRs
CL
)s(G
)s(U
)s(Y
L
C
00
2
0
11
1
+⋅+==
 (4)
Being s the Laplace complex argument. From this 
representation and by applying the final value theorem,  steady 
–state responses of system (2) with step inputs are given by: 
)(GU)s(Y)t(yy CS 00 ⋅===∞==  (5)
The switching frequency is fS = 2 kHz. The relation between 
the switching frequency and the nominal system frequency f0 
becomes 6.28 which is close to the lower limit given in [10]:   
405
0
≤≤
f
f S   (6)
The variation ranges considered for the coil core and the 
load are the following: 
00
00
10
8.0
LLL RRR
LLL
⋅≤≤
≤≤⋅  (7)
Moreover when an ON-OFF PWM controller is used, input 
is of the form: 
⎩⎨
⎧
+<≤−+
−+<≤=
)1())(1(
))(1(0
)(
kTtkdkTforU
kdkTtTkfor
tu
IN
 (8)
Being T = (fS)-1, k an integer, and 1)(0 ≤≤ kd the duty ratio 
(relative pulse width). Steady-state average responses (denoted 
with _S) for a given average dS then become: ( )SCINSS d)(GU)t(yy ⋅+⋅⋅=∞== 100  (9)
Which can be normalized for dS > 0 and it results: ( )
S
SCIN
UNIT_S d
d)(GU
)t(y
⋅+⋅⋅=∞= 100  (10)
Responses from both equations (5) and (9) coincide if U = 
UIN/dS or system (2) includes this scale factor dS into matrix BC 
(which is similar to scale the numerator of GC(s)). These 
average responses do not depend on modulation frequency 
although their ripple amplitude will be smaller the higher this 
frequency. However switching components (transistor T and 
diode D) will increase their heating which is an undesired 
effect. 
III. CONTROL SYSTEM 
Controllability matrix (See[11]) of the given system results: 
( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
=⋅=
LC
LABBQ CCCC 10
01
 (11)
With range(QC) = 2 for all values of L, C ≠∝. Then the 
system results fully controllable.  
If r denotes a reference signal, the control input u is 
composed by a combination of a feed-fordward term and a 
state-space feedback term. It is defined as:  
xKrKu FW ⋅+⋅= 00  (12)
Figure 2 shows a simple state-space control system diagram. 
 
Figure 2. State-space feedback 
 
After choosing the desired eigenvalues of the controlled 
system (roots of the denominator of the closed loop transfer 
function), the feed-forward term KW0 and the gain matrix KF0  
are calculated for nominal values. As example, if desired 
eigenvalues are CRL02010
5−σ=σ , they correspond with a 
desirable characteristic polynomial of the form: 
22
00
2 2510
CRCR
)(
LL
Des +⎟⎟⎠
⎞
⎜⎜⎝
⎛+= λλλφ  (13)
then, the gain matrix KF0 and KW0 are easily obtained as: 
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3
100,1619 02
0
0
0
0
0 =⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−= W
LL
F KCR
L
CR
LK  (14)
If non-nominal conditions (variations in L and RL) are 
considered, the characteristic polynomial of the controlled 
system )( λφ becomes: 
222
0
2
0
0
0
0
02 116911
9
CRLR
RL
LR
RL
CRLR
RL
LL
L
L
L
LL
L ⎟⎟⎠
⎞
⎜⎜⎝
⎛ ++⎟⎟⎠
⎞
⎜⎜⎝
⎛ ++ λλ  (15)
Also, in order to evaluate the steady-state system response, 
the relative average error is defined as SSSr ryr /)( −=ε  
where Sr  denotes the average reference value. 
According to equation (15) if RL = RL0 remains constant and 
only variations in L are considered, the eigenvalues of the 
controlled system
00 R2R1
,σσ are real and vary from the 
nominal ones while relative average error (from equations 
(12),(15) and the last term of denominator from eq.(4)) is: 
011
125
1
1
22
0
0
0
0
=−=⋅
⋅−=
CRL
L
LCK
L
W
rRLε  (16)
Which is not L-dependant. However, if non nominal values 
of RL are considered too, new eigenvalues become from –23.5 
to -5 )( 10 σ⋅CRL  and )( 20 σ⋅CRL from –90.3 to –5 which 
imply larger system frequencies than fS and the controlled 
system will not provide good responses. Under these 
conditions it exhibits a minimum/maximum average relative 
errors which are given by: 
⎩⎨
⎧
=−
==
+⋅
−=ε
0
0
0 10479.0
0
169
251
LL
LL
L
L
r RRif
RRif
R
R
  
(17)
To cancel these errors, a new state variable is created: 
xCryrx C ⋅−=−=ε&  (18)
and the control signal u (See figure 3) is now computed as: 
xKxKu F ⋅+⋅= 00 εεε  (19)
 
 
Figure 3. State-space feedback with a virtual state variable 
 
The new expanded system is then written as: 
ru
B)t(x
)t(x
A
C
)t(x
)t(x
CC
C ⋅⎟⎟⎠
⎞
⎜⎜⎝
⎛+⋅⎟⎟⎠
⎞
⎜⎜⎝
⎛+⎟⎟⎠
⎞
⎜⎜⎝
⎛⋅⎟⎟⎠
⎞
⎜⎜⎝
⎛ −=⎟⎟⎠
⎞
⎜⎜⎝
⎛
0
10
0
0 εε
&
&
 (20)
and using the control law given by equation (19), the 
controlled system (with KFε0  = ( )εε 21 kk ) becomes: 
⎪⎩
⎪⎨⎧ ⋅=
⋅+⋅=
)t(xˆCˆ)t(y
)t(rBˆ)t(xAˆ)t(xˆ
C
CC
&
 (21)
( )
( )100ˆ
0
1
110
1
100
ˆ 210
=
⎟⎟⎠
⎞
⎜⎜⎝
⎛=
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
⋅−
−−= εεε
C
C
L
C
C
B
CRC
L
k
L
k
L
KA
 (22)
By assigning its characteristic polynomial ( )33ˆˆdet xC IA ⋅λ−  
to the polynomial from the desired eigenvalues 
5ˆˆ( 2010 −=σ⋅=σ⋅ CRCR LL  and )1ˆ 30 −=σ⋅CRL  which is:  
33
0
22
0
2
0
3 254511
CR
ˆ
CR
ˆ
CR
ˆ)ˆ(
LLL
Des ⋅+⋅+⋅+= λλλλφ
 (23)
all gains to the control law from equation (19) are obtained:  
⎟⎟⎠
⎞
⎜⎜⎝
⎛ +−−== εε CRCR
L
K
CR
K
LL
F
L
2
0
0
0
0
0
0
10134
10100  (24)
  
IV. STEADY STATE BEHAVIOR 
In this section is analyzed the relative average errors when 
control signal is given by eq.(19). The gains are obtained from 
equation (24) and non-nominal conditions have been included. 
The characteristic polynomial of the matrix CAˆ becomes:  
+⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⋅++= 2
0
03 1101 λλλφ ˆ
CLR
L
R
ˆ)ˆ(
LL
 
2
0
0
2
0
1001110135
LCR
ˆ
LCR
R
CR LL
L
L
+⎟⎟⎠
⎞
⎜⎜⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛ −⋅−+ λ
(25)
where it can be observed that it is different from the desired 
)ˆ(Des λφ . From the new last term of this polynomial, the new 
relative average errors are then cancelled according to: 
011100
1
1
2
0
0 =−=⋅−=ε εε
LCR
LCK
L
r
 
(26)
for all the values of L and RL in the range defined in (7). 
V. SIMULATION RESULTS 
Different simulations have been carried out using the 
linearization technique from [12] with fs = 2 kHz and varying 
the parameters L and RL. Figure 4 depicts some output 
responses when the control law (12) is used for controlling the 
system with different values of L and RL. A step signal is used 
as reference with 1 ms of rise time from 0 to 50V at t1=1 ms.  
 
Fig. 4. Time response of the converter 
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It can be observed how the output has reduced its damping 
and it cannot reach the reference in the so-called worst case 
(RL = 10.RL0 and L = 0.8.L0). The steady-state average ouput 
is 74.72V under this condition while the reference is 50V. 
Both values let obtain a relative average  error 
50/)72.7450( −=εr  = -0.494 which is in accordance with 
equation (17). At the bottom of figure 4 is plotted the PWM 
controller responses for both the nominal and worst cases. 
Figure 5 illustrates the converter responses when the 
proposed control law (19)-(24) is used for controlling the 
voltage outputs with the same reference signal. PWM 
responses for nominal and worst cases are plotted too.  
 
Fig. 5. Time response of the converter 
 
The controlled system exhibits slower responses than the 
previous. This is because the dominant eigenvalue 
desired 3σˆ has been chosen of a value smaller than 
2010 σσ , .Thus the steady-state average errors are cancelled 
for every value of L and RL (even in the worst case) which is 
in accordance with equation (26).  
Finally, in order to show the valuable results of the 
controller proposed, two output responses from two sets of 
parameters have been plotted in figure 6: the nominal one and 
other with the worst case (L=0.8.L0 and RL=10.RL0) when a 
new reference signal is applied. It switchs at t1=1 ms from 0V 
to 75V then it drops to 25V at t2=25 ms and finally, it reaches 
the nominal 50V at t3=50 ms. All the transitions are generated 
with rise/fall times of 1ms.  
 
Figure 6.- Output voltage with variable reference. 
 
The relative average errors are cancelled for all the 
parameter values while the ripple has not increased 
significantly in either both cases (Modulation frequency is the 
same). 
VI. CONCLUSIONS 
A robust control of a Buck converter with an amorphous 
core coil and variable load (voltage and current relation) has 
been presented. It is controlled via PWM under relative small 
modulation frequencies. State-space controllers which use  
coil current and capacitor voltage as state variables can be 
used for controlling this kind of converters with fast responses 
and null steady-state average errors; even in the case that the 
coil inductance decreases its value throughout the time. 
However, if a non-nominal load is used (which implies that 
current drawn from the converter can be very different from 
the nominal) the controller response may not operate properly. 
An additional virtual state variable that increases the order of 
the system combined with a state-space feedback let control 
the converter over the entire range of parameters including a 
decrease in L due to aging and a high increment of the value 
of RL. The well known correspondence between the responses 
of real electronic circuits and simulation results for this kind 
of systems allow us to test the proposed method and validate 
the control proposed. 
ACKNOWLEDGMENT 
This work was supported in part by the Spanish Ministerio 
de Ciencia e Innovación under Grant DPI2011-24113. 
 
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PWM Control of a Buck Converter with an Amorphous Core Coil

Archivo Digital UPM

Annealing effects on microstructure and coercive field of ferritic–martensitic

Discrete sliding-mode control of a PWM inverter for sinusoidal output wave form systhesis with optimal sliding curve”

Effect of Frequency on the Iron Losses of 0.5% and 1.5% Si Nonoriented Electrical Steels”,

Frequency dependence of initial permeability and magnetic loss factor of

Modeling grain size and dislocation density effects on harmonics of the magnetic induction”

PWM Control of a Buck Converter with an Amorphous Core Coil

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