- Email: [email protected]
System Modelling and Control in the Design of DC-DC Converters
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D .. .
14th World Congress ofIFAC
0-7c-04-1
Copyright l[.: 1999 IFAC 14th Triennial World Congn.:ss, Beijing. P.R. China
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-DC CONVERTERS Kw Zeuger * Kai Heikkinen" Idris Gadoll.ra· Teuvo Suntio"· Petri Vallittu "'**
• Helsinki University oj Technology, Control Engineering Laboratory, P.O.Box 5400, FIN-DeDl5 HUT, Finland, Phone: +35894515204, Fax: +358 94515208, Email; [email protected] *'" University of Oulu, Electronics Laboratory, P. 0. Box 444, FIN-90571 , Oulu, Finland H* Elare Oyj, P.O.Box 61, FIN-02211, Espoo, Finland
Abstract: In the paper basic modelling techniques of the Buck converter are presented. The voltage mode control is tested by tuning a PID controller, and its operation is tested by simulation. The theoretically interesting case of constant power load is discussed, and it is shown to lead to an unstable open loop system. The ability of the :(>ID controller to stabilize the system is shown. Copyright ©1999 IFAC. Keywords: Power supplies, DC-DC converters, PID controllers, power system control, constant power load.
1. INTR.ODUCTION
In recent years a lot of research has been done for designing control algorithms for switching power supplies. Among traditional control methods also modern paradigms and technologies have been introduced in the control of DC-DC converters: fuzzy control seems to be the contemporary magic word also in this context. Despite this, the traditional control engineering community seems not to have participated much in the research of this application area, and the development of modern control algorithms for switching power supplies is therefore in somewhat immature stage, which makes it an interesting field of research. The worldwide competition calls for cost efficient controllers, which are easy to implement and flexible enough to be modified for different applications. This can be achieved by developing better control algorithms sharing intelligent features . These algorithms are then implemented in digital signal processors (DSP) instead of using analogue implementation techniques.
A smooth introduction to different models, modelling techniques, and control principles is given by Erickson (1997a). Because of the nonlinear character of the switching power unit small signal models are often used to give approximate models for analysiS and controller design. Some of these modelling techniques are described by Vorperian (1990a, 1990b) and Erickson (1997b). AdvJ.nces in fuzzy control in the context of DC-DC converters ha.ve been reported e.g. by \Vang and Lee (1995), Mattavelli et al. (1997). An example of the use of robust control is given by Chang (1995). An interesting case of modelling a buck converter with a constant power load is discussed by Grigore et al. (1998). In this paper preliminary results of a project are describe d, ill which the voltage mode control of a Buck converter is studied. A small signal model of the converter is developed, and a PID type controller is designed and tuned for the system. A design environment consisting of a commercial program package (MatlabjSimulink) is used, and
7288
Copyright 1999 IF AC
ISBN: 0 08 043248 4
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
14th World Congress oflFAC
S...A1 ch
Open loep vortage and currem wavil.cms
C.
n
-5·
Fig. 1. The Buck converter its applicability in the design phase of switching power supplies is studied.
"\\
41/ ~L' \\ ,/ 2r
2. DIFFERENTIAL MODEL OF THE BUCK CONVERTER
,
"\
\,L
\;
\
I
\
\ /
/\
.J
' \ .j i\
,, 1\
,.".,/
' \ .'// '\
.,/
V
V
~. --L----.J_--'---'--------'--------'-------'-_ J.0275
The circuit describing one of the basic DC-DC converters, the Buck converter, is shown in Fig. l. The PWM (pulse width modulation) switch in the dashed box is used to control the output voltage by changing the duty cycle i.e. by changing the relation between the "ON" and "OFF" times in each switching period. In "OK" state the current through the inductor is growing transferring electric power to the load; in "OFF" state the current flows through the diode, and the induct or current is decreasing. Because the sv-ritching frequency is very high, e.g. 100 kHz, the current and voltage ripples in the inductor and in the load are small. In spite of disturbances in the input voltage and load current it is possible to control the duty cycle in a manner which keeps the output voltage constant with high accuracy. In this paper the voltage mode control is studied, in which only the output voltage, and not the inductor current, is used in the feedback. The parameters to be used in the examples and simulations are taken from one stage of a real switching power supply. The nominal values are as follows: Vi" = 140 V, VD = 54 V, D = 0.386 (nominal duty cycle), L = 100 ~H, C = 1000 ~F, R = 11 n (resistive load), fB=100 kHz (switching frequency, whic.h corresponds to the period lOftS). The dynamics of the system can be expressed by the state equations
i\
I-- -'--~----'---'--'j
0.0275
0.0275
0 . ~7S
0 . ~76
0.0276
0.0215
0.0276
0.0276
1
Fig. 2. Open loop responses to note that regarding vg(t) as the control input the equations form a linear time invariant system representation, which becomes nonlinear however, when the inductor current becmnes zero. In the former case the inductor is in continuous conduction mode, in the latter case in discontinuous conduction mode. Based on the equations (1) and (2) a Simulink model was constructed to simulate the open loop response of the system. The results (Fig.2) show that after a transient phase the voltage achieves the nominal value 54V with a small ripple; the inductor current becomes zero (discontinuous conduction mode) during the initial transient, but remains in the continuous conduction mode thereafter. In the lower figure a small part of the inductor current. waveform can be seen more clearly; the triangular shape agrees with the theory ofihe circuit topology. The time scale in the figure is in seconds (s). Note the large initial transient of the system, which is not acceptable in the use of a power supply. In Fig. 3 it is shown, how the system behaves near the steady state. In steady state analysis it is customary to use the so ealled small ripple approximation, in which the ripple components are removed from the equations. For the DC-values it holds that (Erickson, 1997a) Vo = DV; .. =54 V, h = VoiR =4.9 A. The ripple components can be calculated from
in which v.'l(t) V
(3)
7289
Copyright 1999 IFAC
ISBN: 0 08 043248 4
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
Vo~a.g.? a.nd
14th World Congress ofIFAC
linear small signal model the deviations from values of an operating point are calculated
c. w.en) wa'tlefClms: iritisJ !rS[lS,enj r~mQved
-,----- .
I
55
(Vin(t)} d(t)
'" V/i/V'VvVVIfIMIVWVVV"V'Vc1 52 1 L C
~c--:-----'--~--'--~-"----"--~...L-~-'---~.l.---_L-----' oms D.DC5
CD'
~.~1:
0.02
0.825
CC3
OM
0.045
D.CS
(7) (8)
(iL(t)}=h+iL(t)
I
~
= Vin + Vin(t) = D + JCt)
(vo(t) = Vo
(9)
+ voCt)
(10)
The linearized small signal model then becomes
diL(t)
L~ = DVin(t) - vo(t)
+
~
V'ind(t)
Cdv~~t) = iL(t) - ~Vo(t)
(11) (12)
from which the transfer functions are easy to derive
voCS) 2L-L_ _. _
0026
----'~_
______________1 _ _ _ _
D.C265
__
0.027
~
0.0275
ifs + 1 VineS)
=
____ ~
LCS2 : :
+
0.028
LCs2
Fig. 3. Open loop responses without initial transient
v:c,
+
*8 +
des) 1
The line-to-output and control-to-output transfer functions are
(14)
(4) in which Ts is the switching period. A closer look at the simulation results shows that they are well in accordance with the above approximations. It is interesting to note that the simulation of the basic models can well be done by Matlab/Simulink instead of programs specially designed for the simulation of electric circuits (e.g. Pspice, MicroCap, Saber).
3. SMALL SIGNAL MODEL In order to analyse the dynamic behaviour of the converter(s) for the purpose of controller design, small signal models can be constructed. This well known method (Erickson, 1997a, 1997b) can be applied to equations ( 1) and ( 2), which leads to
L d(i~?)
= d(t)(Vin(t»
Cd(vo(t» dt
= (. %L
- {vo(t)
(t)) _ {vo(t» R
(5) (6)
In the equations the abbreviation () is used to denote the time average of the signal over one switching period; the term d(t) is the duty cycle, the nominal value of which is D. The equations (5) and (6) are non-linear (or timevarying) because of the term d(t); to obtain a
(13)
G
()
vd S
=
1
+
Vin _8_ Qw~
+
(..L)2 WO
VID
+ Q:. + (:)2 1/VLC and the quality =~-~~~~~
1
in which
(15)
factor For the example case wo(fo) =3162 rad/s (503 Hz) and the quality factor Q=34.8=30.8 dB. The poles of the transfer functions are -45.5±i 3162.
Q
=
Wo
RI(w"L)
=
= RvCIL.
4. CONSTANT POWER LOAD The analysis ofthe dynamics of DC-DC converters feeding a constant power load is a particularly interesting problem as noted by Grigore et al. (1998). A converter is often feeding another converter in which case it can be modelled as a power supply with a constant power load. Although it is questionable, how accurate this assumption is in real power supplies, a good starting point in analysis is to consider the constant power load case. The load current then obeys the rule
(16) in which P is the constant output power, which in the operating point is P = VoIo = RoI~.
7290
Copyright 1999 IF AC
ISBN: 008 0432484
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
14th World Congress oflFAC
By considering small deviations io = Io + ~v, ro = Ra + To, Vo = Vo + Vo the load resistance can be written
(17) where i~ has been neglected as a "small" term. Taking two first terms from the Taylor approximation gives
which can be used in the voltage equation i
l
6d
where the second order term has again been neglected. The result shows that in the DC model the load resistance is Ra = Vr? / P, and in the small signal model -Ro. Hence, in the constant power load case the small-signal model transfer functions of the Buck converter become (compare with equation (13)
VoCs)
=
LCS2 _D..£..8
0.015
ed(t)
+ 1 vg(s)
+ LC8~
- ·~s Rn
+
, 1 des)
e(t) (20)
The result is interesting in the sense that it predicts the open loop system to be unstable (the transfer functions have two poles in the right half plane). Simulation results are presented in Fig. 4. The constant power load P o =265 W has been realized in the simulation by changing the resistance of the load continuously. Compared to Fig. 3 it can be noticed that the open loop response is more oscillatory than in t he case of a resistive load. The system is inherently unstable although the oscillations do not grow wit hout limit, because t.he induclor goes to discontinuous conduction mode occasionally.
f
t
u(t)
=
K (e p
+
1 Ti
e(r)dr
ded + Td{ft)
(21)
o
in which u is the controller output, and K, T i , and Td are the tuning parameters (gain, integration time, and derivation time) . The terms e,,(t)
=
bYsp(t) - yet)
0.:13
C.035
= CYsp(t) - yet) = y"p(t) - yet)
(23) (24)
describe the error signal with p ossible weightings, which can be regarded as additio nal tuning parameters of the controller (Y"p is the output reference value, and y is the process output). For example, by choosing b = c = 1 the proportional and derivative parts are affecting the error signal directly; by choosing c = 0 the pea k in the deriV"d.tive part caused by a step change in the reference Value is removed. In commercial products different modifications of the standard PID algorithm are used. One example is
U(s)
= K(bYBp(s) STd
5. PID CONTROL OF THE COKVERTER The standard textbook formula for the PID controller (Astrom and Hagglund, 1995) is
0025
Fig. 4. Open loop responses in the case of constant power load
Ro
Vq
0.32
1
Yes) + -E(s) STi
+1 + sTd/N(cYsP(S)
-Ye s»))
(25)
which is different from the previous algorithm in the sense that an additional lag term has been included in the derivative part. The tuning parameter N has typically va lues b etween 3 and 10 (Astrom and Hagglund, 1995). The converter is modelled by equations (1) and (2), and in accordance with (25) the control volta ge is calculated from
(22)
7291
Copyright 1999 IFAC
ISBN: 0 08 043248 4
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
qesp O"lses D01a1 nedby col"ltl nuo us. PlO controller
'- 'I
~O'I
r\.
I
"I
'---
!!3. 9~
O. :ll~
0,02
0.025
0.""
0, 03:5
O .{].I
i
.------L-
~ """
1 CC"'
""
.r
L o~
J C. C l ~
, ~J.a2
:J . ~:3
003
O ,03~
,
J __
004
0."""
__
:J. C:l
Fig. 5. Closed loop responses
The duty cycle controlling the switch is determined by the PWM block with the additional property that the duty cycle is restricted to the interval [0 0.5]. In Fig. 5 the closed loop responses have been presented. The parameters of the PID controller are K = 500, Ti = 0.001, Ta = 0.0005, and N = 3. The amplitude scale has been chosen accurate enough to show the disturbances caused by two step changes in the input voltage Vin (at time 0 .02 s from 140 V to 120 V, and at time 0.04 s from 120 V to 160 V). The lower figure shows the control voltage. Practical experience shows that for real switching power supplies the above control result is far too optimistic. It is interesting to study reasons for t hat in order to develop the simulation procedures towards a more realistic setting. There are three main reasons why practical controllers cannot achieve as accurate results as above. Firstly, a real PWM switch is not an ideal component: it contains an inherent inaccuracy caused by quantization. Secondly, t he discretization intenrc!l of the PID controller has not been taken into account. Thirdly, the components of the converter are not ideal as assumed in the preceding discussion. They contain non-ideal elements ("parasite effects"), which can to some accuracy be modelled by series resistors. Only the two first problems are considered short ly in t his paper. The PWM switch can be described as an element, which converts the control voltage vc(t) to the duty cycle. The value of the duty cycle is theoretically between 0 and 1; for practical Buck converters the duty cycle must be restricted
14th World Congress oflFAC
to the values below 0.5 to keep the inductor in continuous conduction mode. The duty cycle is formed by using a sawtooth signal with linearly increasing values between 0 and 0.5 during the switching period l011S (100 kHz). The "ON" and "OFF" periods of the real switch in the converter is determined by the time instants, in which the sawtooth signal crosses the V"d.lue of the desired duty cycle. In practice, the precision for this kind of a PWM module is typically about 50 ns, which can also be considered as a quantization nonideality of the P'¥M switch. The accuracy can be estimated by noting that the control voltage range is divided into 100 parts (50 ns/5 ,us=O.Ol). For a peak value of 50 V that would mean an inaccuracy of 0.5 V. If an analog controller of the converter is replaced by a discrete algorithm, a suitable DSP card is normally used. The basic switching frequency (100 kHz in this example case) is too high for commercial DSP technology of today; processor operating frequencies of 10 kHz - 40 kH7, must be considered as a realistic alternative. To keep the price of the eventual product (switching power supply with processor-based voltage controller) cheap, frequencies at the lower end of t he mentioned interval must be considered. The controller algorithm (26) is discretized by using backward approximations in both integral and derivative terms. The sampling frequency is 25 kHz (sampling time 40 l1S). In t h e transfer function notation the controller is then
vc(z)
=
KC1
hz
+ Ti.z _ Ti )(vret(z ) - v o(z»
-K( (Ta/ N
TdZ - Td + h)z _ T d/N)vo(.z) (27)
where h is the sampling period. The results of the simulation are presented in Fig. 6. The tuning parameters of the PID controller are K=3, 1i=O.OOl, Td = O.OOOl, and N=3. The effects of the line and load disturbances are clearly more notable when compared to Fig. 5. In the lower picture of Fig. 6 the noise caused by the nonideal P\\'''M switch can be noted. The variation in the output signal is in this case not as severe as predicted. The same controller has then been used in the constant power load case. The result is shown in Fig. 7 . It can be noticed that the closed loop system is stable (compare to Fig. 4), although the deviations after the disturbances are still too large. Normally, the desired accuracy of the output voltage is ±1 V.
6. CONCLUSIOK The voltage mode control of the Buck converter operating mainly in the continuous conduction
7292
Copyright 1999 IFAC
ISBN: 0 08 043248 4
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
14th World Congress ofIFAC
to be highly nonlinear, and therefore "intelligent" control in required. The results obtained by using an analog or discretized PID controller can be used for comparison and as a starting point. The reported results in the pa.per have been obtained in a project, which aims to replace analog controllers in switching power supplies with digital control algorithms. The validation of the results has been done with Saber, which is an extensive simulation package for the simulation of electrical systems. In the next stage, an additional software, FuzzyTech, will be studied to investigate intelligent control of the converters. All algorithms will be considered for possible implementation in a DSP setup.
7. REFERENCES Fig. 6. Closed loop responses obtained by using a discrete time PID controller and a non-ideal PWM switch
-,-------,---,--1--
C;:.m;;ta ~t pcwer
,,:~-. ,-~--,--
l~ad
di!:;c'ete-
~'U ~nlrol~9f
~I
~
/,.,~I~'~~~·w~_-.-~~-~~'\ ..--,~~~.~.~-. -~'"" \"~~'~-l
",!
~(
"'
02.,Li_-"--_--'-_--'-_--'-_--'-_--'-_--'-_--'-_--'-_--' C
'O.:J0!5
00<,
Q.:J\5
0.002
Q.C25
Chang, C. (1995). Robust Control of DC-DC Converters: The Buck Converter, Proceedings of the IEEE Power Electronics Specialists Conference, pp. 1094-1097.
0.00
0.,030
'0.04
(j,~
(j,C!!
Erickson, R. '.iV. (1997a). Fundamentals of Power Electronics. Chapman & Hall. Erickson, R.. W. (1997b). Advances in Averaged Switch Modeling, Proceedings of the Fourth Brazilian Congress of Power Electronics (COBEP97), Belo Horizonte, BraziL Grigore, V., Hiitonen, J., Kyyra, .J., and T. Suntio. (1998). Dynamics of a Buck Converter with a Constant Power Load, Proceedings of the PESC'98 Conference, Fukuoka, Japan, Vo!. 1, pp. 72-78. Mattavelli, P., Rossetto, L., Spiazzi, G., and P. Tenti. (1997). General-Purpose FU:l:lY Controller for DC-DC Converters. IEEE Transactions on Power Electronics, Vol.12, No.l, pp. 79-86.
Fig. 7. Closed loop responses in the constant power load case; a discrete time PID controller and a non-ideal PWM switch mode has been discussed in the paper. Small signal models were used to tune the PID controller, which was then tested in the case of both resistive load and constant power load. The results showed that the ba.sic converter can very accurately be controlled by the PID control algorithm, and the inherently unstable case of constant power load can be stabilised with this controller. However, -the control results were not good enough in the case of initial transients and large disturbances entering the system. More research is still needed in these aspects. Furthermore, questions like the pa.rasite effects, output power limit, and overcurrent protection have not been considered in the paper. In practice, the control problem turns out
Vorperian, V. (1990a). Simplified Analysis of PWM Converters Using Model of PWM Switch. Part I. Continuous Conduction Mode. IEEE Transactions on Aerospace and Electronic Sy.~tems , Vol. 26, No. 3, pp. 490-496. Vorperian, V. (1990b). Simplified Analysis of PWM Converters Using Model of PWM Switch. Part 11. Discontinuous Conduction Mode. IEEE Transactions on Aerospace and Electronic Systems, Vol. 26, No. 3, pp. 497-505. Wang, F. H., C. Q. Lee. (1995). Comparison of Fuzzy Logic and Current-Mode Control Tech-
niques in Buck, Boost, and Buck/Boost Converters, Proceedings of the IEEE Power Electronics Specialists Conference, Vo!. 2, pp. 1079-1085. Astrom, K . .T., T. Hagglund. (1995). PID Controllers: Theory, Design, and Tuning. Instrument Society of America.
7293
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress ofIFAC
0-7c-04-1
Copyright l[.: 1999 IFAC 14th Triennial World Congn.:ss, Beijing. P.R. China
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-DC CONVERTERS Kw Zeuger * Kai Heikkinen" Idris Gadoll.ra· Teuvo Suntio"· Petri Vallittu "'**
• Helsinki University oj Technology, Control Engineering Laboratory, P.O.Box 5400, FIN-DeDl5 HUT, Finland, Phone: +35894515204, Fax: +358 94515208, Email; [email protected] *'" University of Oulu, Electronics Laboratory, P. 0. Box 444, FIN-90571 , Oulu, Finland H* Elare Oyj, P.O.Box 61, FIN-02211, Espoo, Finland
Abstract: In the paper basic modelling techniques of the Buck converter are presented. The voltage mode control is tested by tuning a PID controller, and its operation is tested by simulation. The theoretically interesting case of constant power load is discussed, and it is shown to lead to an unstable open loop system. The ability of the :(>ID controller to stabilize the system is shown. Copyright ©1999 IFAC. Keywords: Power supplies, DC-DC converters, PID controllers, power system control, constant power load.
1. INTR.ODUCTION
In recent years a lot of research has been done for designing control algorithms for switching power supplies. Among traditional control methods also modern paradigms and technologies have been introduced in the control of DC-DC converters: fuzzy control seems to be the contemporary magic word also in this context. Despite this, the traditional control engineering community seems not to have participated much in the research of this application area, and the development of modern control algorithms for switching power supplies is therefore in somewhat immature stage, which makes it an interesting field of research. The worldwide competition calls for cost efficient controllers, which are easy to implement and flexible enough to be modified for different applications. This can be achieved by developing better control algorithms sharing intelligent features . These algorithms are then implemented in digital signal processors (DSP) instead of using analogue implementation techniques.
A smooth introduction to different models, modelling techniques, and control principles is given by Erickson (1997a). Because of the nonlinear character of the switching power unit small signal models are often used to give approximate models for analysiS and controller design. Some of these modelling techniques are described by Vorperian (1990a, 1990b) and Erickson (1997b). AdvJ.nces in fuzzy control in the context of DC-DC converters ha.ve been reported e.g. by \Vang and Lee (1995), Mattavelli et al. (1997). An example of the use of robust control is given by Chang (1995). An interesting case of modelling a buck converter with a constant power load is discussed by Grigore et al. (1998). In this paper preliminary results of a project are describe d, ill which the voltage mode control of a Buck converter is studied. A small signal model of the converter is developed, and a PID type controller is designed and tuned for the system. A design environment consisting of a commercial program package (MatlabjSimulink) is used, and
7288
Copyright 1999 IF AC
ISBN: 0 08 043248 4
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
14th World Congress oflFAC
S...A1 ch
Open loep vortage and currem wavil.cms
C.
n
-5·
Fig. 1. The Buck converter its applicability in the design phase of switching power supplies is studied.
"\\
41/ ~L' \\ ,/ 2r
2. DIFFERENTIAL MODEL OF THE BUCK CONVERTER
,
"\
\,L
\;
\
I
\
\ /
/\
.J
' \ .j i\
,, 1\
,.".,/
' \ .'// '\
.,/
V
V
~. --L----.J_--'---'--------'--------'-------'-_ J.0275
The circuit describing one of the basic DC-DC converters, the Buck converter, is shown in Fig. l. The PWM (pulse width modulation) switch in the dashed box is used to control the output voltage by changing the duty cycle i.e. by changing the relation between the "ON" and "OFF" times in each switching period. In "OK" state the current through the inductor is growing transferring electric power to the load; in "OFF" state the current flows through the diode, and the induct or current is decreasing. Because the sv-ritching frequency is very high, e.g. 100 kHz, the current and voltage ripples in the inductor and in the load are small. In spite of disturbances in the input voltage and load current it is possible to control the duty cycle in a manner which keeps the output voltage constant with high accuracy. In this paper the voltage mode control is studied, in which only the output voltage, and not the inductor current, is used in the feedback. The parameters to be used in the examples and simulations are taken from one stage of a real switching power supply. The nominal values are as follows: Vi" = 140 V, VD = 54 V, D = 0.386 (nominal duty cycle), L = 100 ~H, C = 1000 ~F, R = 11 n (resistive load), fB=100 kHz (switching frequency, whic.h corresponds to the period lOftS). The dynamics of the system can be expressed by the state equations
i\
I-- -'--~----'---'--'j
0.0275
0.0275
0 . ~7S
0 . ~76
0.0276
0.0215
0.0276
0.0276
1
Fig. 2. Open loop responses to note that regarding vg(t) as the control input the equations form a linear time invariant system representation, which becomes nonlinear however, when the inductor current becmnes zero. In the former case the inductor is in continuous conduction mode, in the latter case in discontinuous conduction mode. Based on the equations (1) and (2) a Simulink model was constructed to simulate the open loop response of the system. The results (Fig.2) show that after a transient phase the voltage achieves the nominal value 54V with a small ripple; the inductor current becomes zero (discontinuous conduction mode) during the initial transient, but remains in the continuous conduction mode thereafter. In the lower figure a small part of the inductor current. waveform can be seen more clearly; the triangular shape agrees with the theory ofihe circuit topology. The time scale in the figure is in seconds (s). Note the large initial transient of the system, which is not acceptable in the use of a power supply. In Fig. 3 it is shown, how the system behaves near the steady state. In steady state analysis it is customary to use the so ealled small ripple approximation, in which the ripple components are removed from the equations. For the DC-values it holds that (Erickson, 1997a) Vo = DV; .. =54 V, h = VoiR =4.9 A. The ripple components can be calculated from
in which v.'l(t) V
(3)
7289
Copyright 1999 IFAC
ISBN: 0 08 043248 4
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
Vo~a.g.? a.nd
14th World Congress ofIFAC
linear small signal model the deviations from values of an operating point are calculated
c. w.en) wa'tlefClms: iritisJ !rS[lS,enj r~mQved
-,----- .
I
55
(Vin(t)} d(t)
'" V/i/V'VvVVIfIMIVWVVV"V'Vc1 52 1 L C
~c--:-----'--~--'--~-"----"--~...L-~-'---~.l.---_L-----' oms D.DC5
CD'
~.~1:
0.02
0.825
CC3
OM
0.045
D.CS
(7) (8)
(iL(t)}=h+iL(t)
I
~
= Vin + Vin(t) = D + JCt)
(vo(t) = Vo
(9)
+ voCt)
(10)
The linearized small signal model then becomes
diL(t)
L~ = DVin(t) - vo(t)
+
~
V'ind(t)
Cdv~~t) = iL(t) - ~Vo(t)
(11) (12)
from which the transfer functions are easy to derive
voCS) 2L-L_ _. _
0026
----'~_
______________1 _ _ _ _
D.C265
__
0.027
~
0.0275
ifs + 1 VineS)
=
____ ~
LCS2 : :
+
0.028
LCs2
Fig. 3. Open loop responses without initial transient
v:c,
+
*8 +
des) 1
The line-to-output and control-to-output transfer functions are
(14)
(4) in which Ts is the switching period. A closer look at the simulation results shows that they are well in accordance with the above approximations. It is interesting to note that the simulation of the basic models can well be done by Matlab/Simulink instead of programs specially designed for the simulation of electric circuits (e.g. Pspice, MicroCap, Saber).
3. SMALL SIGNAL MODEL In order to analyse the dynamic behaviour of the converter(s) for the purpose of controller design, small signal models can be constructed. This well known method (Erickson, 1997a, 1997b) can be applied to equations ( 1) and ( 2), which leads to
L d(i~?)
= d(t)(Vin(t»
Cd(vo(t» dt
= (. %L
- {vo(t)
(t)) _ {vo(t» R
(5) (6)
In the equations the abbreviation () is used to denote the time average of the signal over one switching period; the term d(t) is the duty cycle, the nominal value of which is D. The equations (5) and (6) are non-linear (or timevarying) because of the term d(t); to obtain a
(13)
G
()
vd S
=
1
+
Vin _8_ Qw~
+
(..L)2 WO
VID
+ Q:. + (:)2 1/VLC and the quality =~-~~~~~
1
in which
(15)
factor For the example case wo(fo) =3162 rad/s (503 Hz) and the quality factor Q=34.8=30.8 dB. The poles of the transfer functions are -45.5±i 3162.
Q
=
Wo
RI(w"L)
=
= RvCIL.
4. CONSTANT POWER LOAD The analysis ofthe dynamics of DC-DC converters feeding a constant power load is a particularly interesting problem as noted by Grigore et al. (1998). A converter is often feeding another converter in which case it can be modelled as a power supply with a constant power load. Although it is questionable, how accurate this assumption is in real power supplies, a good starting point in analysis is to consider the constant power load case. The load current then obeys the rule
(16) in which P is the constant output power, which in the operating point is P = VoIo = RoI~.
7290
Copyright 1999 IF AC
ISBN: 008 0432484
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
14th World Congress oflFAC
By considering small deviations io = Io + ~v, ro = Ra + To, Vo = Vo + Vo the load resistance can be written
(17) where i~ has been neglected as a "small" term. Taking two first terms from the Taylor approximation gives
which can be used in the voltage equation i
l
6d
where the second order term has again been neglected. The result shows that in the DC model the load resistance is Ra = Vr? / P, and in the small signal model -Ro. Hence, in the constant power load case the small-signal model transfer functions of the Buck converter become (compare with equation (13)
VoCs)
=
LCS2 _D..£..8
0.015
ed(t)
+ 1 vg(s)
+ LC8~
- ·~s Rn
+
, 1 des)
e(t) (20)
The result is interesting in the sense that it predicts the open loop system to be unstable (the transfer functions have two poles in the right half plane). Simulation results are presented in Fig. 4. The constant power load P o =265 W has been realized in the simulation by changing the resistance of the load continuously. Compared to Fig. 3 it can be noticed that the open loop response is more oscillatory than in t he case of a resistive load. The system is inherently unstable although the oscillations do not grow wit hout limit, because t.he induclor goes to discontinuous conduction mode occasionally.
f
t
u(t)
=
K (e p
+
1 Ti
e(r)dr
ded + Td{ft)
(21)
o
in which u is the controller output, and K, T i , and Td are the tuning parameters (gain, integration time, and derivation time) . The terms e,,(t)
=
bYsp(t) - yet)
0.:13
C.035
= CYsp(t) - yet) = y"p(t) - yet)
(23) (24)
describe the error signal with p ossible weightings, which can be regarded as additio nal tuning parameters of the controller (Y"p is the output reference value, and y is the process output). For example, by choosing b = c = 1 the proportional and derivative parts are affecting the error signal directly; by choosing c = 0 the pea k in the deriV"d.tive part caused by a step change in the reference Value is removed. In commercial products different modifications of the standard PID algorithm are used. One example is
U(s)
= K(bYBp(s) STd
5. PID CONTROL OF THE COKVERTER The standard textbook formula for the PID controller (Astrom and Hagglund, 1995) is
0025
Fig. 4. Open loop responses in the case of constant power load
Ro
Vq
0.32
1
Yes) + -E(s) STi
+1 + sTd/N(cYsP(S)
-Ye s»))
(25)
which is different from the previous algorithm in the sense that an additional lag term has been included in the derivative part. The tuning parameter N has typically va lues b etween 3 and 10 (Astrom and Hagglund, 1995). The converter is modelled by equations (1) and (2), and in accordance with (25) the control volta ge is calculated from
(22)
7291
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SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
qesp O"lses D01a1 nedby col"ltl nuo us. PlO controller
'- 'I
~O'I
r\.
I
"I
'---
!!3. 9~
O. :ll~
0,02
0.025
0.""
0, 03:5
O .{].I
i
.------L-
~ """
1 CC"'
""
.r
L o~
J C. C l ~
, ~J.a2
:J . ~:3
003
O ,03~
,
J __
004
0."""
__
:J. C:l
Fig. 5. Closed loop responses
The duty cycle controlling the switch is determined by the PWM block with the additional property that the duty cycle is restricted to the interval [0 0.5]. In Fig. 5 the closed loop responses have been presented. The parameters of the PID controller are K = 500, Ti = 0.001, Ta = 0.0005, and N = 3. The amplitude scale has been chosen accurate enough to show the disturbances caused by two step changes in the input voltage Vin (at time 0 .02 s from 140 V to 120 V, and at time 0.04 s from 120 V to 160 V). The lower figure shows the control voltage. Practical experience shows that for real switching power supplies the above control result is far too optimistic. It is interesting to study reasons for t hat in order to develop the simulation procedures towards a more realistic setting. There are three main reasons why practical controllers cannot achieve as accurate results as above. Firstly, a real PWM switch is not an ideal component: it contains an inherent inaccuracy caused by quantization. Secondly, t he discretization intenrc!l of the PID controller has not been taken into account. Thirdly, the components of the converter are not ideal as assumed in the preceding discussion. They contain non-ideal elements ("parasite effects"), which can to some accuracy be modelled by series resistors. Only the two first problems are considered short ly in t his paper. The PWM switch can be described as an element, which converts the control voltage vc(t) to the duty cycle. The value of the duty cycle is theoretically between 0 and 1; for practical Buck converters the duty cycle must be restricted
14th World Congress oflFAC
to the values below 0.5 to keep the inductor in continuous conduction mode. The duty cycle is formed by using a sawtooth signal with linearly increasing values between 0 and 0.5 during the switching period l011S (100 kHz). The "ON" and "OFF" periods of the real switch in the converter is determined by the time instants, in which the sawtooth signal crosses the V"d.lue of the desired duty cycle. In practice, the precision for this kind of a PWM module is typically about 50 ns, which can also be considered as a quantization nonideality of the P'¥M switch. The accuracy can be estimated by noting that the control voltage range is divided into 100 parts (50 ns/5 ,us=O.Ol). For a peak value of 50 V that would mean an inaccuracy of 0.5 V. If an analog controller of the converter is replaced by a discrete algorithm, a suitable DSP card is normally used. The basic switching frequency (100 kHz in this example case) is too high for commercial DSP technology of today; processor operating frequencies of 10 kHz - 40 kH7, must be considered as a realistic alternative. To keep the price of the eventual product (switching power supply with processor-based voltage controller) cheap, frequencies at the lower end of t he mentioned interval must be considered. The controller algorithm (26) is discretized by using backward approximations in both integral and derivative terms. The sampling frequency is 25 kHz (sampling time 40 l1S). In t h e transfer function notation the controller is then
vc(z)
=
KC1
hz
+ Ti.z _ Ti )(vret(z ) - v o(z»
-K( (Ta/ N
TdZ - Td + h)z _ T d/N)vo(.z) (27)
where h is the sampling period. The results of the simulation are presented in Fig. 6. The tuning parameters of the PID controller are K=3, 1i=O.OOl, Td = O.OOOl, and N=3. The effects of the line and load disturbances are clearly more notable when compared to Fig. 5. In the lower picture of Fig. 6 the noise caused by the nonideal P\\'''M switch can be noted. The variation in the output signal is in this case not as severe as predicted. The same controller has then been used in the constant power load case. The result is shown in Fig. 7 . It can be noticed that the closed loop system is stable (compare to Fig. 4), although the deviations after the disturbances are still too large. Normally, the desired accuracy of the output voltage is ±1 V.
6. CONCLUSIOK The voltage mode control of the Buck converter operating mainly in the continuous conduction
7292
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ISBN: 0 08 043248 4
SYSTEM MODELLING AND CONTROL IN THE DESIGN OF DC-D ...
14th World Congress ofIFAC
to be highly nonlinear, and therefore "intelligent" control in required. The results obtained by using an analog or discretized PID controller can be used for comparison and as a starting point. The reported results in the pa.per have been obtained in a project, which aims to replace analog controllers in switching power supplies with digital control algorithms. The validation of the results has been done with Saber, which is an extensive simulation package for the simulation of electrical systems. In the next stage, an additional software, FuzzyTech, will be studied to investigate intelligent control of the converters. All algorithms will be considered for possible implementation in a DSP setup.
7. REFERENCES Fig. 6. Closed loop responses obtained by using a discrete time PID controller and a non-ideal PWM switch
-,-------,---,--1--
C;:.m;;ta ~t pcwer
,,:~-. ,-~--,--
l~ad
di!:;c'ete-
~'U ~nlrol~9f
~I
~
/,.,~I~'~~~·w~_-.-~~-~~'\ ..--,~~~.~.~-. -~'"" \"~~'~-l
",!
~(
"'
02.,Li_-"--_--'-_--'-_--'-_--'-_--'-_--'-_--'-_--'-_--' C
'O.:J0!5
00<,
Q.:J\5
0.002
Q.C25
Chang, C. (1995). Robust Control of DC-DC Converters: The Buck Converter, Proceedings of the IEEE Power Electronics Specialists Conference, pp. 1094-1097.
0.00
0.,030
'0.04
(j,~
(j,C!!
Erickson, R. '.iV. (1997a). Fundamentals of Power Electronics. Chapman & Hall. Erickson, R.. W. (1997b). Advances in Averaged Switch Modeling, Proceedings of the Fourth Brazilian Congress of Power Electronics (COBEP97), Belo Horizonte, BraziL Grigore, V., Hiitonen, J., Kyyra, .J., and T. Suntio. (1998). Dynamics of a Buck Converter with a Constant Power Load, Proceedings of the PESC'98 Conference, Fukuoka, Japan, Vo!. 1, pp. 72-78. Mattavelli, P., Rossetto, L., Spiazzi, G., and P. Tenti. (1997). General-Purpose FU:l:lY Controller for DC-DC Converters. IEEE Transactions on Power Electronics, Vol.12, No.l, pp. 79-86.
Fig. 7. Closed loop responses in the constant power load case; a discrete time PID controller and a non-ideal PWM switch mode has been discussed in the paper. Small signal models were used to tune the PID controller, which was then tested in the case of both resistive load and constant power load. The results showed that the ba.sic converter can very accurately be controlled by the PID control algorithm, and the inherently unstable case of constant power load can be stabilised with this controller. However, -the control results were not good enough in the case of initial transients and large disturbances entering the system. More research is still needed in these aspects. Furthermore, questions like the pa.rasite effects, output power limit, and overcurrent protection have not been considered in the paper. In practice, the control problem turns out
Vorperian, V. (1990a). Simplified Analysis of PWM Converters Using Model of PWM Switch. Part I. Continuous Conduction Mode. IEEE Transactions on Aerospace and Electronic Sy.~tems , Vol. 26, No. 3, pp. 490-496. Vorperian, V. (1990b). Simplified Analysis of PWM Converters Using Model of PWM Switch. Part 11. Discontinuous Conduction Mode. IEEE Transactions on Aerospace and Electronic Systems, Vol. 26, No. 3, pp. 497-505. Wang, F. H., C. Q. Lee. (1995). Comparison of Fuzzy Logic and Current-Mode Control Tech-
niques in Buck, Boost, and Buck/Boost Converters, Proceedings of the IEEE Power Electronics Specialists Conference, Vo!. 2, pp. 1079-1085. Astrom, K . .T., T. Hagglund. (1995). PID Controllers: Theory, Design, and Tuning. Instrument Society of America.
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