Digital Control for Switched Mode DC-DC

Buck Converters

Nistor Daniel Trip, Sanda Dale

Faculty of Electrical Engineering and Information

Technology

University of Oradea

Oradea, Romania

dtrip@uoradea.ro, sdale@uoradea.ro

Viorel Popescu

Department of Applied Electronics, Faculty of Electronics

and Telecommunications

University “Politehnica” of Timişoara

Timişoara, Romania

viorel.popescu@etc.upt.ro

Abstract— The paper presents the design of a PI digital controller

used to control the output voltage of a switched mode DC-DC

buck converter. The digital controller is implemented with the

help of an 8 bit microcontroller. To derive the control law of the

PI controller, the switched mode DC-DC converter is modeled as

a linear system using the state variables averaging technique. The

authors also present the simulation and experiments results

related to the theoretical aspects mentioned in the paper.

Keywords-switched mode DC-DC converter, state averaging

method, digital controller, PI digital control law

I. INTRODUCTION

An important problem in power electronics is to control or to design controllers for different kind of switched mode converters. All switched mode power converters are nonlinear systems and for this reason to develop appropriate control circuits it is not easy. Despite of this fact, one can derive an equivalent linear model for a nonlinear system and then to accomplish a controller based on the linear control theory. The control law can be then implemented with specialized analog integrated circuits. The design of the controllers with the help of the analog circuits for the switched mode converters is presented in many valuable scientific papers and books, as well as in design guides, such as [1],[2],[3],[5],[14]. The analog control circuits present some drawbacks as follows: monitor a reduced number of signals to save costs, solve only specific task, requires auxiliary active and passive electronic devices [15], use pulse amplifier as interface for the electronic power switches, shown reduced noise immunity and difficulty to assure further developments or new more complex control functions.

Nowadays, one can observe a trend to change the analog controllers with digital controllers and even with embedded system. This change is possible due to the great progress in the field of digital technologies, for the need of more and more complex control functions [8] and the communications capabilities. The implementation of complex control function with analog circuits is difficult but using a digital programmable device the implementation becomes easier. Moreover, the digital approach offers a more accurate solution for a wide range of temperature. The digital representation or

format of the signals values or results fits and is useful when the conversion system includes a digital graphical user interface or when variables or results have to be stored on memory units. Another advantage of the digital implementation is the fact that one can set many different thresholds for the signals used to monitor the limit conditions: over voltage and / or current, under voltage and over temperature.

In the first part of the paper authors present some consideration regarding the equivalent linear model of a switched mode DC-DC buck converter. The linear model is obtained using the state averaging technique [1], [2]. After this key step of the controller design, the authors present then the implementation of a digital control PI algorithm to control the value of the converter’s output voltage. The parameters of the digital PI controller and the control algorithm are derived and verified with the help of MATLAB and Simulink programming and simulation environment. To test the control algorithm, the authors developed an embedded circuit whose main device is an 8 bit microcontroller, PIC16F887 [11], that contains all specific circuits requested in control application.

The experimental results shown at the end of the paper are in good agreements with the simulation results.

II. OPERATION OF BUCK CONVERTER

The switched mode DC-DC buck converter [1], [5], [6], [7] together with its main signals is shown in Fig.1.

Figure 1. Switched mode DC-DC buck converter

Briefly, the operation principle of the switched mode DC-DC converter is described hereinafter. In the continuous

ugs

D

iD

L iL

iC

C R ui

T1

uC

iR

uD

978-1-4244-8460-7/10/$26.00 ©2010 IEEE 99

conduction mode, the converter pass through two operation modes, since the transistor T1 is operating in switching mode, at a constant period T. In the first operation mode, when the power MOSFET is in its on-state [8] the diode D is in off-state, the operation of the converter can be described with the help of the next system:

−=

+−=

R

ui

dt

duC

Uudt

diL

CL

C

iCL

(1)

This operation mode runs for αT period, α ∈ (0, 1). In the second operation mode, when the transistor T1 changes its

state, the diode turns-on and the converter runs for (1-α)T period according to next system:

−=

−=

R

ui

dt

duC

udt

diL

CL

C

CL

(2)

The state variables of the converters are iL and uC defined in (3), where IL, UC are the steady state components and

^^

, CL ui are alternate components.

+=

+=

^

^

CCC

LLL

uUu

iIi (3)

In steady state operation, one can derive the next relation:

( )1,0, ∈αα= iC UU (4)

where α is the duty cycle of the command voltage ugs.

The converter operates in the continuous conduction mode when the value of the inductance L is greater than Lmin defined as follow [5]:

( )α−= 12

min

RTL (5)

The converter is a non-linear system and to design a PI controller for this kind of converter it is necessary to obtain first its equivalent linear model as it is shown in the next section.

III. LINEAR MODEL OF BUCK CONVERTER

A non-linear system, such as switched mode DC-DC

converters, could be modeled as a linear system using the state

variables averaging technique. According to this method, each

differential equation system (1) and (2) is multiplied with a

weighting coefficient α and (1 - α) respectively. After the

weighting of the two systems (1) and (2), that present the buck

converter in its two operation modes, the equivalent linear

model of the buck converter (6), written in the matrix form, is

obtained then as a term by term sum.

[ ]

=

==

+

=

=

α+

−

−=

c

L

C

LCO

iC

L

iC

L

C

L

u

iC

u

iuu

BUu

iA

ULu

i

RCC

Lu

i

dt

d

10

011

10

(6)

The last relation in (6) represents the output voltage of the converter that is in this case the same with uC. This relation presents a correspondence between the controlled signal and the state variables of the circuit.

The system (6) is important because one can derive transfer functions [4] of the buck converter as follow:

( ) BAsICsH1

2)(−

−= (7)

To obtain the voltage input to output transfer function, it is necessary to introduce a disturbance in the state variables vector as in (3), in the input voltage as well and keep the duty

cycle α constant. Applying the Laplace transform only to the small signal terms and then using (7), one can derive the requested transfer function shown in (8).

LCRCssLC

sH/1/

1)(

21++

⋅α

= (8)

To obtain the control to output voltage transfer function (9), ui is kept constant and disturbances are introduced in the duty cycle and state variables vector (3).

LCRCssLC

UsH i

/1/

1)(

22++

⋅= (9)

IV. DESIGN OF DIGITAL PI CONTROLLER

The structure chosen to control and control the output voltage is a closed loop with PI controller. The basic structure of the control system is given in Fig.2.

Figure 2.

100

In the first step, a continuous PI controller was tuned to meet the performances imposed to the system. The PI algorithm [4], [12], [13] is given by the time-expression (10) and the transfer function in (11).

∫ ε⋅+ε⋅= dttktktc ip )()()( (10)

s

kksH i

pc +=)( (11)

where the tuned parameters are: 165.0=pk and 1.0=ik .

The Simulink block diagram for the PI control structure is given in Fig.3.

Figure 3.

In a second step the PI controller is sampled through ‘Zero-

Order-Hold’ method with sampling frequency 20=f kHz,

hence the sampling period is 3

1020

1h

⋅= . The discrete

transfer function obtained after this step, as in (12), is implemented by the control algorithm (13).

1

)(−

−⋅+⋅=

z

khkzkzH

pip

c (12)

)1()1()()()( −−−ε⋅−⋅+ε⋅= tctkhktktc pip (13)

The PI-control discrete control algorithm is implemented by the Simulink block diagram as in Fig.4. The closed-loop control structure with the PI-controller was implemented in Simulink as in Fig.5.

Figure 4.

Figure 5.

The results obtained through simulation are shown in Fig.6.a for the output voltage U and in Fig.6.b for the command c.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t,[s]

U,[

V]

a.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010.415

0.416

0.417

0.418

0.419

0.42

0.421

t,[s]

c

b.

Figure 6. Simulation results

In above presented figures, the “process” term represents the linear model of the switched mode DC-DC buck converter.

V. ALGORITHM IMPLEMENTATION AND

EXPERIMENTAL RESULTS

To verify the function of the controller and the simulation results presented in the previous section, the authors developed an embedded system based on an 8 bit microcontroller. The main device for this embedded system is PIC16F887 microcontroller with the next main characteristics, whereby some of them used in the experiments: 10 bit resolution ADC and 14 multiplexed analog input channels, 10 bit PWM with 1, 2 or 4 channels and 20 kHz maximum output frequency, enhanced USART module, 2 analog comparators and in circuit serial programming capability. The PWM signals generated by the microcontroller are applied at a high and low side driver IR2010 for power MOSFETs or IGBT. The main power electronic switch T1 is an IRF640 MOSFET and diode D is a Schottky barrier diode SB560. The switching frequency of the buck converter is set at 20 kHz. The other parameters of the

101

buck converter remain as they were mentioned in the previous section: input voltage 12 V, output voltage 5 V, inductance

L = 1 mH, filter capacitor C = 10 µF and output load R = 2.5 ohm. The communication interface with a PC or other embedded system is assured by means of a serial MAX232 IC.

In the next figures are shown the inductor current iL and voltage uD for different values of the input voltage. The duty cycle is derived automatically with the help of the microcontroller and the digital PI control law (13).

Figure 7. Voltage uD and inductor current iL for ui = 12 V;

Ch1: 5V/div, Ch2: 200 mV/div, time base 20 µs/div.

Figure 8. Voltage uD and inductor current iL for ui = 8 V;

Ch1: 5V/div, Ch2: 200 mV/div, time base 20 µs/div.

Figure 9. Voltage uD and inductor current iL for ui = 18 V;

Ch1: 5V/div, Ch2: 200 mV/div, time base 20 µs/div.

VI. CONCLUSIONS

The authors present in this work an implementation method of a digital controller for a switched mode DC-DC buck converter. The first step was to obtain the linear model of the switched mode power supply and then to tune the PI digital circuit to control the output voltage of the converter. The authors mention the advantages of such controllers in the field of power electronics in general and in the field of DC-DC converters in particular. The design method is accompanied by simulation results obtained using MATLAB and Simulink programming and simulation environment. The simulation results were obtained with the help of an embedded system based on PIC16F887 microcontroller. The experimental results validate the theoretical aspects and simulation results. This design method can be applied also for different kind of classic DC-DC converters and even for those with more complex structures.

The main advantage of the proposed system resides in the fact that one can implement other digital control algorithms or control laws for the buck converter without hardware modifications. The design method presented in the paper can be used also in the education process at laboratory courses.

REFERENCES

[1] R.W. Erickson, D. Maksimovic, Fundamentals of power electronics, Second edition, Kluwer Academic Press, 2001.

[2] L. Dixon, Average current mode control of switching power supply, Texas Instruments Inc., 2001.

[3] Wei, Xile , Tsang, K. M. and Chan, W. L., DC/DC Buck Converter Using Internal Model Control, Electric Power Components and Systems, 2009, 37: 3, 320 — 330.

[4] C. Ilaş, Theory of automatic control systems (in romanian), Matrix Rom Publishing House, Bucharest, 2001.

[5] V. Popescu, D. Lascu, D. Negoitescu, Supply sources in telecommunications (in romanian), Publishing House of West, Timisoara, 2002.

[6] D. Alexa, F. Ionescu, L. Gatlan, A. Lazar, Power converters with resonant circuits (in romanian), Technical Publishing House, Bucuresti, 1998.

[7] N.D. Trip, Industrial electronics (in romanian), Publishing House of University of Oradea, Oradea, 2004.

[8] J. Dudrik, P. Bauer, New Methods in Teaching of Power Electronics Converters and Devices, International Journal of Engineering Education, Vol. 24, Issue 5, 2008.

[9] ***, MATLAB 7, Mathematics, Mathworks Inc.

[10] ***, Simulink 7, User’s guide, Mathworks Inc.

[11] PIC16F882/883/884/886/887 - Data Sheet, Enhanced Flash-Based 8-Bit CMOS Microcontrollers with nanoWatt Technology, Microchip Technology Inc., 2007.

[12] T.L. Dragomir, Regulatoare automate, IPT, Timisoara,1986.

[13] K.J. Astrom, B. Wittenmark, Computer controlled system, Prentice Hall, 1997.

[14] J. Dudrik, M. Bodor, V. Ruscin, Zero-voltage and zero-current switching DC-DC converter with controlled output rectifier, 14th International Power Electronics and Motion Control Conference, EPE-PEMC 2010, 39 – 44, Orhid, Macedonia.

[15] T. Beres, M. Olejar, J. Dudrik, Bi-directional DC/DC converter for hybrid battery, 14th International Power Electronics and Motion Control Conference, EPE-PEMC 2010, 78 – 81, Orhid, Macedonia

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