Modeling Transfer Function for Buck Power Converter

Jonel Horea Baciu, lonut Ciocan and Serban LunguApplied Electronics Department, Faculty of Electronics, Telecommunicationsand Information Technology, Technical University of Cluj Napoca, Romania

Ionel.Baciugael.utcluj .ro

Abstract: Switching regulators offer two main advantages compared to the linear regulators. First,switching efficiency is much better than linear. Second, because less energy is lost in the transfer,smaller components and less thermal management are required. A switching power supply has toassure the stability between input and output voltage, the stability between load and output voltage,and a good transient response between different disturbances signals. The paper will outline amathematical methodfor modeling the transferfunctionfor switching power supplies. That model willbe implemented in Simulink and after that, the result will compared with PSpice. This model can beused to design a best controlforpower converters.

1. INTRODUCTION

The switching regulator is increasing in popularitybecause it offers the advantages of higher powerconversion efficiency and increased design flexibility(multiple output voltages of different polarities can begenerated from a single input voltage).

The most commonly used switching converter isthe Buck, which is used to down-convert a DC voltageto a lower DC voltage of the same polarity. This isessential in systems that use distributed power rails(like 24V to 48V), which must be locally converted to15V, 12V or 5V with very little power loss.

If we want to implement a converter we must tohave independence between load, input source andoutput voltage. This thing is realized with a closedloop control. In that case we want to minimize theerror between the fixed value and output variablevalue. The principal designation is to maintain theoutput voltage with a controlled ripple.

2. CLOSED LOOP CONTROL

In most cases, a control loop is a closed loopcircuit. If we want to obtain the transfer function of aclosed loop circuit, we need to calculate the transferfunction of each block. In Fig. 1 the block schematicof a closed control loop is represented. Y(t) represents

the output, or the Process Variable (PV), yM(t) are thedetected values of the PV and r(t) is the Set Point (SP)of the process.

r(t) +_EM} uController-

YMMt

i(t) m(t) y(t)

Process

i, d-

Fig. 1. Control loop block schematic.

In Fig. 2, a closed loop for a buck converter ispresented. The compensation block, the PWM circuitand the process, represented by the buck converter canbe observed.

T LI

Ui D Ic RI R

::N uRIvEn l t) 4vt±

Fig. 2. Example of voltage control loop for buck converter.

The transducer is included in the same block as thecompensation module and it is a simple resistance.

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The presence of a single loop involves a simple modeof analyzing and designing the control loop. Thedrawback of this circuit is the overdue response to thevariation of the input voltage.

In this case, the configuration of this control looplooks like in figure 3. We can observe that thetransducer block, controller block and thecompensation block are included in the same block.This configuration is presented in figure 4.

I.,

Fig. 3. Block diagram of voltage control loop for buckconverter.

oscillatorcontroller.

block, and of vc(t) generated by PID

VXmax [

A

I o1

---.|---- N

Emp

d

_ _ _ b~~~~~~~~~

dl*Tg TS 2Ts 3Ts

Fig.6. PWM generated signal.

If we will presume that the variation of v,(t) is notlarge in a period, result:

d(t)=max

(1)

which is the transfer function for PWM controller.

3. MATHEMATICAL RESULTS

To obtain the transfer function of closed loopcontrol circuit, we must to partition the system inblocks.

vtti

vcltl

Vref

Fig. 4. PID Controller.

The controller algorithm depends according assystem. The first effect what appear is derivative,second is proportional and in the end is integrative.

vasc (t)oscVd (t)

vc (t)

Fig.5. Pulse Width Modulation Block.

The roll of Pulse Width Modulation circuit is togenerate a modulated signal. That signal depends, likewe see in figure 6, of saw tooth voltage, generated by

If we want to design the voltage controlled loopcircuit in Simulink, we must calculate transferfunction. Transfer function is a mathematicalstatement that describes the transfer characteristics ofa system, subsystem, or equipment.

In that case, the small signal equivalent circuit islike in figure 7.

Small signal modeling is a common analysismethod used in electrical engineering to describenonlinear devices in terms of linear equations.

For Buck Converter, which looks like in figure 8,we will obtain, for the transfer function betweencontrol and output, this relation:

HVd (S)= Ui (2)l+s +s2LCR

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L

Ui(s) L D d (s) UiR

- 0

Fig.8. Block Converter.

For the transfer function between input and outputwe will obtain:

Hu, (s) = D

1+sL+S2LCR

(3)

10. The important functional blocks are: compensatorblock, Comparator block, oscillator block, PWMgenerator block (RS block), power supply block.

Blockl1

+15V +Vcc OUT1 *-<<outl

*GND15 -Vcc OUT2 *-<<out2

- OUO(5

Oscilator BlockBlock

+15V 5Vcc

-15V 1-Vcc

out 1 > Vin OUT * -<< R 50

*-Vin

GND

- Com parator Block

Block3

out2> *S_+15V _*+15V R6

-15V O_*-15V OUT_S *~\_ ~~~~~~~~100

-GND

-0 RS Block

Block2

+Vcc * O 15V

-Vcc * -15V

GND*

Supply Block

Ll

The output impedance for Buck Converter is:

sL(4)-u J L

I +s +s2LCR

and is obtained from following circuit:

Fig. 10. Buck Converter with voltage control loop.

The analysis is realized from 0 to 20ms.

Zo ( S)

Fig.9. Output impedance for Buck Converter.

For close loop control system we obtain fromfigure 7 result the following transfer function relation:

Fig. 11. A. Output Voltage; B. OUT_SI Signal.

In this case, the output voltage is controlled near5V. We have a ripple equal to 388.6 mV. Theoscillator frequency is 21 KHz.

We observe the discontinue functionality.

V(s) =HVd(s*HC (s) Vemax

1-HVd(s) Hc(s) H VS+

Vmax+Hvi(s)*Ui(s)+ZO(s)*I(s)

4. EXPERIMENTAL RESULTS

Starting from that relation for transfer function wedesign the voltage control loop on functional blocks.This model realized in PSpice is presented in figure

Fig. 12. A. Inductance current; B. OUT_SI Signal andoutput voltage.

Second case is when output voltage is controllednear 10 V. In that case we have continuous mode,when inductance current don't reach zero level.

1-4244-1218-8/07/$25.00 ©2007 IEEE

(5)

llIllIV-.-- lWWlI IIILIWIWII IILJ uwiii liwii lu iuuii lwlll.

* 4

d 4

FIF] FIFIFIF]u u u u 0

Rl

30

Z.8 (s) =

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R %

c R

I

IXlFig. 13. A. Output Voltage; B. OUT_S I Signal.

We observe, inductance current have -300 mAvalue.

Uov>U UFig. 14. A. Inductance current; B. OUT_SI Signal and

output voltage.

In Simulink case we obtain the following circuit.This structure have based on figure 7.

In this circuit Ramp and Ramp 1 blocks are forsimulate a variation of load current, respective inputvoltage signal.

E(s)WS) lnl ~~~~~d(s ds)

Fi. 1 . PID Controller PWMcrodulitor

Ramp Scope

EP -P uiCs)

RamplPokwer Supply hAlodel

Traductor

Fig. 15. Control loop circuit for Buck Converter.

Power Supply Model contains the followingblocks, like in figure 16. This transfer function isobtained for a circuit with following elements values:L =2 mH, C= 100 uF, R =30 Q; Input voltage Ui50V, and UO = 10V.

Fig. 16. Power Supply Model block.

Fig. 17. Output voltage for Simulink example.

Even if the input voltage signal and load currenthave a variation, output voltage is maintained betweenlimits.

5. CONCLUSIONS

From the presented comparison we can say that:simulating power circuits in PSpice is easier thanusing Simulink and we can take advantage of theparasitic elements already embedded into models.However, simulating the closed loop and moreover,tuning the control algorithm is rather difficult. On theother hand, Simulink will allow better simulation ofcontrol loops and faster algorithm adjustments, still,each phenomenon must be carefully modeled byequations and a lot of effort will be required forcomplex electronic systems.

REFERENCES

[1] D. Petreu>, ,,Electronica Surselor de Alimentare", Ed.Mediamira, Cluj-Napoca, 2002

[2] D. Petreu>, $. Lungu, ,,Surse in Comutatie", Ed.Mediamira, Cluj-Napoca, 1999

[3] V. Popescu, D. Lascu, D. Negoitescu, ,,Surse dealimentare in telecomunicatii", Ed. De Vest, 2002

[4] N. Palaghita, D. Petreu>, C. Fdrca>, ,,Electronica decomanda Ei reglaj" Ed. Mediamira, Cluj-Napoca, 2006.

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