A DQ Synchronous Reference Frame Control for Single-Phase Converters

U.A. Miranda and M. AredesFederal University of Rio de Janeiro

COPPE / Electrical Engineering Programm21945-970 Rio de Janeiro, RJ - BRASIL

Email: ulisses@coe.ufrj.br

L.G.B. RolimFederal University of Rio de Janeiro

Polytechnic School / Department of Electrical Engineering21945-970 Rio de Janeiro, RJ - BRASIL

Email: rolim@dee.ufrj.br

Abstract This paper presents a current control using theDQ synchronous reference frame for single-phase converters.This control method consists in transforming an orthogonal paircomposed by the actual single-phase input current and a fictitiouscurrent, from a stationary to a rotating frame. The steady statecurrent components in DQ frame become DC instead of ACvalues so a zero error current control can be implemented. Asingle-phase PFC boost rectifier is used as an example applicationof this control. To validate the control method simulation andexperimental results are presented.

I. INTRODUCTION

Single-phase converters with input current regulation arewidely used in several applications such as Active PowerFilters, Power Factor Control (PFC) Rectifiers, UninterruptedPower Supplies (UPS), Photo-Voltaic Generation, etc. [1].The current regulation is based on current control loops,however for AC power converters it is not simple to designthis controllers, due their time variant currents and voltages.For DC converters it is quite simple to design linear currentcontrollers with no steady state error, but if the AC controllersare designed the same way as DC controllers, a significantsteady state error in both amplitude and phase may occur.

In three-phase systems the steady-state AC quantities be-come DC by means of the transformation from ABC staticframe to dq synchronous frame. To perform this transformationin a single-phase system it is necessary to create a secondquantity in quadrature with the real one so as to apply thetransformation from the static to the synchronous frame. Inthe technical literature this second quantity is obtained eitherusing the capacitor current feedback [2] , delaying the real oneby 14 of the line period or by means of notch filters tuned attwice the line frequency [3], [4]. Figure 1 shows some of thismethods.

The control presented in this paper is based on the detectionof the frequency and phase of the input voltage by a PLL andthen generates a fictitious input current. The proposed methodneither requires tuned filters nor store samples to produces aquarter cycle delay. Using this method the current control canalso be made unsusceptible to switching noise with a properPLL design. A proper choice for the current peak value alsomake possible the reduction of the computational requirementsin a digital implementation.

II. THE BOOST RECTIFIER

Figure 2 shows the rectifier topology. This is a well knowncircuit and there are many papers describing it [5], [6], [7].The circuit control must be able to regulate the voltage Vdc

(a) Real input delayed by 14

of the line period

(b) Notch filters tuned at twice the line frequency

Fig. 1. Reference and real currents

across the capacitor at the reference value and also drain asinusoidal current with unitary power factor.

Fig. 2. Boost Rectifier

The block diagram of the control is showed in Figure 3. Thereference value V dc is compared with the measured voltageVdc. The error signal is applied to a PI controller and its outputis the amplitude I of the reference current is . The frequencyand phase of the is current are the same of the input voltagevs. They are determined applying vs in a PLL circuit. Oncecalculated the reference current is it must be synthesized bya current controller.

III. DQ SINGLE-PHASE CURRENT CONTROLLERA. DQ Transformation

The transformation to the synchronous frame DQ requirestwo orthogonal components. In three-phase systems the ABC

13770-7803-9033-4/05/$20.00 2005 IEEE.

Fig. 3. Rectifier Control Block Diagram

components are transformed to the orthogonal and stationary frame system and then to the synchronous frame DQ asshown in Figure 4 and equation (1). The inverse transformationis showed in (2).

Fig. 4. Frames

[dq

]=[

cos() sin() sin() cos()

][

](1)

[

]=[

cos() sin()sin() cos()

][

dq

](2)

However this transformation cannot be applied directly inthe single-phase systems, because there is only one variable.This drawback can be solved creating a fictitious input current.

B. Reference ValuesAs showed in the previous section, the desired current is

has phase and frequency obteined from the t signal of thePLL circuit. It is also possible to create a second referencecurrent in quadrature and delayed in relation with is , using thesame t signal. These reference currents are equivalents to thecurrents in the frame, as show in equations (3) and (4).

i = is = I

cos(t) (3)

i = I sin(t) (4)

Performing the DQ transformation taking to the inputvoltage vs as reference, it is equivalent to make = t inequation (1), leads to equations (5) and (6).

Id = I (5)

Iq = 0 (6)

C. Input currents DQ transformationThe input current is is defined as i (7) and the fictitious

input current i is defined as the reference current i (8).This choice for the fictitious input current will result in asimplification, as will be shown.

i = is = I cos(t) (7)i = i

=

sin(t) (8)The values of Id and Iq are obtained using equation (1)

where is the input current (7) and the fictitious current (8).D. Control circuit

The purpose of this control is to regulate the inductor currentis. Considering no losses in the circuit of Figure 2, its averageequation through a switching period is given by (9).

LdIsdt

= Vs Vpwm (9)This equation is converted to dq frame substituting the

variables for its values in equations (10), (11) and (12).

Vs(t) = V cost (10)

Is = Id cos(t) Iq sin(t) (11)

Vpwm = Vpwmd cos(t) Vpwmq sin(t) (12)Splitting the resulting equation in the cosine and sine terms,

leads to the circuit equations for the d (13) and q (14) axis.dIddt

= Iq VpwmdL

+V

L(13)

dIqdt

= Id Vpwmq

L(14)

The equilibrium point (15),(16) is reached when the deriva-tives terms are equal to zero.

Vpwmd = LIq + V (15)

Vpwmq = LId (16)Now the references and the control variables are DC quan-

tities and the controller can be implemented the same way asin the DC converters. Note that were added coupling termsbetween the d and q axis but these terms can be decoupledwith a proper controller design.

The block diagram of the control is showed in Figure 5. Idand Iq currents are compared with their references values andthe PI controllers guarantee no steady state current error. Thegain L provide the decoupling terms and the addition of thepeak voltage V is intend to compensate the countereffect ofthe supply voltage, with respect to the converter output voltagevpwm, which has the same behavior as a disturbance signal.

When transformed to the stationary reference frame (2) the term is discarded and the term is the signal vpwm that isapplied to the single-phase unipolar SPWM.

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Fig. 5. Single-Phase dq Current Control Block Diagram

E. Simplified Control CircuitAs the main concern is the real implementation of this

control, it is useful to reduce the number of mathematicaloperations. A simplification in the current control loop ispossible due the value attributed to the fictitious input currenti (8), as follow.

The steady state current error in dq frame in Figure 5 isgiven by equations (17) and (18).

errord = Id Iderrord = [i cos(t) + i

sin(t)]

[i cos(t) + i sin(t)]errord = (i i) cos(t) + (i i) sin(t) (17)

errorq = Iq Iqerrorq = [i sin(t) i cos(t)]

[i sin(t) i cos(t)]errorq = (i i) sin(t) + (i i) cos(t) (18)

Using equation (8) in equations (17) and (18) results in thesimplified equations (19), (20) for the steady-state errors.

errord = (i i) cos(t) (19)

errorq = (i i) sin(t) (20)The decoupling terms have the objective to improve the

controller dynamic and can be neglect without committing itsoperation. The input voltage feed-forward is added directly tothe vpwm signal. It has exactly the same effect of adding thepeak value in the vpwmd component and has the advantagethat is no more necessary the voltage peak detector.

As result, Figure 6 shows the simplified current controlblock diagram.

Fig. 6. Simplified Single-Phase dq Current Control Block Diagram

IV. RESULTSA. Simulation Results

Simulations of the simplified dq current control have beenmade by using EMTDC/PSCAD software. The simulatedcircuit is shown in Figure 2 and its parameters in Table I.

Switching frequency fs 5kHzInput Voltage vs 220V

Boost inductor L 5mHRated Direct Voltage Vdc 425V

Load RL 50

TABLE ISIMULATED CIRCUIT PARAMETERS

At the simulation instant of 0.5s the rectifier is turned onand the load at 1.5s. The reference and the real currents inthe synchronous dq and in the static reference frames at thisinstants are shown in Figures 7 to 10 . Figure 11 shows theinput voltage and current.

B. Experimental ResultsThe experimental implementation of the simplified dq cur-

rent control was done using the Texas Instruments DSPTMS320F2812. Figure 12 shows the turn on transition mo-ment. Once the capacitor voltage was regulated, as Figure 13shows, the load was applied, this instant is showed inFigure 14. The steady state input voltage ande current areshowed in Figure 15.

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Fig. 7. Id and Iq : Rectifier on

Fig. 8. Id and Iq : Load on

Fig. 9. is: Rectifier on

Fig. 10. is: Load on

V. CONCLUSION

A single-phase synchronous reference frame current controlbased on the PLL circuit was presented. The calculation of thefictitious input current using the frequency and phase detectedby the PLL was successful. It has the advantage that it is notnecessary the use of filters and does not require digital memoryto allocate samples.

And also, the proper choice for the peak value of thiscurrent has made possible a simplification in the control

Fig. 11. Input voltage and current

Fig. 12. Input voltage and current at the beginning of the rectifier switching

Fig. 13. Capacitor Voltage

without compromising its operation. After this simplificationthe control become very simple to be implemented in a digital

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Fig. 14. Input voltage and current when the load is applied

Fig. 15. Steady state Input voltage and current

controller.The control was able to achieve zero steady state current

error. It was validated by the simulation and experimentalresults presented.

ACKNOWLEDGMENTThis project was supported by CAPES/CNPq.

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