A D-Q Frame Controller for a Full-Bridge Single

Phase Inverter Used in Small Distributed PowerGeneration Systems

Arman Roshan*, Rolando Burgos, Andrew C. Baisden, Fred Wang and Dushan Boroyevich

Center for Power Electronics Systems (CPES)Virginia Polytechnic Institute and State University

Blacksburg, VA 24060-0111

Abstract- This paper presents a Direct-Quadrature (DQ)rotating frame control method for single phase full-bridgeinverters used in small hybrid power systems. A secondaryorthogonal imaginary circuit is created to provide the secondphase required for the transformation; thus a DQ model of theinverter is obtained and its controller designed emulating thecontrols of three-phase power converters. The proposed controllerattains infinite loop gain in the rotating coordinate, thusproviding zero steady-state error at the fundamental frequency ofthe converter. The proposed controller is designed and validatedthrough simulations using a DQ-frame average model in Matlaband a detailed switching model in Saber, as well as experimentalresults obtained with a 2.5 kW single phase full-bridge inverterprototype using a DSP/FPGA based digital control system wherethe proposed DQ-frame controller is fully implemented.

I. INTRODUCTION

Today, small distributed power generation (DG) systems arebecoming more common as the need for electric powerincreases. Small DG systems are usually built close to the userand they take advantage of using different energy sources suchas wind turbine and solar panels. Few examples are hybrid cars,solar houses, data centers, or remote hospitals where providingclean, efficient and reliable AC voltage is critical to the loads.This task is most often left to single phase inverters in DGhybrid systems where they are the only interface betweensources and loads. Fig. la depicts a picture of a stand alonehybrid system where different energy sources interface todifferent loads through a single phase inverter. Much has beendone for the control of single phase inverters in the past years;however, due to the requirements of standalone systems andthe nature of the converter, its controller design is still quitedifficult, and especially so if its critical functionality within thesystem is taken into consideration. Part of the challenge is dueto the fact that the load is unknown, further complicating thecontroller design. It is also difficult to achieve goodperformance because of the time-varying nature of theconverter [1][2][5].

* A. Roshan is currently at Delphi Electronics & Safety,(arman.roshan@ delphi.com)

Over the past 15 years many advance control methods wereintroduced, all aiming to control the instantaneous outputvoltage of the single phase inverter with superior dynamicresponse as well as zero steady-state error at the fundamentalfrequency of the converter[l]-[12]. Traditionally, a fast innercurrent loop with a slower outer voltage loop are designed toeliminate the weakly damped LC filter of the inverter [3].While this method results in improved performance of theinverter under linear loads, it deteriorates under nonlinear loads.Comparing to traditional methods, repetitive controllers aremore complex and advanced and they have gained a lot ofattention recently mainly due to their ability of removing theperiodic disturbances; however, they require quite complexcompensation or a continuous knowledge of the load [14].They are also known to be slow and only effective fordisturbances that are of the harmonics of the fundamental.Deadbeat controllers have also been applied for many years.Although this type of controller theoretically can provide thefasted response to transient for digital implementation, itsdisadvantages include high sensitivity to model uncertainties,and parameters as well as the noise on the sensed variablesbecause of its high gain [8]-[10][15][16]. It must be noted thathigher gain makes the controller more sensitive to the noisegenerated in the system which in return degrades theperformance of the controller itself. Harmonic controllers havealso being explored; however they are only applied toharmonic disturbances and their transient response over theirstretched fundamental frequencies [17] [18].

Figure 1. a) Small stand alone hybrid system b) Single phase full-bridgeinverter

This paper aims to apply the well-know DQ control methodof three phase converters to single phase inverters and it isbased by the work presented in [4]. Although the

1-4244-0714-1/07/$20.00 C 2007 IEEE. 641

implementation of DQ regulators requires a minimum of twoindependent phases in the system, it is possible to provide asecond phase for transformation by creating a secondorthogonal imaginary circuit. While the Imaginary circuit hasthe exact same components as the Real circuit, its statevariables are 900 phase shifted respect to their counterpart inthe Real circuit.

This paper proposes the use of differentiation to create thesecond set of phase variables compared to a 900 delay used in[4]. The second orthogonal imaginary circuit can beconstructed from the original real circuit of the inverter bydifferentiating the state variables of the Real circuit. Thismethod has the advantages of removing the 900 delay whichimproves the response of the controller during transients. Thecontroller then is designed and implemented in the DQ framewhere only one operating point is defined for the converter.The DQ transformation provides a time-invariant model of theinverter which makes the control design similar to that of dc-dcconverters. Theoretically, infinite loop gain can be achieved inDQ rotating frame resulting in the elimination of the steady-state error at the fundamental frequency of the inverter byplacing a pole at the origin [4][19][20]. The superiorperformance of the controller was first evaluated by simulationusing a DQ-frame average model as well as a detailedswitching model, and experimentally verified on a 2.5kWhardware prototype using a DSP/FPGA based digital controlsystem.

II. SINGLE PHASE INVERTER PRINCIPLES

A typical single phase full-bridge inverter is shown in Fig.lb where the AC voltage is created by switching the full-bridgein an appropriate sequence. The output voltage of the bridge,Vab is proportional to input dc voltage as well as the duty cycleof the inverter and can be either +Vdc, 0 or -Vd, depending onhow the switches are controlled. It must be noted that theswitches in one leg cannot be on at the same time; otherwise ashort circuit would exist across the dc source. Pulse WidthModulation (PWM) is used to create the proper gating signalsof the switches, controlling the amplitude and frequency of theoutput voltage directly. To create a clean output sinusoidalvoltage, high frequency harmonics such as the switchingharmonics of the voltage are filtered using a low pass filter. Itis worth mentioning that as the switching frequency increases,the size of the filter decreases due to the fact that less filteringis required; however, increasing switching frequency results onincreased switching losses of the converter.

The state space equations of the single phase inverter aregiven in (1) and (2).

[L]

IZ(RL +RC)- RLRc

L(Z + Rjc 1iL(Z +Rj) [V (t)] +[i]v

zc 1 +;

V ZRC Z iF L]

Z+Rc Z+Rc LVCi (2)

where iL and v, are the sate variables, inductor current andcapacitor voltage respectively. Inductor ESL, RL and capacitorESR, Rc are also taken into account. The load is representedwith Z and the full bridge voltage, Vab is defined as

u(t) = 0

l-l

Vab(t) = VdC

Vab(t) = 0

Vab(t) = Vdc

(3)

A simple model of the inverter derived from the aboveequations is shown in Fig. 2, where the full-bridge is replacedby d(t)Vd, assuming that the switching frequency is muchhigher than the fundamental frequency of the converter, andthat the dynamics of the full-bridge can be ignored due to itshigh switching frequency. Also, it is assumed that the dcsource is constant in magnitude.

v, u(t) = d(t)Vd,

Figure 2. A simplified model of the single phase inverter

III. DQ ROTATING FRAME CONTROLLER

A. DQ Rotating Frame Concept

The DQ rotating frame transformation is most often used inthe analysis and control design of three phase converters.Once a DQ model of the converter obtained, all time-varyingstate variables become DC; thus making the analysis easierbecause the converter can be treated as a dc-dc converter. Dueto the limitation of only one available phase in single phaseconverters, this transformation cannot be realized unless asecond phase is created for every state variable in the circuit.To create the required second phase, an imaginary

orthogonal circuit is constructed based on the Real circuitmodel of the inverter. The imaginary circuit has the exact samecomponents and devices as the Real circuit does, however theImaginary circuit state variables are phase shifted by 900 withrespect to their counterparts in the Real circuit. Theseimaginary state variables are obtained by differentiation in thiswork. Specifically, by differentiating the filter voltage andinductor current of the Real circuit.

Let us assume the output voltage of the Real circuit is given

in (4), VR (t) = VR cos(Ct + 0) (4)where VR is the output voltage magnitude, X is the angularfrequency in rad/sec and i is the initial phase angle of thevariable in the system. The corresponding state variable for

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the Imaginary circuit can be attained by differentiating (4).Notice that the gain of (5) must be multiplied by -1/a becauseof the differentiation method used to obtain the sine function.

VI (t)=VI sin(ol + ) (5)

Fig. 3 illustrates the concept of the stationary c,B and rotatingDQ frames relative to some arbitrary state variable shown asvector x . Notice that x can be decomposed into twocomponent vectors Xa and xfi. Then as vector X rotatesaround the center, its components Xa and x theprojections on the oc, axes vary in time accordingly. Let usassume there is a rotating DQ coordinate that rotates with thesame angular frequency and direction as X, then the positionof x relative to its components, Xd and x is the sameregardless of time. It is clear that the xd and xq componentsremain constant over the entire period and only depend on themagnitude of x and its relative phase with respect to the d-qrotating frame. Angle 0 is the rotating angle of the DQ frameand it is defined by (6).

Figure 3. Stationary and rotating frames concept

= '-)d +0 (6)

The switching network of a single phase inverter introducesnonlinearity and discontinuity in the converter operation. Forthe sake of analysis, an average model of the inverter must bedefined before developing its DQ model. An average modelprovides a way to model the low-frequency components of thewaveforms in switching mode converters, accomplished byremoving the switching harmonics through the averaging ofwaveforms over one switching period. Consequently, thedynamics of the converter at much higher frequencies areneglected, since the focus is only on the components that areessential to the converter power flow and its control subsystem.Average models can be used when the fundamental frequencyof operation is significantly smaller than its switchingfrequency, in which case the model captures the effects that arelikely to be of significance for the analysis of the converter andits control subsystem. Another benefit of using average modelsis its fast simulation time compared with the switching model.Oftentimes, it is desired to have an accurate and fast model ofthe plant for control design due to intrinsic iterative processinvolved in the design.The state space average model equations of the inverter are

given in (10) and (11) and the average model is shown in Fig. 4including the capacitor and inductor ESR. It must be noted thatthe source current, ldc is the same as the Real circuit current IRbecause the Imaginary circuit does not exist physically.

IL IL -,

R RC +VC R 1c R d[I R]

dt IL I IL I L (+ Z)J VC I CL LZ( + Z)J d, L

d [VC- R- 1 IL R 1 VC R

dt VC C(1+ Rcl) L ZC(1+R'LvV

(10)

(1 1)

The relationship between stationary and rotating frames isobtained from Fig. 3. Equation (7) defines the transformationfrom stationary to rotating frame, and (8) from rotating frameto stationary frame.

TL cos(cx) sin(x)]- sin(aX) cos(ax)

T-1 =cos(o) - sin(ox)rsin(wt) cos(ox)]

R+|-~~~ =-IVd +

R,

(7)

(8)

The DQ components of the output voltage state variable ofthe inverter are obtained in (9) as an example, where theinductor currents may be obtained similarly. Notice that Vap isthe magnitude of the inverter output voltage or simply VR.

LVd 1 TFVa8COs (a 0)- FCos ()1

Vq J L / (9)

B. Single phase inverter average and DQ models

Figure 4. Single phase full-bridge inverter average model

The DQ model of the inverter can be developed once theaverage model is obtained. The transformation matrices givenin (7) and (8) and the state space equations (10) and (11) of theinverter are used to attain DQ equations which represent theinverter dynamics.

d [Id Vd [Dd o+ ij[Id 1[Vd (

dt Iqj L LDq L-co iqL LVq (12)

|Z( + ) [V9 L (+ ,Z) [I?

d FVd1 1 [Id [0r ][Vd lFvdldt LVq C jI+R LIq c 0 Oj[Vq (13)

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Equations (12) and (13) are the DQ equations which definethe DQ model of the inverter shown in Fig. 5.

Figure 5. Single phase full-bridge inverter DQ model

If the ESR and the ESL are ignored, then (12) and (13) canbe simplified into (14) and (15). Notice that cross-couplingterms are introduced into the model because of thetransformation from stationary to rotating frame. A processcommonly knows as Decoupling can be done in the controllerto decouple these terms easily.

d FVd I IFd~+ 0 CO ~Vd~ FVddtlVqJ Cl/40~ 0)o OVJ0ZCIVqJ (14)

d F'd VdC FDd +F o IFd~ FVd~d0(X0=V, Lf,+ ° 0 -LV (15)dt _IqJ L ~DqJ-C0) OJIqJ L-VqJ 15

Although the inverter has a time-varying nature, its DQmodel derived in this section is non-varying; hence only oneoperating point is defined for the inverter while in steady-stateoperation. To find the DC operating point of the model thefollowing equations are developed using Fig. 5.

Vd (16)

z=

'q WOCVd (17)

DdVd - wLIq + RLK j

VI'

D coLId +RLIq (19)

Vd,To further simplify the analysis, the output voltage of the Q

channel Vq is set to zero by aligning the voltage vector to the Daxis.

C. DQ controller structure

A DQ controller can be designed now that the DQ model ofthe single phase inverter is defined. Fig. 6 represents thecontroller block as well as the power stage of the inverter. TheDQ model of the inverter was constructed in Matlab/Simulinkand the compensators were designed with the aid of Single-Input-Single-Output Tool (SISOTOOL) of Matlab. The DQcontroller design of the single phase inverter is similar to thoseof dc-dc and three phase converters. The controller consists of

two channels, one for D and one for Q. Each channel containscompensators for voltage and current loops.

First, the DQ components of the voltage and current aregenerated by differentiating the state variables of the systemand applying the rotating transformation matrix given in (7).Second, current and voltage compensators are designed usingtwo-pole two-zero compensators to obtain a better performance.Finally, the duty cycles Dd and Dq are transformed back intothe stationary frame using (8), where naturally only the dutycycle of the Real circuit is fed back to the power stage.

Dd dqdL D4 71 17D4

I' IVK

r- ----------d--------- ----1----1

i~ lP1 t V

l~~~~P L v l P1 <0FI~~~~~~~~~~~~~~~~~~~~ 1q l1Figure 6. Single phase inverter and its DQ controller

One important factor is the effect of the digital delay in thesystem that must be considered in the design. Although thereare multiple factors which can affect how much the delay is,they can all be summed under one delay. The total delay in thesystem has to include the conversion time of the ADCs as wellas the DSP processing time and total signal transmission timebetween the hardware and controller board. The largest bulk ofdelay comes from the time required by the DSP to perform allthe calculations necessary for the controller. Although this timecan be minimized by proper coding techniques, there will be atleast an inherent one switching period delay from the time theoutput states are measured until the duty cycles are ready to beupdated for the modulator. Analog controllers have infiniteresponse time and therefore their delay is zero; however theyare severely limited in the functions they can implement.

Fig. 6 illustrates how the total delay in the system wasaccounted for when designing the controller. This delay wasadded to the open-loop DQ model of the inverter before thecontroller was designed. A simple delay is given in (20) whereTs is the switching period.

H(S) delay e (20)

It is generally hard to accurately model the total delay of thesystem, consequently a maximum one to two switching periodsmust be adopted for the total digital delay of the entire system.The maximum delay considered in this design was equal to twoswitching period as the worse case and an 8th order transportdelay in Matlab was used to model this delay.

It must be noted that this delay will not affect the magnitudeof the plant transfer functions; however, it does create a roll-offin the phase. In most cases this reduction in phase causes

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instability in the system due to the fact that the phase of theclosed loop system crosses -180° sooner than what is was

intended in the design. A simple way to counter the negativeeffects of the delay is to reduce the controller gain; howeverreducing the gain will reduces the bandwidth of the close-loopsystem. As the bandwidth decreases, the response time of thesystem decreases, thus creating a design trade-off situation.

The controller also needs to take into account the light-loadoperation of the inverter. The inverter in light-load conditionshas a higher resonant peak at the resonant frequency of thefilter which affects the controller if this is not considered. Forthe purpose of this study, the controller was designed with 10%resistive light-load while its performance was studied later on

with different types of loads, such as nonlinear loads.

As mentioned before, each channel consists of a voltage andcurrent loop. A common configuration is to have a fast innercurrent loop and a slower outer voltage loop, where the currentreference is determined by the error in the outer voltage loop.The current loop not only allows for a faster transient response

and improved THD for nonlinear loads, but it also provides an

inherent current limit used to protect the converter. This limitcan be used to stop the converter if the current exceeds some

predefined boundary when for example a short circuit appears

at the load. The form of the two-pole two-zero compensatorsused for each channel is shown below.

Gc-K (s + NJz )(s + CDz2)Gc Kdq (s + )p(2 ) (21)

The integrator provides zero steady-state error, while thezeros are placed as high as possible to provide as much as

phase margin as possible. The zeros will provide a boost inphase to compensate for the phase drop created by the digitaldelay. The additional pole is used to provide loop-gainattenuation after crossover frequency and it is placed at half ofthe switching frequency.

IV. DQ AND SWITCHING MODEL SIMULAITON RESULTS

A single phase full-bridge inverter is selected to verify theperformance of the proposed DQ controller and the inverterparameters selected for this design are listed in Table. 1.

The switching frequency of the converter was assumed to bea system selected option; thus four different frequencies were

studied. The controller was designed for the lowest frequencygiven in Table. 1 while studied for other switching frequenciesusing its switching model. A current loop bandwidth of 2 to3kHz is desirable for this converter while the bandwidth of thevoltage loop should be a tenth of the current loop bandwidth toeliminate undesired interactions between both loops. Given theabove conditions, the DQ controller was designed with the aidof Matlab/Simulink.

TABLE 1 Single phase full-bridge inverter parameters

Inductor 500 pHCapacitor 22 RFESR IonQESL 100 mQfs 20kHz, 40kHz, 60kHz, lOOkHzVdc 300 VVout 120 VrmsPout 2.5 kW

The switching model of the inverter studied was built inSaber. The controller was created entirely in MAST code tomimic the same C code that would be implemented in the DSP.In this was the discrete transfer functions of the compensatorscreated were later used in the hardware. The controller of themodel updates its duty cycles based on a digital clock runningat different switching frequencies as defined in Table. 1. It isworth mentioning that the state variables, output voltage andinductor currents of the inverter were sampled every switchingperiod. To create a true balanced waveform these signals mustbe sampled in the middle of each switching period. Bysampling each waveform at the middle of the switching period,an average discrete waveform is obtained that can be used tocreate the cosine and sine waveforms required fortransformation. Although parasitic components of the invertersuch as those of switches were ignored in the switching model,it is still a great tool to evaluate the performance and study theproposed controller without the hardware in hand.

The simulation results of the inverter DQ model are shownin Fig. 7 while the results of the switching model are shown inFig. 8 for a load step from full load to no-load and back. As itcan be seen from Fig. 7 and Fig. 8, the voltage overshoot is lessthan 15% while reaching steady-state operation in less thanIms. The inverter was then studied with a 50% step on thereference of the output voltage with the simulation resultshown in Fig. 9.

Figure 7. Output voltage of channel D, Vd and its sinusoidal waveform

under load step from Full-load to no-load (DQ model in Simulink)

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Inverter Parameters

g ; - - ? ; ; ? _mS.

i_1ixJ NIL . .. ; . s \.E: : .. rJ=l. . , .; OL. . . t 5 ...... . . tan 1L : f 5 . 1 .. . : \.8 : . . . i2r.\ iir \ 0 J- \ . . ,1F. . 5, .. t.fuX . s X X T - X X - :; X - - .x s . X S f !

.1 w jf z jr X sr 1i rrS -\ Jr 'l,L,2:.iF '; \_,,\' "i.f_S4@.a . k . _, .... . ,. . _ . ..... . ...... .... . . .. . . k _ , ..................... _F ' 'Sg1 ,,, ., ., .,, ,, ,,, ,,, ,, ,,,g.S MhG ............................. . ...... ........... .................................... ; ............

= . g1W 4>0zf.S;ffi0,.>./:,...g.,..r 8 r f 2} X i. /i iz, , ke t >, S J >& ,¢ - si ,/ X Jg .'iX im d ' ' \' ' ' i ' ' ' 'S - / ., .,.1 .,. , X, at *8

;i ', X i r \ 8 i' /- \-/.' . 'ta'' i < . i < '- ''7 ' - ' <-/ ' 'La i : .. ................. .............. , .. ........ ...... : ...............Si

2SM MRRWG rlggg xa Xgr. m1> > 1JWJ erw W XQm Ww } W> Xmw lm rf EBw Irf

Figure 8. Output voltage, inductor current and duty cycle waveforms of

the inverter under load step from Full-load to no-load

(Switching model in Saber)

Figure 9. Output voltage and reference waveforms of the inverter under50% reference variation (Switching model in Saber)

The proposed controller was also simulated with a nonlinearload. A single phase diode bridge loaded with a parallel RCcircuit used as the load and output voltage THD of 2% wasobserved. Fig. 10 shows the output voltage as well as theinductor current waveforms under nonlinear load operation ofthe converter.

stage. A DSP/FPGA based Universal Controller (UC) board isused to implement the controller design [21]. Fig. 11 shows thehardware setup for the test as well as the output voltage andinductor current under nominal operating conditions. Thecontroller board is sitting on top of the inverter power stage. Itconnects to the ADCs on the power stage through its I/O pinswhile fiber optic cables transmit the gating signals form thecontroller board to the gate drivers of the IGBT switches,required for as a form of isolation, protection and noiseimmunity. The state variables of the inverter are sampled in themiddle of the period and a digital modulator includingdeadtime and deadtime compensation was implemented in theFPGA of the digital controller board. The DQ controller wasprogrammed in the DSP using C-language and its performancewas verified under load steps as well as step commands of thevoltage reference.

Figure 11. Single Phase full-bridge inverter hardware and its output voltage

and inductor current waveforms under nominal operation

The output voltage and reference duty cycle of the Realcircuit are shown in Fig. 12a, while Fig.12b shows the outputvoltage reference stepped down to 50% with the proposed DQcontroller. It can be seen that the output voltage reachessteady-state operation in less than 2ms. Load steps were alsoperformed by stepping it from full-load to no-load and theresults are shown in Fig. 13. Clearly, simulation andexperimental results are in close agreement showing that theproposed DQ controller is capable of providing good dynamicresponse under different conditions.

a) b)Figure 12. a) Output voltage and duty reference of the inverter undernominal operation b) Output voltage and inductor current with

50% output voltage reference step-down

Figure 10. Output voltage and inductor current with a nonlinear load

(Switching model in Saber)

V. EXPERIMENTAL RESULTS

The 2.5 kW full-bridge single phase inverter prototypeshown in Fig. 11 was used to verify the control design. Thepower stage contains the full-bridge, dc link and outputcapacitors. The ADCs and all the sensors are also on the power

Figure 13. Output voltage and inductor current of the inverter under load

step (Full-load to no-Load)

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VI. CONCLUSIONS

A DQ rotating frame controller was proposed in this paper

for a 2.5kW single phase full-bridge inverter used in smallstandalone hybrid power systems. To achieve thistransformation, an imaginary orthogonal circuit was created bydifferentiating the state variables from the original invertercircuit in order to emulate the Q-axis dynamics. In the rotatingframe, the controller design for the single phase inverterbecame easy and equivalent to that of three phase converters,so that an infinite loop gain at the fundamental frequency ofthe system and fast dynamic response could be achieved whileevaluating the inverter DQ controller system under differentloading conditions. The proposed controller was verified usingsimulations and a 2.5 kW experimental prototype controlledwith a DSP/FPGA based digital control system. It is also worthmentioning that the controller design must take into accountthe differentiation of noisy signals, where the implementationof analog and digital filters will help cope with this greatly.

ACKNOWLEDGMENT

This work was supported primarily by the EngineeringResearch Center Program of the National Science Foundationunder NSF Award number EEC-973 1677 and the CPESIndustry Partnership Program.

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