COMPARISON OF QUADRATIC BOOST TOPOLOGIES OPERATING UNDERSLIDING-MODE CONTROL

O. Lopez-Santos1,3, L. Martinez-Salamero2, G. Garcia1, H. Valderrama-Blavi2, T. Sierra-Polanco3

1LAAS - CNRS, Univ de Toulouse - INSA Toulouse, Toulouse, France2Univ de Rovira i Virgili, Tarragona, Spain

3Univ de Ibague, Ibague, Colombiaoswaldo.lopez@unibague.edu.co

Abstract—The recent development of dual-stage pho-tovoltaic micro-inverters motivates the study of DC-DCconverter topologies with high-gain capabilities. Followingthis idea, a comparison of both static and dynamicperformances of the known quadratic boost convertertopologies is tackled in this paper. To achieve an accuratecomparison, we use the same electronic components andthe same control scheme. Considering its advantages, asliding-mode control law has been proposed to reach astable equilibrium point in the switching converters. Thesliding surface uses only the input current following aconstant reference. This current reference is adjustedto obtain an output voltage of 400 V for an inputvoltage between 15 and 25 V and an output powerbetween 20 and 120 W. The local stability of theproposed control law is tested by using the Routh-Hurwitz criteria. The comparison is accomplished usingsimulation results to disclose the best configuration interms of efficiency, dynamic response and applicability inphotovoltaic generation.

Keywords—High-gain DC-DC converters, quadraticboost converter, sliding-mode control.

I. INTRODUCTION

PHotovoltaic micro-inverters are a promising solution inthe future of the distributed generation systems [1]-[2].

These devices allow injecting the power generated by one ortwo photovoltaic modules to the utility. The great differencebetween the output voltage of a photovoltaic module (12 -24 V) and the peak amplitude of the grid voltage waveform(310 V) demands a high voltage gain. To reach this goalavoiding a low frequency transformer in the AC side, twosolution are known: 1) using a power transformer operating athigh frequency, and 2) employing a multiple-stage convertertopology without transformer [3]. Because the introduction ofthe high-frequency transformer increases the size and cost ofmicro-inverters, the transformer-less solutions have attractedthe attention of some researchers [4]. Among these, the dual-stage topologies with a DC-DC stage in cascade with aDC-AC stage have been identified as the most competitivesolutions [5]. Therefore, it is necessary to study candidatetopologies for the DC-DC stage which can provide high-gaintogether with high efficiency and a simple and reliable controlscheme.

Table IGain relationship of the well-known DC-DC convertersBoost Buck-boost Cuk SEPIC Boost-buck

11−D

−D1−D

−D1−D

D1−D

D1−D

Table I reveals that the boost converter have the best gainratio since this is not affected for the duty cycle D in thenumerator [6]. On the other hand, the boost converter shows ahigher efficiency in a great range of its operational conditions[7]. However, to attain a higher gain, it is required a dutycycle near to the unity which reduces the efficiency andreliability of the converter. Then, a proper solution can befound among the topologies with the gain function (1) sincethis is a positive power of the boost converter DC gain whichensures operation in a safer region of the duty cycle.

M(D) =

(1

1 −D

)a

(1)

A recent study shows that among the converters fora=1, a=2 and a=3 (boost, quadratic boost and cubic boostconverter respectively), the quadratic boost converter (QB)have the best efficiency to obtain high gain operating with thevoltage and power levels required by usual micro-inverters[8]. Such work takes into account the topologies which areobtained from the generalization of the cascade converterswith a single active switch introduced in [9]. The workhere presented deals with the comparison of four converterswith the same quadratic gain. The first topology is obtainedwith the series connection of two boost converters with thesame control signal. This topology will be named Cascade-Connected-Double-Switch (C2DS−QB). The second topol-ogy results changing a controlled switch for a diode in theC2DS−QB topology [9]. This topology will be denominatedCascade-Connected-Single-Switch (C2S2 − QB). The thirdtopology has been presented for power factor correctionapplication in [10]. This topology will be called Software-Synthetized-Single-Switch (S4 −QB). The last topology hasbeen proposed for photovoltaic application being obtainedwith the rotation of the commutation cells [11]. This topologywill be named Rotated-Cells-Single-Switch (RCS2 −QB).

A stable and reliable operation of the DC-DC converterscan be obtained using the sliding-mode control approach[12]. Taking advantage of the fact that DC-DC converters arevariable structure systems, the sliding-mode control allowsus covering the control aims with simplicity and robustness.In the case of the high-gain converters, the use of controlstructures based on pulse width modulation (PWM) canlead to modulator saturation since the duty cycle is nearto its maximum limit. Hence, application of the sliding-mode is mainly a more reliable choice. In order to stabilizethe quadratic boost converter topologies, a sliding surfacebased in the input current can be used, allowing to obtainan indirect control of the output voltage [13]. This simplecontrol strategy allows evaluating and comparing the dynamicperformance without considering the influence of an externalloop for output voltage regulation.

The main issues of the four known quadratic convertertopologies and its models are presented in section II. Next, theanalysis of the sliding-mode based control law and its localstability study are tackled in section III. Sections IV and Vshow the results of the comparison in terms of efficiency anddynamic response respectively. Finally, the conclusions showa discussion on the selection of the more suitable topology.

II. CONVERTER MODELING

Figures 1, 2, 3 and 4 show the C2DS−QB, C2S2 −QB,S4 −QB and RCS2 −QB topologies.

Fig. 1. Schematic diagram of Cascaded-Connected Double-SwitchQuadratic Boost converter topology (C2DS −QB)

Fig. 2. Schematic diagram of Cascaded-Connected Single-SwitchQuadratic Boost converter topology (C2S2 −QB)

Fig. 3. Schematic diagram of Software-Synthetized Single-SwitchQuadratic Boost converter topology (S4 −QB)

Fig. 4. Schematic diagram of Rotated-Cell Single-SwitchQuadratic Boost converter topology (RCS2 −QB)

When the switching converters operate in continuousconduction mode (CCM), we have two structures whichcorrespond to the two different states of the power switch.The entire here studied topologies use only the control

u signal, which selects the instantaneous structure of theconverter.

The ideal model of both C2DS −QB and C2S2 −QB isrepresented by (2).

diL1

dt=

vi

L1−vC1

L1(1 − u)

diL2

dt=vC1

L2−vC2

L2(1 − u)

dvC1

dt= −

iL2

C1+iL1

C1(1 − u)

dvC2

dt= −

vC2

R · C2+iL2

C2(1 − u)

(2)

On the other hand, the behavior of the S4 −QB convertercan be expressed by (3).

diL1

dt=

vi

L1u−

vC1

L1(1 − u)

diL2

dt=

vi

L2+vC1

L2−vC2

L2(1 − u)

dvC1

dt= −

iL2

C1+iL1

C1(1 − u)

dvC2

dt= −

vC2

R · C2+iL2

C2(1 − u)

(3)

Finally, the RCS2 − QB converter can be described bythe expression (4).

diL1

dt=

vi

L1−

(vC2 − vC1)

L1(1 − u)

diL2

dt= −

vC1

L2+vC2

L2u

dvC1

dt=iL2

C1−iL1

C1(1 − u)

dvC2

dt= −

vC2

R · C2−iL2

C2u+

iL1

C1(1 − u)

(4)

Taking into account the fact that reasonable values of theparasitic elements do not change dramatically the converterdynamics, these ideal models will be used in the sliding-modeanalysis avoiding unnecessary complexity.

However, a complete model including parasitic resistancesand voltage drops it is necessary to study the static perfor-mance of the converter. So, a piece-wise state-space equationcan be used to easily represent a more complete model. Theexpression (5) corresponds to the circuit structure obtainedwhen the controlled switch and the diode D3 are in ON-state and simultaneously the diodes D1 and D2 are in OFF-state. The expression (6) corresponds to the circuit structureobtained when the controlled switch and the diode D3 are inOFF-state and simultaneously the diodes D1 and D2 are inON-state. It is worth to note the fact that in the C2DS −QBtopology, there exist two controlled switches which have thesame control signal.

x = A1x+B1v for u = 1 (5)

x = A2x+B2v for u = 0 (6)

where x have the states([iL1 iL2 vC1 vC2

]T),and v have the circuit variables considered as inputs([vin vD io

]T), where io models a load disturbance. Thus,the model can be expressed in the compact representation (7).

x = A2x+B2v + [(A1 −A2)x+ (B1 −B2) v]u (7)

In order to simplify the presentation of the converter’smodels, we will show only the corresponding matrices foreach converter topology.

By taking into account the equivalent series resistorin inductors (RL1, RL2) and capacitors (RC1, RC2), theON-resistance of the controlled switch (RM ), the seriesresistance on the diode (RD) and the fixed voltage sourcerepresenting the minimum drop-out voltage of the diode(VD), the matrices for the C2DS −QB converter are:

A1 =

−RL1−RM

L10 0 0

0−RC1−RL2−RM

L21L2

0

0 −1C1

0 0

0 0 0−KoR·C2

A2 =

−RL1−RD−RC1

L1RC1L1

−1L1

0RC1L2

−RC1−RL2−RD−RC2KoL2

1L2

−KoL2

1C1

−1C1

0 0

0KoC2

0−KoR·C2

B1 =

1L1

0 0

0 0 00 0 0

0 0 −1C2

; B2 =

1L1

−1L1

0

0 −1L2

0

0 0 0

0 0 −1C2

while the matrices for the C2S2 −QB converter are given

by:

A1 =

−RL1−RD−RM

L1−RML1

0 0−RML2

−RC1−RL2−RML2

1L2

0

0 −1C1

0 0

0 0 0−KoR·C2

A2 =

−RL1−RC1−RD

L1RC1L1

−1L1

0RC1L2

−RC1−RL2−RD−RC2KoL2

1L2

−KoL2

1C1

−1C1

0 0

0KoC2

0−KoR·C2

B1 =

1L1

−1L1

0

0 0 00 0 0

0 0 −1C2

; B2 =

1L1

−1L1

0

0 −1L2

0

0 0 0

0 0 −1C2

In turn, the matrices for the S4 − QB converter areexpressed as follows:

A1 =

−RL1−RD−RM

L1−RML1

0 0−RML2

−RC1−RL2−RML2

1L2

0

0 −1C1

0 0

0 0 0−KoR·C2

A2 =

−RL1−RD−RC1

L1RC1L1

−1L1

0RC1L2

−RC1−RL2−RD−RC2KoL2

1L2

−KoL2

1C1

−1C1

0 0

0KoC2

0−KoR·C2

B1 =

1L1

−1L1

0−1L2

0 0

0 0 0

0 0 −1C2

; B2 =

0 −1

L10

1L2

−1L2

0

0 0 0

0 0 −1C2

Finally, the matrices for the RCS2 −QB converter are:

A1 =

−RL1−RD−RM

L1−RML1

0 0−RML2

−RC1−RL2−RM−RC2KoL2

−1L2

KoL2

0 1C1

0 0

0−KoC2

0−KoR·C2

A2 =

−RL1−RC1−RD−RC2Ko

L1RC1L1

1L1

−KoL1

RC1L2

−RC1−RL2−RDL2

−1L2

0−1C1

1C1

0 0KoC2

0 0−KoR·C2

B1 =

1L1

−1L1

0

0 0 00 0 0

0 0 −1C2

; B2 =

1L1

−1L1

0

0 −1L2

0

0 0 0

0 0 −1C2

Thus, the complete model of the converters can be easilyimplemented in a MATLAB based algorithm. Also, it is worthpointing out that this state space representation leads to thesame ideal models (2), (3) or (4) introducing a value of zerofor the parasitic resistances and voltage drops in the matrices.

III. SLIDING MODE ANALYSIS

The sliding surface (8) has been defined to compare thedynamic behavior of the quadratic boost topologies on thesame basis. This surface allows obtaining a sliding modewithout introduce a forced dynamic in the state-variables’trajectory.

S(x) = iE − iL1 (8)

Thus, the value of the output voltage is indirectly definedby the value of the current of the inductor L1. This meansthat a constant value of iE corresponds to a constant valueof the output voltage at the equilibrium point. To obtainthe linearized dynamics of the converter around the reachedequilibrium point and show its stability, the following step bystep procedure has been applied for all topologies:

• STEP 1: Applying the equivalent control method toobtain ueq .

• STEP 2: Obtaining the ideal sliding-mode dynamicsby replacing the equivalent control.

• STEP 3: Obtaining the equilibrium point.• STEP 4: Linearization of the ideal sliding dynamics

around the equilibrium point.• STEP 5: Obtaining the characteristic polynomial.• STEP 6: Determination of the stability conditions by

using the Routh-Hurwitz test.

A. C2DS −QB and C2S2 −QB topologiesThe C2DS − QB and C2S2 − QB topologies show the

same ideal dynamics since its model is the same. We find thefollowing equivalent control:

ueq =vC1 − vin

vC1

(9)

By replacing (9) in (2) assuming that the time derivativesare zero, the equilibrium point (10) is obtained.

iL1 = Iref , iL2 =

(vin

R

) 14

(Iref )34

vC1 = (Iref · R)14 (vi)

34 , vC2 = (Iref · vi · R)

12

(10)

After linearizing and applying the Laplace transform, wefind the following characteristic polynomial:

P (s) = s3+

K2mC2 + C1

RC1C2

s2+

2K2mR2C2 + K4

mL2 + R2C1

K2mR2L2C1C2

s+4

RL2C1C2

(11)

where Km = vC2/vC1 = vC1/vin. By applying the RouthHurwitz stability test, the following condition is obtained:

K2m

(K2

mL2 + C2R2) (K2

mC2 + C1)

+R(K2

mC2 − C1)2> 0 (12)

which is always valid. Hence, these topologies are locallystable in the sliding surface (8).

B. S4 −QB topologyWe find the following equivalent control:

ueq =vC1

vC1 + vin(13)

By replacing (13) in (3) assuming that the time derivativesare zero, the equilibrium point (14) is obtained.

iL1 = Iref , iL2 =

(vin

R

) 14

(Iref )34

vC1 = (Iref · R)14 (vin)

34 − vin, vC2 = (Iref · vi · R)

12

(14)

Fig. 5. Efficiency of the quadratic boost converters: a) C2DS −QB, b) C2S2 −QB, c) S4 −QB, and d) RCS2 −QB.

After linearizing and applying the Laplace transform, wefind the following characteristic polynomial:

P (s) = s3+

K2mC1 + C2

K2mRC1C2

s2+

2K2mR2C2 + K4

mR2C1 + L2

K2mR2L2C1C2

s+4

RL2C1C2

(15)

where Km = (vC1 + vin) /vC2 = vin/ (vC1 + vC1). By applyingthe Routh Hurwitz stability test, the following condition isobtained:

R2[(K2

mC1 −RC2)2

+ C22 + 2K2

mC1C2]

+K2mC1 + C2L2 > 0

(16)which is always satisfied. Hence, this topology is locally

stable in the sliding surface (8).

C. RCS2 −QB topologyWe find the following equivalent control:

ueq =vC2 − vC1 − vin

vC2 − vC1

(17)

By replacing (17) in (4) assuming that the time derivativesare zero, the equilibrium point (18) is obtained.

iL1 = Iref , iL2 =

(vi

R

) 14

(Iref )34

vC1 = (Iref · vi · R)12 − (Iref · R)

14 (vi)

34 , vC2 = (Iref · vi · R)

12

(18)

After linearizing and applying the Laplace transform, wefind the following characteristic polynomial:

P (s) = s3+

C2 + M1C1

K2mRC1C2

s2+

L2 + 2R2C2 + M2R2C1

L2R2C1C2

s+4

RL2C1C2(19)

where M1 = K2m + Km + 1, M2 = K2

m + 3Km + 2 andKm = vin/ (vC2 − vC1) = (vC2 − vC1) /vC2. By applying the

Routh Hurwitz stability test, the following condition isobtained:

R2[(K2

mC1 −RC2)2

+ C22 + 2K2

mC1C2]

+K2mC1 + C2L2 > 0

(20)which is always accomplished. Hence, this topology is

locally stable in the sliding surface (8).To summarize, the four studied topologies shows local

stability with the proposed sliding-mode control. Now, it ispossible to compare the converter’s dynamic by applying loaddisturbances and current reference changes.

IV. EFFICIENCY COMPARISON

To compare the efficiency of the four quadratic boostconverter topologies, the complete model described by (5)-(7), has been implemented using the values in Table II,following the expressions presented in [8]. Some differencesappear in the ON and OFF transition losses. On the otherhand, we use a hysteresis comparator to introduce a finiteswitching frequency and the related ripples in the convertervariables. Thus, our analysis includes AC and DC conductionlosses and switching losses.

The efficiency has been computed covering a power rangebetween 20 and 120 W and an input voltage range between15 and 25 V. The results are shown in figure 5. It is possibleto observe that the maximum efficiency in all topologiesis obtained at the maximum input voltage. Moreover, alltopologies show a pronounced detriment of the efficiencywhen the power increases and the input voltage decreases.However, it is worth to mention that this condition is notusual in photovoltaic applications since a low voltage impliesto some extent a low power.

Table IISimulation parameters

ELEMENT PARAMETER VALUE

Inductor L1L1 470 µHRL1 0.028 Ω

Inductor L2L2 4.7 mHRL2 1.14 Ω

Capacitor C1C1 9 µFRC1 0.0019 Ω

Capacitor C2C2 9 µFRC2 0.0019 Ω

Diode IDT16S60C RD 0.05 ΩvD 0.8 VQrr 38 nC

MOSFET STW45NM50 RM 0.08 Ωtf 35 nstr 23 ns

Cgate 3.7 nF

Resistive load R 1000-8000 Ω

Hysteresis band ±δ ±0.25 A

Among the studied topologies, the C2DS −QB topologyshows the best performance having an efficiency almostconstant and higher than 90% and an efficiency peak of93.6%. The C2S2 − QB shows the an efficiency peak of87% and an efficiency around 80% over the entire range. TheS4 − QB topology has a constant efficiency behavior overthe entire range of operation conditions with an efficiencypeak of 94%. In comparison with the C2DS −QB topology,only slight differences can be identified. The RCS2 − QBtopology shows the best efficiency in the range of highpower and low input voltage. This topology has a constantefficiency around 90% for output powers higher than 40 W.From elsewhere, our analysis shows that the efficiency of theC2S2 − QB topology is slightly lower than the other threetopologies. The minimum efficiency for all topologies wasfound with a high output power and a low input voltagecondition. However, as was mentioned earlier the worstefficiency takes place in a condition of output power andinput voltage which is not consistent with the operation of thephotovoltaic modules.

It is worth noting that the differences between the singleswitch topologies are introduced by the switching lossesbecause of the differences of the power switches arrangementin the circuits. Additional work is currently addressedincluding experimental results.

V. DYNAMIC RESPONSE COMPARISON

As can be seen in (11), (15) and (19), the quadratic boosttopologies operating with the sliding surface (8) have a thirdorder resultant dynamics. Some slight differences can benoted in the coefficients of the characteristic polynomials.This implies different locations of the system poles, whichmeans that the transient response can be also different. Tohighlight them, we have analyzed four conditions: 1) start-up,2) current reference change, 3) output load disturbance, and4) input voltage disturbance. The measurements were takenusing simulations with the complete model of the convertersand parameters in table II.

The comparison is accomplished by reaching the sameequilibrium point in all converters by means of a changeof reference or a disturbance. Therefore, because ofthe converter’s performance and the absence of voltage

regulation, the amplitude of the applied excitation resultsdifferent. The results are presented separately.

A. Start-up transient responseAt 0 s, all converters are in zero state. Next, the converters arepowered with an input voltage of 20 V entering in the star-uptransition. The C2DS − QB uses a current reference of 5.3A, the C2S2 −QB and the S4 −QB use a current referenceof 5.8 A, and the DCSS-QB uses a current reference of 5.85A. The output load is a resistor of 1600 Ω (100 W @ 400V). It is possible to observe in figure 6 that the C2DS −QB,C2S2−QB and S4−QB topologies reach the steady state at0.04 s while the DCSS-QB topology reaches the steady stateat 0.06 s.

Fig. 6. Transient response under start-up condition

B. Step change on the output loadOnce in the first desired equilibrium point (vC2 = 400 V), astep load disturbance is applied at 0.08 s. The excitations havebeen adjusted to reach a new equilibrium point in vC2 = 300V. The amplitude of the step disturbance is the 0.137 A inthe C2DS −QB, 0.14 A in the C2S2 −QB, S4 −QB andRCS2 − QB topologies. It is possible to observe in figure 7that the C2DS −QB, C2S2 −QB and S4 −QB topologiesreach the new equilibrium point in 0.03 s while the DCSS-QBtopology reaches the steady state in 0.05 s.

Fig. 7. Step change on the output load

C. Step change on the current referenceOnce in the second desired equilibrium point (vC2 = 300 V)a step change in the current reference is applied at 0.16 s. Thechange has been adjusted to reach a new equilibrium point invC2 = 350 V. The new value of the current reference is 6.81A for the C2DS −QB topology, 7.55 A for the C2S2 −QB

and S4 − QB topologies, and 7.64 A for the RCS2 − QB

topology. It is possible to observe in figure 8 the fact thatthe settling times are similar with respect to the other appliedexcitations.

Fig. 8. Step change on the current reference

D. Step change on the input voltageOnce in the third desired equilibrium point (vC2 = 350 V)a step change in the input voltage is applied at 0.225 s. Theamplitude of the disturbance has been adjusted to come backto the first equilibrium point (vC2 = 400 V). The new valueof the input voltage is 24.3 V for the C2DS −QB topology,23.85 V for the C2S2 − QB, S4 − QB and RCS2 − QBtopologies. It is possible to observe in figure 9 the fact thatthe settling times are similar with respect to the other appliedexcitations.

Fig. 9. Step change on the input voltage

To summarize, the four quadratic boost convertertopologies shows a stable first order dynamic in all evaluatedcases. A slight difference becomes evident in the RCS2−QBtopology, which shows a slower response.

VI. CONCLUSIONS

A comparison of four quadratic boost converter topologieshas been presented in this paper. The efficiency analysisshows that the best topology is the cascade connection oftwo conventional boost converters (C2DS − QB). However,the operation with a single active switch is a very attractivekind of the other topologies although there efficiency can beslightly lower. Among these topologies, the S4 − QB topol-ogy shows an efficiency comparable with the C2DS − QBtopology. However, its input has a pulsating nature becauseof the connection of the diode D3 in the input port. Thisfact can affect the life expectance of the overall system since

the main source is a photovoltaic module in parallel with anelectrolytic capacitor.

The dynamic analysis reveals that the DCSS-QB topologyhas a slower response in comparison with the othertopologies. However, the dynamic response of C2DS −QB,C2S2 − QB and S4 − QB topologies is similar in all thecases, which forces us to select the more suitable topologyavailing ourselves of the conclusions of the efficiencycomparison. More work will be addressed to includeexperimental results in this research.

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