DC-DC Boost Converter Operating - General Methodology

Part I Modeling of Power Electronic Converters

6.3 General Methodology

6.4.2 DC-DC Boost Converter Operating

Given a DC-DC boost power stage operating in dcm, one must deduce its averaged dynamic model that represents output voltage variation as a function of averaged control input (duty ratio). The circuit diagram is presented in Fig. 6.13, where inductor resistance and capacitor equivalent series resistance are neglected.

This case study is placed in the same context as the example from Sect.6.3.2.

Therefore, in order to deduce its ROAM one must consider the equivalent averaged current that feeds the output capacitor and express it as a function of output voltage.

0.05 0.1 0.15

Fig. 6.12 TCR behavior atβ0step change: (a) the effective voltage, v u, applied to the inductor;

(b) output of ROAM (ist(t)) vs. output of switched model (i(t)) 0 0.01 0.02 0.03 0.04 0.05

ix20[A] i_stx20[A]

a b

Fig. 6.11 TCR behavior atβ0¼ 135: (a) output of ROAM (ist(t)) vs. output of switched model (i(t)); (b) inductor current spectrum

The switched model is given by (see for reference Eq. (3.11) of Sect. 3.2.3, Chap.3):

_iL ¼1

LðE vCð1 uÞ riLÞ 1þ sgn ið ÞL

2 _vC ¼1

C iLð1 uÞ vC

R 0

1 A, 8>

(6.29)

with

sgnð Þ ¼iL 1 ifiL> 0

1 if iL 0

andu being the T-periodical switching function taking values in the set {0;1} and havingα as duty ratio, i.e., hui0¼ α.

Discontinuous-conduction operation is characterized by the existence of three time subintervals within a switching periodT, which describe the time evolution of the inductor current (see Fig.6.14), similar to the ones of the buck-boost converter (depicted in Fig.6.6). The assumption of output voltagevC being constant at the evolution time scale of currentiLis adopted. The expression ofimaxcan be written in two ways corresponding to the two triangles; thus

imax¼E

L T1¼vC E L T2, hence

T2¼ E

vC E T1: (6.30)

The averaged value of the diode current over aT-length time window can be expressed using the gray filled area of Fig.6.14, namely,

h i0¼imax T2

2T , E

i_L L

v_C

C R

i_D

u Fig. 6.13 Boost power

stage diagram

6.4 Case Studies 165

in which Eq. (6.30) is replaced; therefore,

h i₀¼ E²Tα²

2L vð C EÞ: (6.31)

Now let us return to the second equation from (6.29), where the inductor current is expressed depending on the diode current, i.e.,iL(1 u) ¼ iD; this equation is further averaged and relation (6.31) is used for replacing the average value ofiD:

Ch ivC_ 0 ¼ E²Tα²

2L vh iC 0 E h ivC 0

R : (6.32)

Relation (6.32) represents the ROAM of the boost converter operating in discontinuous-conduction mode. One can note that this relation describes the nonlinear dynamic dependence of the averaged output voltagehvCi0on the duty ratioα. This model corresponds to the diagram depicted in Fig.6.15.

Relation (6.32) can be linearized around an equilibrium point in order to simplify further analysis and control law design. The equilibrium operating point is computed by zeroing the derivative in Eq. (6.32) for a fixed value of the duty ratioαe. t i_L

slope

E (E − vC)

slope

T₁= αT T₂ T

i_D i_L, iD

t u

i_max

L L

Fig. 6.14 Current waveforms of boost power stage operating in dcm

v_C 0

0 C i_D Fig. 6.15 Boost power

stage – equivalent circuit corresponding to ROAM

In this way, by supposing that voltageE is constant, the steady-state value ofhvCi0

results from solving a second-order polynomial equation, namely, 2L vh iC ²0e 2EL vh iC 0e E²RTα²e¼ 0:

The above equation has two solutions; the positive one is chosen:

h i_0e¼E 2þE

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ2RTα²e

L r

: (6.33)

In order to proceed with the linearization around the operating point given by (6.33), the following change of variable is used:

v¼ vh iC 0 E: (6.34)

Note that_v ¼ _h ivC 0. Using this and introducing notation (6.34) into (6.32), one obtains

C_v ¼E²Tα²

2Lv vþ E R or, equivalently,

C _v² ¼E²T L α²2

Rv²2E

R v: (6.35)

Equation (6.35) is further linearized around the equilibrium point given by (6.33). Notatione denoting small variation around the chosen equilibrium point is adopted. Note thatve¼ hvCi0e E and ev ¼ gh ivC 0. Then straightforward compu-tation leads to

_ev ¼E²Tαe

LCve eα 2

RCev E RCve

ev,

which, according to notation (6.34), is equivalent to gvC_

h i0 ¼ E²Tαe

LC vh iC 0e E eα 2h ivC 0e E

RC vh iC 0e E gh ivC ₀: (6.36) Relation (6.36) represents the linearized ROAM of the boost converter operating in discontinuous conduction. One can see that the output voltage dynamic is of first order in relation to the duty ratio, having ^E²^RT^α^e

L 2 vðh iC0eEÞas gain and^RCð^{h i}^v^C^0e^EÞ

2h ivC0eE as time constant.

6.4 Case Studies 167

Next, simulations of the boost converter operating in dcm have been carried out using MATLAB^®-Simulink^®. The input voltage is constant, E¼ 5 V, the switching frequency is 2000 kHz. The switches are perfect and the circuit parameters are L¼ 0.5 mH, C ¼ 100 μF and R ¼ 150 Ω. The ROAM from (6.32) is assessed against its linearized version from (6.36) and the switched model of the circuit – see Fig.6.16.

Figure6.16shows the capacitor voltage evolution as the duty ratio is changed from 0.2 to 0.3 in step. Consequently, the average voltage evolves between 11.51 and 15.73 V – see expression (6.33). As the linearized model (6.36) is expressed in variations, one must add the steady-state value given by expression (6.33) in order to compare model outputs.

Bode diagrams of the linearized model (6.36) have also been plotted for various equilibrium points. Figure6.17ashows how Bode diagrams change as the operating point changes for duty ratios between 0.1 and 0.4 at constant load. Figure 6.17b shows how Bode diagrams change as operating point changes for load resistances between 50 and 200Ω. Note that system bandwidth decreases and steady-state gain increases slightly in both cases: when equilibrium duty ratio increases and when load resistance increases.

0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 11

averaged model v_C ₀ v_C Liniarized model

v_C 0

Fig. 6.16 Boost dcm modeling results

Fig. 6.17 Bode diagrams for various equilibrium points of the boost converter operating in dcm:

(a) curve family for variousαe; (b) curve family for various load resistances R

6.5 Conclusion

At this point let us formulate some summary remarks about the necessity and utility of the reduced-order averaged model. There are at least three cases where the use of ROAM is recommended:

• if the converter operates in discontinuous conduction;

• when the converter has both alternating and DC variables, and one wishes to neglect the former;

• if the converter dynamic is altered by means of a control structure, i.e., some of the variables are rendered significantly faster than others and hence can be neglected.

The mechanism of effectively reducing the system order consists in emphasizing an algebraic relationship between some fast variables and some slow variables, which makes the differential equations of the fast variables drop out. The result is a reduced-order model.

Reduced-order averaged modeling has the strength of being easy to compute and easy to use for control purposes – due to elimination of undesirable variables; hence it is widely applicable. Its drawbacks are related to ignoring the dynamics that can be qualified as “internal”, exposing the control law designer to instability problems that are not predictable by this kind of modeling.

A final remark is that the ROAM provides the most suitable mechanism for modeling converters operating in discontinuous conduction. This model can be used in other cases, especially when a simpler model is needed, but it is recommended that one first analyze the dynamics likely to be neglected.

Problems

Problem 6.1. Linearized ROAM of buck-boost converter in dcm

Let us consider the buck-boost circuit in Fig.6.5. It is required to obtain the small-signal ROAM.

Solution. In order to derive the small-signal ROAM, Eq. (6.19) – representing the nonlinear ROAM of the buck-boost converter in discontinuous conduction – should be linearized around an equilibrium point:

Ch ivC_ 0 ¼ E²T 2L vh iC 0

α²h ivC 0

R , (6.37)

where notations have the meanings as introduced in the example detailed in Sect. 6.3.2. Equation (6.37) describes the first-order averaged dynamic of the capacitor voltage in relation to the input represented by duty ratio α. Supposing

Problems 169

that voltage E is constant, the equilibrium point results from zeroing the time derivative in Eq. (6.37) for a fixed valueαe; hence

2L vh iC ²0e¼ E²RTα²e, which gives

h i0e¼ Eαe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ= 2LRT ð Þ

p : (6.38)

Equation (6.37) can be written as the dynamic equation of the variablehvCi02

depending linearly onα²:

C _

h i²0

¼E²T

L α²2h ivC ²0

R : (6.39)

Lete be the notation dedicated to denoting small variations around a given equilibrium point. Notationsh ivC 0¼ vh iC 0eþ gh ivC 0 andα ¼ αeþ eα are adopted;

hence,h ivC_ ₀ ¼ _h igvC ₀ and _α ¼ _eα . Linearization uses the first-order Taylor series approximation of function x², namely x² x02

+ 2x0(x x0). Thus, Eq. (6.39) gives by linearization around the equilibrium operating point (6.38)

gvC_

h i₀ ¼ E²Tα²e

2LC vh iC 0e

þ E²Tαe

LC vh iC 0e

eα h ivC 0e

RC 2

RC gh ivC ₀, where_{2LC v}^E²^T^α²^e

h iC0e^{h i}^v_RC^C^0e ¼ 0 according to Eq. (6.38). Therefore:

gvC_

h i₀ ¼ E²Tαe

LC vh iC 0e

eα 2

RC gh ivC ₀: (6.40) Equation (6.40) corresponds to a first-order linear dynamic system having as input small variations of duty ratio eα and as output small variations of averaged capacitor voltage gh ivC ₀. That is, the ROAM dynamic of a buck-boost converter around a given equilibrium point is of first order, linear, having_{2L v}^E²^RT^α^e

h iC0eas gain and^RC₂ as time constant.

Problem 6.2

Consider the system in Fig.5.30from Problem 5.1 in Chap.5. It is about a circuit used to convey the energy between a variable DC source and a single-phase strong AC grid via a DC link, the reactive power being zero.

Given the system’s large-signal model developed in the solution to Problem 5.1, obtain a small-signal reduced-order averaged model used for control purpose by taking into account the following considerations.

The system must have the possibility of varying the input power; this is done by controlling the DC-source current by means of control inputu1. The primary control purpose is to ensure the power flux between the primary DC source and the AC grid while preserving the rated operating conditions in terms of voltage and current; this is done by means of control inputu2. To this end, a widely employed cascaded control structure can be used (Astro¨m and Ha¨gglund1995), in which the DC-link voltage controller imposes the set-point of the grid inductor current. Therefore, the control structure is composed of two loops: the inner loop used for controlling the grid-side inductor current and the outer loop for controlling the DC-link voltage.

Solution. As shown in the solution of (b) in Problem 5.1, computation of the small-signal model and of its associated transfer functions may be a difficult way to obtain necessary information for designing control laws. Depending on each particular control objective, some remarks may be useful in order to introduce simplifying assumptions before starting control design procedures. In our case, let us focus on the regulation of the average value of the DC-link voltage,vDC. As AC voltage is stiff, the (variable) output power can be controlled by means of the inductorL2

current (the output current loop). This supposes a variable DC current drain from the DC-link. In order to maintain the value ofvDCwithin operating limits, a voltage loop should also be used. In fact, the invariance ofvDCguarantees the input–output power balance. Figure6.18presents briefly this control approach.

v₀

i_d

v^∗_DC= cst i_L1

α βd

Current control Voltage

control

i ^∗_L1 0

i ^∗_d Current

control

L₂ L₁

V sin ωt R C

Gate driver

Gate driver i_L10

i_d v₀ 0= vDC

Dynamics neglected

Fig. 6.18 Simplified diagram of cascaded control structure for the DC-AC converter given in Fig.5.30(Chap. 5). Auxiliary elements – filters, i_dcomputation using PLL, etc. – have been omitted

Problems 171

Let us review the result from Problem 5.1 (Eq.5.83), preserving the significance of notations introduced there:

d ih iL1 0

describing the averaged large-signal model of the circuit, where vDC¼ hv0i0, βd¼ hu2i1andid¼ hiL2i1.

First note that variableiL1is controlled independently; it represents a perturbation for the remainder of the system and its dynamic is not of interest here.

The dynamic of currentidis rendered significantly faster than the dynamic of vDCby means of the inner control loop, i.e.,idequalsid*practically instantaneously in relation to the dynamic of vDC. Therefore, one can assume that variable id

influencesvDCby means of its steady-state regime, characterized by zeroing the derivative of the corresponding dynamic equation in relations (6.41) and taking into account neglecting the resistance of inductorL2,r¼ 0. Thus, the steady-state value ofβdis reached, which depends on the value ofvDC, namelyβd _ st¼ V/vDC. At its turn, the latter relation is used in the dynamic equation ofvDCto obtain

Relation (6.42) represents thelinear dynamical equation of the DC-link voltage squared. This relation also represents thenonlinear ROAM of the circuit because the resulting system order has been decreased by one.

One can use relation (6.42) to design linear control laws to regulatevDC2(instead ofvDC) by usingidas control input, knowing that Pin _ DC is a perturbation. The plant transfer function used to this end is

H_i_d_!v²

DCð Þ ¼s v²_DC id

¼ V R

s^RC₂ þ 1: (6.43)

If the transfer fromidtovDCis sought, then linearization of Eq. (6.42) around an equilibrium operating point vDCe is necessary. This operating point obeys the following relation:

2Pin DCe

C 2

RCv²_DCeV

Cide¼ 0: (6.44)

As in the previous solved problem, the linearization makes use of the first-order Taylor series approximation of function x², namely x² x02

+ 2x0(x x0).

Variables of interest can respectively be written as the sum of their equilibrium values and their small variations:

vDC ¼ vDCeþ gvDC

id¼ ideþ eid:

Therefore, linearization of Eq. (6.42) yields

2vDCe _gvDC ¼2Pin DCe

C 4vDCe

RC vgDC 2v²_DCe RC V

CideV

Ceid, (6.45) where the power supplying the DC link has been supposed a sufficiently slowly variable, i.e., constant:Pin DCeg ¼ 0. By substituting relation (6.44) into (6.45) one obtains

2vDCe _gvDC ¼ 4vDCe

RC gvDC V Ceid, from which the transfer function results as

Hei_d!f^vDC

ð Þ ¼s vgDC

eid

¼ V R 4vDCe

s^RC₂ þ 1: (6.46) Equation (6.46) gives the small-signal ROAM that expresses the linear first-order dynamic of the DC voltage as depending on current id. Its utility consists in providing the basis for the design of a DC-link voltage regulator that imposes theid

value in response tovDCevolution (see Fig.6.18).

Note that transfer functions given by relations (6.43) and (6.46) have the same time constant – meaning that they correspond to the same dynamic – but different gains (in the latter case this depends on the chosen equilibrium point).

The following problems are proposed to the reader to solve.

Problem 6.3. Modeling a buck power stage operating in dcm

Let us consider the buck converter with capacitive output filter having the circuit diagram presented in Fig.2.10of Chap.2. The converter operates in discontinuous-inductor-current mode. The following points should be addressed.

(a) Deduce the converter ROAM by taking the duty ratio as control input and the output voltage as controlled variable.

Problems 173

(b) Determine the conversion ratio between the input and the output voltages.

Problem 6.4. Modeling a flyback converter operating in dcm

Answer the same questions as in Problem 6.3 for the case of a flyback converter having the circuit diagram given in Fig.4.24a of Chap.4.

Problem 6.5. ROAM of a series-resonance supply of a diode rectifier

Let us consider the circuit composed of a voltage inverter, a resonant tank and a diode rectifier given in Fig.6.19.

The voltage inverter is assumed to operate in full wave at a frequency close to the resonance frequency of the alternative circuit. Address the following points.

(a) Deduce the switched model.

(b) Explain why the classical first-order harmonic approach is suitable for studying the resonant circuit.

(c) Write the ROAM of this converter by taking the output voltage vC0 as a controlled variable and the inverter frequency as a control input variable.

(d) Deduce the small-signal model of the previously obtained ROAM.

(e) Give the equivalent diagram corresponding to the ROAM of the converter.

References

Astro¨m KJ, Ha¨gglund T (1995) PID controllers: theory, design and tuning, 2nd edn. Instrument Society of America, Research Triangle Park

Bacha S, Rognon JP, Hadj Said N (1995) Averaged modelling for AC converters working under discontinuous conduction mode – application to thyristors controlled series compensators.

In: Proceedings of European Power Electronics Conference – EPE 1995. Sevilla, Spain, pp 1826–1830

Chetty PRK (1981) Current injected equivalent circuit approach to modeling switching DC-DC converters. IEEE Trans Aerosp Electron Syst 17(6):802–808

Chetty PRK (1982) Current injected equivalent circuit approach to modelling and analysis of current programmed switching DC-to-DC converters (discontinuous inductor conduction mode). IEEE Trans Ind Appl 18(3):295–299

C´ uk S, Middlebrook RD (1977) A general unified approach to modelling switching dc-to-dc converters in discontinuous conduction mode. In: Proceedings of IEEE Power Electronics Specialists Conference – PESC 1977. Palo Alto, California, USA, pp 36–57

L C

H H

C₀ R v_C0

v_C i_L

Fig. 6.19 Series-resonance supply of a diode rectifier

Erikson RW, Maksimovic´ D (2001) Fundamentals of power electronics, 2nd edn. Kluwer, Dordrecht

Maksimovic´ D, C´ uk S (1991) A unified analysis of PWM converters in discontinuous modes.

IEEE Trans Power Electron 6(3):476–490

Middlebrook RD (1985) Topics in multiple-loop regulators and current mode programming.

In: Proceedings of IEEE Power Electronics Specialists Conference – PESC 1985. Toulouse, France, pp 716–732

Middlebrook RD (1989) Modelling current-programmed buck and boost regulators. IEEE Trans Power Electron 4(1):36–52

Mohan N, Undeland TM, Robbins WP (2002) Power electronics: converters, applications and design, 3rd edn. Wiley, Hoboken

Sun J, Grotstollen H (1992) Averaged modeling of switching power converters: reformulation and theoretical basis. In: Proceedings of IEEE Power Electronics Specialists Conference – PESC 1992. Toledo, Spain, pp 1166–1172

Sun J, Grotstollen H (1997) Symbolic analysis methods for averaged modeling of switching power converters. IEEE Trans Power Electron 12(3):537–546

Sun J, Mitchell DM, Greuel MF, Krein PT, Bass RM (2001) Averaged modeling of PWM converters operating in discontinuous conduction mode. IEEE Trans Power Electron 16 (4):482–492

Vorpe´rian V (1990) Simplified analysis of PWM converters using model of PWM switch. Part II:

discontinuous conduction mode. IEEE Trans Aerosp Electron Syst 26(3):497–505

References 175

Control of Power Electronic

Converters

Chapter 7 General Control Principles of Power Electronic Converters

This chapter aims at offering a synthetic perspective over control goal formulation and control design methods for power electronic converters in the context of their specific features and constraints. It concerns intrinsically variable-structure systems with fast nonlinear dynamics that are potentially subject to significant noise distur-bance because of modulated control signals. Such plants are challenging from the control perspective and so a large palette of control methods will be explored.

7.1 Control Goals in Power Electronic Converter Operation

Generally speaking, power electronic converters are key elements in power systems. Besides conveying electrical power with high efficiency, they offer the possibility of controlling internal variables in order to ensure both safe operation and output regulation. In the quasi-totality of their applications, power electronic converter operation requires a form of control intended not only for attaining the operating objective but also for safety. Control specifications are quite diversified, depending on the particular converter role.

As an example, a DC-DC converter operating in a switch-mode power supply feeding a certain variable load may need duty ratio adjustments in order to ensure a constant output voltage for the entire operating range (voltage regulation). Alter-nately, a grid-tie inverter fed by a renewable energy system must output a desired AC current to the grid in order to satisfy certain power transfer requirements, thus behaving as a controlled current source. Active power filters must provide the necessary current/voltage spectrum in order to cancel undesired harmonic content produced by a polluting load, all by maintaining the grid-load power balance.

Figure 7.1shows a generic application of power electronics control. In most cases the control algorithm outputs the duty ratio in response to the circuit state/

output evolution. The duty ratio needs modulation (conversion to the ON/OFF

S. Bacha et al.,Power Electronic Converters Modeling and Control: with Case Studies, Advanced Textbooks in Control and Signal Processing, DOI 10.1007/978-1-4471-5478-5_7,

179

signal) to be applied to the power switch base driver. To this end, various types of modulation may be used, of which the most frequently employed are pulse-width and sigma-delta modulations (in their analog or digital version) and space-vector modulation (SVM) (Leon et al.2010). Hysteretic modulators or constant turn-on modulators can also be used (Corradini et al. 2011). However, the diagram in Fig.7.1is not general, as there are control laws that directly output the two-state control signal (e.g., the sliding-mode control).

In either case, the controller must respond to significant system perturbation, that is the load variation, according to certain a priori specified control objectives. As usual in control systems technology, these objectives include output regulation/

tracking, internal variable dynamics compensation, limitation of variables to admis-sible values, etc. For purposes of zero steady-state error, integral control action is mandatory in most cases, at least in the outer control loops.

Generally speaking, converter control design is focused on imposing desired low-frequency (macroscopic or otherwise equivalent) behavior with respect to the specified requirements (Kassakian et al. 1991). In this context, the converter

In document Power Electronic Converters Modeling and Control.pdf (Page 183-200)