Sub-cuestión tercera - ESTADO DE LA CUESTIÓN

2. ESTADO DE LA CUESTIÓN

4.3. Sub-cuestión tercera

Conclusion

Averaged models can form the basis for obtaining normalized models by making suitable variable changes in both state variables and time (Sira-Ramı´rez and Silva-Ortigoza2006). Discrete-time averaged models, useful for digital control design, can also be obtained based upon sampled-data models (Maksimovic´ and Zane2007).

Among the advantages of the averaged model in particular is the facility of building and implementing one, as well as its good approximation precision when the state variables are lightly rippled. The other way round, this model begins to lose its precision when good filtering conditions are no longer satisfied. Moreover, the model is not useful for converters having one or multiple AC stages (zero average value of certain variables). It also exhibits difficulties in the discontinuous-conduction case, as illustrated by the presented case study. Thus, as the concerned variable becomes zero at each switching cycle, on a nonzero time interval it cannot be expressed as a large-signal recurrent form, and therefore it cannot be approximated by the averaged model.

Finally, there are some questions that the scientific community has not yet answered (Krein and Bass1990). The main ones are:

• What conditions guarantee a satisfactory approximation?

• Is the averaged model trajectory a sliding average along the original circuit?

• Are averaged model and original system dynamic characteristics identical?

• Can the original solution be recovered starting from the approximated solution?

• Is the approximation valid for large signals?

0.028 0.03 0.032 0.034 0.036 0.038 -80

-70 -60 -50 -40 -30 -20

0.0280 0.03 0.032 0.034 0.036 0.038 5

10 15 20 25

30 [A] i_L

t[s]

α = 0.3 α = 0.2

[V] v_C 0

v_C

α = 0.3 α = 0.2

t[s]

i_L 0 v_C0

i_L0

a b

Fig. 4.23 Evolutions of state variables in discontinuous-conduction case, obtained at the bound-ary between ccm and dcm (2 kHz switching frequency): (a) interrupted current; (b) first-order behavior of the capacitor voltage

• What is the lower limit of the ratio between the switching frequency and the smallest converter time constant for which the averaged model remains valid?

• Can the model be used in closed loop?

Some of these questions have partial answers; the reader is referred to the relevant literature. Taking into account the above issues, alternative solutions to the averaged model have been developed, such as the steady-state first-order-harmonic model (Sanders et al. 1990; Kazimierczuk and Wang 1992), the reduced-order averaged model (ROAM) (Chetty 1982; Sun and Grotstollen 1992), or generalizations of the averaged model.

Some of these solutions, such as developed by Krein et al. (1990), bring greater precision, but at the price of implementing and building a complexity difficult to accept.

The first-order harmonic dynamic approach solves the problem of AC variables presence, but it cannot solve the discontinuous conduction problem. In addition, this latter technique does not add precision in the DC-DC conversion case, although it provides a more complex solution. Nevertheless, modeling in the first-order harmonic dynamic sense finds a natural application in the field of power converters having both DC and AC stages (rectifiers, inverters, resonant power sources, etc.). The disconti-nuous conduction problem has no solution (in the modeling sense), unless the model of the concerned variable is eliminated. This can be achieved by using the ROAM technique. These techniques will be approached in the following chapters.

Problems

Problems 4.1, 4.2, and 4.3 are given with solutions. Problems 4.4, 4.5, and 4.6 are left as exercises.

Problem 4.1. Flyback converter

Let us consider the flyback converter in Fig.4.24a, allowing isolated non-inverting boost topology via a transformer with ration.

It is supposed that the transformer has negligible primary and secondary winding resistances and leakage inductances, whereas it has nonzero core reluctance. These assumptions lead to transformer modeling as a “two-winding inductor”, i.e., represented by a magnetizing inductance referred to the primary winding coupled

i_L v_C

C R

H u

1: n

E E v_C

1: n D

H L u

C R

a b

Fig. 4.24 Flyback converter: (a) electrical circuit; (b) electrical circuit including model of the transformer

82 4 Classical Averaged Model

with an ideal transformer, as in Fig.4.24b(Erikson and Maksimovic´2001). It is required to address the following points.

(a) Obtain the averaged model and the corresponding equivalent diagram by using Algorithm 4.1.

(b) Compute the steady-state model.

(c) Considering that input voltageE varies (representing a disturbance input), deduce the small-signal state-space model and draw the associated equivalent diagram.

(d) Using the previously obtained small-signal model, get expressions for the transfer functions representing the influence from the duty ratio (control input) and the input voltageE (disturbance input) to the state variables.

Solution. (a) The averaged model is deduced from the switched model. This latter results from an analysis of the circuit operation; it has two configurations: switch H turned on and diode D blocked (switching function u takes value 1), and switch H turned off and diode D conducting (switching function u takes value 0) (see Fig.4.25).

When the diode is blocked (u ¼ 1), the transformer’s primary and secondary currents are zero. When switchH is turned off (u¼ 0) the magnetizing inductor is decoupled from the input source, the primary voltage is vC/n and the secondary current isiL/n. Hence, the system has two state variables, current iLand voltagevC, whose dynamics are described by the following relations:

u¼ 1 :

Equations (4.41) can be expressed by a single set, which represents the switched model of the flyback converter:

i_L

Fig. 4.25 Two configurations of flyback converter, corresponding respectively to the two states of the switch network composed ofH and D: (a) H turned on and D blocked (u¼ 1); (b) H turned off andD in conduction (u¼ 0)

L _iL¼ 1 uð Þ vC

The averaged model results directly from model (4.42) by replacing switching functionu by its average, i.e., the duty ratio denoted byα. The equivalent diagram of the averaged model can be drawn as shown in Fig.4.26. The three subcircuits are linked by two couplings, each of which acts as an AC + DC transformer, with ratio α (input voltage side) and (1 α)/n (load side).

(b) The steady-state model results by zeroing the derivatives in Eq. (4.42).

Equilibrium values of the state variables are denoted by subscripte:

vCe¼ n Ee αe

The first equation of (4.43) shows that the output voltage has expression similar to that of the buck-boost case, except it is positive and contains a supplementary multiplying factorn.

(c) The small-signal model may result from perturbation and linearization. Let us replace the perturbed input and state variables into Eq. (4.42): α ¼ αeþ eα , E

¼ Eeþ eE , iL¼ iLeþ eiL and vC¼ vCeþ evC . By using (4.43) and neglecting products of small variations, one obtains, after some simple algebra:

L _eiL ¼ 1 αe

These lead to the small-signal AC equivalent diagram in Fig.4.27.

αiL

Fig. 4.26 Equivalent diagram of flyback converter averaged model

84 4 Classical Averaged Model

Equations (4.44) allow state-space matrix representation as in Eq. (4.45), emphasizing the small-signal state vector x¼ eiL veC

, the small-signal input vector u¼ eα eE T

, the state matrix A and the input matrix B:

_eiL (d) Setting the output vector to be identical to the state vector, i.e., y x, results in output matrix C being the 2 2 identity matrix.

Based upon the matrix state representation in (4.45) and on the definition of the output matrix C, one can compute the transfer matrix H(s), containing the transfer functions of the four input-to-output channels. To this end, the following well-known formula can be used:

H sð Þ ¼ C sI Að Þ¹ B:

After performing the computation and knowing that elementHij(s) of matrix H (s) represents the Laplace image of the transfer from input j to output i, the expression of the transfer matrix can be written as

H sð Þ ¼ H_α!i_Lð Þ Hs E!iLð Þs H_α!v_Cð Þ Hs E!vCð Þs

, (4.46)

where the transfer functions are

v_C 0

Fig. 4.27 Small-signal AC equivalent diagram of flyback converter

H_α!i_Lð Þ ¼ ks αiL T_αi_Lsþ 1

with the different gain and time constant notations standing for T0¼ RC

Equations (4.47) and (4.48) indicate that the small-signal model is a linear-parameter-varying one because its parameters depend on the steady-state operating point, as expected. Note also that all the four transfer functions have a pole at the origin. As in the case of the buck-boost converter, the capacitor voltage exhibits nonminimum-phase behavior in response to duty ratio variation, as shown by the associated transfer function H_α!v_Cð Þ having right-half-plane zero (sees Eq. (4.47)). Based upon expressions (4.47) of the transfer functions, one can draw the corresponding frequency responses in the form of Bode diagrams.

Problem 4.2. Single-ended primary-inductor converter (SEPIC)

Let us consider the single-ended primary-inductor converter (SEPIC) in Fig.4.28, allowing noninverting up/down voltage conversion using two uncoupled inductors.

The circuit is driven by a single binary switching function and hence has two configurations (Sira-Ramı´rez and Silva-Ortigoza2006).

The following points must be addressed.

(a) Obtain the averaged model and the corresponding equivalent diagram.

(b) Compute the steady-state model.

D Fig. 4.28 Electrical circuit

of SEPIC

86 4 Classical Averaged Model

(c) Assuming that input voltageE varies (being a disturbance input), deduce the small-signal state-space model and draw the associated equivalent diagram.

(d) Using the previously obtained small-signal model, get the expression of the transfer function representing the influence from the duty ratio (control input) to the voltagevC2as output variable.

Solution. (a) The averaged model is deduced from the switched model, which results from analyzing the two configurations of the circuit: for switchH turned on and diode D blocked (switching function u takes value 1 – Fig.4.29a), and for switchH turned off and diodeD conducting (switching function u takes value 0 – Fig.4.29b).

The circuit is described by four state variables, the two inductor currentsiL1and iL2, and the two capacitor voltagesvC1andvC2. Operation of the two configurations can be merged to yield the switched model given in Eq. (4.49).

− Fig. 4.29 The two

configurations of the SEPIC power stage, corresponding respectively to the two states of the switchH:

(a) H turned on (u¼ 1);

Fig. 4.30 Equivalent diagram of SEPIC averaged model

L1iL1_ ¼ 1 uð Þ vð ^C1þ vC2Þ þ E

The averaged model results directly from model (4.49), in which one replaces the switching function u by its average, i.e., the duty ratio denoted by α. The equivalent diagram of the averaged model is shown in Fig.4.30.

(b) By zeroing the derivatives in Eq. (4.49) one obtains the algebraic relations allowing computation of the steady-state values of state variables (marked with subscripte) as follows:

iL1e¼ αe

Equations (4.50) show that the steady-state value ofvC1equals the input voltage E, with the steady-state output voltage vC2ecorresponding to the noninverting buck-boost topology.

Thus, the perturbed input and state variablesα ¼ αeþ eα , E ¼ Eeþ eE , iL1¼ iL1e

Fig. 4.31 Small-signal equivalent diagram of SEPIC

88 4 Classical Averaged Model

Eq. (4.49). After computations, by using Eq. (4.50) and neglecting products of small variations, one obtains Eq. (4.51), whose associated equivalent diagram is given in Fig.4.31.

L1fiL1_ ¼ 1 αð eÞ fvC1 1 αð eÞ fvC2 þ vð C1eþ vC2eÞeα þeE Equations (4.51) can further be put into the matrix form:

_x ¼ A x þ B u, (4.52)

where x¼ fiL1 fiL2 vfC1 vfC2

is the small-signal state vector, u¼ eα eE T

is the small-signal input vector and state matrix A and input matrix B are

A¼

Relations (4.52) and (4.53) define the SEPIC small-signal state-space model.

(d) Using the matrix state representation in (4.52), the matrix definitions in (4.53) and the definition of the output matrix C as the four-by-four identity matrix, one can compute the transfer matrix H(s) as

H sð Þ ¼ C sI Að Þ¹ B, (4.54) which contains the transfer functions of the eight input-to-output channels. Apply-ing relation (4.54) requires in this case the computation of the inverse of a four-by-four matrix in analytical form, which is quite difficult. In order to get the expression of the transfer function of the channel from the duty ratioeα (first input) to the output voltagevfC2 (fourth state variable), the output vector must be set as y vC2, hence the output matrix must be set as C¼ 0 0 0 1½ ^T. Therefore, computing only the fourth row of matrix H(s) is sufficient to get the transfer function sought for.

A quite laborious but simple computation finally produces the duty-ratio-to-output-voltage transfer function,

H_α!v_C2ð Þ ¼s b3s³þ b2s²þ b1sþ b0

s⁴þ a3s³þ a2s²þ a1sþ a0

, (4.55)

where the different parameters of numerator and denominator are, respectively

b3¼ iL1eþ iL2e

Analyzing the numerator of expression (4.55), one can identify the presence of unstable zeros, characterizing the nonminimum-phase behavior of the output volt-age in response to duty ratio variations, similar to the buck-boost and flyback converter cases.

Problem 4.3. Nonideal Boost Converter

Let us consider the boost converter in Fig.4.32, where the inductor is modeled as a pure inductanceL ¼ 2 mH and a series resistance RL¼ 0.5 Ω due to copper losses.

The output capacitor has C¼ 100 μF and an equivalent series resistance, RC

¼ 0.05 Ω. Both input voltage and load resistor are variant around their respective rated valuesE¼ 5 V and R ¼ 10 Ω. It is required to solve the following points.

(a) Deduce the small-signal state-space model and draw the associated equivalent diagram taking into account that both input voltageE and load resistance R vary (they will be represented as disturbance inputs).

(b) Get the expressions of the three following transfer functions: from duty ratio to output voltage, from input voltage to output voltage and from load resistance to output voltage, respectively, by using the previously obtained diagram.

(c) Compute the steady-state model and draw the steady-state characteristics and input–output efficiency curve with respect to the duty ratio.

Fig. 4.32 Nonideal boost power stage

90 4 Classical Averaged Model

(d) Draw the zero-pole diagrams for each of the three above mentioned influence channels using the MATLAB^®-Simulink^®software.

(e) Simulate the averaged nonlinear (large signal) model and assess the results for small variations around the steady-state operating point corresponding to the full load; make comparisons with the zero-pole diagram.

Solution. (a) For the sake of simplicity, in this example the bracketsh · i0will be dropped; therefore, any variable encountered will denote in fact the corresponding average. Following the developments in Sect. 4.5.3 of Chap.4, the large-signal averaged state-space model may be written as

L _iL¼ E v0ð1 αÞ RLiL

C _vC ¼ iLð1 αÞ v0

R v0¼ CRC _vCþ vC: 8>

(4.56)

Next, variables describing the steady-state (equilibrium) operating point and rated values bear the subscripte. With notationαe

0 ¼ 1 αe, in the equilibrium point it holds that

Ee v0e α⁰e RLiLe ¼ 0 iLe α⁰_ev0e

¼ 0 v0e¼ vCe: 8>

(4.57)

In order to obtain the small-signal model around the considered operating point, one must differentiate the model (4.56):

v_0e⋅ α L R_L

i_Le⋅ α

C R_C

R_e v₀

v_C

i_S

α^′e ⋅ v0

α^′e :1 α^′e . iL

i_L

∼

∼ ∼

∼

Fig. 4.33 Small-signal model of the nonideal boost circuit: equivalent diagram

L _eiL ¼ Eeþ eE vð 0eþ ev0Þ 1 αð e eαÞ RL iLeþ eiL

Using relations (4.57) in the system of (4.58) and neglecting small variations, one obtains the small-signal model of the considered boost power stage:

L _eiL ¼ eE þ v0e eα α⁰e ev0 RL eiL

where the variation of the load current due to the load variation has been denoted by eiS ¼ v0e=R²e eR . The associated equivalent diagram is given in Fig.4.33.

This diagram shows the influence in variations of all the exogenous variables over system outputve0: output results from superposition of all input variables. In order to extract a certain transfer function corresponding to one of these influence channels, one must nullify all other input variations.

(b) The duty-ratio-to-output-voltage transfer function is obtained by putting eE

¼ 0 and eiS ¼ 0 in Eq. (4.59) or in Fig.4.33. The result is presented in Fig.4.34a.

Further, the inductor and the voltage source in Fig.4.34amay be pushed through the transformer; the circuit in Fig.4.34bresults, where the new inductor current has

Fig. 4.34 (a) Small-signal model of nonideal boost circuit: duty-ratio-to-output-voltage influence; (b) push of inductor and voltage source through transformer

92 4 Classical Averaged Model

been denoted by ei . Using Kirchhoff’s laws, one solves this circuit by expressing output voltage variationsve0 as a function of the duty ratio variationseα :

Combining Eq. (4.60) one finds that

Fig. 4.35 (a) Small-signal model of the nonideal boost circuit: load-current-to-output-voltage influence; (b) push of inductor through transformer

Similarly, the output-current-to-output-voltage transfer function may be obtained by putting eE ¼ 0 and eα ¼ 0 in Eq. (4.59) or in Fig.4.33. The result is shown in Fig.4.35a. Further, the inductor in Fig.4.35amay be pushed through the transformer, resulting in the circuit in Fig.4.35b. One aims at expressing the output voltage variations ve0 as a function of output current variations eiS and output impedance:

v0 ¼ eiS Rejj RCþ 1 sC

jj s L α⁰e²

þRL

α⁰e²

: (4.63)

Simple algebra gives the required transfer function,Hv0iSð Þ ¼ es ^VeI_S⁰ð Þ^{ð Þ}s^s:

Hv₀i_Sð Þ ¼s ReðCRCsþ 1Þ Rð Lþ LsÞ CL Rð eþ RCÞs²þ α ⁰e²ReRCCþ CRLðReþ RCÞ þ L

sþ α⁰e²Reþ RL

: (4.64) Computation of the input-voltage-to-output-voltage transfer function results similarly and it is left to the reader. Solutions to the questions proposed in (c), (d) and (e) are also left to the reader, as are the solutions to the following problems.

Problem 4.4. Noninverting Buck-boost Converter

The circuit in Fig. 4.36 has a switching network composed of four switches (two transistors and two diodes). As the transistors are operated synchronously v_C D₁

C R

H₁

H₂ D₂ i_L

+ Fig. 4.36 Noninverting

buck-boost power stage

v_C D₁

C R

H₁

H₂ D₂

i_L

+ Fig. 4.37 Watkins–Johnson

power stage

94 4 Classical Averaged Model

with the same binary switching function,u ∈ {0,1}, this switching network leads to two circuit configurations.

By takingiLandvC(see Fig.4.30) as state variables (a) obtain the switched model (bilinear form);

(b) obtain the averaged model and the corresponding equivalent diagram;

(d) using the perturb-and-linearize method, deduce the small-signal state-space model and draw the associated equivalent diagram (E is considered constant);

(e) using the previously obtained small-signal model, get the expression of the transfer function representing the influence from the duty ratio (control input) and from the input voltageE (disturbance input) to the output voltage;

(f) analyze the system at point (d) for nonminimum-phase behavior and how its poles and zeroes migrate as the load resistor varies.

Problem 4.5. Watkins–Johnson Converter

Given the circuit in Fig.4.37, answer the same requirements as in Problem 4.4.

In addition, it is required to simulate numerically (for example, using MATLAB^®) the switched and the averaged models in the following case: input voltageE ¼ 5 V, inductance L ¼ 5 mH, output capacitor C ¼ 100 μF and output resistorR¼ 10 Ω. Compare the system behavior for duty ratios larger than 0.5 with the behavior for duty ratios smaller than 0.5. Draw the Bode diagram of the transfer from the duty ratio to the output voltage forαe¼ 0.6 and αe¼ 0.4.

Problem 4.6. Quadratic Buck Converter

Given the circuit in Fig.4.38, answer the same requirements as in Problem 4.4.

References

Ben-Yakoov S (1993) Average simulation of PWM converters by direct implementation of behavioral relationships. In: Proceedings of the eighth annual Applied Power Electronics Conference and Exposition – APEC 1993. San Diego, California, USA, pp 510–516 Brown AR, Middlebrook RD (1981) Sampled-data modeling of switching regulators.

In: Proceedings of the IEEE Power Electronics Specialists Conference – PESC 8. Boulder, Colorado, USA, pp 349–369

H D₁

D₂

D₃

L₁ L₂

i_L1

C₁

C₂ R

E v_C2

i_L2

+ v_C1+

u Fig. 4.38 Quadratic buck

power stage

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

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

Kazimierczuk MK, Wang S (1992) Frequency-domain analysis of series resonant converter for continuous conduction mode. IEEE Trans Power Electron 7(2):270–279

Krein PT, Bass RM (1990) Geometric formulation and classification methods for power electronic systems. In: Proceedings of the IEEE power electronics specialists conference. San Antonio, Texas, USA, pp 499–405

Krein PT, Bentsman J, Bass RM, Lesieutre B (1990) On the use of averaging for the analysis of power electronic systems. IEEE Trans Power Electron 5(2):182–190

Lehman B, Bass RM (1996) Extension of averaging theory for power electronic systems.

IEEE Trans Power Electron 11(4):542–553

Maksimovic´ D, Zane R (2007) Small-signal discrete-time modelling of digitally controlled PWM converters. IEEE Trans Power Electron 22(6):2552–2556

Maksimovic´ D, Stankovic´ AM, Thottuvelil VJ, Verghese GC (2001) Modeling and simulation of power electronic converters. Proc IEEE 89(6):898–912

Middlebrook RD (1988) Small-signal modelling of pulse-width modulated switched-mode power converters. Proc IEEE 76(4):343–354

Middlebrook RD, C´ uk S (1976) A general unified approach to modelling switching converter power stages. In: Proceedings of the IEEE power electronic specialists conference. Cleveland, Ohio, USA, pp 18–34

Perard J, Toutain E, Nougaret M (1979) Modelling of energy converters by an equivalent circuit (in French: Mode´lisation des convertisseurs d’e´nergie par un sche´ma equivalent). L’Onde E´ lectrique 59(12)

Rim CT, Joung GB, Cho GH (1988) A state-space modelling of non-ideal DC-DC converters.

In: Proceedings of the IEEE Power Electronics Specialists Conference Averaged Model PESC 1988. Kyoto, Japan, pp 943–950

Rodriguez FD, Chen JE (1991) A refined nonlinear averaged model for constant frequency current mode controlled PWM converters. IEEE Trans Power Electron 6(4):656–664

Sanders SR, Verghese GC (1991) Synthesis of averaged circuit models for switched power converters. IEEE Trans Circuit Syst 38(8):905–915

Sanders SR, Noworolski JM, Liu XZ, Verghese GC (1990) Generalized averaging method for power conversion circuits. In: Proceedings of the IEEE power electronics specialists confer-ence. San Antonio, Texas, USA, pp 333–340

Sira-Ramirez H, Silva-Ortigoza R (2006) Control design techniques in power electronics devices.

Springer, London

Sun J, Grotstollen H (1992) Averaged modelling of switching power converters: reformulation and theoretical basis. In: Proceedings of the IEEE/PESC 1992 Power Electronics Specialists Conference. Toledo, Spain, pp 1165–1172

Tymerski R, Li D (1993) State-space models for current programmed pulsewidth-modulated converters. IEEE Trans Power Electron 8(3):271–278

Verghese GC, Bruzos CA, Mahabir KN (1989) Averaged and sampled-data models for current

In document Eu legislative framework on maternity leave: adoption of a multi-social risk approach? (página 56-61)