HYBRID MODELLING OF THE SECOND-ORDER PWM CONVERTERS WITH DCM

ELENA NICULESCU Department of Electronics and Instrumentation

University of CraiovaAl. I. Cuza Street, No. 13, Craiova

ROMANIA

Abstract: - A hybrid technique to derive the characteristic coefficients of the second-order PWM converters with discontinuous conduction mode (DCM) is developed. The starting point is the large-signal averaged model of the three-terminal PWM switch and a new duty-ratio constraint, and the end result is the canonical full-order (ac small signal) model of converter. The ac small-signal model of elementary PWM converters expressed in the terms of the characteristic coefficients represents the canonical full-order (ac small signal) model of converter. All the output characteristic coefficients are complex functions and their singularities appear in the input-to-output voltage and control-to-output voltage transfer functions of the power stage: the second high-frequency pole (buck, boost and buck-boost converters) and the right-half-plane (RHP) zero (boost and buck-boost converters). Derivation of the characteristic coefficients of converters allows using the full-order canonical model (ac small signal) in any form: mathematical or with equivalent circuit or functional block diagram. All these three equivalent forms of the full-order model can correctly capture the fast dynamics associated with the inductor current and predict the converter dynamics up to around one third of the switching frequency.

Key-Words: - Averaged PWM switch model, PWM converters with DCM, characteristic coefficients

1 IntroductionProper mathematical models of the PWM converters with discontinuous conduction mode (DCM) operation are essential for the analysis and design of converters in a variety of applications.

DCM operation can occurs in dc/dc converters at light load or it can be preferred by designers in order to avoid the reverse recovery problem of the diode. Also some designers consider the DCM operation as o possible solution to the right-half-plane (RHP) zero problem that encounters in boost and buck-boost topologies [1], [2].

The major results concerning the modelling of the PWM converters with DCM have been presented in [3]. There the authors made a re examination of all existing averaged models and proposed new full-order averaged models in both mathematical and circuit forms. These new models can correctly predict the small-signal responses up to one third of the switching frequency and are more accurate than all previous performed models. The duty ratio constraint that defines the parameter is considered to be the key of an accurate prediction of high-frequency behaviour of PWM converter. This parameter gives the conducting time of the diode (

) that is the decay interval of the inductor current [2].

Despite of their limitations concerning the high-frequency phenomena characterisation, the averaging techniques are simple and more likely to give tractable mathematical and circuit models. The derivation of an averaged model of the PWM converters with DCM involves two major steps: the establishing the constraint concerning the parameter

and the averaging process of the variables. The major differences among the various models of a PWM converter with DCM appear in the frequency responses of converter. From this point of view, the dynamic models of converters can be either reduced-order or full-order models. Let be the second-order PWM converter case (buck, boost, buck-boost converters). In the reduced-order models, the input-to-output voltage and control-to-output voltage transfer functions have a single low-frequency pole in the left-half-plane (LHP). In contrast with these, the singularities of the transfer functions from the full-order models are two poles in the LHP, one at low-frequency and other at high-frequency, and one zero in the RHP (boost and buck-boost converters).

The following general notations and symbols are used in the paper: the small-signal variations of the quantities are written with lowercase letters with a tilde above them, while their Laplace transformations will be written as explicit functions of the complex frequency . The average values of currents and voltages are written with lowercase letters with a line above them. The symbol with circle is used for an independent source (of current or voltage) and that with rhombus for a controlled source. Other notations: - inductor current; - peak inductor current; - absorbed current; - injected current; - input (or line) voltage; - capacitor voltage; - output voltage; - load resistance of the stage; - capacity of output capacitor; - continuous voltage conversion ratio (

); - switching frequency; -conduction parameter through inductor (

). The three distinct time intervals that characterise the DCM operation of a second-order converter are highlighted in the Fig. 1 where: is the transistor-on duty ratio, is the diode-on duty ratio and is the transistor and diode-off duty ratio. For constant switching frequency, .

Fig. 1. Inductor current waveform of second-order PWM converter with DCM

The state-space averaging (SSA) method and the injected-absorbed-current (IAC) method are modelling techniques dedicated to switching dc-dc converter representation by means of continuous approximate models [2], [6]. Both original SSA and IAC methods lead to reduced-order models for a PWM converter with DCM. In this operating mode, one considers that the inductor does not carry any information from cycle to cycle, because the inductor current resets to zero in every switching cycle (Fig. 1).

In particularly, the paper shows that the full-order canonical model with characteristic coefficients of switching cell with DCM can be obtained by means of hybrid modelling procedure having as starting point the three-terminal PWM switch cell of Vorperian [4]. Using a new duty-ratio constraint the full-order models of second-order PWM converters with DCM have been derived. The paper is organised as follows. In Section 2, the large-signal averaged models of second-order PWM converters for DCM operation are reported. In Section 3, after a short review of characteristic coefficient derivation, the full-order dynamic models described by characteristic coefficient are developed. Finally, a comparison between the models performed and the full-order models of PWM converters developed by means of the modified state-space averaging method is made.

2 Large-signal averaged models for DCM operation

In order to perform large-signal averaged models for PWM converters with DCM, some efforts have been made on the modelling direction of the PWM switching cell. For instance, in the IAC method, the large-signal behaviour of the PWM switching cell is described by help of the average absorbed and injected currents: and (Fig. 2). The relations

(1)

(2)

define these functions that are derived from the current waveforms [6]. For the second-order PWM converters, these functions are given in the Table 1.

Fig. 2. Representation of PWM switching cell in IAC method

Table 1. Average injected and absorbed currents

ipk

Tsd1Ts d2Ts d3Ts0

C

RPWM

switchingcell

OvvI

Ii Ji

d1

Buck

Boost

Buck-

boost

This modelling method uses a volt-second balance relation of the inductor to define the duty-ratio constraint. After replacing the parameter with its expression given in the Table 2, the functions of the absorbed and injected currents become:

(3)

. (4)

Table 2. The duty-ratio constraint

Buck

Boost

Buck-Boost

An averaged switch model that can replace the PWM switch cell with DCM operation has been established. The switch PWM cell includes the switch, the diode and the inductor as shown in Fig. 3a, and that is common in different topologies [3], [4]. The averaged model for DCM operation of this three-terminal PWM switch cell is shown in Fig. 3b where:

(5)

. (6)

In the relation (5), the sign plus corresponds to buck and buck-boost converters and the sign minus corresponds to boost converter.

Taking into account that the actual average value of the inductor current for DCM operation is by the form [3]

, (7)

the duty-ratio constraint has the general expression

. (8)

a.

b.

Fig. 3. a. Three-terminal PWM switch cell; b. averaged model of the PWM switch cell

Replacing the three-terminal PWM switch cell by its averaged model into the elementary PWM converter diagrams, the equivalent circuits from Fig. 4 are obtained. This equivalent circuits together expressions of current and voltage controlled sources from the Table 3, and with the duty-ratio constraint given in (8), represent the large-signal averaged models of the second-order PWM converters with DCM.

Table 3. Voltages of averaged model of PWM switch cell

ip

ia iL

a c z

p

L

SD

a Li

c

p

L

+-

Vi z

Vv

Buck Boost Buck-Boost

In order to derive the characteristic coefficients from the large-signal averaged models of converters, the state-output averaged relations are written too, considering three output quantities: the absorbed and injected currents, and the output voltage. From the equivalent circuits given in the Fig. 4, the relations summed in the Table 4 are obtained.

Fig. 4. Large-signal averaged models of second-order PWM converters: a. buck; b. boost; c. buck-boost Table 4. State-output averaged relations

Buck Boost Buck-Boost

The Kirchhoff’s Laws applied on the three equivalent circuits from the Fig. 4 together the state-output relations and the duty-ratio constraint (8) lead to the following large-signal models of the second-order PWM converters:

1) Buck converter

(9)

(10)

(11)

(12)

(13)

2) Boost converter

(14)

(15)

(16)

(17)

(18)

3) Buck-Boost converter

(19)

C

+- RvI

a

c pz LLi

Vi

+ - Vv

cv+-

b.

C

-+ RvI

+-

a c p

z

LLi

+ -

cv

Vi Vv

c.

a.

a Li

C

+- RvCvI

+-

c

p

L

+-

Vi z

Vv

(20)

(21)

(22)

. (23)

Using the standard linearization techniques, a steady-state (dc) model and a small-signal model can be derived from the equations (9)-(23).

3 AC small-signal models3.1. Review of characteristic coefficient derivationModelling switching dc-dc PWM converters with the help of the injected-absorbed-current (IAC) method is quite simple. There, the obtaining small-signal model described through characteristic coefficients can be summed up in three steps [5], [6].

1. The absorbed and injected currents are averaged over one cycle and putted in the forms (1) and (2). The parameter is eliminated by help of the duty-ratio constraint given in the Table 2, and the functions of the absorbed and injected currents become (3) and (4).

2. The linear model of the cell is found by evaluating small increments of (3) and (4):

(24)

(25)

In the small-signal assumption, the small increments are considered to be equal to the ac components of quantities to which the Laplace transform can be applied: ; ; ; ;

. The partial derivatives are evaluated for a given operating point.

3. Take the Laplace transform of (24) and (25), and it obtains

.

(26)

Relation (26) represents the canonical dynamic (ac small-signal) model of any converter. Its form rest unchanged regardless converter topology, controlled quantity and control type. Moreover, this mathematical linear dynamic model can be directly transposed into an equivalent circuit with fixed topology or into functional block diagram [6], [7]. There, six characteristic coefficients are highlighted, namely: three input ( , , ) and, respectively, three output averaged characteristic coefficients (, , ). All these coefficients are real for second-order PWM converters with DCM [6].

Finally, from Fig. 2, it can be shown that is related to . After replacing the injected current with its (26) relation, the two transfer functions of converter can be derived, namely [8]:

- line-to-output voltage transfer function

; (27)

- control-to-output voltage transfer function

. (28)

The dynamic model of second-order PWM converters with DCM, given by the previous transfer functions is a reduced-order model, like as that obtained through original SSA method. These transfer functions describe a first-order system, because all three output characteristic coefficients are real even in the non-ideal converter case.

3.2. Derivation of characteristic coefficients for full-order model

After perturbation and linearization steps applied on the large-signal average models, given by equations (9)-(23), the steady-state (dc) models and ac small-signal models of elementary PWM converters are obtained. The steady-state (dc) models have the same forms as those shown in [3] and they not reply here.

The dynamic model (ac small signal) described by the linearized state equations has the general form:

, (29)

where the matrices A and B are:

, .

The components of the matrices are function on the switching frequency, duty ratio, line voltage, load resistance, inductance, capacity, dc voltage conversion ratio or zero as follows.

1)Buck converter:

; ; ;

; ; ; ; .

2)Boost converter:

; ; ;

; ; ;

; .

3)Buck-Boost converter:

; ; ;

; ; ;

; .

Apply the Laplace transform to state equations and state-output relations. After solving the first equation set, the state-output relations become:

.

(30)

The following results have been found for the elementary PWM converters with DCM:

1) Buck converter. All the input characteristic coefficients are real and the output characteristic coefficients are complex:

; ;

; ; ;

; ; ;

.

2) Boost converter. All the characteristic coefficients are complex:

; ; ;

; ;

; ;

; ;

; ; ;

; .

3)Buck-Boost converter. The input characteristic coefficients are real or null, and the output characteristic coefficients are complex:

; ; ;

; ;

; ;

; ; ;

; .

Using the output characteristic coefficients, the input-to-output voltage and control-to-output voltage transfer functions are by the following forms:

1) For buck converter

; (31)

. (32)

2) For boost and buck-boost converters

; (33)

. (34)

The input-to-output voltage and control-to-output voltage transfer functions can be directly obtained from the dynamic model of PWM converter given by the equation (29). Following this way, the transfer functions are expressed in the terms of the components of matrices A and B, having the general forms:

; (35)

. (36)

The model with characteristic coefficients was chosen because its mathematical form or the topology of synthesized equivalent circuit is invariable. This feature greatly simplifies and accelerates comparative investigations of different configurations ranging from single cells to closed-loop switching regulators with input filters and cascaded connections of regulators [6], [10]. Once found the expressions of these coefficients, all small-signal properties of converter can be computed: input impedance, output impedance, input-to-output voltage and control-to-output voltage transfer functions of open or closed-loop regulator.

4 Model verificationThe full-order models of the second-order PWM converters performed in the terms of the characteristic coefficients have been compared with the new full-order averaged models presented in [3]. These later models are derived through modified SSA method with the new duty-ratio constraint. The comparison is made at large and small-signal level. So, the averaged large-signal models obtained in this paper (Section 2) are identical with these given by the relation (26) (31) from [3]. The simulation results compared with the measured frequency response from [3] shows that the full-order dynamic models derived by means of the presented procedure and those taken as reference term [3] dovetail. Therefore, the full-order models of second-order PWM converters with DCM described by help of characteristic coefficients derived by this way is accurate up to around one third of the switching frequency.

5 ConclusionA hybrid technique to model the second-order PWM converters with discontinuous conduction mode (DCM) is developed. The starting point is the large-signal averaged model of the three-terminal PWM switch and a new duty-ratio constraint, and the end result is the canonical full-order (ac small signal) model of converter described by means of the characteristic coefficients.

The ac small-signal model of elementary PWM converters expressed in the terms of the characteristic coefficients represents the canonical full-order (ac small signal) model of converter. All the output characteristic coefficients are complex functions and their singularities appear in the input-to-output voltage and control-to-output voltage transfer functions of the power stage: the second high-frequency pole and the right-half-plane (RHP) zero (boost, buck-boost). Besides the singularities of the characteristic coefficients, which are complex functions, appear in the transfer function of the PWM converter. As it can be seen from the expressions (31) (34) of the transfer functions and those of the characteristic coefficients, the high-frequency pole of transfer functions is very closed of the pole , and the RHP zeroes from and

are recovered as RHP zeroes in transfer functions.

Using the old duty-ratio constraints given in the Table 2, the reduced-order models result that do not include the second high-frequency pole or the RHP zero.

The full-order canonical model (ac small signal) of second-order PWM converters with DCM, described by the characteristic coefficients, can correctly capture the fast dynamics associated with the inductor current and predict the converter dynamics up to around one third of the switching frequency. This model can be used in any form: mathematical or with equivalent circuit or functional block diagram.

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