Small-Signal Properties of the Modified Boost Topology
ELENA NICULESCU
Department of Electronics and Instrumentation
University of Craiova
Al. I. CuzaStreet, No. 13, Craiova, RO-1100
ROMANIA
Abstract: The paper proposes a full-order dynamic model of the modified PWM boost converter with discontinuous conduction mode (DCM). This canonical model uses the characteristic coefficients to describe the small-signal dynamic behaviour of converter. The derivation of these coefficients is based on the small-signal model of PWM switch with DCM. The dynamic model is then used in the analysis of open-loop small-signal input, output and transfer properties of converter to yields theoretical predictions. The small-signal model of converter with DCM corresponds to a fourth-order system, offering good theoretical predictions of the low-frequency dynamics of converter.
Key-words: Modified PWM boost converter with DCM, Small-signal properties
1 Introduction
The modified boost topology proposed by Wang et al., just like the well-known Cuk and Sepic converters, can achieve continuous input current no matter the operating mode, by coupling the two inductors [1], [2]. Therefore, the input-current harmonics are much less than the pulse-width-modulation (PWM) boost converter and EMI is much better. Consequently, it can achieve small size converter and continuous and ripple-free input current. The modified boost topology contains one extra capacitor and inductor with respect to the conventional boost topology. Compared to Cuk and Sepic converters, the modified boost topology holds several advantages of the conventional boost topology such as lower switch and diode-voltage stresses [2]. In order to short the name of converter, this one will be called Wang PWM converter in this paper. A steady-state analysis of this converter with DCM and the condition for ripple-free input-current are shown in [2], but a small-signal dynamic model or measured and predicted frequency responses of this converter with DCM have not yet been reported.
The Wang PWM converter belongs to the fourth-order PWM converter family, like as Cuk, Sepic and Zeta converters. Concerning the modelling of this PWM converter family with complex behaviour, as it results from the reported papers, most efforts were oriented to model Cuk converter. Various methods such as the state-space averaging method (SSA method), injected-absorbed-current method (IAC method), current-injected equivalent-circuit approachhave been used there [3], [4], [7], [9]. The measured frequency responses of Cuk converter indicated a fourth-order system behavior regardless of operating mode (continuous or discontinuous). Using one of these modelling methods, it obtains a full-order small-signal model of converter with CCM and a reduced-order small-signal model of converter with DCM. An analysis of Cuk PWM converter with DCM based on the PWM switch model in DCM was presentedin [5]. The carried-out transfer functions of converter and the good agreement between the experimental and predicted results confirm the validity of the switch model and the fact that the Cuk converter with DCM is still a fourth-order system [5].
The paper proposes a full-order dynamic model of the ripple-free input-current PWM boost converter with discontinuous conduction mode. This canonical model describes the small-signal dynamic behaviour of converter by means of the characteristic coefficients and corresponds to a fourth-order system. The derivation of these coefficients is based on the small-signal model of PWM switch with DCM. The dynamic model is then used in the analysis of open-loop small-signal input, output and transfer properties of converter to yield theoretical predictions.
We favoured this model because the relationships between the Laplace transforms of small-signal perturbations of terminal currents and voltages, and control quantity remain invariant. So, no matter which converter the model with characteristic coefficients is implemented in, this linear model really being a canonical dynamic model [7].
In Section II, the small-signal equivalent circuit of this converter with DCM obtained by replacement of the transistor and diode with the small-signal PWM switch in DCM is shown. Starting from this small-signal equivalent circuit of converter, the characteristic coefficients are derived in Section III. The open-loop small-signal transfer, input and output properties of modified boost converter with DCM are expressed in the terms of characteristic coefficients in Section IV.
In generally, the well-known conventions and notations are used in this paper. For instance, we denote by the interval during which the transistor is turned on and the diode is off, by the interval during which the diode is turned on and the transistor is off, and by the interval during which both the transistor and diode are off. For a constant switching frequency , . The capitalized letters are used for steady-state quantities and the small letters with tilde above them for small-perturbations of quantities.
2 Small-signal equivalent circuit of Wang converter with DCM
The diagram of the Wang PWM converter with coupled inductors is given in Fig. 1 and the small-signal model of the three-terminal PWM switch in DCM is shown in Fig. 2 [2], [5]. The absorbed current () is the current , and the injected current () is the diode current ().
The small-signal equivalent circuit of Wang converter with DCM is obtained by replacement of the two switches (transistor and diode) with the model of the three-terminal PWM switch in DCM as in Fig. 3. We take the effects of magnetically coupling into account by the effective inductance and :
, ,
where denotes the coupling coefficient and denotes the turns ratio. For simplicity of expressions, the parasitic element effects are supposed to be negligible except the equivalent series resistance (ESR) of the two capacitors denoted as and .
Fig.1. The diagram of the Wang PWM converter
Fig.2. The small-signal model of the three-terminal PWM switch in DCM
After the determination of the characteristic coefficients, the small-signal equivalent circuit of Wang converter with DCM will be replaced with its small-signal canonical dynamic model shown in the Fig. 4. The steady-state (dc) analysis of converter yields the following results [2]:
,,,,, , , .
The small signal parameters of PWM switch with DCM are evaluated at the operating point as follows [5]:
, ,
,,
Fig.3. The small-signal equivalent circuit of Wang converter with DCM
Fig.4. The small-signal canonical dynamic model of Wang converter with DCM
Fig. 5. Equivalent circuit of converter for input impedance
Fig. 6. Equivalent circuit of converter for output impedance
.
3 Derivation of the characteristic coefficients
Let be the admittance of the external series elements connected between the points “” and “”,
.
Let be the admittance of the elements connected in parallel in the output node, namely the output filter capacitor and the load resistance,
.
The following equations directly result from the circuit:
(1)
(2)
(3)
(4)
.(5)
After some algebra, we can express the absorbed current at input and the injected current into the output node solely as function of the perturbations of the input voltage, duty ratio and output voltage:
, (6)
,(7)
where:
,
,
,
,
,,
with
,,,,
,,,,,,.
Also, the coefficients , , are functions on the effective inductances, capacities, ESR, small-signal parameters of PWM switch and operating point of converter.
The input and output characteristic coefficients are directly identified in (6) and (7) as multipliers of , and . These last equations can be written in the form
.(8)
(9)
The characteristic coefficients are polynomial ratios of various degrees in variable as it can be seen in (6) and (7).
4 Small-signal properties of converter
The most general expressions of input, output and transfer properties of converter such as the input and output impedance, and the input-to-output and control-to-output transfer functions can be found by means of the model with characteristic coefficients.
4. 1 Open-loop transfer functions
The relationship between the output voltage and the injected current given by (9) leads to
.(10)
The relationship (10) contains two transfer functions of the open-loop PWM Wang converter, namely:
- the input-to-output voltage transfer function
;
- the control-to-output voltage transfer function
.
Taking into account the expressions of the output characteristic coefficients, we found the following expressions for the transfer functions:
(11)
and
(12)
where
,
.
The coefficients of the characteristic polynomial are functions on the effective inductances, capacities, ESR, small-signal parameters of PWM switch and operating point of converter.
The open-loop transfer functions of the Wang PWM converter with DCM correspond to a fourth-order system, as it indicates by its diagram and (11) and (12). The expressions of the transfer functions show that their zeros are even the zeros of characteristic coefficients and . So, if these coefficients do not contain right-half plane (RHP) zeros, neither the transfer functions do not contain them.
Using the method of time constants as explained in [9], the numerator and denominator polynomial can be factored approximately in an analytical form. The operating conditions and circuit parameters of Wang converter with DCM given in [2] yield the following forms:
, (13)
, (14)
where , .
As it can be seen from (13), the quadratic in the numerator of the input-to-output transfer function corresponds to complex left-half-plane zeros. The factored result (14) indicates that we have three zeros of the control-to-output transfer function in the right-half plane, namely: a complex low-frequency pair well separated from a high-frequency real zero. Both transfer functions contain an extra low-frequency real zero, if . For the operating conditions given in [2], the nulls of the denominator can be found by other means of dealing.
The numerator and denominator of the transfer functions of Wang PWM converter are expressed as analytical functions of the parameter elements and consequently the analysis is design-oriented. They yield insight into how the element values can be chosen such that given specified pole frequencies are obtained [10].
4.2 Open-loop input and output impedance
The input and output small-signal properties of Wang PWM converter with DCM, such as the input and output impedance, can be expressed too in the terms of characteristic coefficients. Their expressions can be easily found starting from the definitions and using the corresponding small-signal equivalent circuits of converter like as in the linear amplifier analysis.
We denote with the internal impedance of line voltage source . The impedance includes both impedance and :
.
So, using the input-impedance definition
and the small-signal equivalent circuits of converter given in Fig. 5, the following formula of this input small-signal property results:
.(15)
The formula (15) of the open-loop input impedance of Wang PWM converter with DCM is the general expression of input impedance of switching cell in the terms of characteristic coefficients. This one is common to all converters described by means of characteristic coefficients, regardless of their topology, operating mode and control type.
After replacement of characteristic coefficients with their expressions, we found
.(16)
Setting s at zero, the input resistance of the coupled-inductor Wang PWM converter with DCM is given as .
Next, we denote with the equivalent impedance
.
Starting from the output-impedance definition
and using the small-signal equivalent circuits of converter given in Fig. 6, we found
.(17)
Once again, the formula (17) that expresses the open-loop output admittance of Wang PWM converter with DCM is the general expression of output admittance of switching cell described in terms of characteristic coefficients. The output resistance of the coupled-inductor Wang PWM converter with DCM is found as .
The magnetically coupling of inductors modifies the operating point of converter and therefore the small-signal parameters of PWM switch model. For the converter with separate inductors, they are obtained by setting at zero. In these conditions, , , and . So, the expressions of the characteristic coefficients and small-signal properties of converter hold their formula and only the polynomial coefficients change the values there.
5 Conclusion
The small-signal full-order canonical model in terms of characteristic coefficients that take the effects of coupling between inductors into account was derived for Wang PWM converter with DCM. The model with characteristic coefficients was chosen because this model yields the most general expressions of input, output and transfer properties of converter such as the input and output impedance, and the input-output and control-output transfer functions. Such model can be directly used with a general-purpose program package such that the overall dynamic properties and stability can be analysed and designed.
The derivation procedure is based on the small signal PWM switch model in DCM. The small-signal characteristics of Wang PWM converter with DCM are obtained like as the small-signal characteristics of linear amplifiers.
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