Part I Modeling of Power Electronic Converters
6.3 General Methodology
6.3.2 Example with Discontinuous-Conduction Mode:
0 ¼ Rf þ Re
Lf
if
0þ 1 Lf
Ud0, (6.14)
which leads to the equivalent circuit presented in Fig.6.4.
Remarks. The order of the resulting model has been reduced by two. The final model is first-order, thus easier to use. Since the control input of the converter is the switching frequency ω of the current-source inverter, model (6.14) is highly nonlinear: it depends onRe, so finally on a nonlinear function ofω (see (6.13)).
The dynamics of the AC variables are totally ignored, and sometimes this could have non-negligible consequences.
6.3.2 Example with Discontinuous-Conduction Mode:
Buck-Boost Converter
This example deals with a buck-boost converter (Fig. 6.5) operating in discontinuous-conduction mode (dcm). In the literature it has been shown that the dcm significantly affects converter behavior (C´ uk and Middlebrook 1977;
Vorpe´rian1990). Circuit-oriented analysis on DC-DC converters has proved that in general the poles of its second-order dynamic are not complex-conjugated anymore (as in continuous conduction), but placed on the real axis (Maksimovic´
and C´ uk1991). The one corresponding to inductor energy accumulation is located in high frequency close to the switching frequency (Erikson and Maksimovic´
2001). Therefore, the output voltage response will consequently become Lf
Rf Ud0
Re If
ve Fig. 6.4 Reduced-order
averaged model of a current-source inverter used for induction heating
aperiodical – see for reference Fig.4.23b in Chap.4. When focusing on controlling the output voltage, such a system can easily be assumed as first-order (because of neglecting the HF pole). System-oriented analysis of these aspects can be found in Sun et al. (2001).
As inductor current becomes zero periodically in dcm operation, the classical average modeling no longer applies. So, in this example the average (LF) behavior is expressed mathematically by considering that the output capacitor is fed by an equivalent current generator (Chetty1981,1982).
To solve the modeling problem, one follows the step procedure synthesized in Algorithm 6.1, namely:
• compute the average value of current iD through diode D as a function of capacitor voltage, which is equivalent to solving the high-frequency subsystem;
• average the dynamic equation of output voltagevC;
• replace the previously computed value of currentiDin the dynamic equation of the averaged output voltage;
• obtain the classical averaged model of the resulting system (which will be in this case of first order).
The switched model of the circuit from Fig. 6.5 operating in continuous-conduction mode was already given in Eq. (4.34) of Sect. 4.6 (Chap. 4). The operation in dcm can be described by either adding a supplementary equation – expressing the constraint of current zeroing – or by modifying the dynamic equation of the inductor current, as shown below:
_iL ¼1
LðEuþ vCð1 uÞ riLÞ 1þ sgn ið ÞL
2 _vC ¼1
C iLð1 uÞ vC
R 0
@
1 A, 8>
>>
>>
<
>>
>>
>:
(6.15)
where
sgnð Þ ¼iL 1 ifiL> 0
1 if iL 0
E vC
u H
iL L
D
C R
+ iD Fig. 6.5 Electrical circuit
of a buck-boost converter
6.3 General Methodology 157
and u is the T-periodical switching function taking values in the set {0;1} and havingα as duty ratio. The other notations have the usual meaning assigned to circuit element characteristics. Note that the model given in (6.15) is general – it does not adopt any simplifying assumption, except for what is stated in Sect.2.2of Chap.2.
Now, the search for an equivalent low-frequency model justifies the adoption of an approximation: as the switching frequency is large enough with respect to the output voltage evolution, one can assume that the currentiLdecreases linearly in time when the switching functionu is equal to zero. Omitting inductor resistance r, the discontinuous-conduction time evolution of currentiLcan be viewed in Fig.6.6.
Three phases can be identified in this figure:
• from 0 toT1¼ αT, the switch H being turned on, the current increases linearly and slopeE/L> 0, according to the first equation from (6.15) withu ¼ 1, which corresponds to the energy being accumulated in the inductorL;
• during the time interval denoted byT2, after turning off the current switchH, the current decreases linearly and slopehvCi0/L< 0 (note that hvCi0< 0) according to the first equation from (6.15) with u ¼ 0, which corresponds to using the previously accumulated inductor energy to charge the capacitorC;
• finally, during the last time subinterval the current remains zero until the switch H is again turned on.
One can see in Fig.6.5that in dcm the effective conversion ratio is no longer equal toα and depends on the circuit operating point (subinterval T2is a variable ofvC).
The goal is to transform the initial model into a model of an equivalent average current generator supplying a capacitor C and replacing the averaged current through diodeD, as shown in Fig.6.7(Chetty1982).
t iL
slope
E 0
slope vC
T1= αT T2
T
iD
iL, iD
0
t u
0
imax
1 0
= α u L L
Fig. 6.6 Time evolution of currentiLin discontinuous conduction (buck-boost case)
The averaged output current – computed by assuming hvCi0 constant and denoted by h iiD 0 st – corresponds to area of the gray filled surface in Fig. 6.6 divided by the time intervalT. Thus, noting that the height of the filled triangle is imax¼ T1 E/L, one obtains
iD
h i0 st¼ E
2LTT1T2: (6.16)
The value ofT2is found from Fig.6.6by noting that in the filled triangle it is true thatimax/T2¼ hvCi0/L; by further replacing imax ¼ T1 E/L one obtains
T2¼ E vC
h i0T1: (6.17)
By replacing the value ofT2as obtained from (6.17) andT1¼ α T into (6.16), it can be seen that the averaged values ofiDandvCare relatedalgebraically:
iD
h i0 st¼ E2Tα2
2L vh iC 0, (6.18)
thus indicating as a matter of fact (ifhvCi0is considered constant on a switching period), thatthe two state variables exhibit the same dynamic. This remark justifies saying that the equivalent low-frequency converter dynamics operating in dcm are of first order instead of second order, as explained next.
Let us consider the second equation from (6.15), emphasizing the diode current iD:
C _vC ¼ iD vC=R:
The latter relation is averaged, the average value of iD being replaced by Eq. (6.18):
Ch i_vC 0¼ E2T
2L vh iC 0α2h ivC 0
R : (6.19)
Equation (6.19) is nonlinear and represents the ROAM of the buck-boost converter operating in discontinuous-conduction mode. It also shows a first-order linear dynamic between the squared duty ratioα2and the output voltage squared hvCi02.
vC 0
C R
0_st
iD Fig. 6.7 Reduced-order
averaged model of a buck-boost converter
6.3 General Methodology 159
Remarks. In Fig.6.8is given the converter output voltage evolution in response to a step of the duty ratio during operation in dcm (for reference see Fig.4.23b in Sect.
4.6of Chap.4). This figure allows a comparison of the ROAM with the switched model output.