Part I Modeling of Power Electronic Converters
3.2 Modeling Methodology
3.2.2 General Algorithm
Some a priori knowledge about converter operation is indispensable in deriving the switched model. A preliminary analysis is necessary, aimed at one or more of the following:
• checking different waveforms of the studied converter;
• providing the sequence of configurations taken during a switching time interval;
• giving the set of equations that mathematically describe each configuration (an example of how this can be done automatically can be found in Merdassi et al.2010).
Nowadays, commercial computer-aided design software products are available to assist the user in performing the above actions.
The manner of obtaining a switched model is not unique. Irrespective of the method employed, the following procedure is general and can be applied in most of applications.
Remark. In the case of only two circuit configurations, the two respective validation functions are complementary, i.e.,h1¼ 1 h2. Hence, one can reasonably take u¼ h1∈ {0;1} in the case of converters and u ¼ 2h1 1 ∈ {1; 1} in the case of DC-AC converters. In cases with more than two configurations there is no general rule that applies.
For cases where the number of possible configurations is quite large, directly applying Algorithm 3.1 requires significant time and effort. An alternative proce-dure may be based on identifying so-called switched (or else intermediary) variables and reaching the bilinear form more easily. The steps are detailed in Algorithm 3.2.
As an example, in the case of DC-DC converters the switches are implemented by means of transistors and diodes, connected in the so-calledswitch network (Erikson and Maksimovic´2001). Depending on their position in the converter circuit, both of these elements can switch both current and voltage, as shown in Fig.3.2.
3.2 Modeling Methodology 31
Algorithm 3.1.
Obtaining the switched model of a given power electronic converter based upon analysis of all possible configurations
#1. Either collect the waveforms and corresponding circuit configurations or simply study the waveforms provided by an adequate software package.
#2. Choose the state variables, by either directly taking capacitor voltages and inductor currents or by taking an appropriate combination of these variables.
#3. (a) Write the expressions of the derivatives of the above state variables for each configuration.
(b) Identify the transition conditions between configurations and write the model in a compact manner by making the control variables (switching functions) appear explicitly.
#4. Make an equivalent topological (or exact) diagram of the studied con-verter where the different coupling terms appear. This step is optional.
Algorithm 3.2.
Deducing the switched model of a given power electronic converter based upon identifying the switched variables
#1. Choose the state variables by either directly taking capacitor voltages and inductor currents or by taking an appropriate combination of these variables.
#2. Following the initial analysis, identify the switched variables; e.g., these can be the transistor voltages and the diode currents, or vice versa. Write their mathematical expressions depending on states of switches (on/off) and on state variables.
#3. Write Kirchhoff’s voltage laws for expressing the derivatives of inductor currents and Kirchhoff’s current laws for obtaining the derivatives of capacitor voltages.
#4. Introduce the switching functions u and write the switched variables as functions of u.
#5. Replace the switched variables in the state-space equations developed at step #3. Obtain the bilinear form.
3.2.3 Examples
Example 1. Power electronic converter for induction-based heating
Let us consider the power electronic converter given in Fig.3.3. It is composed of a diode rectifier, an LR filter with inductanceLfand resistancer and a current source
inverter supplying a resonant circuit composed of a capacitorC and an induction-based heater of inductanceL and resistance R.
The current source inverter operates in full wave; it is conceived such that switches T1and switches T2operate complementarily (turning on T1determines T2being turned off and inversely).
In order to obtain the exact (switched) model of this structure by using Algo-rithm 3.1 presented in Sect.3.2.2, one can proceed as follows.
Given sufficient initial information about the process, one can begin directly with the second step of Algorithm 3.1.
vH = v1 = v2(1−u)
iH = i1 = i2u
iD = i2 = i1(1–u)
vD = v2 = v1u i1 D
H
H
D u u
u u
i2
v1 v2
v1 v2
i1 i2
a
b
Fig. 3.2 Example of switch network showing switched variables: (a) boost converter case, (b) buck converter case
T1
T2
T1 T2 V
Lf
r if
Ud0 ve
+ L
R C vC
iS iL
Fig. 3.3 Power electronic converter for induction-based heating
3.2 Modeling Methodology 33
#2. This is a third-order dynamic system having as state variables filtering inductance current, capacitor voltage and load inductance current. Thus, the state vector is
x¼ i½ f vC iLT: Voltage Ud0 is given by Ud0¼ 3 ffiffiffi
p6
=π V , where V is the root mean square (RMS) value of the three-phase grid voltages (Mohan et al.2002).
Conservation laws are expressed by the following equations:
_if ¼ 1
#3. The intermediary variablesveandiSmust be expressed as being dependent on state variables.
The two validation functionsh1andh2, corresponding respectively to the two configurations taken by the current source inverter, are defined as follows:
h1¼ 1 if switchesT1are closed
0 otherwise, h2 ¼ 1 if switchesT2are closed 0 otherwise:
By noting thath1andh2are complementary, it follows that ve¼ vC and iS¼ if ifh1¼ 1 and h2¼ 0 ve¼ vC and iS¼ if ifh2¼ 1 and h1¼ 0:
System (3.5) is written under the equivalent form given by (3.2), namely:
_x ¼ Að 1xþ B1EÞ h1þ Að 2xþ B2EÞ h2, (3.6)
Representation (3.6) must be condensed into the bilinear form by using a single switching function, which is sufficient in order to represent two configurations (p ¼ 1 is the largest integer satisfying 2p N ¼ 2).
If the switching functionu is chosen such that u¼ 2h1 1, then a unique representation yields
_x ¼ A x þ B x u þ d, (3.8)
where functionu takes values within the discrete set {1; 1} and
A¼A1þ A2
2 , B¼A1 A2
2 , d¼ B1Ud0:
Once the above expressions have been established, corresponding matrices are given by Eq. (3.9).
One can note that the topological model obtained as Eqs. (3.8) and (3.9) is bilinear, i.e., it contains products of form x · u; this is the case of the quasi-totality of power electronic converters.
The equivalent circuit translating exactly the switched model (3.8) is given in Fig.3.4, where several features can be identified:
• the obtained circuit is simplified in relation to the original converter;
• the obtained circuit reveals the coupling terms by means of so-calledcoupled sources, which are represented by diamonds.
A¼
For experienced users, the equivalent circuit in Fig.3.4can be directly obtained by making use of certain rules of transformation that allow transposition of switch operation of the original circuit (Fig.3.3) into operation of coupled sources.
R Lf L
r if iL
Ud0 u . vC u . if vC
C Fig. 3.4 Equivalent circuit
of the switched model of converter presented in Fig.3.2
3.2 Modeling Methodology 35
Example 2. Boost DC-DC converter
The example of an ideal boost converter (Fig.3.5) was chosen to illustrate how the switched model may be found in the case of continuous-conduction mode (ccm).
A discussion concerning the case of discontinuous conduction is also presented.
Case of continuous-conduction mode
As stated earlier, one can obtain the switched model of any power electronic converter and its bilinear form in two ways:
(a) by listing all its possible configurations and finding a general structure which leads to the bilinear form, according to Algorithm 3.1;
(b) by applying the method of emphasizing the variables exhibiting switched-time evolution, as described in Algorithm 3.2.
Each of these methods will be detailed next in the case of the boost converter operating in continuous-conduction mode.
(a) Using the list of all possible configurations
According to preliminary analysis, the boost converter operating in continuous conduction can take two configurations, as shown in Fig.3.6: cases (a) and (b) cor-respond to switchH being turned on (h1¼ 1) and turned off (h2¼ 1) respectively.
As this is a DC-DC case having two configurations, switching functionu can be taken such thatu¼ h1¼ 1 h2, as discussed in Sect.3.2.2.
The state variables are the inductor currentiLand the capacitor voltagevC. The state-space equations corresponding to the two circuit configurations are listed below.
u¼ 1 : _iL ¼ E=L
_vC ¼ vC= RCð Þ, u¼ 0 : _iL¼ E=L vC=L _vC¼ iL=C vC= RCð Þ:
(3.10) iL
L D
C R
E vC
u
H vH
iD Fig. 3.5 Boost power stage
iL L
E vC E vC
C R C R
L iL
u = 1 u = 0
a b
Fig. 3.6 The two possible configurations of circuit in Fig.3.4
Equations from (3.10) can be condensed into a single form by employing Eq. (3.2), in which validation functions h1and h2are expressed using switching functionu, namely h1¼ u and h2¼ 1 u:
from which one can derive
_iL¼ 1 uð ÞvC=L þ E=L which can be used directly for simulation purposes. Equation (3.11) allows one to obtain the bilinear form:
_iL
(b) Identifying the switched variables
The first step – choosing the state variables – is the same as in the previous procedure; iL andvC are the state variables. In the case presented, the switched variables are the transistor voltagevHand the diode currentiDwhich can be written as functions of state variables (see Fig.3.5):
vH¼ 0 ifH is turned on
vC ifH is turned off and iD¼ 0 ifH is turned on iL ifH is turned off:
The equations defining the circuit behavior are obtained by applying Kirchhoff’s voltage law for expressing diL/dt and Kirchhoff’s current law for dvC/dt (see Fig.3.6). That is,
L _iL¼ E vH
C _vC¼ iD vC=R:
(3.13) Next, the switched variables must be expressed as depending on a suitably defined switching function. If the switching function is introduced as
u¼ 1 ifH is turned on 0 ifH is turned off,
3.2 Modeling Methodology 37
then the switched variables are written as:
vH¼ vCð1 uÞ iD¼ iLð1 uÞ:
(3.14) By substituting Eq. (3.14) into Eq. (3.13), one obtains
_iL¼ 1 uð Þ vC=L þ E=L _vC ¼ 1 uð Þ iL=C vC=
RC ,
which is identical to Eq. (3.11) and from which one can derive the bilinear form expressed by (3.12).
Irrespective of the method used in deducing the bilinear form, the exact equiva-lent circuit of the boost converter in ccm results from Eq. (3.11) and it is presented in Fig.3.7.
The two dependent sources define a coupling between the circuit’s input and output. The behavior is like that of an ideal DC transformer with the variable ratio controlled by an external action.
The system represented by Eq. (3.12) can be simulated using dedicated software.
Figure3.8shows the behavior of the ideal boost power stage at step-variation of duty ratio as simulated in Simulink®. To this end, the system is fed by a switched functionu(t), obtained by PWM modulation of the duty ratio. Note the ripple in the state variables due to the switching of the circuit Eq. (3.10) with the modulation
vC
(1−u)iL
R
iL L (1−u)vC
E C
Fig. 3.7 Exact equivalent diagram of the boost power stage operating in ccm
0 0.02 0.04 0.06 0.08 0.1 0
5 10 15
0.08 0.082 0.084 0.086 0.088 2
4 6 8 10 12 14
t [s] t [s]
vC[V]
vC[V]
iL[A]
iL× 2[A]
Zoom
a b
Fig. 3.8 Dynamic behavior of the switched model of the ideal boost converter
frequency. Also, the time response is variable with the operating point (vC,iL), indicating that the system is nonlinear. For the same reason, the ripple has variable magnitude, depending on the operating point. Non-minimum-phase behavior can also be observed.
Case of discontinuous-conduction mode
Now consider that the boost power stage operates in discontinuous-conduction mode (dcm). This means that the switching period is large enough with respect to inductance to allow the inductor current to become zero (see Fig.3.9a). This occurs when the load current value determines a mean value for the inductor current, smaller than its ripple (Vorpe´rian1990; Sun et al.1998).
An analysis of the circuit for this case shows that within a switching period there are three time subintervals, each corresponding to a circuit configuration. Intervals T1andT2are associated with the circuit topologies from Figs.3.6a, b, respectively.
In the third subinterval, T3, the converter remains in the configuration shown in Fig.3.8b, but currentiLis interrupted. State equations describing the three circuit configurations are given by Eq. (3.15).
T1 T2 T3
_iL ¼ E=L _vC¼ vC= RCð Þ,
_iL¼ E=L vC=L _vC¼ iL=C vC= RCð Þ,
_iL¼ iL¼ 0 _vC ¼ vC= RCð Þ:
(3.15)
As the number of switching functionsp must obey the relation 2p N ¼ 3 (see Sect.3.1.2), a second switching function must be introduced in order to obtain a unified switched model.
To conclude, the first switching function is an external independent action, whereas the second switching function depends on an internal state, iL. They can be defined as follows:
u1¼ 1 ifH is turned on
0 ifH is turned off and u2¼1þ sgn ið ÞL
2 ,
where
sgnð Þ ¼iL 1 ifiL> 0 ccmð Þ
1 if iL 0 dcmð Þ:
iL
t T1 T2 T3
αT T 0
E vC
C R
L iL
a b
Fig. 3.9 Boost converter operating in dcm:
(a) inductor current evolution; (b) circuit configuration corresponding to dcm (iL¼ 0, time subintervalT3)
3.2 Modeling Methodology 39
Equations (3.15) can be condensed into a single unified model showing the two switching functions:
_iL¼ u2 E=L 1 uð ð 1Þ vC=LÞ _vC ¼ u2 1 uð 1Þ iL=C vC=
RC :
(3.16) Model (3.16) does not fit the bilinear form (3.3) because here products of switching functions (u1 u2) appear (trilinear form). Moreover, this form cannot be directly used for control purposes.