2. EU LAW ON MATERNITY LEAVE
2.2. Maternity protection for self-employed workers: the Self- Self-Employment Directive 1986 265Self-Employment Directive 1986265
Re uh i1
þ Im Gh ia 1
Im uh i1
: (5.21)
5.3.4 Switching Functions
The switching functions commonly used exhibit a square-waveform time evolution in the case of symmetrical-switching converters; these functions may depend on state variables or on external inputs. The expressions of their first-order sliding harmonics are useful in developing Eqs. (5.17), (5.18), (5.19), (5.20), and (5.21).
5.3.4.1 Case of Switching Functions Depending on Time
To this class belong the switching functions of forced-commutated inverters, for example.
Let us consider the functionu(t) represented in Fig.5.5, that can be described analytically byu(t)¼ sgn(sin(ωt)).
Using Definition 4.2 particularized to the first-order harmonic gives
h iu 1¼ 2
πj: (5.22)
If this function is phase-shifted in relation to an arbitrarily set origin, its analytical expression is u(t)¼ sgn(sin(ωt + δ)) and its expression in the sense of the first-order harmonic becomes
h iu 1¼ 2
πjejδ: (5.23)
+1
−1
0 π 2π ωt
u(t) Fig. 5.5 Square-wave
switching function
5.3.4.2 Case of Switching Functions Depending on a State Variable
An example of switching function belonging to this class is a diode rectifier. In this case the sign of the AC current shows which branch of the rectifier is in conduction at a given moment. This function is expressed analytically as u(t)¼ sgn(x(t)), wherex is an AC state variable. If variable x is phase-shifted by an angle φ in relation to the phase origin, then the following relation holds:
h iu 1¼ 2
πjejφ: (5.24)
5.3.4.3 Case of Switching Functions Depending on State Variables and Time
A thyristor rectifier is an example belonging to this class because the branch in conduction depends in the meantime on the sign of the AC variable in question and on the moment when the firing order is applied. This dependence may be expressed analytically asu(t)¼ sgn(x(t)) ν(δ), where x is an AC state variable and ν() is a delay function.
Under the same assumption of aφ phase shift of variable x, one obtains
h iu 1¼ 2
πjejðφδÞ: (5.25)
5.3.4.4 Case of Multilevel Switching Functions This may be the case of a three-phase inverter.
For this type of switching function one can always write expressions as a finite sum of elementary square-waveform functions. Thus, for the example, from Fig. 5.6 function u(t) can be put into the form of a three-term sum containing elementary switching functions, which are phase-shifted by 2π/3 each in relation to each other. Therefore, one can write
+2
−2
0 π 2π ωt
u(t) +4
−4 Fig. 5.6 Example of a
multilevel switching function
106 5 Generalized Averaged Model
u tð Þ ¼ 2u1ð Þ ut 2ð Þ ut 3ð Þ:t Then computation ofhui1leads to
h iu 1¼ 2
πj2 e2πj=3 e2πj=3
¼ 6 πj:
5.4 Methodology of Averaging
As in the case of the classical averaged model, building of the generalized averaged model can be performed either graphically, by means of the topological diagram, or analytically, based upon the topological exact model.
5.4.1 Analytical Approach
Without loss of generality, a given power electronic converter can be described by Eq. (3.2) presented in Chap.3:
_x ¼XN
i¼1
Aixþ Bie
ð Þhi:
A more condensed form describes the converter under a general bilinear form, where matrices A and B are constant:
_x ¼ Ax þXp
n¼1
unðBxþ bÞ þ d,
where thep switching functions are emphasized.
If the converter has in the meantime AC stages as well as DC stages, one can apply the Fourier series expansion, comprising the fundamental and the DC-component values of the concerned state variables.
Thus, for the DC components one obtains by averaging d
dth ix 0¼ A xh i0þXp
n¼1
B uh nxi0þ uh nbi0þ dh i0,
whereas for the fundamental component the following holds (first-order sliding averaging):
d
dth ix 1¼ jω xh i1þ A xh i1þXp
n¼1
B uh nxi1þ uh nbi1þ dh i1:
The mathematical development is further pursued with computation of terms hdi0 and hdi1 and with obtaining the expressions of the coupled terms hunxi1, hunbi1,hunxi0andhunbi0.
By collecting the above expressions, model (5.26) expressed as below represents the generalized averaged model of a given power electronic converter:
d
dth ix 0¼ A xh i0þXp
n¼1
B uh nxi0þ uh nbi0þ dh i0
d
dth ix 1¼ jω xh i1þ A xh i1þXp
n¼1
B uh nxi1þ uh nbi1þ dh i1: 8>
>>
><
>>
>>
:
(5.26)
The above computation steps, which may appear complicated at a first sight, can be quickly simplified given that the AC variables have null average values and that the DC variables are sufficiently filtered.
A supplementary step is, however, necessary; in this case it is the separation of complex terms into their respective real and imaginary parts.
5.4.2 Graphical Approach
When one uses the graphical approach to deduce the GAM, the starting point is the equivalent exact diagram. The algorithm to follow is resumed below.
Correct use of steps of the Algorithm 5.1 will lead to the same result as that obtained by employing the analytical approach, that is, to model (5.26).
Algorithm 5.1 Building the GAM by using the graphical approach
#1 Build the equivalent exact diagram.
#2 Identify the DC stage and the AC stage of the converter
#3 Within the DC stage:
• replace state variables and coupled products by their sliding averages;
• leave passive elements as they are.
#4 Within the AC stage:
• replace state variables and coupled products by their respective first-order sliding harmonics;
• compute equivalent impedance of inductances and add it in series with them;
(continued)
108 5 Generalized Averaged Model
Algorithm 5.1 (continued)
• compute equivalent impedance of capacitors and add it in parallel with them.
• leave resistances as they are.
#5 Get the expressions of coupled terms.
#6 Based upon the generalized averaged diagram thus obtained, write the characterizing equations.
General Remarks. Irrespective of the approach used, the Fourier series expansion can go beyond the fundamental. In this way, several equivalent generalized aver-aged diagrams are obtained, which are coupled. Each of these diagrams is valid for any given order harmonic.