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Fusion of Neural Networks, Fuzzy Systems and Genetic Algorithms: Industrial Applications
by Lakhmi C. Jain; N.M. Martin CRC Press, CRC Press LLC ISBN: 0849398045 Pub Date: 11/01/98 |
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For example, a controlled rectifiers voltage independent variable is the input terminals ac mains, and its dependent variable the output terminals dc voltage. On the contrary, the independent current variable is the output terminals dc current, and the dependent variable the input terminals ac line current. Clearly, two transfer functions are defined. One relates input and output voltages, and the other one input and output currents. However, these voltage and current transfer functions are a single converter transfer function. This can be proved for every converter shown in Table II.
Figure 2 Schematic of a three-phase rectifier, including ac mains (Va, Vb, Vc) and a current-source load (Io). Input line currents (Ia, Ib, Ic) and output dc voltage (Vo) are also shown.
Let us consider the system shown in Figure 2, comprising the ac mains, a three-phase rectifier, and a current-source load. If an ideal converter is considered, i.e., with zero losses and no energy storage elements, then the following relation can be established considering the input and output instantaneous power.
Expanding (3) yields the next expression,
Using matrix notation, (4) can be rearranged as
By inspection of (5), the input independent voltages are multiplied by a matrix to obtain the output dependent voltage. Hence, (5) can be rewritten using definition (2) and Table II in the following way:
Where matrix H is defined by Equation (7a and b), and matches the voltage transfer function characteristics.
It should be noticed that matrix H is defined even if the load current io is zero (7a). Therefore, as the line currents ia, ib, and ic approach zero, the load current io approaches zero, thus defining the limit shown in (8).
The current transfer function can be defined according to (2) and Table II as the quotient between the input line currents (dependent variable) and the output current (independent variable). By observing (7a), matrix H has that form, and thus matches the converters current transfer function characteristics. Consequently, matrix H is the converters transfer function, and relates the input and output electric variables as shown in (9).
Figure 3 Electric variables of three-phase rectifier shown in Figure 2. a) Input phase voltages (va, vb, vc), b) rectifier transfer function element Ha (phase a), c) output dc voltage vo and load current io, d) input line current ia.
Figure 3 shows these relations graphically for the system shown in Figure 2, using a diode bridge as a rectifier. Figure 3a) shows the independent input phase voltages of the ac mains (220 Vrms). Figure 3b) shows the phase a transfer function component Ha. It should be noticed that this term is simply the normalized line current (7a). Figure 3c) shows the dependent output voltage vo obtained using (9), and the independent load current io. Finally, Figure 3d) shows the dependent input line current ia obtained using (9).
The DFC is used as a case study in this chapter. Figure 4 shows a simplified converter-load system used for modeling purposes. Using the transfer function [20], and the electrical variables classifications given in Table II, the converters input and output voltages and currents relationships can be written as shown in (10) considering Figure 4.
Figure 4 Schematic of a simplified XDFC drive, comprizing the input voltage source (Vr, Vs, Vt), input capacitive filter, the XDFC converter using ideal switches, and a three-phase delta connected load.
where,
Expanding (10) yields the following two equations.
Elements hij of transfer function H can assume values only in { -1,0,1 } to assure that Kirchoffs voltage law is satisfied in (10).
Matrix H can be written as a function of the converters switches as
This equation is deduced by referring to Figure 1 and Table I. Each element hij in H can be determined by observing how the input phase voltage Vi is reflected to the corresponding output line voltage Vj. For example, element h11 reflects input phase voltage Vr positively to the output line voltage Vab through switch S1, and reflects phase voltage Vr negatively to the output line voltage Vab through switch S2, and does not affect Vab at all with switch S3. Thus, element h11 is defined by (14),
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