Influence of converters on the grid

Power-electronics components based on semiconductor switches con- stitute a nonlinear load for the grid (Czarnecki, 1987; Emanuel, 1999; Kassakian et al., 1991; Mohan et al., 2003; Zinov’ev, 2012). Consider the operation of a converter with a single-phase sinusoidal grid. The charac- teristic electromagnetic processes are illustrated in Figure 1.5. Figure 1.5a shows the grid voltage u and the current i consumed by the converter. The instantaneous power, shown in Figure 1.5b, is p(θ) = u(θ) ⋅ i(θ), where θ = ωt. When p > 0, energy is transmitted from the grid to the converter and then to the load. When p < 0, energy is returned from the load to the grid.

The instantaneous power p1a(θ) (Figure 1.5b) includes a constant com- ponent. This means that the active power transmits the active component of the fundamental of current i to the load. Conversely, no constant com- ponent is seen in p1r(θ) or ph(θ). Therefore, the reactive current of the fun- damental and higher harmonics is not involved in the transmission of active power and is responsible for the useless oscillation of the energy between the grid and the converter.

where ν is the fundamental factor of current I (the ratio of the effective values of the fundamental to the total current: ν = I1/I), which character- izes the nonsinusoidal form of current i and the distortion power T; φ is the phase shift of current i1 relative to the grid voltage u (Figure 1.5a), characterizing the reactive power Q.

Schottky power diodes

The operational principle of the Schottky diode is based on the interac- tion between a metal and a depleted layer of a semiconductor, and the contact between which has rectifier properties in certain conditions. Schottky diodes are based on n− silicon with electron conductivity. The highly doped n+ substrate has a donor concentration of 5 × 1018–5 × 1019 cm−3 and its thickness is 150–200 μm; this is determined by the thick- ness of the initial silicon plate. The presence of a highly doped substrate considerably reduces the resistance of the diode and ensures satisfactory ohmic contact with the metallized cathode layer. The active n− base of the Schottky diode has a lower impurity concentration (3 × 1015 cm−3 ); its thickness wB is determined by the diode’s working voltage and is in the range from a few microns to tens of microns. To minimize extreme ava- lanche breakdown and to increase the electric field strength in the base, the diode includes a system of guard rings with a p−n junction, whose depth is a few microns (Figure 2.2).

The voltage drop at the junction in the Schottky diode is less than that for a diode with a p−n junction, whereas the reverse currents are greater (Melyoshin, 2005). The forward voltage in the Schottky diode consists of two main com- ponents: the voltage at the junction and the voltage at the active-region resistance in the n− base of the diode

​Differential and matrix equations—Switching function

As already noted, in terms of mathematical description, a power electronic device must be regarded as a nonlinear control plant with a variable or switchable structure. Each structure defined by a particular combination of switch states is linear, on account of the electrical engineering laws employed. In linear description, each variable and its derivatives appear in an equation offering first-order description of the properties of the device’s structure. The variables employed usually include the circuit currents and voltages. The processes in the circuit are described by linear integro- differential equations. If one variable is enough to describe the electrical processes, such con- trol plants are said to be one-dimensional and are described by an nth- order nonhomogeneous linear differential equation (where n is the largest derivative employed) a x t a x t n a x f t b e i n n n i n n i i e d i i d d d + + − + = + − 1 − 1 1 0 ( ) . (3.16) Here i is the number of the structure (i = 1,..., I), x the indepen- dent variable, a a a i i n i 0 1 , ,..., the coefficients of the equation describing the behavior of x, and be i the coefficient of the supply voltage for structure i. Superscript i indicates that the corresponding coefficient or equation describes structure i.

We may also write Equation 3.16 in the Cauchy form, that is, as a sys- tem of first-order equations d d d d d d d d x t x x t x x t x x t a a x a n n n ni n i n i = = = = − − − − − − − − 1 1 2 2 1 1 1 1 1 , , , a x a a x a f t b a e n i i n i n i i e i n i i 1 0 1 − + ( ) + . (3.17) We may write Equation 3.17 in the vector-matrix form by means of the state vector X x x xn n x T =| | 1 2 ... − −1 , the supply voltage vector ( ) E e i i T =| | 0 0...0 , the external-disturbance vector ( ) F f i i T =| | 0 0...0 , and the state matrix Ai d d X t A X a F b a E i n i i e i n i i = + + 1 , where A a a a a a a a a i i n i i n i n i n i n i n i = − − − − − − 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 2 1 ... ... ... . (3.18) The space in which the one-dimensional system is described is known as the derivative space. In such systems, the state variable is usually the output variable: y = x. If the behavior of the structure is described by means of several state vari- ables, the system is said to be multidimensional. Its dimensionality is deter- mined by the number of independent variables required. Such structures are usually described by vector-matrix equations (Kwakernaak and Sivan, 1972) d d X t A X B E D F i i i i i = + + , (3.19) Y = CX,

where X is the state vector, consisting of independent variables that describe the behavior of the control plant, Ei is the supply voltage vec- tor, consisting of independent supply voltages, ( ) E e e e e i i i K i K T i =| | 1 2... −1 , Fi is the external disturbance vector, ( ) F f f f f i i i L i L T i =| | 1 2... −1 , Yi is the output variable vector, ( ) Y y y y y i i i M i M T i =| | 1 2... −1 , and Ai , Bi , C, and Di are matrices characterizing the features of the structure of the control plant. In general, the number of independent variables may be different from the number of output variables. In other words, the matrix C may be rectangular. We call Equation 3.18 the equation of state and Equation 3.20 the output vari- able equation. To obtain the mathematical model for the power electronic device as a whole, rather than for each individual structure, we use switch- ing functions. In that approach, the power switch (or a combination of switches) is described by a threshold or step function or by a sign func- tion (Figure 3.4). The argument α of the switching function, which determines the con- ditions in which the switch is turned on and off, may be a function of the time. In that case, the threshold function is Ψm k k k k k k k t t t t t t t t t = ∈ ∈ + + + + + 1 0 0 1 2 1 2 3 , ( , ),( , ), , , ( , ), ( , ), if if ... ..., ⎧ ⎨ ⎩ (3.21) and the sign function is Ψm k k k k k k k t t t t t t t t t = ∈ − ∈ + + + + + 1 0 1 1 2 1 2 3 , ( , ),( , ), , , ( , ),( , ) if if ... ,..., ⎧ ⎨ ⎩ (3.22) –1 +1 +1 (a) (b) 0 α 0

Note that the topology of the power electronic device will deter- mine whether the switch state 0 or –1 will be used. In physical terms, the threshold switching function corresponds to unipolar switching of a dc supply voltage, when the power supply is connected or not. In bipolar switching, when different polarities of the supply voltage are connected, we use the sign switching function (e.g., relay control). The following algebraic relation between these two types of switching functions may be established:

Rotating Cartesian coordinate system (d, q)

In many cases, it is expedient to use a rotating Cartesian coordinate sys- tem in the analysis of the processes within the power electronic compo- nent and the design of appropriate control laws. The rate of rotation Ω of the system is chosen equal to that of one of the device’s variables. For example, for electrical circuits, it is expedient to use the synchronous grid frequency. The behavior of synchronous machines (especially salient-pole machines) is often described using the Park equations, which are written in a coordinate system that rotates at the speed of the motor’s driveshaft (Leonhard, 2001). In that case, the electromagnetic processes in the syn- chronous machine are described by differential equations with constant coefficients, rather than periodic coefficients; this simplifies the analysis. For induction motors, we use a coordinate system rotating at the speed of the rotor flux (Leonhard, 2001). This simplifies the formula for the motor’s electromagnetic torque, which takes the form of the product of two variables. In most cases, the axis from which the rotation of the coordinate sys- tem (d, q) is measured is the A-axis of the three-phase motionless coor- dinate system or, equivalently, the α-axis of the motionless Cartesian coordinate system (α, β) (Figure 3.7). Conversion of vector X from the coordinate system (α, β) to the sys- tem (d, q) may be based on the formula

Structural analysis usually reduces to finding the current or voltage varia- tion in the branches of the equivalent circuit—that is, to finding the solu- tion of Equation 3.16 for a one-dimensional control plant or of Equations 3.19 and 3.20 for a multidimensional one (Section 3.2.1). If the structure is described by linear differential equations with con- stant coefficient, an effective method of solution is to convert the initial time functions into functions of a Laplace variable by Laplace transforma- tion (Doetsch, 1974). The main benefit of such transformation is that the differentiation and integration of initial functions are replaced by alge- braic operations on their Laplace transforms. In other words, we need to only solve simple algebraic equations, rather than complex differential equations. Inverse Laplace transformation to the time domain entails dividing the algebraic solution into the sum of simple fractions, finding their solutions from Table 3.1 for inverse Laplace transformation, and then superimposing the linear differential equations obtained. The Laplace variable is denoted by s. The direct Laplace transform of the variable x(t) takes the form

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