Page 6 - Complementarity and Variational Inequalities in Electronics

List of Figures

Fig. 1.1 Ideal diode model. 3
Fig. 1.2 Clipping circuit 1: Diode as shunt element. 5
Fig. 1.3 Clipping circuit 1: Ideal diode as shunt element, E = 1. 6
Fig. 2.1 Epigraph of a convex function. 7
Fig. 2.2 A lower semicontinuous function. 8
Fig. 2.3 Affine function y = (x 0 ) + w i (x − x 0 ) with w i ∈ ∂ (x 0 ) (i = 1,2,3). 8
Fig. 2.4 (x) =|x| and ∂ (x). 9
2
Fig. 2.5 (x) = max{0,x − 1} and ∂ (x). 10
Fig. 2.6 For any w ∈ R, there exists v ∈ R such that i (v) < wv, and thus
∂ i (0) =∅ (i = 1,2). However, for any given w ∈ R, we see that
1 (v) + 2 (v) ≥ wv,∀v ∈ R. Thus ∂( 1 + 2 )(0) = R. 12
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Fig. 2.7 Computation of (z). 14
Fig. 2.8 x → ∂ (x) and z → ∂ (z). 15
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Fig. 2.9 Normal cone N K (x i ) = ∂ K (x i ) of K at x i (i = 1,2,3). 16
o
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Fig. 2.10 A cone K and the corresponding polar cone K =−K . 18
Fig. 2.11 Electrical Device. 19
Fig. 2.12 A maximal monotone set-valued function F with D(F) =]−∞,1] and the
0
minimal section β of F. 20
Fig. 2.13 The function ϕ as defined in (2.3). 20
Fig. 2.14 Practical diode model. 23
Fig. 2.15 Complete diode model. 24
Fig. 2.16 Zener diode model. 26
Fig. 2.17 Ideal Zener diode model. 27
Fig. 2.18 Practical Zener diode model. 27
Fig. 2.19 Varistor. 29
Fig. 2.20 Transistor P–N–P. 30
Fig. 2.21 Transistor N–P–N. 31
Fig. 3.1 Clipping circuit 1: Practical diode as shunt element, V 1 = 0.1, V 2 =−90,
E = 1. 37
Fig. 3.2 Rectifier–Stabilizer circuit. 37
Fig. 3.3 Rectifier circuit. 38
Fig. 3.4 Stabilizer circuit. 38
Fig. 3.5 Rectifier circuit. 40
Fig. 3.6 Stabilizer circuit. 41
Fig. 3.7 Ideal Zener Diode. 41
Fig. 4.1 K and K ∞ . 46
Fig. 4.2 K and K ∞ . 48
Fig. 4.3 K and K ∞ . 49

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