APPLICATION TO TRANSIENT ANALYSIS OF ELECTRICAL CIRCUITS

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1 EECE 552 Numerical Circuit Analysis Chapter Nine APPLICATION TO TRANSIENT ANALYSIS OF ELECTRICAL CIRCUITS I. Hajj

2 Application to Electrical Circuits Method 1: Construct state equations = f(x, t) Method 2: Apply directly to Tableau Equations, which are mixed algebraic-differential equations: f(x,, t) = 0

3 Special Case for Constructing State Equations For circuits suitable for nodal analysis (no inductors, no current variables; if voltage sources exist, transform them into current sources using Norton equivalent circuit), and if there is a capacitive path from every node to ground, the nodal equations generate state equations.

4 Example

5 Linear Circuit ia = gava, ib = gbvb

6 State Equations More generally, the MNA equations of a circuit containing linear capacitors and inductors and linear and nonlinear resistors can be written in the form If C is nonsingular,

7 General Case: f(x,,t) = 0 apply or get

8 Tableau Equations of General Linear and Nonlinear Circuits Element Characteristics (a) Resistors (linear and nonlinear, including independent sources)

9 Capacitors

10 Inductors

11 memrisitve systems

12 memcapacitive systems

13 meminductive systems

14 Mem Devices: Memristors

15 Memcapacitors

16 Meminductors

17 Companion Models and Stamps No need to construct state equations. Apply integration formula to differential operator and derive a companion model. Derive a stamp to incorporate elements into the linearized circuit equations

19 Capacitor 'companion' models using lmf. Consider an implicit lmf: Apply to capacitor characteristic equations

20 Or

21 Nonlinear Capacitor Stamp Implicit lmf

22 Capacitor 'companion' model using B.E. = (x n - x n-1 )/h Linear Capacitor: q = Cv => i = dq/dt = Cdv/dt i n =

23 Linear Capacitor (B.E) MNA Stamp: v n = v i v j

24 Nonlinear Capacitor (voltage-controlled): B.E. (B.E) β 0 = 1, s c,n = f c (v c,n-1 )/h

25 Nonlinear Capacitor (voltage-controlled): B.E. β 0 = 1, s c,n = f c (v c,n-1 )/h (changes at every time point); a c,n and b c,n change at every iteration point at time point t n

26 Trapezoidal Rule

27 Linear Capacitor: T.R.

28 Linear Capacitor Stamp T.R. ; i 0 = 0

29 Nonlinear Capacitor: T.R.

30 Nonlinear Capacitor: T.R. β 0 = 1/2, s c,n = 2f c (v c,n-1 )/h + i c,n-1 (changes at every time point); a c,n and b c,n change at every iteration point at time point t n

31 2 nd Order Backward Differentiation Formula BDF Linear Capacitor: q = Cv, i = dq/dt

32 Linear Capacitor Stamp: BDF

33 Charge-Controlled Capacitor (1) (2)

34 Stamp of Two-Terminal Charge-Controlled Capacitor: B.E. and ENA

35 Charge-and-Voltage Controlled Capacitor

36 Stamp of Two-Terminal Charge-and-Voltage- Controlled Capacitor: B.E. and ENA

37 Inductor

38 Linear Inductor: B.E.

39 Stamp of Linear Inductor: B.E.

40 Linear Inductor: T.R.

41 Stamp of Linear Inductor: T.R.

42 Linear Inductor: 2 nd Order BDF

43 Nonlinear Current-Controlled Inductor Ф L = f L (i L ); v L = (dф L )/(d t ) Ф L,n = (1/h) Ф L,n - (1/h) Ф L,n-1 (B.E) = (1/h)f L (i L, n ) (1/h)f L (i L, n-l )

44 Stamp of Linearized Current-Controlled Inductor For B.E.: β 0 = 1, s L,n = f L (i L,n-1 )/h

45 Nonlinear Inductor: T.R.

46 Stamp of Linearized Current-Controlled Inductor: T.R. β 0 = 1/2, s L,n =2 f L (i L,n-1 )/h + v L,n-1

47 Flux-Controlled Inductor

48 Stamp of (Linearized) Flux-Controlled Inductor: B.E. (ENA)

49 Flux-and-Current Controlled Inductor

50 Stamp of (Linearized) Flux-and-Current Controlled Inductor: B.E. (ENA)

51 Current-Controlled Memrisitive System

52 Stamp of (Linearized) Current Controlled Memristive system: B.E. (ENA)

53 Charge-Controlled Memristor

54 Stamp of (Linearized) Current Controlled Memristive system: B.E. (ENA)

55 Transient Analysis Flowchart

56 Example t = 0

57 At t = 0 - => put C = 0, L = 0

58 Example (nonlinear circuit)

59 Given: Assume B.E. is used:

60 Suppose Newton's method is applied to find the solution at time tn; and Taylor series is used to linearize all nonlinear elements at Newton iteration point at tn.

61 Vc = V3 Linearization

64 (1) How to estimate LTE in practice Use Finite Difference Interpolation:

65 Note: LTE decreases with h, but when h decreases, round-off error increases.

66 (2) The timestep h is determined from the LTE estimate

67 (3) Specify two bounds: Upper bound B u Lower bound B l (i) If B l < LTE < B u, keep same h (ii) If LTE > B u > reject x n, decrease h and re-compute x n (iii) If LTE < B 1, accept x n, increase next h

68 (4) Decrease or increase h so that

69 (5) B u and B 1 can be either constants, or variables as functions of x (t) (6) Use voltage across each capacitor and current in each inductor to estimate LTE. Choose the largest one to determine h.

70 (7) Specify minimum time step h min, based on computer word length do not choose h < h min even if LTE is still too large. (8) At time t = 0, after initial conditions are found, start with one-step formula (B.E. or T.R.) and with h min for at least two steps before computing LTE and increasing h. After that, h can change and multistep formulas can be used.

71 (9) If the time response at any point in the circuit becomes discontinuous in t or its time derivative becomes discontinuous, restart with one-step formula and h min after that time point. (10) In solving for x n at t n using Newton's method, x n-1 is usually chosen as an initial guess. However, another initial guess can be used, such as obtained (or predicted) by an explicit formula:

72 Companion models using explicit formulas Explicit Formulas:

73 is available, then x n = x n-1 + h f(x n-1, t n-1 ) can be computed explicitly at time t n in terms of previous solutions and time step h. If state equations are not available, apply formula to capacitor and inductor equations to derive "explicit" companion models as follows.

74 Linear Cap

75 Nonlinear Cap

76 Nonlinear Cap (cont.)

77 Linear Inductor

78 Nonlinear Inductor

79 Nonlinear Inductor (cont.)

80 Example

81 Loops of Capacitors (and voltage sources)

82 Last three rows are dependent.

83 Cutset of Inductors (and current sources)

84 The last two rows are dependent.

86 TIME DOMAIN SENSITIVITY

87 Linear Case

90 i-th equation:

100 Nonlinear Case

101

102

103

104

105

106

107

108

109

110

111

NONLINEAR DC ANALYSIS

ECE 552 Numerical Circuit Analysis Chapter Six NONLINEAR DC ANALYSIS OR: Solution of Nonlinear Algebraic Equations I. Hajj 2017 Nonlinear Algebraic Equations A system of linear equations Ax = b has a