Power Systems Control Prof. Wonhee Kim. Modeling in the Frequency and Time Domains
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1 Power Systems Control Prof. Wonhee Kim Modeling in the Frequency and Time Domains
2 Laplace Transform Review - Laplace transform - Inverse Laplace transform 2
3 Laplace Transform Review 3
4 Laplace Transform Review 4
5 Laplace Transform Review: - Solution of a Differential Equation Example 2.3) 5
6 System Modeling in Frequency Domain r(t) c(t) r(t) c(t) From input to output: Convolution t * 0 c g r t g r t d 6
7 System Modeling in Frequency Domain Laplace transform r(t) c(t) R(s) C(s) t * 0 c g r t g r t d G s Rs C s Transfer function C s R s G s 7
8 System Modeling in Frequency Domain R(s) C(s) 8
9 System Modeling in Frequency Domain Laplace transform r(t) c(t) R(s) C(s) Example 2.4) dc t 2ct r t dt sc s C s R s 2 Cs 1 Rs s 2 G s 9
10 System Modeling in Frequency Domain Example 2.5) G s C s 1 R s s 2 1 C sg s Rs Rs s 2 1 Rs s 1 1 1/ 2 1/ 2 s s 2 s s 2 G s Rs C s e c t t 10
11 Electrical Network Transfer Functions 11
12 Electrical Network Transfer Functions Example 2.6)
13 Electrical Network Transfer Functions Example 2.6) For the capacitor: For the resistor: For the inductor: Impedance: i t Cdv t / dt I s C scv s
14 Operational Amplifier 14
15 Operational Amplifier 15
16 Operational Amplifier 16
17 Translational Mechanical System Transfer Functions 17
18 Rotational Mechanical System Transfer Functions 18
19 Translational Mechanical System Transfer Functions Example 2.16) 19
20 Nonlinearities 20
21 Nonlinearities - Linearization 21
22 Nonlinearities - Linearization 22
23 Nonlinearities - Linearization 23
24 Some Observation Example Loop equation: I s Vs Ls R 24
25 Some Observation Loop equation: Input State variable I s Vs Ls R State equation: Output 25
26 Some Observation Example Loop equation: Not first order differential equation! 26
27 Some Observation Loop equation: Input State equation: State variable i t, q t Output 27
28 Some Observation Loop equation: Input State equation: State variable v t, v t C R 28
29 Some Observation State variable: or Input State variables must be linearly independent! Linearly dependent: ex) State variable definition v t, i t R v t, v t C i t R, q t 29
30 Some Observation State variable: or Input State variables must be linearly independent! Linearly dependent: ex) State variable definition v t, i t R v t, v t C i t R, q t 30
31 Some Observation 31
32 Some Observation State-space equation 32
33 Some Observation x v v R C Fig. Graphic representation of state space and a state vector 33
34 State-space Equation State-space equation State equation Output equation This representation of a system provides complete knowledge of all variables of the system at any t t 0. 34
35 State-space Equation 2nd order single-input single-output state-space equation: The choice of state variables for a given system is not unique. 35
36 State-space Equation How do we know the minimum number of state variables to select? Typically, the minimum number required equals the order of the differential equation describing the system. State variable i t, q t 36
37 Converting from Transfer Function to State Space Differential equation State variables and state equation (Phase variable form): 37
38 Converting from Transfer Function to State Space Differential equation State-space equation (Phase variable form): Converting is not unique! 38
39 Converting from Transfer Function to State Space Example 2.4) 39
40 Converting from Transfer Function to State Space Example 3.4) 40
41 Converting from State Space to a Transfer Function State-space equation Laplace transform assuming zero initial conditions Transfer Function 41
42 Converting from State Space to a Transfer Function Example 3.6) 42
43 Converting from State Space to a Transfer Function Example 3.6) 43
44 Linear System and Nonlinear System 1) Linear systems: - Linear time-invariant (LTI) system x t Ax t Bu t tc td t y x u - Linear time-varying (LTV) system x t A t x t B t u t tc ttdtt y x u 2) Nonlinear systems: - Nonlinear time-invariant system x t f x t,u t t g tt y x,u - Nonlinear time-varying nonlinear system x t f x t,u t, t t y t g x t,u, t Y s T s C si A 1 B D U s Y s T s C si A 1 B D U s Y s T s C si A 1 B D U s Y s T s C si A 1 B D U s 44
45 Linear System and Nonlinear System 1) Linear systems: - Linear time-invariant (LTI) system x t Ax t Bu t tc td t y x u - Linear time-varying (LTV) system x t A t x t B t u t tc ttdtt y x u 2) Nonlinear systems: - Nonlinear time-invariant system x t f x t,u t t g tt y x,u - Nonlinear time-varying nonlinear system x t f x t,u t, t t y t g x t,u, t x 2 x u y x x 2t x u y x 3 x x u y x 3 t x x e u y x 45
Taking the Laplace transform of the both sides and assuming that all initial conditions are zero,
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Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)