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5. Modeling of Electrical Systems

EE3511: Automatic Control Systems. 5. Modeling of Electrical Systems. Dr. Ahmed Nassef. Learning Objective. To derive mathematical models (ODE models or transfer functions) of simple electric circuits involving resistors, capacitors, inductors, current sources and voltage sources.

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5. Modeling of Electrical Systems

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  1. EE3511: Automatic Control Systems 5. Modeling of Electrical Systems Dr. Ahmed Nassef Salman bin Abdulaziz University

  2. Learning Objective • To derive mathematical models (ODE models or transfer functions) of simple electric circuits involving resistors, capacitors, inductors, current sources and voltage sources. Salman bin Abdulaziz University

  3. Modeling Procedure • Represent the system by a set of idealized elements • Define a set of suitable variables • Write the appropriate element laws • Write the appropriate interaction laws • Combine element and interaction laws to write the model of the system • If the model is nonlinear, select an appropriate equilibrium condition and linearize the system Salman bin Abdulaziz University

  4. VariablesThe equations are expressed in terms of one or more of the following types of variables • Other variables of interest ( charge, flux, flux linkage ) Salman bin Abdulaziz University

  5. Notation i1 i2 Circuit element The current entering any circuit element is the same as the current leaving the element and therefore only one of them is shown i1 Circuit element Salman bin Abdulaziz University

  6. Notation i + _ Circuit element The direction of the current is the direction of movement of positive ions entering the element. Electrons (negative charges) moves opposite to the indicated direction Salman bin Abdulaziz University

  7. Element LawsResistor Salman bin Abdulaziz University

  8. Element LawsCapacitor Salman bin Abdulaziz University

  9. Element LawsCapacitor Salman bin Abdulaziz University

  10. Element LawsInductor Salman bin Abdulaziz University

  11. Element LawsInductor Salman bin Abdulaziz University

  12. Element LawsSources 3A 6V + − 6V + − Salman bin Abdulaziz University

  13. Interaction Laws • Interaction laws describe the way different elements are interconnected • important interaction laws • Kirchhoff’s Current Law (KCL) • Kirchhoff’s Voltage Law (KVL) Salman bin Abdulaziz University

  14. Kirchhoff’s Current Law (KCL) • The algebraic sum of all currents at any node is zero at all times. Current entering the node i1 i2 i3 Current leaving the node Salman bin Abdulaziz University

  15. Applications of KCL The current through the capacitor is the same as the current through the resistor The current through the capacitor is the same as the sum of currents through the two resistors Salman bin Abdulaziz University

  16. Kirchhoff’s Voltage Law (KVL) • The algebraic sum of voltages across all the elements that make up a loop is zero Salman bin Abdulaziz University

  17. Applications of KVL The voltage across the capacitor is the same as the voltage across the resistor Salman bin Abdulaziz University

  18. Element Laws (t -Domain) Salman bin Abdulaziz University

  19. Element Laws (S -Domain)Resistor Salman bin Abdulaziz University

  20. Element Laws (S -Domain)Capacitor Salman bin Abdulaziz University

  21. Element Laws (S -Domain)Capacitor Salman bin Abdulaziz University

  22. Element Laws (S -Domain) Inductor Salman bin Abdulaziz University

  23. Element Laws (S -Domain) Inductor Salman bin Abdulaziz University

  24. Element Laws (S -Domain) (With zero initial condition) Salman bin Abdulaziz University

  25. Obtaining The Model (Method 1)Loop Equation Method • Label loop currents • Express current through all elements in terms of one or more loop currents • Use KVL and element's laws to write the model Salman bin Abdulaziz University

  26. Obtaining The Model (Method 2) Node Equation Method • Label all voltage nodes • Express voltage across all elements in terms of node voltages • Use KCL and element's laws to write the model Salman bin Abdulaziz University

  27. Resistive Circuits • You can simplify the resistive circuit by • replacing resistors in parallel by a single resistor • replacing resistors in series by a single resistor • Remember that a resistive circuit does not have any dynamics and it can be modeled by a static model. Salman bin Abdulaziz University

  28. Resistor in series Equivalent resistance of resistors in series is the sum of resistances Salman bin Abdulaziz University

  29. Resistors in parallel + e − + e − Salman bin Abdulaziz University

  30. Current Divider Salman bin Abdulaziz University

  31. Voltage Divider + e − Salman bin Abdulaziz University

  32. Transfer Function (TF) The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero. Transfer Function G(s) Salman bin Abdulaziz University

  33. Example 1Find eo in terms of ei Two resistors in parallel Replace them by one resistor Salman bin Abdulaziz University

  34. Example 1Find eo in terms of ei Two resistors in series Replace them by one resistor Salman bin Abdulaziz University

  35. Example 1Find eo in terms of ei Two resistors in parallel Replace them by one resistor Salman bin Abdulaziz University

  36. Example 1Find eo in terms of ei Two resistors in series Replace them by one resistor Salman bin Abdulaziz University

  37. Example 1Find eo in terms of ei Two resistors in parallel Replace them by one resistor Salman bin Abdulaziz University

  38. Example 1Find eo in terms of ei This can be seen as voltage divider Salman bin Abdulaziz University

  39. Example 1Find eo in terms of ei This can be seen as voltage divider Salman bin Abdulaziz University

  40. Example 1Find eo in terms of ei This can be seen as voltage divider Salman bin Abdulaziz University

  41. Example 2 Derive an input-output model Input ei (t)output e0 (t) Salman bin Abdulaziz University

  42. Example 2 i Salman bin Abdulaziz University

  43. Example 2 i Salman bin Abdulaziz University

  44. Example 2 (alternative method) Convert the circuit to its Laplace form and use KVL and KCL as the resistive circuits i Salman bin Abdulaziz University

  45. Example 3: Find TF model of the circuit i1 i2 Salman bin Abdulaziz University

  46. Example 3: Transfer the circuit to Laplace and use KVL i1 i2 Salman bin Abdulaziz University

  47. Example 4 (Homework)Derive a state-variable model Find a model of the circuit Input ii (t)output e0 (t) Salman bin Abdulaziz University

  48. Steady State Response to Constant input • Steady state response to constant input is the response after very long time. • The steady state behavior is governed by a static model obtained from resistive circuit by • Replacing capacitors by open circuits • Replacing inductors by short circuits • Solve to get the steady state response. Salman bin Abdulaziz University

  49. Steady State Response to Constant input • Method 1: After deriving the I/O model, replace derivatives by zero and solve the algebraic model to get steady state response. • Method 2: Replace Capacitors by open circuits and replace inductors by short circuits and obtain the model from the resistive circuit. Solve to get the steady state response. Both give the same answer Salman bin Abdulaziz University

  50. Example 5 + 24 − + 24 − Salman bin Abdulaziz University

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