5. Modeling of Electrical Systems EE3511: Automatic Control Systems 5. Modeling of Electrical Systems Dr. Ahmed Nassef EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Learning Objective To derive mathematical models (ODE models or transfer functions) of simple electric circuits involving resistors, capacitors, inductors, current sources and voltage sources. EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University 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 EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Variables The equations are expressed in terms of one or more of the following types of variables Other variables of interest ( charge, flux, flux linkage ) Variables Unit symbol Electromotive Force (Voltage) Volt e, v Current Ampere i EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University 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 EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University 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 EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Element Laws Resistor EE3511_L5 Salman bin Abdulaziz University

Element Laws Capacitor EE3511_L5 Salman bin Abdulaziz University

Element Laws Capacitor EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Element Laws Inductor EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Element Laws Inductor EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Element Laws Sources 3A 6V + − 6V + − EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Interaction Laws Interaction laws describe the way different elements are interconnected important interaction laws Kirchhoff’s Current Law (KCL) Kirchhoff’s Voltage Law (KVL) EE3511_L5 Salman bin Abdulaziz University

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 EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University 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 EE3511_L5 Salman bin Abdulaziz University

Kirchhoff’s Voltage Law (KVL) The algebraic sum of voltages across all the elements that make up a loop is zero EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Applications of KVL The voltage across the capacitor is the same as the voltage across the resistor EE3511_L5 Salman bin Abdulaziz University

Element Laws (t -Domain) EE3511_L5 Salman bin Abdulaziz University

Element Laws (S -Domain) Resistor EE3511_L5 Salman bin Abdulaziz University

Element Laws (S -Domain) Capacitor EE3511_L5 Salman bin Abdulaziz University

Element Laws (S -Domain) Capacitor EE3511_L5 Salman bin Abdulaziz University

Element Laws (S -Domain) Inductor EE3511_L5 Salman bin Abdulaziz University

Element Laws (S -Domain) Inductor EE3511_L5 Salman bin Abdulaziz University

Element Laws (S -Domain) (With zero initial condition) EE3511_L5 Salman bin Abdulaziz University

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 EE3511_L5 Salman bin Abdulaziz University

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 EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University 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. EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Resistor in series Equivalent resistance of resistors in series is the sum of resistances EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Resistors in parallel + e − + e − EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Current Divider EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Voltage Divider + e − EE3511_L5 Salman bin Abdulaziz University

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) EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei Two resistors in parallel Replace them by one resistor EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei Two resistors in series Replace them by one resistor EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei Two resistors in parallel Replace them by one resistor EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei Two resistors in series Replace them by one resistor EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei Two resistors in parallel Replace them by one resistor EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei This can be seen as voltage divider EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei This can be seen as voltage divider EE3511_L5 Salman bin Abdulaziz University

Example 1 Find eo in terms of ei This can be seen as voltage divider EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Example 2 Derive an input-output model Input ei (t) output e0 (t) EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Example 2 i EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Example 2 i EE3511_L5 Salman bin Abdulaziz University

Example 2 (alternative method) Convert the circuit to its Laplace form and use KVL and KCL as the resistive circuits i EE3511_L5 Salman bin Abdulaziz University

Example 3: Find TF model of the circuit EE3511_L5 Salman bin Abdulaziz University

Example 3: Transfer the circuit to Laplace and use KVL EE3511_L5 Salman bin Abdulaziz University

Example 4 (Homework) Derive a state-variable model Find a model of the circuit Input ii (t) output e0 (t) EE3511_L5 Salman bin Abdulaziz University

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. EE3511_L5 Salman bin Abdulaziz University

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 EE3511_L5 Salman bin Abdulaziz University

Salman bin Abdulaziz University Example 5 + 24 − + 24 − EE3511_L5 Salman bin Abdulaziz University