University:
California State University, Northridge
Course:
ME 584 | Modeling and Simulation of Dynamic Systems
Academic year:

2010
Views:

228

Pages:

24
Author:

Alfonso Mullins

chp4 1 Modeling Electrical Systems chp4 2 Agenda • • • • Basic Electrical Elements Passive Circuit Analysis Active Circuit Analysis Case Study: A Speaker Model • Active Learning: Pair-share Exercises, Case Study chp4 4 Electrical System • Composed of resistors, capacitors, inductors, transistors, amplifiers, power supplies – Passive circuits: respond to applied voltage or current and do not have any amplifiers – Active circuits: made of transistors and/or amplifiers, require active power source to work • Basic quantities – – – – – Charge q [coulomb] = 6.24x1018 electrons Current i [ampere] = dq/dt Voltage e [Volt] = dw/dq Energy or Work w [joule] Power p [watt] = e x i = dw/dt chp4 5 Units and Representations for Common Electrical Quantities Chap4 6 General Model Structure for Mechanical System Chap4 General Model Structure for Electrical System 7 chp4 8 Equivalent Circuit for Spring-MassDashpot Systems chp4 9 Resistance • Resistance behavior is between insulator and conductors, allowing a predictable restriction of electron flow • Power dissipated = • Resistance – A: cross section are of wire – l: length of wire – ρ: resistivity of material chp4 10 Capacitance • Capacitor stores electrons on 2 parallel plates separated by an insulating dielectric material in an electric field or • Energy stored in capacitor • Capacitance – A: area of plates – D: spacing between plates – ϵ:permittivity of the dielectric chp4 11 Inductance • Inductance relates voltage induced to time rate of change of magnetic field • Faraday’s law: where φ is the magnetic flux, φ = Li • Energy stored: • Inductance: where A= wire cross section area, l = wire length, n = number of turns, μm =permeability of magnetic circuit chp4 12 Impedance • Impedance Z: instantaneous ratio of voltage difference to current • Impedance of common circuit elements – Resistive: Zr = R (not dynamic) – Capacitive: Zc = 1/CD – Inductive: ZL=LD Where D= differential operator d/dt, 1/D = integrator operator chp4 13 Ideal and Non-Ideal Sources Voltage Source Ideal Non-Ideal Battery Current Source chp4 14 Open and Short Circuits • An open circuit is any element through which current cannot flow • A short circuit is any element across which there is no voltage chp4 R C L Series and Parallel Impedance Combinations 15 chp4 16 Laws for Passive Circuit Analysis chp4 Techniques for Passive Circuit Analysis for Classical Deriving Differential Equations 1. Draw circuit schematic and label components (e.g., R1, R2, C1, L1…) 2. Assign voltage at each node (e.g., e1, e2) 3. Assign current in each component (e.g., i1, i2, ..) and show positive current direction with arrows 4. Write equation for current for each component (e.g., iR1 = (e1-e2)/R1 or iC1 = CDe1 ) 5. Write node equations for each significant node (not connected to voltage or current source) 6. Substitute component equations into node equations and reduce results to a single differential equation with output and input variables 17 chp4 18 Example 1: Voltage Divider Evaluate e1 chp4 19 Example 2: Resistor Circuit • Calculate the amount of power dissipated in resistor R3 in the circuit shown below • Solution: chp4 20 Example 3: Pair-Share Exercise Seven-Resistor Circuit • The resistive circuit shown consists of a voltage source connected to a combination of seven resistors. The output is voltage e0. Find the equivalent resistance Req of the sevenresistor combination and evaluate e0. chp4 21 Example 3: Pair-Share Exercise Seven-Resistor Circuit Apply voltage divider equation repeatedly chp4 22 Example 4: Dual RC Circuit • Write the modeling equations for circuit (a) • Derive the differential equation in the form [τD+1]e1=Ge0 • What are the mathematical expressions for time constant τ and gain G? • For circuit (b), is the differential equation for this circuit a product of two RC’s circuit, that is, [τ1D+1][τ2D+1]e2=e0 ? chp4 23 Example 4: Dual RC Circuit (a) iR1 iC1 chp4 24 Example 4: Dual RC Circuit iR2 iR1 iC1 (b) iC2 chp4 25 Example 4: Dual RC Circuit

Modeling Electrical Systems