A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. CHAPTER 3 ELECTRICAL CIRCUITS / NETWORKS 3.1DEFINITIONS AND FORMULA An Electrical or Electric Circuit and/or Network is an interconnection of electrical elements: passive and active elements.It is used to transfer (electrical) energy from one point to another.Network elements include voltage and current sources, resistors, capacitors, and inductors. ThemostbasicquantityinanelectriccircuitistheElectricCharge.Charge(Q)isan electrical property of the atomic particles of which matter consists, measured in Coulombs. ACoulomb(C);namedafterFrenchscientist,inventor,andmilitary engineer,CharlesAugustindeCoulomb;istheamountofchargethatflows throughanycross-sectionofawire/conductorinone(1)secondifthereisa steady current of one (1) Ampere in the wire or conductor. 1 electron = 1.6022 x 10-19 C1 C = 6.2415 x 1018 electrons ElectricCurrent(I)isthetimerateofchangeofcharge,measuredin Amperes(AmporA).Itrepresentsthequantityofelectricitythatflowsina given time (t) in seconds. The unit, Ampere, was named after French Mathematician, Physicist, and Professor,AndreMarieAmpere,whoformulatedsomefundamentallawsof electricity and magnetism. 1 Ampere = 1 Coulomb/second = 6.2415 x 1018 electrons/second tQI =t I Q =IQt = Two Types of Current: 1.Direct Current (dc) is a current that remains constant with time. 2.Alternating Current (ac) is a current that varies sinusoidally with time. Voltage or Potential Difference is the energy required to move a unit charge through an element, measured in Volts (V). APotentialDifferenceofone(1)Voltexistsbetweentwopointsifone(1) Joule (J) of energy or work, W, is exchanged in moving one (1) Coulomb of charge betweenthetwopoints.Theunitofmeasurement,Volt,waschosentohonor Italian Physicist and Professor of Physics, Count Alessandro Volta. QWV = VQ W =VWQ = A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. Potentialisthevoltageatapointwithrespecttoanotherpointintheelectricalsystem.Typically, the reference point is ground, which is at Zero Potential. Potential Difference is the algebraic difference in potential (or voltage) between two points ofanetwork.AVoltageDroporRise isthe terminologyusedwhenconsideringtheeffectofthe direction of current. B A ABV V V = Electromotive Force is the force that establishes the flow of charge (or current) in a system due to the application of a difference in potential. Two Classifications of Electrical Elements or Devices: 1.Active Elements or Devices Voltage and Current Sources 2.Passive Elements or Devices Resistors, Inductors, and Capacitors DC Voltage Sources: 1.Batteries (Chemical Action) 2.Generators (Electromechanical) 3.Power Supplies (Rectification) Sinusoidal AC Voltages are available from a variety of sources.The common source is the typicalconvenienceoutlet,whichprovidesanacvoltage thatoriginatesatapowerplant;sucha power plant is most commonly fueled by water power, oil, gas, or nuclear fusion.In each case, an AC Generator or Alternator is the primary component in the energy-conversion process. DCCurrentSourceswillsupply,ideally,afixedcurrenttoanelectrical/electronicsystem, even though there may be variations in the terminal voltage as determined by the system. Ideal Independent Source is an active element that provides a specified voltage or current that is completely independent of other circuit elements. An Ideal Dependent or Controlled Source is an active element in which the source quantity is controlled by another voltage or current. DependentSourcesareusedinmodelingelementssuchastransistors,operational amplifiers, and integrated circuits. Dependent Sources: 1.Voltage-Controlled Voltage Source (VCVS) 2.Current-Controlled Voltage Source (CCVS) 3.Voltage-Controlled Current Source (VCCS) 4.Current-Controlled Current Source (CCCS) 3.2RESISTANCE Materials, in general, have the characteristic behavior of resisting the flow of electric charge.This physical property or ability to resist current is known as Resistance and is represented by the symbol R. The unit of measurement is Ohm, symbolized by the capital Greek letter Omega (), is named after German Physicist, Georg Simon Ohm. A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. Specific Resistance or Resistivity, symbolized by the Greek letter Rho (), is the resistance offered to a current if passed between the opposite faces of the unit cube of the material. British Units: Resistivity, , of Common Elements and Alloys @ 20oC, in ohm-cirmil/ft Silver9.9Lead132 Annealed Copper or simply Copper10.37Mercury577 Hard-drawn Copper10.65Brass42 Pure Gold14German Silver199 Aluminum17Manganin265 Magnesium28Lucero280 Tungsten33Advance294 Zinc36Constantan302 Nickel47Excello550 Cast Iron54Nichrome600 Platinum60Nichrome II660 Commercial Iron75Chromel625-655 SI Units: Resistivity, , of Common Elements and Alloy @ 20oC, in # x 10 8 ohm-m Silver1.64Tungsten4.37 Annealed Copper or simply Copper1.72Brass6.16 Hard-drawn Copper1.77Electrolytic Iron9.95 Pure Gold2.44Lead22 Aluminum2.83Mercury96 ALR = |.|

\| =cirmilfeetfeetcirmil ohmohm ||.|

\| =meter squaremetermetermeter square ohmohmAL V =22AVVLR = = where: - Resistivity or Specific Resistance of the conductor L - Length of the material A - Cross-sectional area V - Volume of the conductor material Formostconductors,anincreaseintemperaturecorrespondstoanincreaseinresistance due to the increased molecular movement within the conductor, which hinders the flow of charge. ((

++=121 2t Tt Tt at R t at R InferredAbsoluteTemperature,T,istheabsolutevalueofthetemperatureatwhichthe resistance of a material becomes zero. Inferred Absolute Temperatures, T, for Metals, in oC Aluminum236Silver243 Annealed Copper234.5Soft Steel218 Hard-drawn Copper242Tin218 Iron180Tungsten202 Nickel147Zinc250 A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. TemperatureCoefficientofResistance,designatedbytheGreekletterAlpha(),ata specified temperature (t), is the change in resistance (in ohm) per ohm per degree Celsius. 11t T1+= ( ) | |1 2 1 1 2t t 1 t at R t at R + = Temperature Coefficient of Resistance, , of Various Elements and Alloys, in /oC Iron20oC0.005Lead0oC0.00422 Tungsten0oC0.0048Nichrome20oC - 500oC0.00015 Aluminum0oC0.00446Manganin15oC - 35oC0.000015 Gold20oC0.0034Permanickel30oC - 500oC0.0036 Copper20oC0.00393Copel20oC - 100oC0.00002 Silver20oC0.0038Grade A Nickel0oC - 100oC0.005 3.3CAPACITANCE Apassiveelementconstructedsimplyoftwoparallelplatesseparatedby aninsulatingmaterialiscalledaCapacitor.Capacitanceisameasureofa capacitors ability to store chargeon its plates, its storage capacity.A capacitor has a capacitance of one (1) Farad if one (1) Coulomb of charge is deposited on the plates by a potential difference of one (1) Volt across the plates. Theunit,Farad(F),isnamedafterMichaelFaraday,anEnglishChemist, Physicist, and Electrical experimenter.The Farad, however, is generally too large a measure for capacitance for most practical applications, so the microfarad (10-6) or picofarad (10-12) are more commonly used. VQC =CQV = VC Q = Diagram shown below is courtesy of: http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/capcon.html#c1 A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. Permittivity,designatedbytheGreekletterEpsilon(),isaphysicalquantitythat describeshowan electric fieldaffectsand isaffectedbyadielectricmedium.It isdeterminedby the ability of a material to polarize [negative charges move toward one side and positive charges move toward the other] in response to the field; thereby reducing the total electric field inside the material. Thus, permittivity relates to a material's ability to transmit or permit an electric field. Permittivity of Free Space or Vacuum:=omF10361mF10 8542 . 89 12 = Permittivity of Any Other Medium or Dielectric: Relative Permittivity of the Dielectric: o r/ = ; dimensionless or unitless. Relative Permittivity, r , (Average) of Various Dielectrics (unitless or dimensionless) Vacuum or Free Space1.0Mica5.0 Air1.0006Porcelain6.0 Teflon2.0Bakelite7.0 Paraffin Paper2.5Glass7.5 Rubber3.0Distilled Water80.0 Transformer Oil4.0Barium-Strontium Tantalum Ceramic7500.0 For Parallel-Plate Capacitors:Cylindrical Capacitor or Coaxial Cable: 3.4INDUCTANCE Inductance is a property of an electric circuit by which a changing magnetic fieldcreatesanElectromotiveForce,orVoltage,inthatcircuitorinanearby circuit.It is the ability of a coil of conductor to oppose any change in Current. Inductance,measuredinHenry(H),afterJosephHenry,anAmerican Physicist,Mathematician,andProfessorofNaturalPhilosophy.One(1)Henryis equivalent to 1 Volt-second per Ampere. Self-Inductance of a Solenoid: (Diagram shown below is courtesy of: http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/indsol.html#c1) where: N is the number of [coil] turns is the permeability of the material core A is the cross-sectional area of the core l is the length of the solenoid is flux I is current carried by the solenoid ||.|

\|=12rrlnL 2C dAC=IA NL2 = =mF;o r =A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. Permeability,ormorespecifically,MagneticPermeability,symbolizedbyGreekLetterMu (),isaconstantofproportionalitythatexistsbetweenmagneticinductionandmagneticfield intensity. Permeability of Free Space or Vacuum:=omH10 4mH10 25664 . 17 6 = Permeability of Any Other Material: Relative Permeability of the Material:; dimensionless or unitless MaterialRelative Permeability, rApplication Ferrite U 608UHF chokes Ferrite M33750Resonant circuit RM cores Nickel (99% pure)600- Ferrite N413000Power circuits Iron (99.8% pure)5000- Ferrite T3810000Broadband transformers Silicon GO steel40000Dynamos, mains transformers Supermalloy1000000Recording heads Materialsthatcausethelinesoffluxtomovefartherapart,resultinginadecreasein magnetic flux density compared with a vacuum, are called diamagnetic. Materials that concentrate magneticfluxbyafactorofmorethan1butlessthanorequalto10arecalledparamagnetic. Materialsthatconcentratethefluxbyafactorofmorethan10arecalledferromagnetic.The permeabilityfactorsofsomesubstanceschangewithrisingorfallingtemperature,orwiththe intensity of the applied magnetic field. Diagram shown below is courtesy of: http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/indcon.html#c1 mH;o r =o r =A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. 3.5OHMS LAW Ohms Law states that the voltage (V) across a resistor (R) [or impedance (Z)], is directly proportional to the current (I) flowing through the resistor (R) [or impedance (Z)]. GermanPhysicist,Mathematician,andProfessorofPhysics,GeorgSimon Ohmiscreditedwithfindingtherelationshipbetweencurrentandvoltagefora resistor. IR V =RVI = IVR =

IZ V = ZVI =

IVZ = 3.6KIRCHHOFFS LAWS GustavRobertKirchhoff,aGermanPhysicist,contributedtothe fundamentalunderstandingofelectricalcircuits,spectroscopy,andtheemission ofblack-bodyradiationbyheatedobjects.Hecoinedtheterm"blackbody" radiation in 1862, and two sets of independent concepts in both circuit theory and thermal emission are named Kirchhoff's Laws, after him. KirchhoffsVoltageLaw:Thealgebraicsumofvoltagesaroundaclosed loopequalstozero.Also,thesumofvoltagerisesorgainsisequaltosumof voltage drops around a closed path. Kirchhoffs Current Law: The algebraic sum of currents entering/leaving a node is zero. Also, the sum of the currents entering a node is equal to the sum of currents leaving the node. 3.7CONNECTIONS IN A CIRCUIT ABranchrepresentsasingleelement:activeorpassiveelement.Also,itisthepath terminated by nodes at both ends. A Node is the point of connection between two or more branches. A Loop is any closed path in a circuit. Two or more elements are in series if they exclusively share a single node and consequently carry the same current. Twoormoreelementsareinparalleliftheyareconnectedtothesametwonodesand consequently have the same voltage across them. *Thediagramsaboveshowexamplesforseries,parallel,and series-parallelconnections,evenifthecircuitcontainsonly resistances and a source voltage. A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. 3.8ELECTRICAL ELEMENT VOLTAGE-CURRENT RELATIONS Resistor:Inductor: ) t ( i R ) t ( vR R= ) t ( idtdL ) t ( vL L=R) t ( v) t ( iRR= ) 0 ( i dt ) t ( vL1dt ) t ( vL1) t ( it0L tL L+ = = Capacitor: C) t ( q) t ( vC= ) t ( qdtd) t ( iC= ) t ( vdtdC ) t ( iC C= ) 0 ( q dt ) t ( i ) t ( qt0C+ = ) 0 ( v dt ) t ( iC1dt ) t ( iC1) t ( vt0C tC C+ = = 3.9MATHEMATICAL MODELING USING INTEGRA-DIFFERENTIAL EQUATIONS 1.Using KirchhoffsLaws,employnetworkanalysismethodssuchasBranchCurrentMethod, Maxwells Mesh Loop Method, or Nodal Method, write the voltage and/or current equations. 2.Substitute the corresponding Integra-Differential formula for voltages and/or currents. 3.Take the Laplace Transformation of the resulting Integra-Differential equations. 4.Whenever possible, solve for individual solution of every unknown voltage and/or currents, expressing the answers in terms of time. Example: Using Branch Current Method ) t ( i ) t ( i ) t ( i2 1 T+ =) t ( v ) t ( v ) t ( v ) t ( v EC 2 R L 1 R+ = + =) 0 ( v dt ) t ( iC1) t ( i R ) t ( idtdL ) t ( i R Et0C 2 R 2 L 1 R 1+ + = + = ) t ( i ) t ( i ) t ( iL 1 R 1= = ) t ( i ) t ( i ) t ( iC 2 R 2= =) t ( idtdL ) t ( i R E1 1 1+ = ) 0 ( v dt ) t ( iC1) t ( i R Et02 2 2+ + = A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. Take the Laplace Transform of the equations.If the initial conditions are not provided, assume all initial conditions to be zero. LetL{ } ) s ( I ) t ( i1 1=andL{ } ) s ( I ) t ( i2 2= | | ) 0 ( i ) s ( sI L ) s ( I RsE1 1 1 + =s) 0 ( vs) s ( IC1) s ( I RsE22 2+ + =) s ( LsI ) s ( I RsE1 1 1+ =Cs) s ( I) s ( I RsE22 2+ =| | ) s ( I Ls RsE1 1 + = ) s ( ICs1 Cs R) s ( ICs1RsE222 2((

+=((

+ =| |((

+=+=LRs LsELs R sE) s ( I1 11 | | 1 Cs REC1 Cs R sECs) s ( I2 22+=+=LRsBsALRs sLE) s ( I1 11++ =((

+=C R1sREC R1s C REC) s ( I22222+=((

+= Partial Fraction Expansion would yield:Inverse Laplace Transform would yield: LRsREsRE) s ( I11 11+ =C Rt222eRE) t ( i= Amp Inverse Laplace Transform would yield:Total Current:) t ( i ) t ( i ) t ( i2 1 T+ =Lt R1 111eRERE) t ( i =

Amp C Rt2Lt R1 1T21eREeRERE) t ( i + =Amp The Voltages are: Lt RLt R1 11 R 1 1 R1 11Ee E eRERER ) t ( i R ) t ( v =((

= =Volts Lt RLt R11Lt R1 11 L1 1 1Ee eLRRE0 L eREREdtdL ) t ( idtdL ) t ( v =(((

||.|

\| =((

= =Volts C RtC Rt22 2 2 2 R2 2Ee eRER ) t ( i R ) t ( v =(((

= =Volts ( )(((

= =(((

= = 0 C Rtt0C Rt22t0C Rt2t02 Ce e E e C RREC1dt eREC1dt ) t ( iC1) t ( v2 2 2 C RtC2Ee E ) t ( v =Volts A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. Example: Using Maxwells Mesh Loop Method ) t ( v ) t ( v ) t ( v EC L R+ + = ) 0 ( v dt ) t ( iC1) t ( idtdL ) t ( i R Et0C L R+ + + = ) t ( i ) t ( i ) t ( i ) t ( iC L R= = =) 0 ( v dt ) t ( iC1) t ( idtdL ) t ( i R Et0+ + + = Take the Laplace Transform of the equation. Assuming that at t = 0, i = Io, where Io is a constant quantity, and at t = 0, v = 0. LetL{ } ) s ( I ) t ( i = | |s) 0 ( vs) s ( IC1) 0 ( i ) s ( sI L ) s ( I RsE+ + + =| |Cs) s ( ILIo ) s ( sI L ) s ( I RCs) s ( IIo ) s ( sI L ) s ( I RsE+ + = + + =) s ( ICs1Ls RCs) s ( I) s ( sI L ) s ( I RsLIos ELIosE((

+ + = + + =+= +) s ( ICs1 LCs RCssLIos E2((

+ +=+) s ( IC1 RCs LCsE LIos2((

+ += +| | | |((

+ +((

+=((

+ ++=+ ++=LC1sLRs LLIoEs LIoLC1sLRs LCE LIos C1 RCs LCsE LIos C) s ( I2 22 (((

|.|

\| +|.|

\| +((

+ + =|.|

\| +(((

|.|

\|+ ++=+ +((

+=2 2 2 222L 2RLC1L 2RsL 2Rs IoL 2IoRLEL 2RLC1L 2RsLRsLEIosLC1sLRsLIoEs Io) s ( I(((

|.|

\| + |.|

\| +((

++(((

|.|

\| + |.|

\| +=2 2 2 2L 2RLC1L 2RsL 2Rs IoL 2RLC1L 2RsL 2IoRLE) s ( ItL 2RLC1cos Ioe tL 2RLC1sin eL 2RLC1 L 2IoRLE) t ( i2L 2Rt 2L 2Rt2)`(((

|.|

\| +)`(((

|.|

\|(((

|.|

\||.|

\|= Amp A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. Example: Using Nodal Method ) t ( i ) t ( i ) t ( i2 1 T+ =) t ( vdtdC ) 0 ( i dt ) t ( vL1R) t ( vCt0LR+ + = ) t ( v E ) t ( v ) t ( v ) t ( vb b a R = = ) t ( v ) t ( v ) t ( v ) t ( v ) t ( vb 0 b C L= = = ) t ( vdtdC ) 0 ( i dt ) t ( vL1R) t ( v Ebt0bb+ + = Take the Laplace Transform of the equation. Note that vo(t) is the nodal voltage at node o. Assuming that at t = 0, v = Vo, where Vo is a constant quantity, and at t = 0, i = 0. LetL{ } ) s ( V ) t ( vb b=| | ) 0 ( v ) s ( sV Cs) 0 ( is) s ( VL1) s ( VsER1bbb + + =((

| | CVo ) s ( CsVLs) s ( VVo ) s ( sV CLs) s ( VR) s ( VRsEbbbb b + = + = ) s ( V CsLs1R1) s ( CsVLs) s ( VR) s ( VCVoRsEb bb b((

+ + = + + = + ) s ( VRLsRLCs R LsRsRCsVo Eb2((

+ +=+ ) s ( VLR Ls RLCsRCsVo Eb2((

+ += +| | | |((

+|.|

\|++=((

+|.|

\|++=+ ++=LC1sRC1s RCRCsVo ELC1sRC1s RLCRCsVo E LR Ls RLCsRCsVo E L) s ( V2 22 b (((

|.|

\| + |.|

\| + |.|

\| + +=|.|

\| +(((

|.|

\|+ |.|

\|++=2 2 2 22bRC 21LC1RC 21sRC 2VoRC 21s VoRCERC 21LC1RC 21sRC1ssVoRCE) s ( V(((

|.|

\| + |.|

\| +|.|

\| ++(((

|.|

\| + |.|

\| +=2 2 2 2bRC 21LC1RC 21sRC 21s VoRC 21LC1RC 21sRC 2VoRCE) s ( VtRC 21LC1cos Voe tRC 21LC1sin eRC 21LC1RC 2VoRCE) t ( v2RC 2t 2RC 2t2b)`(((

|.|

\| +)`(((

|.|

\|(((

|.|

\||.|

\|= Volts A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 3 CATADMAN, LIZETTE IVY G. 3.10PROBLEM SET 1.Let E be constant and all initial conditions be zero. Solve for the series current and voltages across the R and L. 2.Let E be constant and initial condition is at t = 0, q = Qo. Hint: Replace current in terms of charge. Solve for the charge in the series circuit, series current, and voltages across the R and C. 3.Let Vs(t) = 8 cos 9t Volts R1 = 2 ohmsR2 = 4 ohms R3 = 7 ohmsC = 5 F L1 = 6 HL2 = 3 H All initial conditions are assumed to be zero. Solve for the loop currents, I1(s) and I2(s). 4.Let Vs(t) = 12 cos 3t Volts E = 10 Volts R1 = 4 ohmsR2 = 6 ohms L1 = 2 HL2 = 5 H C1 = 3 FC2 = 7 F All initial conditions are assumed to be zero. Solve for the loop currents, I1(s) and I2(s). SOURCES / REFERENCES Alexander,CharlesK.andMatthewN.O.Sadiku.FundamentalsofElectricCircuits,Second Edition. Singapore: McGraw-Hill Co., 2003. Boylestad,RobertL.IntroductoryCircuitAnalysis,EightEdition.USA:Prentice-HallInc., 1997. Hostetter, Gene H., Clement J. Savant Jr., and Raymond T. Stefani. Design of Feedback Control Systems, 2nd Edition. USA: Saunders College Publishing, 1989. http://www.swehs.co.uk/docs/xpert.html