Chapter 6 Electrical Systems and Electromechanical Systems

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1 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Chapter 6 Electrcal Systems and Electromechancal Systems 6. INTODUCTION A. Bazoune The majorty of engneerng systems now have at least one electrcal subsystem. Ths may be a power supply, sensor, motor, controller, or an acoustc devce such as a speaker. So an understandng of electrcal systems s essental to understandng the behavor of many systems. 6. ELECTICAL ELEMENTS Current and Voltage Current and voltage are the prmary varables used to descrbe a crcut s behavor. Current s the flow of electrons. It s the tme rate of change of electrons passng through a defned area, such as the cross-secton of a wre. Because electrons are negatvely charged, the postve drecton of current flow s opposte to that of electron flow. The mathematcal descrpton of the relatonshp between the number of electrons ( called charge q ) and current s dq dt = or q( t) = d t The unt of charge s the coulomb (C) and the unt of current s ampere (A), whch s one coulomb per second. Energy s requred to move a charge between two ponts n a crcut. The work per unt charge requred to do ths s called voltage. The unt of voltage s volt (V), whch s defned to be joule per coulomb. The voltage dfference between two ponts n a crcut s a measure of the energy requred to move charge from one pont to the other. Actve and Passve Elements. actve or passve. Crcut elements may be classfed as Passve Element: an element that contans no energy sources (.e. the element needs power from another source to operate); these nclude resstors, capactors and nductors Actve Element: an element that acts as an energy source; these nclude batteres, generators, solar cells, and op-amps. /0

2 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Current Source and Voltage Source A voltage source s a devce that causes a specfed voltage to exst between two ponts n a crcut. The voltage may be tme varyng or tme nvarant (for a suffcently long tme). Fgure 6-(a) s a schematc dagram of a voltage source. Fgure 6-(b) shows a voltage source that has a constant value for an ndefnte tme. Often the voltage s denoted by E or V. A battery s an example of ths type of voltage. A current source causes a specfed current to flow through a wre contanng ths source. Fgure 6-(c) s a schematc dagram of a current source e ( t ) E ( t ) Fgure 6. (a) Voltage source; (b) constant voltage source; (c) current source esstance elements. The resstance of a lnear resstor s gven by where = e s the voltage across the resstor and s the current through the resstor. The unt of resstance s the ohm ( Ω ), where volt ohm= ampere esstances do not store electrc energy n any form, but nstead dsspate t as heat. eal resstors may not be lnear and may also exhbt some capactance and nductance effects. PACTICAL EXAMPLES: Pctures of varous types of real-world resstors are found below. e e Wrewound esstors Wrewound esstors n Parallel /0

3 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Wrewound esstors n Seres and n Parallel Capactance Elements. Two conductors separated by a nonconductng medum form a capactor, so two metallc plates separated by a very thn delectrc materal form a capactor. The capactance C s a measure of the quantty of charge that can be stored for a gven voltage across the plates. The capactance C of a capactor can thus be gven by q C = e where q s the quantty of charge stored and e c s the voltage across the capactor. The unt of capactance s the farad ( F ), where ampere-second coulomb farad = = volt volt Notce that, snce = dq dt and ec or Therefore, = q C, we have de = C c d t c dec = dt C ec = dt + e C Although a pure capactor stores energy and can release all of t, real capactors exhbt varous losses. These energy losses are ndcated by a power factor, whch s the rato of energy lost per cycle of ac voltage to the energy stored per cycle. Thus, a small-valued power factor s desrable. PACTICAL EXAMPLES: Pctures of varous types of real-world capactors are found below. t 0 c ( 0) C e c 3/0

4 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Inductance Elements. If a crcut les n a tme varyng magnetc feld, an electromotve force s nduced n the crcut. The nductve effects can be classfed as self nductance and mutual nductance. Self nductance, or smply nductance, L s the proportonalty constant between the nduced voltage e L volts and the rate of change of current (or change n current per second) d dt amperes per second; that s, L = d e L dt The unt of nductance s the henry (H). An electrcal crcut has an nductance of henry when a rate of change of ampere per second wll nduce an emf of volt: volt weber henry = = ampere second ampere L e L The voltage e L across the nductor L s gven by d el = L d t Where L s the current through the nductor. The current ( ) t e t t ( ) = d + ( 0) L L L L 0 L L t can thus be gven by Because most nductors are cols of wre, they have consderable resstance. The energy loss due to the presence of resstance s ndcated by the qualty factor Q, whch denotes the rato of stored dsspated energy. A hgh value of Q generally means the nductor contans small resstance. Mutual Inductance refers to the nfluence between nductors that results from nteracton of ther felds. PACTICAL EXAMPLES: Pctured below are several real-world examples of nductors. 4/0

5 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems TABLE 6-. Summary of elements nvolved n lnear electrcal systems Element Voltage-current Current-voltage Voltage-charge Impedance, Z(s)=V(s)/I(s) Capactor t v( t) = ( τ ) dτ c 0 dv( t) ( t) = C v ( t) = q( t) dt c Cs esstor v ( t) = ( t) ( t) = v( t) dq( t) v( t) = dt Inductor t d( t) v( t) = L ( t) = dt v( τ ) dτ L 0 d q( t) v( t) = L Ls dt The followng set of symbols and unts are used: v(t) = V (Volts), (t) = A (Amps), q(t) = Q (Coulombs), C = F (Farads), = Ω (Ohms), L = H (Henres). 6.3 FUNDAMENTALS OF ELECTICAL CICUITS Ohm s Law. Ohm s law states that the current n crcut s proportonal to the total electromotve force (emf) actng n the crcut and nversely proportonal to the total resstance of the crcut. That s e = were s the current (amperes), e s the emf (volts), and s the resstance (ohms). Seres Crcut. The combned resstance of seres-connected resstors s the sum of the separate resstances. Fgure 6- shows a smple seres crcut. The voltage between ponts A and B s where e = e + e + e3 e =, e =, e = 3 3 5/0

6 ME 43 Systems Dynamcs & Control Thus, e = + + Chapter 6: Electrcal Systems and Electromechancal Systems 3 The combned resstance s gven by = In general, n = = 3 A e e e e 3 B Fgure 6- Seres Crcut Parallel Crcut. For the parallel crcut shown n fgure 6-3, 3 e 3 Snce = + + 3, t follows that] Fgure 6-3 Parallel Crcut e e e =, =, = 3 3 e e e e = + + = 3 where s the combned resstance. Hence, or = /0

7 ME 43 Systems Dynamcs & Control In general 3 = = n = = Chapter 6: Electrcal Systems and Electromechancal Systems 3 3 Krchhoff s Current Law (KCL) (Node Law). A node n an electrcal crcut s a pont where three or more wres are joned together. Krchhoff s Current Law (KCL) states that or The algebrac sum of all currents enterng and leavng a node s zero. The algebrac sum of all currents enterng a node s equal to the sum of all currents leavng the same node Fgure 6-4 Node. As appled to Fgure 6-4, krchhoff s current law states that or = = Enterng currents Leavng currents Krchhoff s Voltage Law (KVL) (Loop Law). Law (KVL) states that at any gven nstant of tme Krchhoff s Voltage or The algebrac sum of the voltages around any loop n an electrcal crcut s zero. The sum of the voltage drops s equal to the sum of the voltage rses around a loop. 7/0

8 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Fgure 6-5 Dagrams showng voltage rses and voltage drops n crcuts. (Note: Each crcular arrows shows the drecton one follows n analyzng the respectve crcut) A rse n voltage [whch occurs n gong through a source of electromotve force from the negatve termnal to the postve termnal, as shown n Fgure 6-5 (a), or n gong through a resstance n opposton to the current flow, as shown n Fgure 6-5 (b)] should be preceded by a plus sgn. A drop n voltage [whch occurs n gong through a source of electromotve force from the postve to the negatve termnal, as shown n Fgure 6-5 (c), or n gong through a resstance n the drecton of the current flow, as shown n Fgure 6-5 (d)] should be preceded by a mnus sgn. Fgure 6-6 shows a crcut that conssts of a battery and an external resstance. E B r A Fgure 6-6 C Electrcal Crcut. Here E s the electromotve force, r s the nternal resstance of the battery, s the external resstance and s the current. Followng the loop n the clockwse drecton( A B C D ), we have e + e + e = 0 or From whch t follows that AB BC CA E r = E = + r 0 8/0

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13 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Transfer Functons of Cascade Elements. Consder the system shown n Fgure 6.8. Assume e s the nput and e o s the output. The capactances C and E s E s. C are not charged ntally. Let us fnd transfer functon o ( ) ( ) e C C eo Fgure 6-8 Electrcal system The equatons of ths system are: Loop + ( ) dt = e (6-7) C 3/0

14 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Loop ( ) dt + + dt = 0 (6-8) C C Outer Loop dt = e o (6-9) C Takng LT of the above equatons, assumng zero I. C s, we obtan I ( s) + I ( s) I ( s) = E ( s) Cs (6-0) I ( s) I ( s) I ( s) I ( s) 0 Cs + + = (6-) Cs I ( s) = Eo ( s) (6-) Cs From Equaton (6-0) I ( s) + I ( s) I ( s) = E ( s) C s C s I ( s) Substtute I ( s ) nto Equaton (6-) E E s I s C s ( ) + ( ) ( ) + ( ) = = C s + C s + C s C se s I s ( s) = ( ) ( ) E s C C s + C + C + C s + o = s C C ( C + C + C ) + s + C C C C whch represents a transfer functon of a second order system. The characterstc polynomal (denomnator) of the above transfer functon can be compared to that of a second order system s + ζω s + ω. Therefore, one can wrte n n ωn = and ζω = C C or ( ) ω ( C C ) ( C + C + C ) n C C ( ) C + C + C C + C + C ζ = = C C n Complex Impedance. In dervng transfer functons for electrcal crcuts, we frequently fnd t convenent to wrte the Laplace-transformed equatons drectly, wthout wrtng the dfferental equatons. Table 6- gves the complex mpedance of the bascs passve elements such as resstance, an nductance L, and a capactance C. Fgure 6-9 shows the complex mpedances Z and Z n a seres crcut whle Fgure 6-9 shows the transfer functon 4/0

15 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems between the output and nput voltage. emember that the mpedance s vald only f the ntal condtons nvolved are all zeros. The general relatonshp s E( s) = Z( s) I s corresponds to Ohm s law for purely resstve crcuts. (Notce that, lke resstances, mpedances can be combned n seres and n parallel) ( ) Fgure 6-9 Z Z e e e E( s) Z = Z + Z = I ( s) Electrcal crcut Dervng Transfer Functons of Electrcal Crcuts Usng Complex Impedances. The TF of an electrcal crcut can be obtaned as a rato of complex mpedances. For the crcut shown n Fgure 6-0, assume that the voltages e and e o are the nput and output of the crcut, respectvely. Then the TF of ths crcut can be obtaned as Z ( s) I ( s) ( ) ( ) E ( ) Z ο s Z( s) = = E ( s) Z ( s) I s + Z ( s) I s Z ( s) + Z ( s) e (nput) Z e o (output) For the crcut shown n Fgure 6-, Fgure 6-0 Electrcal crcut Z = Ls +, Z = Cs Eο ( s) Hence, the transfer functon, s E ( s) ( ) ( ) Eο s Z s = = Cs = E ( ) s Z( s) + Z( s) LCs + Cs + Ls + + Cs Z L e (nput) Z C e o (output) Fgure 6- Electrcal crcut 5/0

16 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems 6.5 ANALOGOUS SYSTEMS Systems that can be represented by the same mathematcal model, but that are physcally dfferent, are called analogous systems. Thus analogous systems are descrbed by the same dfferental or ntegrodfferental equatons or transfer functons. The concept of analogous s useful n practce, for the followng reasons:. The soluton of the equaton descrbng one physcal system can be drectly appled to analogous systems n any other feld.. Snce one type of system may be easer to handle expermentally than another, nstead of buldng and studyng a mechancal system (or a hydraulc system, pneumatc system, or the lke), we can buld and study ts electrcal analog, for electrcal or electronc system, n general, much easer to deal wth expermentally. Mechancal-Electrcal Analoges Mechancal systems can be studed through ther electrcal analogs, whch may be more easly constructed than models of the correspondng mechancal systems. There are two electrcal analoges for mechancal systems: The Force-Voltage Analogy and The Force Current Analogy. Force Voltage Analogy and the electrcal system of Fgure 6-4(b). Consder the mechancal system of Fgure 6-4(a) e L C Fgure 6-4 Analogous mechancal and electrcal systems. The equaton for the mechancal system s d x dx m + b + kx = p (6-4) dt dt where x s the dsplacement of mass m, measured from equlbrum poston. The equaton for the electrcal system s d L + + dt = e dt C In terms of electrcal charge q, ths last equaton becomes 6/0

17 ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems d q dq dt dt C L + + q = e (6-5) Comparng equatons (6-4) and (6-5), we see that the dfferental equatons for the two systems are of dentcal form. Thus, these two systems are analogous systems. The terms that occupy correspondng postons n the dfferental equatons are called analogous quanttes, a lst of whch appear n Table 6- TABLE 6- Mechancal Systems Force p (Torque T ) Mass m (Moment of nerta J ) Vscous-frcton coeffcent b Sprng constant k Dsplacement x (angular dsplacement θ ) Velocty ẋ (angular velocty θ ) Force Voltage Analogy Electrcal Systems Voltage e Inductance L esstance ecprocal of capactance, C Charge q Current Force Current Analogy the textbook Page The student s advsed to read ths secton from 6.6 MATHEMATICAL MODELING OF ELCTOMECHANICAL SYSTEMS To control the moton or speed of dc servomotors, we control the feld current or armature current or we use a servodrver as motor-drver combnaton. There are many dfferent types of servodrvers. Most are desgned to control the speed of dc servomotors, whch mproves the effcency of operatng servomotors. Here we shall dscuss only armature control of a dc servomotor and obtan ts mathematcal model n the form of a transfer functon. 7/0

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