Chapter 6 Electrical And Electromechanical Systems - KFUPM

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical SystemsChapter 6Electrical Systems andElectromechanical SystemsA. Bazoune6.1INTRODUCTIONThe majority of engineering systems now have at least one electrical subsystem. Thismay be a power supply, sensor, motor, controller, or an acoustic device such as a speaker. Soan understanding of electrical systems is essential to understanding the behavior of manysystems.6.2ELECTRICAL ELEMENTSCurrent and Voltage Current and voltage are the primary variables used todescribe a circuit’s behavior. Current is the flow of electrons. It is the time rate of change ofelectrons passing through a defined area, such as the cross-section of a wire. Becauseelectrons are negatively charged, the positive direction of current flow is opposite to that ofelectron flow. The mathematical description of the relationship between the number ofelectrons ( called charge q ) and current i isi dqdtorq ( t ) i dtThe unit of charge is the coulomb (C) and the unit of current is ampere (A), which is onecoulomb per second.Energy is required to move a charge between two points in a circuit. The work perunit charge required to do this is called voltage. The unit of voltage is volt (V), which isdefined to be joule per coulomb. The voltage difference between two points in a circuit is ameasure of the energy required to move charge from one point to the other.Active and Passive Elements.Circuit elements may be classified asactive or passive. Passive Element: an element that contains no energy sources (i.e. the elementneeds power from another source to operate); these include resistors,capacitors and inductors Active Element: an element that acts as an energy source; these includebatteries, generators, solar cells, and op-amps.1/20

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical SystemsCurrent Source and Voltage Source A voltage source is a device thatcauses a specified voltage to exist between two points in a circuit. The voltage may be timevarying or time invariant (for a sufficiently long time). Figure 6-1(a) is a schematic diagram ofa voltage source. Figure 6-1(b) shows a voltage source that has a constant value for anindefinite time. Often the voltage is denoted by E or V . A battery is an example of this typeof voltage.A current source causes a specified current to flow through a wire containing thissource. Figure 6-1(c) is a schematic diagram of a current sourcee (t )Ei (t )Figure 6.1(a) Voltage source; (b) constant voltage source; (c) current sourceResistance elements.The resistanceR whereeRis the voltage across the resistor andof resistance is the ohmiRof a linear resistor is given byeRiis the current through the resistor. The unit( Ω ) , whereohm RvoltampereieRResistances do not store electric energy in any form, but instead dissipate it as heat. Realresistors may not be linear and may also exhibit some capacitance and inductance effects.PRACTICAL EXAMPLES: Pictures of various types of real-world resistors are found below.Wirewound ResistorsWirewound Resistors in Parallel2/20

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical SystemsWirewound Resistors in Series and in ParallelCapacitance Elements.Two conductors separated by a nonconductingmedium form a capacitor, so two metallic plates separated by a very thin dielectric materialform a capacitor. The capacitance C is a measure of the quantity of charge that can be storedfor a given voltage across the plates. The capacitance C of a capacitor can thus be given byC whereqis the quantity of charge stored andof capacitance is the faradqecec is the voltage across the capacitor. The unit( F ) , whereCampere-second coulombfarad voltvoltNotice that, sincei dq dtandec q Ciec, we havedecdti Cordec 1i dtCTherefore,t1e c i dt e c ( 0 )C 0Although a pure capacitor stores energy and can release all of it, real capacitors exhibitvarious losses. These energy losses are indicated by a power factor , which is the ratio ofenergy lost per cycle of ac voltage to the energy stored per cycle. Thus, a small-valued powerfactor is desirable.PRACTICAL EXAMPLES: Pictures of various types of real-world capacitors are foundbelow.3/20

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical SystemsInductance Elements.If a circuit lies in a time varying magnetic field, anelectromotive force is induced in the circuit. The inductive effects can be classified as selfinductance and mutual inductance.Self inductance, or simply inductance, L is the proportionality constant between theinduced voltage e L volts and the rate of change of current (or change in current per second)di dtamperes per second; that is,L eLdi dtThe unit of inductance is the henry (H). An electrical circuit has an inductance of 1 henrywhen a rate of change of 1 ampere per second will induce an emf of 1 volt:Lhenry voltweber ampere second ampereThe voltage e L across the inductorLiLeLis given byeL LWhereidi Ldtis the current through the inductor. The currenti L (t )can thus be given byti L (t ) 1e L dt i L ( 0 )L 0Because most inductors are coils of wire, they have considerable resistance. The energy lossdue to the presence of resistance is indicated by the quality factor Q , which denotes the ratioof stored dissipated energy. A high value of Q generally means the inductor contains smallresistance.Mutual Inductance refers to the influence between inductors that results from interaction oftheir fields.PRACTICAL EXAMPLES: Pictured below are several real-world examples of inductors.4/20

ME 413 Systems Dynamics & ControlTABLE 6-1.ElementChapter 6: Electrical Systems and Electromechanical SystemsSummary of elements involved in linear electrical systemsVoltage-currentCapacitorv (t ) 1ti (τ )dτc 0Resistorv(t ) R i (t )Inductorv (t ) Ldi (t )dtCurrent-voltagei (t ) Cdv(t )dtVoltage-charge1v (t ) q (t )cImpedance,Z(s) V(s)/I(s)1Csi (t ) 1v (t )Rv (t ) Rdq (t )dtRi (t ) 1 tv(τ ) dτL 0v(t ) Ld 2 q (t )dt 2LsThe following set of symbols and units are used: v(t) V (Volts), i(t) A (Amps), q(t) Q(Coulombs), C F (Farads), R Ω (Ohms), L H (Henries).6.3FUNDAMENTALS OF ELECTRICAL CIRCUITSOhm’s Law.Ohm’s law states that the current in circuit is proportional to thetotal electromotive force (emf) acting in the circuit and inversely proportional to the totalresistance of the circuit. That isi wereiis the current (amperes),eRe is the emf (volts), and Ris the resistance (ohms).Series Circuit. The combined resistance of series-connected resistors is the sumof the separate resistances. Figure 6-2 shows a simple series circuit. The voltage betweenpoints A and B ise e1 e 2 e 3wheree1 i R 1 ,e2 i R2,5/20e3 i R 3

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical SystemsThus,e R1 R 2 R 3iThe combined resistance is given byR R1 R 2 R 3In general,nR Rii 1R1AiR3R2ie1ie2e3BeFigure 6-2Parallel Circuit.Series CircuitFor the parallel circuit shown in figure 6-3,ii1eR1R2Figure 6-3i1 Sincee,R1i2 e,R2i3i2R3Parallel Circuiti3 i i 1 i 2 i 3 , it follows that]i whereReeee R1 R 2 R 3 Ris the combined resistance. Hence,1111 R R1 R 2 R 3or6/20eR3

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical SystemsR 1R1R 2R 3 111R1R 2 R 2 R 3 R 3R1 R1 R 2 R 3In generaln11 R i 1 R iKirchhoff’s Current Law (KCL) (Node Law).A node in anelectrical circuit is a point where three or more wires are joined together. Kirchhoff’s CurrentLaw (KCL) states thatThe algebraic sum of all currents entering and leaving a node is zero.orThe algebraic sum of all currents entering a node is equalto the sum of all currents leaving the same node .i3i1i5i4i2Figure 6-4Node.As applied to Figure 6-4, kirchhoff’s current law states thati1 i 2 i 3 i 4 i 5 0ori1 i 2 i 3 Entering currentsi4 i5 Leaving currentsKirchhoff’s Voltage Law (KVL) (Loop Law).Kirchhoff’s VoltageLaw (KVL) states that at any given instant of timeThe algebraic sum of the voltages around any loop in an electrical circuit is zero.orThe sum of the voltage drops is equal to the sum of the voltage rises around a loop.7/20

ME 413 Systems Dynamics & ControlFigure 6-5Chapter 6: Electrical Systems and Electromechanical SystemsDiagrams showing voltage rises and voltage drops in circuits. (Note: Each circulararrows shows the direction one follows in analyzing the respective circuit)A rise in voltage [which occurs in going through a source of electromotive force from thenegative terminal to the positive terminal, as shown in Figure 6-5 (a), or in going through aresistance in opposition to the current flow, as shown in Figure 6-5 (b)] should be precededby a plus sign.A drop in voltage [which occurs in going through a source of electromotive force from thepositive to the negative terminal, as shown in Figure 6-5 (c), or in going through a resistancein the direction of the current flow, as shown in Figure 6-5 (d)] should be preceded by aminus sign.Figure 6-6 shows a circuit that consists of a battery and an external resistance.BiEARrCFigure 6-6Electrical Circuit.Here E is the electromotive force, r is the internal resistance of the battery, R is theexternal resistance and i is the current. Following the loop in the clockwisedirection( A B C D ) , we have e A B e BC eCA 0orE iR ir 0From which it follows thati 8/20ER r

ME 413 Systems Dynamics & Control6.4Chapter 6: Electrical Systems and Electromechanical SystemsMATHEMATICAL MODELING OF ELECTRICAL SYSTEMS9/20

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical Systems10/20

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical Systems11/20

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical Systems12/20

ME 413 Systems Dynamics & ControlChapter 6: Electrical Systems and Electromechanical SystemsTransfer Functions of Cascade Elements.Considerthesystemshown in Figure 6.18. Assume ei is the input and eo is the output. The capacitances C1 andC2 are not charged initially. Let us find transfer function Eo ( s ) Ei ( s ) .i1 i2R1ei i1C1Figure 6-18i2R2C2eoElectrical systemThe equations of this system are:Loop1R1i1 1 ( i1 i2 ) dt eiC1(6-17)13/20

ME 413 Systems Dynamics & ControlLoop2Outer LoopChapter 6: Electrical Systems and Electromechanical Systems11 ( i1 i2 ) dt R2i2 i2 dt 0C1C21 i2 dt eoC2(6-18)(6-19)Taking LT of the above equations, assuming zero I. C’s, we obtain1 I1 ( s ) I 2 ( s ) Ei ( s )C1s 11 I1 ( s ) I 2 ( s ) R2 I 2 ( s ) I2 ( s ) 0C1sC2 s1I 2 ( s ) Eo ( s )C2 sR1 I1 ( s ) (6-20)(6-21)(6-22)From Equation (6-20)R1 I1 ( s ) 11I1 ( s ) I 2 ( s ) Ei ( s )C1sC1s1I2 ( s )C sE ( s ) I 2 ( s )C1s 1 iR1C1s 1R1C1s 1C1sEi ( s ) I1 ( s ) Substitute I1 ( s ) into Equation (6-21)Ei ( s )Eo ( s ) 1R1C1 R2C2 s ( R1C1 R2C2 R1C2 ) s 121 R1C1 R2C2RC R(12 C2 R1C2 )s2 1 1s R1C1 R2C2R1C1 R2C2which represents a transfer function of a second order system. The characteristic polynomial(denominator) of the above transfer function can be compared to that of a second ordersystem s 2 2ζωn s ωn2 . Therefore, one can writeωn2 or1R1C1R2C2ζ and2ζωn ( R1C1 R2C2 R1C2 )R1C1 R2C2( R1C1 R2C2 R1C2 ) ( R1C1 R2C2 R1C2 )2ωn ( R1C1R2C2 )2 R1C1 R2C2Complex Impedance.In deriving transfer functions for electricalcircuits, we frequently find it convenient to write the Laplace-transformed equations directly,without writing the differential equations.Table 6-1 gives the complex impedance of the basics passive elements such asresistance R , an inductance L , and a capacitance C . Figure 6-19 shows the compleximpedances Z1 and Z 2 in a series circuit while Figure 6-19 shows the transfer function14/20