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# Electrical Principles ECET 3000

GPA 3.57

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This 28 page Class Notes was uploaded by Mrs. Maud Gutkowski on Tuesday October 20, 2015. The Class Notes belongs to ECET 3000 at Southern Polytechnic State University taught by Jeff Wagner in Fall. Since its upload, it has received 14 views. For similar materials see /class/225431/ecet-3000-southern-polytechnic-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Southern Polytechnic State University.

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Date Created: 10/20/15

ECET 3000 Electrical Principles Introduction to Transformers Chapter 14 Simple ACsupplied Magnetic Circuit If a voltage is applied to a magnetic source coil that is coupled to a magnetic core a current will ow in the coil which in turn will create a magnetic eld within the core Vt v Simple ACsupplied Magnetic Circuit If an AC source is applied to the coil the resultant field strength may be determined from quM I Va Np T Faraday s Law Simple ACsupplied Magnetic Circuit Given m I Mo I 39 E V0 COSamp I V Npgi E lt I I 3 The field will be x E Ma NLjvmdr NLHEVRMS cosatdt 5 Np Asinmt RMSsinat a Simple ACsupplied Magnetic Circuit Thus a sinusoidal o quotquot Wquot quotquotquot quotx voltage source will lg i E create a sinusoidal quot 0 PgIF E eld that is phase 5 gt E shifted by 90 quot I39 Va J5 VRMS cosa I U MtJ qgtRMS sinmtJ qgtRMS cosat 90 Simple ACsupplied Magnetic Circuit If the core has a nite gt Mt permeability u a ig 5 E magnetization current will m NpgiD E be drawn into the coil to lt I g E sustain the field The E E magnetizing current can quot be solved from NP z39t MtSR where SR is the reluctance ofthe field path core Simple ACsupplied Magnetic Circuit If the core is made from a I gt Mt linear material chonstant ig 39 and thus has a constantSR m then the current will be I directly proportional to the E magnetic eld as de ned c gt Nplt c gt by MtSR 0 N F Simple ACsupplied Magnetic Circuit Thus a linear core will l quotquotMquot quotquotquot quot result in the following lg current waveform quot 0 MpgF when expressed in 3 terms of the original voltage parameters m W t 90 Np a Simple ACsupplied Magnetic Circuit n Mt The current expression 39 c gt t N may be rewritten as V f4 v z39t JERMS cosat 90 where I S39SR VRMS VRMS M N2 a sz aJL a Simple ACsupplied Magnetic Circuit Note that substituting m l quotquot quotquotMquot quotquotquot quotI I I l N 2 lt l39 i S L vt Npg E into the previous E i equation provides a quot relationship for the selfinductance of a coil with respect to the magnetic circuit parameters Mutually Linked Coils Given the previously defined magnetic circuit what would happen if we added a second coil that is also coupled to or wrapped around the magnetic core Mutually Linked Coils If we assume that the field stays within the core then the entire field created by the source coil will ow down through the second coil Since this field is sinusoidal we can also apply Faraday s Law to the second coil it C V N p lt lt VVVV Z In Mutually Linked Coils To do so properly we need to first define voltage polarities The coil voltages and respective polarities are shown in the figure below Note that the voltage on the second coil opposes the original source voltage as defined by Lenz s Law Lenz s Law Any induced effect will always oppose its source 1 quotwe lg f ltgt t N c 339 c I gt Vquot Pggt Ep Eslt gtNS v gt I i 39 39 Mutually Linked Coils Thus the voltage on the second coil is defined by ES 2 NS dana dt Mutually Linked Coils Yetdq j is the same for both coils thus the following relationship may be derived by solving for wax in both coils and equating the results Es 3 iamp p N E s 1 quotwe 393 J 7 lt gt ltgt c r vt Npltgt Ep ESltTgtNS lt ltgt c r 39 I 39 r I Ideal Transformer The voltage relationship f gt Mt l 399 39 39 i1 lt i quot 0 Nplt Ep Eslt39 N i gt 24 i i 39 l Ideal Transformer Simplified physical drawing of the model it quot gt M quot I I lt V N r lt Ideal Transformer Circuit Simplified physical drawing of the model Tp Ts We gtEs Nquot N E E 5 NF a a E Turns Ratlo NS Ideal Transformer Circuit Current Relationship Tp 395 E N R7 Es I Zload 7p p a ES NS NPNS NitCDtiR NPiptCDMtiRNsixt N T N i P P 5 Ideal Transformer Circuit Simplified physical drawing of the model Tp ls vp c Zload Nquot N Ideal Transformer Definitions Primary Coil side E the side of the transformer into which power ows source side Tp ls vp 0 Zload N N Secondary Coil side E the side of the transformer from which power is delivered load side Ideal Transformer Definitions High Voltage Side E the side of the transformer with the larger voltage magnitude coil with the larger number of turns Tp ls vp c Zload Nquot N Low Voltage Side E the side of the transformer with the smaller voltage magnitude coil with the smaller number of turns Ideal Transformer Example Problem A 200400 volt source is connected to the primary winding of a transformer having a turns ratio of A A load impedance of Zload4Q is connected to the secondary winding Determine the source current as well as the load voltage and current a M Ts lp 200z0 Q EpC Es 49 Nquot N source voltage and power g v56 Power System Example Problem A load impedance of Zload IQ requires a supply voltage of 100400 volts The load is far from the actual voltage source requiring a long wire to connect the load to the source The wire has a total effective resistance of IQ Determine the power consumed by the load as well as the required Power System Example Problem A load impedance of Zload IQ requires a supply voltage of 100400 volts The load is far from the actual voltage source requiring a long wire to connect the load to the source The wire has a total effective resistance of IQ A transformer is also placed at each end of the long wire as shown in the figure Determine the power consumed by the load as well as the required source voltage V5 and power Power System Example Problem ECET 3000 Electrical Principles Resistors Inductors amp Capacitors in SteadyState AC Circuits Resistors Resistors are deVices that resist or oppose the ow of current in an electric circuit A potential force voltage must exist across a resistor for a current to ow through the resistor Electrical energy is consumed by the resistor during this process and is converted into thermal energy causing the resistor to heatup van quotquot quotquot quotA iRt Resistor VI Relationship Whether supplied by a DC or an AC source the voltagecurrent relationship for a resistor remains the same as defined by Ohm s Law van VR DC R W mm Ram quotquotquotquotquotquotquotquot iRt JD Resistor Power The rate of instantaneous energy consumption power by the resistor may is defined by pRt VRt39iRt iRt 50 van quotquot quotquot quotA For steadystate DC power is constant as defined by PR 2 VR I DC 39 RDC Resistor Power AC Power is timevarying If the voltage and current are defined In by their RMS magnitudes as follows vRa 4514 sinat 1R z 451 sinat note they are iiiphase Then the resultant power will be quot mm 50 van pRt 2VM IMsin2atVWIMl cos2at Resistor Power For AC systems it is typically the average value for power that is considered PRAC VVRMS IRMS iRt 50 van quotquot quotquot quotA This result is similar to resistor power in a DC circuit provided that the RMS values of the AC voltage and current are utilized Resistor Power It is for this reason that RMS magnitudes are often referred to as effective values iRt quot mm 50 van In other words an AC source whose RMS magnitude equals the magnitude of a given DC source will supply the same average power to a resistor as PM 2 VW Jim DC the source PR VR 1 DC 39 RDC Inductors An inductor is basically a coil of wire in which a timevarying magnetic eld is created proportional to the magnitude of the inductor s current VLt iLt 39 quotquotquot quotquot A If also timevarying the magnetic field will induce a counteremf voltage across the coil proportional to the field s rate of change The induced voltage will oppose the source voltage thus limiting the rate of change in the current Inductors Energy is required to initially create the magnetic field in the inductor but once it is created the field requires no energy to maintain it M The electrical energy is basically converted into a magnetic form iLt 39 quotquotquot quotquotquotquotquotA If the field strength is decreased due to a decrease in current the energy is converted back to electrical and released back into the circuit Inductor VI Relationship Unlike resistors which have a linear VI relationship inductors have a nonlinear VI relationship where the voltage is proportional to the rate of change of the current diL t dt iLt lt 3 I quotquotquot quotquot A vLtL iL t jVL tdt an tdt 10 0 Inductor VI Relationship If the current is sinusoidal in nature z39Lt JEJW sinat Then the resultant inductor voltage can be defined by iLt 39 VLt quotquotquot quotquotquotquotquotA vLtLLJ ILMcoscota Elm aLsinat90 Inductor VI Relationship Thus given the inductor s current and induced voltage iLt JEJLM sina t vLa 1LmmLsinmt90 iLt lt 3 I quotquotquot quotquot A It can be seen that the inductor s voltage is phase shifted ahead of the current by 90 and has an RMS magnitude defined by V I aL LRMS LRMS Inductor VI Relationship If the inductor s voltage and current are expressed by their phasor values VL V 490 LRMS iLt 39 VLt L 1LM40 quotquotquot quotquotquotquotquotA A phasor Vl relationship may be expressed using complex numbers as IE ltwLz90 12 mm Inductive Reactance Although the inductor s phasor Vl relationship looks similar to Ohm s Law the inductor does not resist current in the traditional sense VLt iLt 39 quotquotquot quotquot A Instead the inductors reaction is to oppose any change in current s magnitude such that the inductor s voltage is proportional to the rate of change in the current instead of the current s magnitude Inductive Reactance The nature of the inductor s reaction is to also cause a 90quot phase shift between its voltage and current waveforms iLt 39 VLt quotquotquot quotquotquotquotquotA The expression ij accounts for both the inductor s rate of change and phase shift reactions to current ow Inductive Reactance The nature of the inductor s reaction is to also cause a 90quot phase shift between its voltage and current waveforms VLt iLt 39 quotquotquot quotquot A The expression ij accounts for both the inductor s rate of change and phase shift reactions to steadystate AC current flow and thus is termed inductive reactance Inductive Reactance Reactance is typically denoted using the variable X Thus inductive reactance may be defined as JXL wL iLt 39 VLt quotquotquot quotquotquotquotquotA Such that it satisfies the inductor s Vl relationship in phasor form V IL jXL Capacitors A capacitor is basically two plates of conductive material that are placed parallel to each other and separated by a thin layer of insulation Vct When subjected to an extemally supplied voltage current will ow into the capacitor to charge the plates one positive and the other negative to create the electric field between the plates required by VCjEd1 b ict quot A Capacitors The rate at which charge is stored in the capacitor ie current ow is proportional to the rate of change of the applied voltage ict vct C 4 Since current must ow into the capacitor to buildup the voltage the created voltage will lag behind the applied current Thus a capacitor tends to oppose a change in voltage Capacitor VI Relationship have a nonlinear Vl relationship but with the voltage and current terms reversed compared to the inductor dvc I dz Similar to inductors capacitors also I ict quotquot A vct C iCtC 1 1 vCt 25 10mm EjzczdzVo 700 0 Capacitor VI Relationship As with the inductor we can solve the Vl relationship ict Given voltage W 5 ch sinw 0 0 vct A We can solve for the current dVCO 1Ct CT CJ VCRMS cosata J VCMmCsinat90 Capacitor VI Relationship Thus given the capacitor s voltage and resultant current W quot A W I ch sinwr m ict J5ch wC sinat90 0 It can be seen that the capacitor s current is phase shifted ahead of the voltage by 90 and has an RMS magnitude defined by I V aC CRMS CRMS Capacitor VI Relationship Thus similar to inductors if the capacitor s voltage and current are expressed by their phasor values VC V 40 ict 0 vct TC Icmz90 A phasor VI relationship may be expressed using complex numbers as 7C ZTCL49OOZTCC QC 0 quotquotquot quotquot 39 1 Capacitive Reactance Capacitive Reactance is the term used to describe the capacitors reaction to a steadystate AC voltage accounting for its opposition to a change in voltage magnitude and the resultant 90 phase shift between the voltage and current waveforms ict quot A vct Capacitive Reactance Thus capacitive reactance may be 4 defined as W 1 1 E X 2 39 vct C J C j wC J wC Such that it satisfies the inductor s Vl relationship in phasor form fC ZINC39j XC SteadyState AC Circuit Analysis Since both inductive reactance and capacitive reactance satisfy VI relationships similar to Ohm s Law when the voltages and currents are expressed as phasors all of the basic DC circuit theory derived from Ohm s Law may also be applied to steady state AC circuit analysis

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