Transcription of Introduction: ModelingElectricCircuits - usna.edu
1 ODEsandElectricCircuits1 Introduction: ModelingElectricCircuitsInt roduction:ModelingElectricCircuitsOneof the topicsthatyou will studyin your Physicsand ElectricalEngineeringcoursesat the Naval Academy is thatof single-loop [a] ThedevicelabeledEMF,which standsforelectromotiveforce, can be a batteryor to the circuit .[b] ThedevicelabeledCis acapacitorwhich storescharge.[c] ThedevicelabeledRis aresistorwhich opposesthe ow of current.[d] ThedevicelabeledLis aninductorwhich opposeschangein the ow of the the circuitis a switch, shown switch is closed,acurrentI(t) runsthroughthe circuitwhich changesthe chargeQ(t) at the measurements are, theirsymbols, and (sec)chargeQ(t)CoulombscurrentI(t)Ampere s(amps)electromotive force(EMF)E(t)Volts (V)Whenthe current is owing,thereis a correspondingvoltagedropacrosseach of devicesymbols, units,and :ModelingElectricCircuitsODEsandElectric Circuits2 Introduction: ModelingElectricCircuitsCir cuitDevicesdevicenamesymbolunitsvoltaged ropcapacitorCFarads(F)EC=1C QresistorROhms( )ER=R IinductorLHenries(H)EL=L dIdtAt any timet, the currentI(t) and the chargeQ(t) are relatedby two important rules:1.
2 Thecurrent equalsthe rateof changeof the chargewithrespect to time(1 amp= 1coulomb/sec):I(t) =dQ(t) 's Law:(theelectricalequivalent of Newton'ssecondlaw of motion)The sum of the voltagedropsaroundthe closedcircuitis equalto the voltagesupplied:EL+ER+EC=EorLdIdt+R I+1CQ=ESinceI=dQdt, we getLd2 Qdt2+RdQdt+1CQ=Ewhich is a secondorderordinarydi erential equation| orODE| for the chargeQ(t).Thereare two specialcasesthatwe can solve Circuits:If thereis no inductor,we can investigatea rstorderODEfor thechargeQRdQdt+1CQ=EODE sandElectricCircuits2 Introduction: ModelingElectricCircuitsODE sandElectricCircuits3 Introduction: ModelingElectricCircuitsR-L Circuits:If thereis no capacitor,we can investigatea rstorderODEfor thecurrentILdIdt+R I=EAsolutionto an ODEis any functionthatsatis esthe we know the initialstateof the circuit | namelythe chargeor current measurements | thenwe have aninitialconditionorIC.
3 An ODEplusan IC is : ModelingElectricCircuits
