Euler’s Identitycos sinjejθθθ=+Euler’s IdentityjeπSubstitutingwe obtain :in Euler's Identity,θπ=cos sinjejπππ=+10j=− +⇒ Most beautiful relationship in Number theoryReal Exponentials•A real exponentials is uniquelyidentified by 3 parameters:• Sum of 2 or more real exponentialsof the same time constantτresults in another real exponential with time constantτ.0,(),and()xt xtτ∞[]0()0() () () ()ttxt xt xt xt eτ−−∞∞=+−00ttnniiiike k eττ−−===∑∑Complex Exponentials•A complex exponentials is uniquely identified by 3 parameters:• Sum of 2 or more complex exponentials (sinusoids) of the same frequencyωresults in another complex exponential (sinusoids) with frequencyω.,,andAωθ()()cos( ) sin( )jtxt AeAt jAtωθωθωθ+==++ +()00jtiinnjjtiiiiAe Ae eωθθω+===∑∑[][]cos( ) sin( )At jAtωθωθ=++ +()jjtAe eθωPhasor Diagramtcos( )Itωθ+()Rit()Rvt+−0θθω−0()Rvtcos( )Itωθ+R() cos( )Rvt RI tωθ=+Phasor DiagramtRIRV+−0θθω−0()Rvtcos( )Itωθ+RjRRVRIRIeθ==jIIeθ=Resistor Current is ∴ with in phaseResistor Voltage.RVRIPhasor Diagram()Lit()Lvt+−0θL()()LLdi tvt Ldt=sin( )LI tωωθ=− +cos( )2LI tπωωθ=++tθω−0()Lvtcos( )Itωθ+cos( )Itωθ+90ωDPhasor DiagramLILV+−0θL2()()jLLVjLI LIeπθωω+==Inductor Current s lag∴Inductor Voltage by 90 .Dtθω−0()Lvtcos( )Itωθ+jIIeθ=90ωDLVLI90DPhasor Diagram()Cit()Cvt+−0θC1() ()CCvt itdtC=∫sin( )ItCωθω=+cos( )2ItCπωθω=+−tθω−0()Cvtcos( )Itωθ+cos( )Itωθ+90ωDPhasor DiagramCICV+−0θC()21jCCIVIejC Cπθωω−==Capacitor Current leads∴tθω−0()Cvtcos( )Itωθ+jIIeθ=90ωDCapacitor Voltage 90 .D90DCVCIPhasor DiagramCICV+−0θC()21jCCIVIejC Cπθωω−==Capacitor Current leads∴tθω−0()Cvtcos( )Itωθ+jIIeθ=Capacitor Voltage 90 .D90D0p<0p>0p<0p>0p<0p>0p<0p>Instantaneous Power0() ()()tpt v i dτττ=∫CVCI()Cvt+−0θC() cos( )vt RI tωθ=+tθω−0cos( )Itωθ+90ωDLR()Rvt+−()Lvt−+()vtLVRVIcos( )2LI tπωωθ+++cos( )2ItCπωθω++−90ωD() cos( )2CIvt tCπωθω=+−() cos( )2Lvt LI tπωωθ=++() cos( )Rvt RI tωθ=+cos( )Itωθ+CVCV+−0θC()LR CV ZV jV IVω=++=tθω−0cos( )Itωθ+90ωDLRRV+−LV−+VLVRVI90ωD() cos( )2CIvt tCπωθω=+−() cos( )2Lvt LI tπωωθ=++() cos( )Rvt RI tωθ=+jIIeθ=V()Zjω(()vtCVCV+−0θCR CLV VV V=++LRRV+−LV−+VLVRVIjIIeθ=VCV1R IjCLjωω=+
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