Euler s Identity j e cos j sin Euler s Identity Substituting in Euler s Identity we obtain e j cos j sin 1 j 0 e j Most beautiful relationship in Number theory Real Exponentials x t x t x t0 x t e t t0 A real exponentials is uniquely identified by 3 parameters x t0 and x t Sum of 2 or more real exponentials of the same time constant results in another real exponential with time constant n ki e i 0 t n ki e i 0 t Complex Exponentials x t Ae j t A cos t jA sin t A complex exponentials is uniquely identified by 3 parameters A and Sum of 2 or more complex exponentials sinusoids of the same frequency results in another complex exponential sinusoids with frequency n Ae i 0 i j t i n j i j t Ai e e i 0 Ae e j j t A cos t j A sin t Phasor Diagram I cos t iR t vR t R v R t R I cos t 0 I cos t vR t 0 t Phasor Diagram I I e j IR VR R VR V R RI R R I e j Resistor Current is in phase with Resistor Voltage IR 0 I cos t vR t 0 t Phasor Diagram iL t vL t L I cos t diL t v L t L dt L I sin t L I cos t 0 2 I cos t vL t t 0 90D Phasor Diagram I I e IL VL L j VL 90D V L j L I L L I e IL j 2 0 Inductor Current lag s Inductor Voltage by 90 D I cos t vL t t 0 90D Phasor Diagram I cos t iC t vC t C 1 vC t iC t dt C I sin t C I cos t C 2 0 I cos t vC t t 0 90D Phasor Diagram I I e j IC VC C IC D 90 1 VC j C I j 2 e IC C 0 VC Capacitor Current leads Capacitor Voltage 90 D I cos t vC t t 0 90D Phasor Diagram I I e j IC VC C IC D 90 1 VC j C I j 2 e IC C 0 VC Capacitor Current leads Capacitor Voltage 90 D Instantaneous Pow er I cos t p t t v i d 0 vC t 0 t p 0 p 0 p 0 p 0 p 0 p 0 p 0 p 0 vR t I cos t R L C vL t vC t v t VC L I cos t C cos t 2 2 I cos t v R t R I cos t v L t L I cos t vC t I C cos t 2 2 t 0 I 0 v t R I cos t I VR VL 90D 90D V VR I I e j R L C VL VC VR VL V I 0 VC V V R V L VC Z j I v R t R I cos t v L t L I cos t I cos t v t I 2 cos t C 2 t 0 vC t 90D 90D Z j V I I e j VR R L C VL VC VR VL V I 0 VC V V R V L VC 1 R j L j C Z j I Impedance I
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