Electrical Overview Ref Woud 2 3 I Q C 1C Q charge N B this is a long note and repeats much of what is is the text C 1 coul 2 50 t min 60 s t time s 1s A 1A I current work done per unit charge potential difference two points aka electromotive force EMF Power U t I t V 1V U volts 1V 1 A 1 W 1V 1 A 1 watt 2 51 source resistance inductance capacitance resistance resistance R 1 ohm 1 Ohm s law U t I t R power in a resistor Power U t I t I t R friction in mechanical system 2 52 2 2 1 1A 1 W inductance 2 53 mass of inertia in mechanical system H 1H inductance L henry 1 H t d U t L I t dt H A s 1V U t dt I t L or V s H 1A 2 54 0 d P U I L I I dt H A A s 2 55 1W t t I d inductive energy stored Eind P t dt L I I dt L I dI dt 0 0 I 1 2 L I dI I L 2 0 0 2 A H 1 J capacitance spring in mechanical system F 1F capacitance C farad 1 F t d I t C U t dt 2 56 F V s 1A I t dt U t C or A s F 1V 2 57 0 d P U I C U U t dt F V t t U d capacitive energy stored Ecap P t dt C U U t dt C U dU dt 0 0 0 V s 1W U 0 C U dU 1 2 2 U C 2 58 2 59 2 V F 1 J 1 11 13 2006 Kirchhoff s laws first number of currents sum of currents towards node 0 Ii t 0 i 1 second 2 60 direction specified sum of voltages around closed path 0 number of voltages Ui t 0 i 1 2 61 series connection of resistance and inductance U t Um cos t imposed external Um amplitude of voltage V 1V frequency Hz 1 t time min 60 s I t Im cos t resulting current assumed also harmonic Im amplitude of current 2 62 1 s A 1 amp 2 63 phase lag angle it is useful to represent this parameters as vectors using complex notation where the values are represented by the real parts Uz t Um cos t Um sin t i Iz t Im cos t Im cos t i Imaginary parts of Uz t Iz t plotting set up Uz t Iz t 1 0 5 0 0 0 5 1 Real parts of Uz t Iz t U t I t over R voltage drop will be U t R I t R I cos t R m over L voltage drop will be UR t R Im cos t cos cos d UL t L I t L Im sin t dt d L I t Im L sin t Im L cos t 2 dt 2 11 13 2006 sin 2 cos cos cos sin sin 0 cos 1 sin 2 2 2 cos or UzR t R Im cos t R Im sin t i in complex vector notation UzL t Im L cos t Im L sin t i 2 2 Imaginary parts of Uz t UzR t UzL t plotting set up Uz t UzR t UzL t 1 0 8 0 6 0 4 0 2 0 0 2 0 0 2 0 4 0 6 0 8 1 Real parts of Uz t UzR t UzL t U t UR t UL t at this point these vectors are shown with two unknowns included I m and i e directions are correct relatively given and magnitudes arbitrary given I m Kirchoff s second law U t UR t UL t R Im cos t L Im sin t Um cos t R Im cos t L Im cos t 2 this can be solved for and Im after expanding the rhs into sines and cosines and setting cos cos and sin sin easier if think in terms of vectors 3 11 13 2006 Imaginary parts of Uz t UzR t UzL t Uz t UzR t UzL t UzL t rel to UzR t UzR t UzL t 1 0 8 0 6 0 4 0 2 0 0 2 0 0 2 0 4 0 6 0 8 1 Real parts of Uz t UzR t UzL t U t UR t UL t for UzR t zL t to Uz t magnitude and angle must be Um R Im 2 L Im 2 UzR t R Im cos t i R Im sin t R L Im 2 2 Uz t Um cos t i Um sin t UzL t L Im sin t i Im L cos t or and Im Um R L 2 2 L R atan using these relationships in the plot plotting set up 4 11 13 2006 Uz t UzR t UzL t UzL t rel to UzR t UzR t UzL t Imaginary parts of Uz t UzR t UzL t etc 1 0 8 0 6 0 4 0 2 0 0 2 0 0 2 0 4 0 6 0 8 1 Real parts of Uz t UzR t UzL t etc U t UR t UL t etc N B angle may not appear as right angle due to scales shown as lag positive value with negative sign capacitor lead approach text similar for Capacitance imposed external U t Um cos t Um amplitude of voltage V 1V frequency Hz 1 1 s 2 62 min 60 s t time this is different from text lag phase angle vs lead angle used resulting current assumed also harmonic I t Im cos t current assumed to have lag angle this approach taken to allow common treatment of L and C in circuits Im amplitude of current V 1V phase lag angle complex vector representation set up with real part expressed as cos Uz t Um cos t Um sin t i Iz t Im cos t Im sin t i plotting set up 5 11 13 2006 Imaginary parts of Uz t Iz t voltage and current at omega t positive lag phase angle Uz t Iz t 1 0 5 0 0 0 2 0 4 0 6 0 8 1 Real parts of Uz t Iz t U t I t voltage across capacitor from above t 2 57 t I cos t Im sin t I t Im m dt dt UC t cos t C C 2 C C 0 0 using complex vector notation Uz t Um cos t Um sin t i UzC t Im C Iz t Im cos t Im cos t i cos t Im sin t i 2 C 2 UzR t R Im cos t R Im sin t i Kirchoff s …