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20electri_theory - Ref Woud 2.3 I= Q t t = time Electrical...

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Electrical Overview Ref: Woud 2.3 N.B. this is a long note and repeats much of what is is the text Q Q = charge C = 1 C C = 1 coul I = t (2.50) t = time min = 60 s s = 1 s I = current A = 1 A work done per unit charge = potential difference two points U = volts V A 1 W 1V 1 = = 1 V aka electromotive force (EMF) 1V 1 source, resistance, inductance, capacitance resistance Power U t ( ) I t ( ) A 1 watt = = (2.51) friction in mechanical system resistance = R = Ω 1 Ω ohm = 1 Ω Ohm's law (2.52) U t ( ) I t ( ) R = 2 2 power in a resistor ... (2.53) Power U t ( ) I t ( ) I t ( ) R 1 Ω (1A) = 1 W = = inductance mass of inertia in mechanical system V s H 0 U I L I H A inductance = L H 1 H henry 1 H = = t A U t ( ) d dt (2.54) U t ( ) L I t ( ) 1 V or ... I t ( ) d 1 A t = = = = L H s (2.55) A d I P 1 W = = = dt s t I I t L I L I L I 0 0 0 0 capacitance spring in mechanical system 1 2 I d I (2.56) inductive_energy_stored E ind P t ( ) d d dI dI L t t = = = = 2 dt 2 A H 1 J = capacitance = C F = 1 F farad 1 F = t A s 0 d F V V V U I I t ( ) d dt U t ( ) P F = (2.57) I t ( ) C 1 A or ... U t ( ) C U d 1 V t = = = = C F s U t ( ) 1 W = = dt s t U U t 1 2 U (2.58), (2.59) d C U C U C U 0 0 0 0 1 11/13/2006 capacitive_energy_stored E cap P t ( ) d U t ( ) d dU dU C t t = = = = 2 dt 2 V F 1 J =
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Kirchhoff's laws first ... number_of_currents sum_of_currents_towards_node = 0 I i ( ) t = 0 (2.60) second ... i = 1 sum_of_voltages_around_closed_path = 0 direction specified number_of_voltages U i ( ) t = 0 (2.61) i = 1 series connection of resistance and inductance ... imposed ... external U t ( ) := U m cos ( ω⋅ t ) U m = amplitude_of_voltage V = 1 V (2.62) 1 ω = frequency Hz = 1 s t = time min = 60 s resulting current assumed also harmonic I t ( ) := I m cos ( ω⋅ t φ ) I m = amplitude_of_current A = 1 amp (2.63) φ = 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 ( ) := U m cos ( ω⋅ t ) + U m sin ( ω⋅ t ) i Iz t ( ) := I m cos ( ω⋅ t φ ) + I m cos ( ω⋅ t φ ) i plotting set up 0 0.5 1 Uz(t) Iz(t) Imaginary parts of Uz(t), Iz(t) 0 0.5 1 Real parts of Uz(t), Iz(t) = U(t), I(t) over R voltage drop will be ... U R ( ) t := ( ) ( −ω ) t ( ) = ( ) ( ) ( −α ) R I t R I m cos + φ U R t R I m cos ω⋅ t − φ cos α = cos over L voltage drop will be ... U L ( ) t := L d I t ( ) L I m sin ( −ω ) t + φ ⋅ω dt L d I t ( ) = I m ⋅ω⋅ L sin ( ω⋅ t φ ) = I m ⋅ω⋅ L cos π + ω⋅ t φ dt 2 2 11/13/2006
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π → − ( ) π π ( ) π ( ) ( ) ( ) cos + α sin α cos + α = cos cos α sin sin α = 0 cos α 2 2 2 2 or ... 1 sin α in complex (vector) notation ...
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