20electri_theory

# 20electri_theory - Ref Woud 2.3 I= Q t t = time Electrical Overview Q = charge C = 1C min = 60 s A = 1A U = volts 1V 1 A = 1 watt C = 1 coul s = 1s

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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 A1 W 1V 1 = = 1 V aka electromotive force (EMF) source, resistance, inductance, capacitance resistance Power U t () It () A 1 watt = = (2.51) friction in mechanical system resistance = R = Ω 1 Ω ohm = 1 Ω Ohm's law (2.52) Ut () It ()R = 2 2 power in a resistor . .. (2.53) Power U t ()It R 1 Ω (1A) = 1W = = inductance mass of inertia in mechanical system Vs H 0 U I LI HA inductance = L H 1 henry 1 H = = t A d dt (2.54) () L It 1V or . .. d 1A t = = = = L H s (2.55) A d I P = = = dt s t I I t 0 0 0 0 capacitance spring in mechanical system 1 2 I d I (2.56) inductive_energy_stored E ind Pt () d d dI dI L t t = = = = 2 dt 2 A H 1J = capacitance = C F = F farad 1 F = t As 0 d FV V V U I d dt P F = (2.57) () C 1A or . .. CU d 1V t = = = = C F s = = dt s t U U t 1 2 U (2.58), (2.59) d 0 0 0 0 1 11/13/2006 capacitive_energy_stored E cap dU dU C t t = = = = 2 dt 2 V F =

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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 Ut () := 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 It := I m cos ( 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 ( t ) + U m sin ( t ) i Iz t := I m cos ( t φ ) + I m cos ( 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 RI m cos + φ U R t R I m cos t − φ cos α = cos over L voltage drop will be . .. U L t := L d LI m sin ( −ω ) t + φ ⋅ω dt L d = I m ⋅ω⋅ L sin ( t φ ) = I m L cos π + ω⋅ t φ dt 2 2 11/13/2006
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 . .. Uz R t := R I m cos ( ω⋅ t − φ ) + RI m sin ( t − φ ) i π π Uz L t := I m ⋅ω⋅ L cos + ω⋅ t − φ + I m L sin + ω⋅ t − φ ⋅ i 2 2 plotting set up 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 φ i.e. directions are correct relatively given φ

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## This note was uploaded on 02/24/2012 for the course MECHANICAL 2.611 taught by Professor Davidburke during the Fall '06 term at MIT.

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20electri_theory - Ref Woud 2.3 I= Q t t = time Electrical Overview Q = charge C = 1C min = 60 s A = 1A U = volts 1V 1 A = 1 watt C = 1 coul s = 1s

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