Electrical OverviewRef: Woud 2.3 N.B. this is a long note and repeats much of what is is the textQQ = charge C = 1C C = 1 coul I = t (2.50) t = time min = 60s s = 1s I = current A = 1A work done per unit charge = potential difference two points U = volts V A1W 1V 1⋅== 1V aka electromotive force (EMF) 1V 1source, resistance, inductance, capacitance resistance ⋅Power U t() It()⋅ A 1 watt = = (2.51) friction in mechanical systemresistance = R =Ω 1 Ω ohm = 1 Ω Ohm's law (2.52)Ut() It()R⋅= 2 2power in a resistor ... (2.53)Power U t()It() It()⋅ ⋅R 1Ω (1A) = 1W= = ⋅⋅⌠⎮ ⎮⌡ ⋅⋅ inductance mass of inertia in mechanical system VsH ⋅⋅ 0 ⋅UILI HA⋅inductance = L H 1 H henry 1 H = = t A Ut()d dt (2.54)Ut() L It() 1V or ... It() d 1At= ⋅===LHs ⎛⎜⎝ ⎞⎟⎠ (2.55)AdIP 1W= = = dt s t I It ⋅ ⌠⎮⌡⋅ ⌠⎮⌡⋅⋅ ⌠⎮ ⎮⌡ ⌠⎮⌡LI LI LI0 0 0 0 capacitance spring in mechanical system ⎛⎜⎝ ⎞⎟⎠ 12IdI (2.56)inductive_energy_stored Eind Pt()d d dI dI ⋅Lt t= = = = ⋅→2dt 2A ⋅H 1J= capacitance = C F = 1 F farad 1 F = t ⋅ ⋅⋅ ⌠⎮ ⎮⌡ As0 d FVV ⋅ V U IIt()⋅ d dt Ut() P ⋅F = (2.57)It() C ⋅⋅ 1A or ... Ut() CUd 1Vt====CFs Ut() 1W== dt s t U Ut 12U(2.58), (2.59)⋅ ⌠⎮⌡⋅ ⌠⎮⌡⋅⋅ ⌠⎮ ⎮⌡ ⌠⎮⌡dCU CU CU0 0 0 0 1 11/13/2006 capacitive_energy_stored Ecap Pt()d Ut() d dU dU ⋅Ct t= = = = ⋅→2dt 2V ⋅F 1J=Kirchhoff's laws first ... number_of_currents sum_of_currents_towards_node = 0 ∑ ⎡⎣Ii()t ⎦⎤ = 0 (2.60) second ... i = 1 sum_of_voltages_around_closed_path = 0 direction specified number_of_voltages ∑ ⎡⎣Ui()t⎤⎦ = 0 (2.61) i = 1 series connection of resistance and inductance ... imposed ... external Ut() := UmUm⋅cos(ω⋅t) Um = amplitude_of_voltage V = 1 V (2.62) 1ω = frequency Hz = 1 s t = time min = 60 s resulting current assumed also harmonic It() := ImIm⋅cos(ω⋅t − φ) Im = 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() := UmUm⋅cos(ω⋅t)+ Um ⋅sin(ω⋅t)⋅i Iz t() := ImIm⋅cos(ω⋅t − φ)+ Im ⋅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 ... UR()tt := ⋅ () → ⋅ ⋅ ⎡(−ω)⋅t ⎤() = ⋅ ⋅( ) () (−α)R I t RImcos⎣ + φ⎦ URt R Imcos ω⋅t − φ cos α = cosover L voltage drop will be ... UL()tt := L⋅ d It() → LI⋅ m ⋅sin⎡⎣(−ω)⋅t + φ⎦⎤⋅ω dt L⋅ d It() = −Im ⋅ω⋅L⋅sin(ω⋅t − φ) = Im ⋅ω⋅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 αin complex (vector) notation ... UzR()t := R II⋅ mm⋅cos(ω⋅t − φ)+ RI⋅ m ⋅sin(ω⋅t − φ)⋅i ⎛π⎞ ⎛π⎞UzL()t := ImIm⋅ω⋅L⋅cos ⎜ + ω⋅t − φ⎟+ Im ⋅ω⋅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 Im and φ i.e. directions are correct relatively given φand magnitudes arbitrary given Im () URt UL()t → ⋅ ⎡⎣− t + φ⎤⎦ ⋅ sin⎡⎣− t + φ⎤⎦⋅Kirchoff's second law ... Utt := () + RIm ⋅cos ( )ω ⋅ + LIm ⋅ ( )ω ⋅ ω ( ) ( ) ⎛π⎞Um ⋅cos ω⋅t = RI⋅ m ⋅cos ω⋅t − φ + LI⋅ m ⋅ω⋅cos ⎜ + ω⋅t − φ⎟ ⎝2 ⎠ this can be solved for φand Im after expanding the rhs into sines and cosines and setting cos = cos and sin = sin easier if think in terms of vectors 0 0.2 0.4 0.6 0.8 1 Uz(t) UzR(t) UzL(t)Imaginary parts of Uz(t), UzR(t), UzL(t) 3 11/13/20060.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 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) 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 = Uz t() → ⋅( )+ ⋅ ⋅sin(ω⋅t)Um cos ω⋅t iUm R +(L⋅ω)2 ⋅Im UzR()t → ⋅ ⋅ ⎡⎣(−ω + φ⎤⎦− ⋅ ⋅ ⋅sin⎡⎣ )⋅t +φ2 R Im cos )⋅t iRIm (−ω= (RI⋅ m)2 +(LI⋅ m ⋅ω)2 =Um UzL()t → ⋅ ⋅sin⎡⎣ )⋅t + φ⎤⎦⋅ω + ⋅ ⋅ ⋅ ⋅ ⎡⎣(−ω)⋅t + φ⎦⎤L Im (−ω iIm ω L cos Um or ... Im = R2 +(L⋅ω)2 ⎛ L⋅ω⎞and ... φ = atan ⎜⎟ ⎝ R ⎠ using these relationships in the plot ... plotting set up 4 11/13/2006Real 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 φ shown as lag (positive value with negative sign) 0 0.2 0.4 0.6 0.8 1 0.2 0 0.2 0.4 0.6 0.8 1 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. capacitor lead approach (text) similar for Capacitance imposed ... external Ut() := UmUm⋅cos(ω⋅t) Um = amplitude_of_voltage V = 1 V (2.62) 1ω = frequency Hz = 1 min = 60 s s t = time this is different from text: lag phase angle vs. lead angle used resulting current assumed also harmonic It() := ImIm⋅cos(ω⋅t − φ) Im = amplitude_of_current V = 1 V current assumed to have lag angle. this approach taken to allow common treatment of L and C in circuits φ = phase_lag_angle complex (vector) representation, set up with real part expressed as cos Uz t() = Um ⋅cos(ω⋅t)+ Um ⋅sin(ω⋅t)⋅i Iz t() = Im ⋅cos(ω⋅t − φ)+ Im ⋅sin(ω⋅t − φ)⋅i plotting set up 5 11/13/2006voltage and current at omega*t positive lag phase angle 0 0.5 1 Uz(t) Iz(t)Imaginary parts of Uz(t), Iz(t) 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) (2.57) UC()t= ⌠⎮ t It()dt = ⌠⎮ t Im ⋅cos(ω⋅t − φ) dt = Im ⋅sin(ω⋅t − φ) = Im ⋅cos ⎛⎜ω⋅t − φ − π⎞⎟ ⎮C ⎮ C C⋅ω C⋅ω ⎝ 2 ⎠⌡⌡0 0 using complex (vector) notation Uz t() := UmUm⋅cos(ω⋅t)+ Um ⋅sin(ω⋅t)⋅i Iz t() := ImIm⋅cos(ω⋅t − φ)+ Im ⋅cos(ω⋅t − φ)⋅i ImIm⎛ π⎞ Im ⎛ π⎞UzC()t := ⋅cos⎜ω⋅t − φ −⎟+ ⋅sin⎜ω⋅t