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Electrical Signals and Trigonometry

Electrical Signals and Trigonometry. Alan Murray. V L. I L. Agenda. Trigonometry and waves i.e. any old waves! Amplitude, phase and frequency Representing electrical signals Voltage, current Lead, lag, phase shift i.e. not any old waves “CIVIL” mnemonic to remember which way around. 3.

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Electrical Signals and Trigonometry

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  1. Electrical Signals and Trigonometry Alan Murray

  2. VL IL Agenda • Trigonometry and waves • i.e. any old waves! • Amplitude, phase and frequency • Representing electrical signals • Voltage, current • Lead, lag, phase shift • i.e. not any old waves • “CIVIL” mnemonic to remember which way around Alan Murray – University of Edinburgh

  3. 3 1 1 2 4 2 Sines and CosinesFrequency/Wavelength y x y = cos(x) y x y = cos(2x) y x y = cos(4x) Alan Murray – University of Edinburgh

  4. y0 -π/2 = -90° y0 2y0 Sines and CosinesAmplitude and Phase y = y0cos(x) y = y0sin(x) or y = y0cos(x-90°) or y = y0cos(x-π/2) y = 2y0sin(x) or y = 2y0cos(x-90°) or y = 2y0cos(x-π/2) Alan Murray – University of Edinburgh

  5. y = y0cos(x) Ф 180° = π y = y0cos(x+Ф) Ф y = y0cos(x+180°) or y = y0cos(x+π) or y = -y0cos(x) 180° =π Sines and CosinesAmplitude and Phase Clickertime Sinusoids simulation Alan Murray – University of Edinburgh

  6. T=1/f V0 And in Circuits … • V = V0 sin (ωt + Ф) • a sinusoidal variation of voltage with time • Amplitude = V0 • Volts • Time = t • seconds • Frequency = f • cycles/second or Hz • Angular freq. = ω (= 2πf) • radians/second • Phase = Ф • radians or ° • Ф=0 in this example • So [ωt] = [radians/second * seconds] = [radians] • [ωt + Ф] = [radians + radians] • and thus sin(ωt + Ф) makes sense and works! V t Alan Murray – University of Edinburgh

  7. IR VR VR IR Resistors are trivial (!) .. • VR = V0sin(ωt) • IR = I0 sin(ωt) • V & I are in phase • VR = RIR • Ohm’s Law • R is the “impedance”of a resistor • V0 sin(ωt) = RI0sin(ωt) • Ohm’s Law • V0 = RI0 • Ohm’s Law, amplitudes only Alan Murray – University of Edinburgh

  8. V = V0 sin(ωt) π/2 I = I0cos(wt) = I0sin(ωt+90°) ... but Voltage and Current (AC) in a Capacitor I Vs V Animate this slide! Alan Murray – University of Edinburgh

  9. IC VC 90° IC time t Capacitors are not trivial .. • VC = V0sin(ωt) • IC = I0cos(ωt) = I0sin(ωt+90°) • ICleads VC by 90° • VC = ZCIC • This is Ohm’s Law • ZC replaces R • Or XC replaces R – see later! • ZC = Capacitor “impedance” • V0sin(ωt) = ?I0sin(ωt+90°) • Ohm’s Law, amplitudes only • V0= ?I0 • What is the capacitor’sequivalent of resistance … • What is “ZC “? • And how does ZC describe the +90° phase shift? VC Alan Murray – University of Edinburgh

  10. VL 90° IL Inductors are equally awkward IL • VL = V0sin(ωt) • IL = I0sin(ωt-90°) • ILlags VL by 90° • VL = ZLIL • This is Ohm’s Law • ZL replaces R • Or XL replaces R – see later! • ZL = Inductor “impedance” • V0sin(ωt) = ?I0sin(ωt-90°) • Ohm’s Law amplitudes only • V0= ?I0 • What is the inductor’sequivalent of resistance … • What is “ZL “? • And how does ZL describe the -90° phase shift? VL Alan Murray – University of Edinburgh

  11. A moment of pedantry … • VR = RIR • R is clearly a resistance • VC = ZCIC • ZC is officially an impedance • IC leads VC by 90° • VC = VC0sin(ωt) • IC =ICOsin(ωt+90°) • ZC describes the magnitude relationship between V and I • ZCmust also describe the 90° phase shift in some magical way • more anon … • However, if we work with amplitudes only • VCO = XCICO • Then XC is the capacitor’s reactance • XC describes the relationship between the magnitudes of V and I • XC does not describe the phase shift Ф • That is described by the sin(ωt), sin(ωt+Ф) • This will make more sense later … Alan Murray – University of Edinburgh

  12. CIVIL in a Capacitor, current I leads Voltage (by 90°) CIVIL Voltage leads current I (by 90°) in an Inductor L mnemonic ... "CIVIL" Alan Murray – University of Edinburgh

  13. here's the phase lag Capacitor Reactance, XC(remember,ω = 2πf ) DC AC • A capacitor blocks DC signals • f = 0, ω = 2πf = 0 • A capacitor passes AC signals • f > 0, ω = 2πf > 0 • Ohm's Law for amplitudes is V0 = XCI0 • It turns out that • VO = 1 IO for a capacitor (blocks DC, passes AC) ωC • so XC = 1 ωC • DC … f=0 ω=0 • XC = ∞ … block • AC … f>0 w<∞ • XC < ∞ … pass • If I = I0 sin(ωt), then • V = 1 x I0sin(ωt-90°) ωC Alan Murray – University of Edinburgh

  14. here's the phase lead Inductor Reactance, XL(remember,ω = 2πf ) • An inductor passes DC signals • f = 0, ω = 2πf = 0 • An inductor blocks AC signals • f > 0, ω = 2πf > 0 • Ohm's Law for amplitudes is VL0 = XLILO • It turns out that • VO = ωL IO for an inductor (blocks AC, passes DC) • so XL = ωL • DC … f=0 ω=0 • XC = 0 … pass • AC … f→∞ w →∞ • XC → ∞ … block • If I = I0sin(ωt), then • V = ωL x I0sin(ωt+90°) Clickertime Alan Murray – University of Edinburgh

  15. Fill in the blanks … Alan Murray – University of Edinburgh

  16. Starting too sound messy, isn’t it? Summary so far,if I =I0 sin(ωt) • VR= R xIO sin(ωt) • VC= 1 xIO sin(ωt - 90°)ωC • VL= ωL xIO sin(ωt + 90°) • All obey Ohm’s Law • with different constants between V and I • R, 1/ ωC and ωL • in capacitors, I leads VC by 90° • in inductors, VL leads I by 90° Alan Murray – University of Edinburgh

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