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L3 - Mathematical representation Consider the following...

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Mathematical representation Consider the following waveform: ( ) cos( ) e t E t ω φ = + e(t) t t φ ω = . t φ ω = − ∆ Note the sign convention where -ve phase difference is a positive time difference. (Things that come later have phase < 0) Here's a different way of showing the same thing. At any fixed point in time: φ E R i Phasor. 1
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Properties of phasors The phase angle φ is the angle between E and the Real axis. The amplitude at time t is the projection of E onto R. Oscillatory motion appears as counter-clockwise rotation at a constant angular frequency, ω . The usual variation in amplitude is seen as the variation in E r , the projected length of the vector of fixed length |E| onto R. When φ = 0, E r = |E| When φ = 90 o or π /2, E r = 0 When φ = 180 o or π , E r = -|E| An alternative to: ( ) ( ) cos e t A t ω φ = + is: ( ) e t A t ω φ = + If the convention is adopted to draw the diagram at t =0, then we can write, e A φ = .
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  • Fall '07
  • Pottebaum
  • Complex number, fixed length, Complex Number Notation, complex exponential Euler, -ve phase difference, Ei Er

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