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Sinusoids

Sinusoids. Period: T Time necessary to go through one cycle Frequency: f = 1/ T Cycles per second Radian frequency: w = 2 p f Amplitude: V M. Phase. Leading and Lagging Phase. x 1 (t) leads x 2 (t) by q -  x 2 (t) lags x 1 (t) by q - . Complex Numbers. x is the real part

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Sinusoids

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  1. Sinusoids • Period: T • Time necessary to go through one cycle • Frequency: f = 1/T • Cycles per second • Radian frequency: w = 2pf • Amplitude: VM

  2. Phase

  3. Leading and Lagging Phase x1(t) leads x2(t) by q- x2(t) lags x1(t) by q-

  4. Complex Numbers • x is the real part • y is the imaginary part • z is the magnitude • q is the phase imaginary axis y z q real axis x

  5. More Complex Numbers • Polar Coordinates: A = z q • Rectangular Coordinates: A = x + jy • There is a good chance that your calculator • will convert from rectangular to polar and from • polar to rectangular.

  6. Arithmetic With Complex Numbers • We need to be able to perform computation with complex numbers. • Addition • Subtraction • Multiplication • Division

  7. Addition • Addition is most easily performed in rectangular coordinates: A = x + jy B = z + jw A + B = (x + z) + j(y + w)

  8. Imaginary Axis A + B B A Real Axis Addition

  9. Subtraction • Subtraction is most easily performed in rectangular coordinates: A = x + jy B = z + jw A - B = (x - z) + j(y - w)

  10. Imaginary Axis B A Real Axis A - B Subtraction

  11. Multiplication • Multiplication is most easily performed in polar coordinates: A = AMq B = BMf A B = (AM  BM)  (q + f)

  12. Multiplication Imaginary Axis A B B A Real Axis

  13. Division • Division is most easily performed in polar coordinates: A = AMq B = BMf A / B = (AM / BM)  (q - f)

  14. Division Imaginary Axis B A Real Axis A /B

  15. Phasors • A phasor is a complex number that represents the magnitude and phase of a sinusoid:

  16. Complex Exponentials • A complex exponential is the mathematical tool needed to obtain phasor of a sinusoidal function. • A complex exponential is ejwt= cos wt + j sin wt • A complex number A = z q can be represented A = z q = z ejq = z cos q + j z sin q • We represent a real-valued sinusoid as the real part of a complex exponential. • Complex exponentials provide the link between time functions and phasors. • Complex exponentials make solving for AC steady state an algebraic problem

  17. Complex Exponentials (cont’d) What do you get when you multiple A withejwt for the real part? Aejwt = z ejqejwt = z ej(wt+q) z ej(wt+q) = z cos (wt+q) + j z sin (wt+q) Re[Aejwt] = z cos (wt+q)

  18. Sinusoids, Complex Exponentials, and Phasors • Sinusoid: z cos (wt+q)= Re[Aejwt] • Complex exponential: Aejwt = z ej(wt+q), A= z ejq, • Phasor for the above sinusoid: V = z q

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