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    Homework Phasors Problem 1 Solution: P (A Cos (+) ) Ae

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    Yimy Garcia
    The document discusses phasor analysis techniques including: 1) Converting cosine signals to phasors by extracting the magnitude and phase shift. 2) Converting phasors back to cosine signals by placing the magnitude in front of "cos(ωt+" and the phase shift after. 3) Using phasors to sum cosine signals with different phases and frequencies, then converting back to a single cosine signal. 4) Taking the derivative of a cosine signal and converting it to a phasor, which is jω times the original phasor.

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    Homework Phasors Problem 1 Solution: P (A Cos (+) ) Ae

    Uploaded by

    Yimy Garcia
    23 views3 pages
    The document discusses phasor analysis techniques including: 1) Converting cosine signals to phasors by extracting the magnitude and phase shift. 2) Converting phasors back to cosine signals by placing the magnitude in front of "cos(ωt+" and the phase shift after. 3) Using phasors to sum cosine signals with different phases and frequencies, then converting back to a single cosine signal. 4) Taking the derivative of a cosine signal and converting it to a phasor, which is jω times the original phasor.

    Original Description:

    solucionario

    Original Title

    ECE1250_NC_HWPhasorp1soln

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    The document discusses phasor analysis techniques including: 1) Converting cosine signals to phasors by extracting the magnitude and phase shift. 2) Converting phasors back to cosine signals by placing the magnitude in front of "cos(ωt+" and the phase shift after. 3) Using phasors to sum cosine signals with different phases and frequencies, then converting back to a single cosine signal. 4) Taking the derivative of a cosine signal and converting it to a phasor, which is jω times the original phasor.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    23 views3 pages

    Homework Phasors Problem 1 Solution: P (A Cos (+) ) Ae

    Uploaded by

    Yimy Garcia
    The document discusses phasor analysis techniques including: 1) Converting cosine signals to phasors by extracting the magnitude and phase shift. 2) Converting phasors back to cosine signals by placing the magnitude in front of "cos(ωt+" and the phase shift after. 3) Using phasors to sum cosine signals with different phases and frequencies, then converting back to a single cosine signal. 4) Taking the derivative of a cosine signal and converting it to a phasor, which is jω times the original phasor.

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    1250 Homework Phasors Problem 1 Solution

    EX: The phasor transform converts a cosinusoid to a complex number that captures the
    magnitude and phase shift:
    P[A cos(ω t + φ )] = Ae jφ

    The inverse phasor transform converts a complex number into a cosinusoid:


    P −1[Ae jφ ] = A cos(ω t + φ )

    The phasor transform of the derivative of a cosinusoid is jω times the phasor


    transform of the cosinusoid:
    ⎡ A cos(ω t + φ ) ⎤
    P⎢ ⎥ = jω Ae jφ
    ⎣ dt ⎦
    a) Find P[2.4 cos(2k t + 30°)] .

    b) Find P −1[ j] .
    c) Use phasors to calculate 3cos(ω t) + 4 sin(ω t) . (Convert to phasors, sum the
    phasors, then convert back to a single sinusoid.)
    ⎡d ⎤
    d) Find P ⎢ [3cos(ω t) + 4 sin(ω t)]⎥ .
    ⎣ dt ⎦

    SOL'N: a) All that one does to find the phasor of the signal is extract the value of A
    and the value of φ from the cosine expression. The A is placed in front of
    ej, and φ is placed behind the j in the exponent.

    P[2.4 cos(2k t + 30°)] = 2.4e j 30°

    Note: if the time-domain waveform has units, e.g., volts, then we attach
    those units to the phasor, too. Here, we have no units.

    b) Typically, we convert complex numbers to polar form before we take the


    inverse phasor transform. From the standpoint of the engineer in a lab, a
    cosine waveform with a phase shift value is easier to comprehend, so the
    polar form is preferable.

    It is straightforward to take the inverse transform of a polar-form phasor.


    All that one does to find the inverse phasor is locate the value of A and the
    value of φ in the phasor. The A is placed in front of "cos(ωt+", and the φ
    is placed after the "+". A closing parenthesis is added to complete the
    expression.
    If the phasor is a complex number in rectangular form, a + jb, the inverse
    phasor transform is the sum of the inverse phasor transform of a and the
    inverse phasor transform of jb. The inverse phasor transform of positive
    real number a is a cosine waveform with magnitude a and zero phase shift.
    A minus sign in front of the phasor becomes a ±180° phase shift of the
    cosine waveform. The inverse phasor transform of a positive imaginary
    number, jb, is a cosine waveform of magnitude b shifted by –90°.
    Equivalently, the inverse phasor of may be written as –bsin(ωt).

    P −1[a + jb] = a cos(ω t) + b cos(ω t − 90°) = a cos(ω t) − bsin(ω t)

    c)
    d)

    ANS: 1. a) 2.4e j 30°

    b) − sin(ω t) (The value of ω cannot be determined.)

    ⎛ −4 ⎞
    j tan −1⎜ ⎟
    −1 ⎝ 3⎠
    c) P [3 − j4 = 5e ] = 5 cos(ω t − 53°)

    d) j5ω e− j53°

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