P1 - A ∠ θ ÷ B ∠ φ = A / B ∠ θ- φ PHASORS A...

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PHASORS REVIEW OF COMPLEX ALGEBRA Imaginary Numbers - = 1 1 j 1 1 j j = - j j 1 1 1 × = - Addition Examples: j 5 + j 6 = j 11 -j 4 + j 5 = j 1 Subtraction Examples: j 5 - j 7 = -j 2 -j 2 - j 9 = -j 11 Multiplication Examples: 6 × j 2 = j 12 j 6 × j 2 = -12 Division Examples: j j 6 2 3 = 6 2 3 1 3 j j j = = - Complex Numbers P1
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A complex number is a real number plus an imaginary number. Multiplication of complex numbers: (2 + j4) × (3 + j5) = (2 × 3) + ( j 2 )(4)(5) + (j4 × 3) + (j5 × 2) = 6 + j10 + j12 -20 = -14 + j22 Division of Complex Numbers: ( 29 ( 29 ( 29 ( 29 10 24 6 4 10 24 6 4 6 4 6 4 156 104 6 4 3 2 2 2 + + = + × - + × - = + + = + j j j j j j j j A complex number can be represented by a point on a complex plane. Complex Numbers in Exponential Form P2 2 4 6 8 10 12 -6 -4 -2 j6 j4 j2 -j2 -j4 Real Axis Imaginary Axis 6, -j4 -4, -j4 10, j2 2, j4
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Euler’s Identity: e j θ = cos θ + j sin θ Converting from exponential form to rectangular form: A e j θ = A cos θ + j A sin θ Complex Number expressed in Polar Form A e j θ = A θ = A cos θ + j A sin θ Multiplication of Complex Numbers: A e j θ × B e j φ = AB e j( θ + φ ) or A θ × B φ = AB θ + φ Division of Complex Numbers:
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Unformatted text preview: A ∠ θ ÷ B ∠ φ = A / B ∠ θ- φ PHASORS A phasor is a complex number associated with a phase-shifted sine wave. P3 A cos θ A sin θ A θ Real Imaginary For v = 3 sin ( ϖ t + 20 ° ) the corresponding phasor is: 3 ~ V = ∠ 20 ° For i = 5 sin (120 π t - 50 ° ) the corresponding phasor is: 5 I ~ = ∠-50 ° Phasor Multiplication: 3 ∠ 20 ° × 5 ∠-50 ° = 15 ∠-30 ° Phasor Division: 3 ∠ 20 ° 5 ∠-50 ° = 3 5 ∠ 20 °-(-50 ° ) = 0.6 ∠ 70 ° Phasor Addition: 3 ∠ 30 ° + 2 ∠-15 ° = A ∠ θ = ? 3 ∠ 30 ° = 3 cos 30 ° + j3 sin 30 ° = 2.598 + j1.5 2 ∠-15 ° = 2 cos (-15 ° ) + j2 sin (-15 ° ) = 1.932 - j 0.518 (2.598 + j1.5) + (1.932 - j 0.518) = 4.53 + j 0.982 P4 Indicates phasor Phase shift A = ( 4.53 2 + 0.982 2 ) 1/2 = 4.64 tan θ = 0.982 / 4.53 = 0.2168 θ = tan-1 0.2168 = 12.2 ° Funicular Diagram: EXAMPLES 9.1 ∠ 20 ° (4 + j7) = ? P5...
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This note was uploaded on 05/21/2011 for the course CGN 3710 taught by Professor Bloomquist during the Summer '10 term at University of Florida.

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P1 - A ∠ θ ÷ B ∠ φ = A / B ∠ θ- φ PHASORS A...

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