Week 4 - Complex Number, Phasors Intro to AC
Page 1
Title: Electrical Technology, Week 4: Complex Number, Phasors & Intro to AC
Date: 27/07/2009
Institution: Universiti Teknologi Malaysia (UTM)
Website: www.utm.my
Page 2: Complex Numbers
Definition: A complex number consists of a real part and an imaginary part, e.g., 2 + j3.
Representation: Complex numbers can be represented on the Argand Diagram:
Real Part: Corresponds to the x-coordinate
Imaginary Part: Corresponds to the y-coordinate
Phasor Rotation: The 'j' operator indicates a 90º counterclockwise rotation of a phasor (instead of 'i', used by mathematicians).
Page 3: Cartesian Form
Modulus: The distance from the origin to a point (x, y) can be calculated using:
Pythagoras’ Theorem: (|z| = \sqrt{x^2 + y^2})
Cartesian Form: Written as ( z = x + jy ) where (x) is the real part and (jy) is the imaginary part.
Page 4: Polar Coordinates
Polar Form: Complex numbers can be expressed in polar coordinates (r, θ):
Magnitude (r): The distance from the origin to the point.
Angle (θ): The angle between the position vector and the positive x-axis.
Notation: ( z = r ∠ θ )
Page 5: Euler's Formula
Relationship: Relationship between Cartesian and polar coordinates:
Euler's Formula: ( z = re^{jθ} = r(cosθ + jsinθ) )
Expression: ( x = r ext{cos}θ ), ( y = r ext{sin}θ )
Page 6: Complex Conjugates
Conjugate Definition: If ( z = x + jy ), then the conjugate is ( z* = x - jy ).
Properties:
Same real value but opposite sign for the imaginary part.
Example: The conjugate of 2 + j3 is 2 - j3.
Arithmetic: Addition and subtraction of complex numbers is straightforward in Cartesian form.
Page 7: Polar Operations
Multiplication and Division: Preferably done in polar form:
Multiplication: ( E_1 imes E_2 = r_1r_2∠(θ_1 + θ_2) )
Division: ( E_1 / E_2 = r_1 / r_2∠(θ_1 - θ_2) )
Page 8: Alternating Current (AC)
Switch from Constant to Time-Varying Circuits: Focus on circuits energized by varying voltage or current sources, particularly sinusoidal forms.
Importance: Crucial for generation, transmission, distribution, and consumption of electric energy under sinusoidal steady state conditions.
Page 9: Sinusoidal Source
Definition: A sinusoidal source produces voltage or current that varies sinusoidally over time.
Advantages:
Common in nature.
Easy to generate, transmit, and mathematically handle.
Page 10: General Form of a Sinusoid
Expression:
(v(t) = V_m ext{sin}( ext{ω}t))
Where (V_m) is the amplitude, (ω) is the angular frequency in radians per second.
Page 11: Period of a Sinusoid
Definition: The period (T) of a sinusoid is the time taken to complete one full cycle:
Formula: (T = rac{2π}{ω})
Page 12: Frequency of a Sinusoid
Frequency (f): The reciprocal of the period (T):
Formula: (f = rac{1}{T}) with unit Hertz (Hz).
Relation: (ω = 2πf)
Page 13: Phase Lag and Lead
Formulation: General representation of a sinusoid can include a phase term:
(v(t) = V_m ext{sin}(ωt + φ))
Phase Differences: Shifts sinusoid along the time axis:
Positive phase angle shifts left; negative phase angle shifts right.
Page 14: Trigonometric Identities
Comparison: Use sine or cosine forms with positive amplitude for easy comparison:
Useful Identities:
Conversions between sine and cosine functions given specific phase shifts.
Page 15: Example Problems (1)
Find amplitude, phase, period, and frequency of ( v(t) = 12 ext{cos}(50t + 10°) )
Given ( v(t) = 5 ext{sin}(4nt - 60°)), calculate its properties.
Calculate phase angle between (V_1 = -10 ext{cos}(ωt + 50°)) and (V_2 = 12 ext{sin}(ωt - 10°)).
Express the followings as phasors:
(a) (v = 7 ext{cos}(2t + 409°) V)
(b) (-4 ext{sin}(10t + 10°) A)
Find sinusoids corresponding to given phasors.
Page 16: Example Problems (2)
Sinusoidal Current Case:
Maximum amplitude = 20 A, period = 1 ms yields:
Frequency: (f = 1/T = 1000 Hz)
Angular Frequency: Required calculations.
Expression for i(t) using cosine function with angle in radians.
Rms Value of current calculation as related above.