1.3 Phasor representation of sinusoidal signals

Phasors are a game-changer in AC circuit analysis. They turn tricky sinusoidal signals into easy-to-handle rotating vectors in the complex plane. This clever trick lets us work with magnitudes and phases instead of time-varying functions.

Using phasors, we can add, subtract, multiply, and divide AC signals like a breeze. It's all about complex numbers, polar and rectangular forms, and Euler's formula. This approach makes solving AC circuit problems way simpler than dealing with time-domain equations.

Phasor Representation

Complex Numbers and Phasors

• Phasors represent sinusoidal signals as rotating vectors in the complex plane
• Complex numbers consist of real and imaginary parts expressed as $a + jb$
• Imaginary unit j defined as $j = \sqrt{-1}$
• Phasors utilize complex numbers to describe magnitude and phase of sinusoidal signals
• Magnitude corresponds to the length of the phasor vector
• Phase angle represents the rotation of the phasor from the positive real axis

Polar and Rectangular Forms

• Polar form expresses complex numbers using magnitude and angle: $A∠θ$
• Magnitude A calculated as $A = \sqrt{a^2 + b^2}$
• Angle θ determined by $θ = \tan^{-1}(b/a)$
• Rectangular form represents complex numbers as $a + jb$
• Conversion from polar to rectangular: $a = A \cos(θ)$ and $b = A \sin(θ)$
• Conversion from rectangular to polar: $A = \sqrt{a^2 + b^2}$ and $θ = \tan^{-1}(b/a)$
• Polar form simplifies multiplication and division of complex numbers
• Rectangular form facilitates addition and subtraction of complex numbers

Phasor Analysis

Phasor Diagrams and Phase Shift

• Phasor diagrams visually represent multiple sinusoidal signals in the complex plane
• Vectors in phasor diagrams rotate counterclockwise at angular frequency ω
• Phase shift indicates the time difference between two sinusoidal signals
• Leading phase shift occurs when one signal reaches its peak before another
• Lagging phase shift happens when one signal reaches its peak after another
• Phase shift measured in degrees or radians
• Positive phase shift denotes a leading signal, negative phase shift indicates a lagging signal

Euler's Formula and Complex Exponentials

• Euler's formula relates complex exponentials to trigonometric functions: $e^{jθ} = \cos(θ) + j\sin(θ)$
• Enables conversion between trigonometric and exponential forms of sinusoidal signals
• Sinusoidal function expressed as $A\cos(ωt + φ) = Re\{Ae^{j(ωt + φ)}\}$
• Complex exponential form simplifies mathematical operations on sinusoidal signals
• Facilitates analysis of AC circuits by allowing algebraic manipulation of phasors
• Phasor representation of a sinusoid: $A∠φ = Ae^{jφ}$
• Time-domain signal recovered from phasor using $A\cos(ωt + φ) = Re\{A∠φ \cdot e^{jωt}\}$