# Chapter 2: Part 1 Phasors and Complex Numbers - PowerPoint PPT Presentation

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Chapter 2: Part 1 Phasors and Complex Numbers. COMPLEX NUMBER SYSTEM CNS is a means for expressing phasor quantities and for performing mathematical operations with these quantities. CNS provides a way to mathematically express a phasor quantity

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Chapter 2: Part 1 Phasors and Complex Numbers

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## Chapter 2: Part 1Phasors and Complex Numbers

• COMPLEX NUMBER SYSTEM

• CNS is a means for expressing phasor quantities and for performing mathematical operations with these quantities.

• CNS provides a way to mathematically express a phasor quantity

• And allows phasor quantities to be addes, subtracted, multiplied and divided.

• PHASOR

• A convenient and graphic way to represent sinusoidal voltages and currents in terms of their magnitude and phase angle.

• They provide a way to diagram sine waves and their phase relationship with other sine waves.

Introduction to Phasors

The Complex Number System

Rectangular and Polar Forms

Mathematical Operations.

### Overview of Phasors and Complex Number Systems

• The fundamental idea about phasor analysis is that circuits that have sinusoidal sources can be solved much more easily if we use a technique called transformation.

• In a transform solution, we transform the problem into another form. Once transformed, the solution process is easier. The solution process uses complex numbers, but is otherwise straightforward.

• The solution obtained is a transformed solution, which must then be inverse transformed to get the answer.

• It is surprising that a process that uses three steps is faster and easier than a process that uses one step, but the steps are so much easier, it is still true.

### Introduction to Phasors: What is a Phasor

Useful for representing sine waves in terms of their

• Magnitude and Phase Angle

• For analysis of reactive circuits

VECTOR

A quantity with both magnitude and direction.

Eg: Force, Velocity, Acceleration

PHASOR

Similar to Vector but, generally refers to quantities that vary with time.

Eg: Voltage, Current e.tc

### Introduction to Phasors: Representation of a Sine Wave

• A full cycle of a sine wave can be representaed by rotation of a phasor through 360°.

• The instantaneous value of the sine wave at any point is equal to the vertical distance from the tip of the phasor to the horizontal axis.

### Introduction to Phasors: Phasors and Sine Wave Formula

• The instantaneous value can be expressed as the hypotenuse times the sine of the angle θ.

v = VPSinθ

### Introduction to Phasors: Angular Velocity of a Phasor

• One cycle of sine wave is traced out when a phasor is rotated through 360°.

• The faster it is rotated, the faster the sine wave is traced out.

• Thus, the period and frequency are related to the velocity of rotation of the phasor.

• The velocity of rotation is called the Angular Velocity and denoted by ω.

### Introduction to Phasors: Positive and Negative Phasor Angles

• The position of a phasor at any instant can be expressed as

• Positive angle (Counter Clockwise, θ)

• Or, Equivalent Negative angle (Clockwise, θ-360°)

### Introduction to Phasors: Phasor Diagrams

• A phasor diagram can be used to used to show the relative relationship of two or more sine waves of the same frequency.

• Every phasor in the diagram will have the same angular velocity because they represent sine waves of identical frequency.

• The length of the each phasor arm is directly related to the amplitude of the wave it represents, and

• the angle between the phasors is the same as the angle of phase difference between the sine waves.

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