Chapter 2: Part 1 Phasors and Complex Numbers

1. 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.

2. 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.

3. 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

4. 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.

5. 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

6. 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.

7. 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°)

8. 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.

9. Introduction to Phasors: Phasors of V and I in Resistor, apacitor, Inductor

10. Suppose: i w i 0 wt t w i i 0 wt w i i wt 0 Phasors for L,C,R ß