L-13(NKD)(ET) ((EE)NPTEL) - Module 4 Single-phase AC...

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Module 4 Single-phase AC Circuits Version 2 EE IIT, Kharagpur
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Lesson 13 Representation of Sinusoidal Signal by a Phasor and Solution of Current in R-L-C Series Circuits Version 2 EE IIT, Kharagpur
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In the last lesson, two points were described: 1. How a sinusoidal voltage waveform (ac) is generated? 2. How the average and rms values of the periodic voltage or current waveforms, are computed? Some examples are also described there. In this lesson, the representation of sinusoidal (ac) voltage/current signals by a phasor is first explained. The polar/Cartesian (rectangular) form of phasor, as complex quantity, is described. Lastly, the algebra, involving the phasors (voltage/current), is presented. Different mathematical operations – addition/subtraction and multiplication/division, on two or more phasors, are discussed. Keywords : Phasor, Sinusoidal signals, phasor algebra After going through this lesson, the students will be able to answer the following questions; 1. What is meant by the term, ‘phasor’ in respect of a sinusoidal signal? 2. How to represent the sinusoidal voltage or current waveform by phasor? 3. How to write a phasor quantity (complex) in polar/Cartesian (rectangular) form? 4. How to perform the operations, like addition/subtraction and multiplication/division on two or more phasors, to obtain a phasor? This lesson forms the background of the following lessons in the complete module of single ac circuits, starting with the next lesson on the solution of the current in the steady state, in R-L-C series circuits. Symbols i or i(t) Instantaneous value of the current (sinusoidal form) I Current (rms value) Maximum value of the current m I I Phasor representation of the current φ Phase angle, say of the current phasor, with respect to the reference phasor Same symbols are used for voltage or any other phasor. Representation of Sinusoidal Signal by a Phasor A sinusoidal quantity, i.e. current, t I t i m ω sin ) ( = , is taken up as an example. In Fig. 13.1a, the length, OP, along the x-axis, represents the maximum value of the current , on a certain scale. It is being rotated in the anti-clockwise direction at an angular speed, m I , and takes up a position, OA after a time t (or angle, t θ = , with the x-axis). The vertical projection of OA is plotted in the right hand side of the above figure with respect to the angle . It will generate a sine wave (Fig. 13.1b), as OA is at an angle, with the x-axis, as stated earlier. The vertical projection of OA along y-axis is OC = AB = Version 2 EE IIT, Kharagpur
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θ sin ) ( m I i = , which is the instantaneous value of the current at any time t or angle . The angle is in rad., i.e. t ω = . The angular speed, is in rad/s, i.e. f π 2 = , where is the frequency in Hz or cycles/sec. Thus, f ft I t I I i m m m 2 sin sin sin = = = So, OP represents the phasor with respect to the above current, i.
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This note was uploaded on 08/03/2010 for the course ELECTRICAL EE212 taught by Professor Shetty during the Spring '10 term at International Institute of Information Technology.

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L-13(NKD)(ET) ((EE)NPTEL) - Module 4 Single-phase AC...

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