Single-phase AC Circuits

Version 2 EE IIT, Kharagpur

Lesson 13

Representation of Sinusoidal Signal by a Phasor and Solution of Current in R-L-C Series Circuits Version 2 EE IIT, Kharagpur

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 Im −

Current (rms value) Maximum value of the current Phasor representation of the current Phase angle, say of the current phasor, with respect to the reference phasor

I

φ

Same symbols are used for voltage or any other phasor.

Representation of Sinusoidal Signal by a Phasor

A sinusoidal quantity, i.e. current, i (t ) = I m sin ω t , is taken up as an example. In Fig. 13.1a, the length, OP, along the x-axis, represents the maximum value of the current I m , on a certain scale. It is being rotated in the anti-clockwise direction at an angular speed, ω , and takes up a position, OA after a time t (or angle, θ = ω t , 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 θ . It will generate a sine wave (Fig. 13.1b), as OA is at an angle, θ with the x-axis, as stated earlier. The vertical projection of OA along y-axis is OC = AB = Version 2 EE IIT, Kharagpur

i (θ ) = I m sin θ , which is the instantaneous value of the current at any time t or angle θ . The angle θ is in rad., i.e. θ = ω t . The angular speed, ω is in rad/s, i.e. ω = 2 π f , where f is the frequency in Hz or cycles/sec. Thus, i = I m sin θ = I m sin ω t = I m sin 2πft So, OP represents the phasor with respect to the above current, i. The line, OP can be taken as the rms value, I = I m / 2 , instead of maximum value, Im . Then the vertical projection of OA, in magnitude equal to OP, does not represent exactly the instantaneous value of I, but represents it with the scale factor of 1 / 2 = 0.707 . The reason for this choice of phasor as given above, will be given in another lesson later in this module.

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Generalized case

The current can be of the form, i (t ) = I m sin (ω t − α ) as shown in Fig. 13.1d. The phasor representation of this current is the line, OQ, at an angle, α (may be taken as negative), with the line, OP along x-axis (Fig. 13.1c). One has to move in clockwise direction to go to OQ from OP (reference line), though the phasor, OQ is assumed to move in anti-clockwise direction as given earlier. After a time t, OD will be at an angle θ with OQ, which is at an angle ( θ − α = ω t − α ), with the line, OP along x-axis. The vertical projection of OD along y-axis gives the instantaneous value of the current,

i = 2 I sin (ω...