Electrical Engineering ⇒ Topic : Significance of Operator j
Significance of Operator j
Just as the symbol + indicates the operation of adding two numbers, similarly j indicates an operation of rotating the phasor through 90° in CCW direction. The operator j does not change the magnitude of the phasor. For example, jr means that Vis 90° CCW from OX-axis (i.e. it is lying along OY axis) while j2V (= -V) means that Vis turned through 180° CCW from OX-axis i.e. it is lying along OX'-axis. As seen in Fig. (a), j will be associated with a phasor lying along °Y-axis and j with that lying along OY'-axis.
The utility off becomes more apparent if we consider a phasor V displaced 0° counter-clockwise from OX-axis as shown in Fig. (b). This phasor is not lying along the rectangular co-ordinate axes. However, it can be resolved into two components viz, the horizontal component 'a' along Xaxis and the vertical component 'b' along Y-axis. It can be seen that vertical component is displaced 90° CCW from OX-axis. Therefore, mathematically, we can express this component as j b, meaning that component b is displaced 90° CCW from the component a (i.e. OX-axis).
The reader may note that a +j b is the mathematical form of the phasor V. This form describes the magnitude as well as the phase angle of the phasor. The following points may be noted carefully
The quantity a +j b is called a complex number or complex quantity. It is because it consists of a real component (a) and an imaginary component (jb).
To call the component lying along Y-axis (i. e. the component associated with j or -j) as imaginary merely because the value off j (= √-1 ) cannot be determined is really *unfortunate. The so called imaginary components can be represented graphically on the Y-axis just as real components can be represented on the X-axis. Electrical engineers rightly call the horizontal and vertical components as in-phase and quadrature components respectively.
Note. In this book, bold-faced letters (e.g. V, I etc.) will be used to represent the phasor completely,including both magnitude and direction. However, only the magnitude of the phasor will be represented by the same letter in the ordinary type (e.g. I; I etc.). However, while representing a phasor in writing, a dot or arrowhead may be used. e.g. V may be written as V and V so on.
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