In a graph all the real numbers can be represented as points on a line positive to the right of the origin and negative to left. For representing the imaginary numbers however we introduce the j factor as an operator. We can convert an imaginary number into a real number by the use of this j operator. For example 1 x j x j = 1 x -1= -1. Thus +1 has been converted into -1 by the operator j which has the value of square root of minus one. The fig.(a) shows that +1 can be converted into -1 by turning the line 01 through 180 degrees.
The operation of multiplying +1 by j is equivalent to turning the line 01 through 90 degrees in the anticlockwise direction.
The operation of multiplying +1 by j x j is equivalent to turning the line 01 through 180 degrees in the anticlockwise direction.
The operation of multiplying +1 by j x j x j is equivalent to turning the line 01 through 270 degrees in the anticlockwise direction.
The operation of multiplying +1 by j x j x j x j is equivalent to turning the line 01 through 360 degrees in the anticlockwise direction.
The operation can also be discussed in the same way for clockwise direction also. Thus we find that all imaginary numbers may be represented graphically by drawing a line through the origin at right angles to the line on which the real numbers are represented. The complex number 4+2j may thus be represented on the diagram by the point A or by the line OA as shown in fig. (b).
Applications of j operator in AC circuits
For an AC circuit containing ohmic resistance, the vector diagram consists of two lines of magnitudes namely voltage E and current I in the same direction. On the Argand diagram they are represented by two distances along the positive axes of real numbers.
For an AC circuit containing only inductance, the applied emf leads the current by 90 degrees and are related to each other by E = wLI. On the Argand diagram the current I is represented along the real positive axis and the voltage E is represented along the positive imaginary axis of the magnitude wLI. The voltage E is therefore fully represented by the equation as E = jwLI.. Thus by including the j operatorin the anticlockwise direction, it very well explains that the voltage E leads current I by 90 degrees.
For an AC circuit containing only capacitance, the applied emf lags the current by 90 degrees and are related to each other by E = I/wL. On the Argand diagram the current I is represented along the real positive axis and the voltage E is represented along the negative imaginary axis of the magnitude I/wL. The voltage E is therefore fully represented by the equation as E = I/jwL.. Thus by including the j operator in the clockwise direction, it very well explains that the current I leads voltage E by 90 degrees.
The operation of multiplying +1 by j x j is equivalent to turning the line 01 through 180 degrees in the anticlockwise direction.
The operation of multiplying +1 by j x j x j is equivalent to turning the line 01 through 270 degrees in the anticlockwise direction.
The operation of multiplying +1 by j x j x j x j is equivalent to turning the line 01 through 360 degrees in the anticlockwise direction.
The operation can also be discussed in the same way for clockwise direction also. Thus we find that all imaginary numbers may be represented graphically by drawing a line through the origin at right angles to the line on which the real numbers are represented. The complex number 4+2j may thus be represented on the diagram by the point A or by the line OA as shown in fig. (b).
Applications of j operator in AC circuits
For an AC circuit containing ohmic resistance, the vector diagram consists of two lines of magnitudes namely voltage E and current I in the same direction. On the Argand diagram they are represented by two distances along the positive axes of real numbers.
For an AC circuit containing only inductance, the applied emf leads the current by 90 degrees and are related to each other by E = wLI. On the Argand diagram the current I is represented along the real positive axis and the voltage E is represented along the positive imaginary axis of the magnitude wLI. The voltage E is therefore fully represented by the equation as E = jwLI.. Thus by including the j operatorin the anticlockwise direction, it very well explains that the voltage E leads current I by 90 degrees.
For an AC circuit containing only capacitance, the applied emf lags the current by 90 degrees and are related to each other by E = I/wL. On the Argand diagram the current I is represented along the real positive axis and the voltage E is represented along the negative imaginary axis of the magnitude I/wL. The voltage E is therefore fully represented by the equation as E = I/jwL.. Thus by including the j operator in the clockwise direction, it very well explains that the current I leads voltage E by 90 degrees.
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