derek181
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I am just wondering why or how we introduce the J operator in analyzing ac circuits. I want more of a proof for this.
AC circuits -- Why we introduce the J operator in analyzing them
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Are you referring to the j used in phasor notation?
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It makes analysis of circuits easier. In an inductor the current lags voltage by 90°, in a capacitor the current leads voltage by 90°. These neatly correspond to the j and -j axes, while voltage takes the positive x axis.derek181 said:
Quite likely you already knew that, and were hoping for something more elucidatory?
Falcifer
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AC electricity involves waves and frequencies, and if we are to model it mathematically it means doing calculations with angles; phase angles. The most powerful tool to help with this is the complex number plane.
It is created by extending the 1-dimensional number line of real numbers into the 2-dimensional plane of complex numbers, with real numbers on the horizontal axis, and imaginary numbers on the vertical that ascend as multiples of j, the square root of -1. Any value on this plane is has a real part and an imaginary part depending on where it lies from the origin, so it is a vector with two components; it has magnitude and direction. The magnitude is the length of the line drawn from the origin to the point of the value in the plane (a hypotenuse) and the direction is given by the angle of the line from the real axis.
You don't have to consider complex numbers to model AC (you can use basic trigonometry), but it does makes the algebra a lot simpler. I can't go to into a detailed proof of this but you might get it if you consider that j when raised to increasing integer powers: 0, 1, 2, 3, 4.. and so on rotates between four different values (corresponding to right, up, left, down), i.e. 1, j, -1, -j, 1...and so on.
With just the real numbers you only get two directions, right and left (i.e. multiplying -1 the same way goes -1, 1, -1, 1 etc.) along a line of one dimension, and you can't think of angles when you only have one dimension!
It is created by extending the 1-dimensional number line of real numbers into the 2-dimensional plane of complex numbers, with real numbers on the horizontal axis, and imaginary numbers on the vertical that ascend as multiples of j, the square root of -1. Any value on this plane is has a real part and an imaginary part depending on where it lies from the origin, so it is a vector with two components; it has magnitude and direction. The magnitude is the length of the line drawn from the origin to the point of the value in the plane (a hypotenuse) and the direction is given by the angle of the line from the real axis.
You don't have to consider complex numbers to model AC (you can use basic trigonometry), but it does makes the algebra a lot simpler. I can't go to into a detailed proof of this but you might get it if you consider that j when raised to increasing integer powers: 0, 1, 2, 3, 4.. and so on rotates between four different values (corresponding to right, up, left, down), i.e. 1, j, -1, -j, 1...and so on.
With just the real numbers you only get two directions, right and left (i.e. multiplying -1 the same way goes -1, 1, -1, 1 etc.) along a line of one dimension, and you can't think of angles when you only have one dimension!
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There go those whacky electrical engineers again! Talking about "j" when they mean "i" and measuring things in degrees that aren't angles!
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Yes, I noticed.HallsofIvy said:
A mentor should move the thread to the engineering homework forum.Apologies for any indignation elicited.
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What can they do? The notation "i" is occupied for current ... :DHallsofIvy said:
ehild