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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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In summary, the j operator is used in phasor notation to represent current and voltage in AC circuits. It makes analysis of circuits easier by allowing for the use of complex numbers.
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SteamKing
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Are you referring to the j used in phasor notation?
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NascentOxygen
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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?
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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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HallsofIvy
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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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NascentOxygen
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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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ehild
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What can they do? The notation "i" is occupied for current ... :DHallsofIvy said:
ehild
FAQ: AC circuits -- Why we introduce the J operator in analyzing them
1. What is the purpose of introducing the J operator in analyzing AC circuits?
The J operator, also known as the imaginary unit, is introduced in AC circuit analysis to represent the phase shift between voltage and current in a circuit. This allows for a more accurate and comprehensive analysis of the behavior of AC circuits.
2. How does the J operator affect the calculations in AC circuit analysis?
The J operator is used to convert sinusoidal functions into complex exponentials, which makes it easier to perform calculations involving AC circuits. It also helps to simplify the equations and make them more manageable.
3. Can AC circuits be analyzed without using the J operator?
Yes, it is possible to analyze AC circuits without using the J operator, but it would be much more complex and time-consuming. The J operator greatly simplifies the calculations and provides a more intuitive understanding of AC circuits.
4. How does the J operator relate to impedance in AC circuits?
In AC circuits, impedance is represented by a complex number that includes both resistance and reactance. The reactance component is directly related to the frequency of the AC signal, and the J operator is used to represent this reactance in the impedance equation.
5. Are there any other reasons for using the J operator in AC circuit analysis?
Aside from simplifying calculations and representing reactance in impedance, the J operator also helps to explain the behavior of AC circuits in terms of phasors and phasor diagrams. This allows for a more visual understanding of AC circuits and their properties.