AC circuits -- Why we introduce the J operator in analyzing them

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Discussion Overview

The discussion revolves around the introduction of the J operator in the analysis of AC circuits, focusing on its mathematical justification and implications for circuit analysis. Participants explore the use of phasor notation and the representation of phase relationships in AC electricity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks a proof for the introduction of the J operator in AC circuit analysis.
  • Another participant clarifies that the J operator refers to the j used in phasor notation.
  • A participant explains that the J operator simplifies circuit analysis by correlating the phase relationships of current and voltage in inductors and capacitors to the j and -j axes.
  • It is noted that AC electricity involves waves and frequencies, necessitating the use of complex numbers to handle phase angles effectively.
  • One participant describes the complex number plane and its utility in representing AC circuit values as vectors with magnitude and direction.
  • Another participant mentions that while complex numbers simplify algebra in AC analysis, basic trigonometry can also be used.
  • Several participants express skepticism or humor regarding the terminology used by electrical engineers, particularly the use of "j" instead of "i" for the imaginary unit.

Areas of Agreement / Disagreement

Participants express differing views on the terminology and notation used in AC circuit analysis, particularly regarding the use of "j" versus "i." The discussion includes both technical explanations and humorous commentary, indicating a lack of consensus on the appropriateness of the terminology.

Contextual Notes

Some participants highlight the potential confusion arising from the use of "j" in place of "i" for the imaginary unit, especially in the context of electrical engineering where "i" is commonly used to denote current. There is also mention of the limitations of using only real numbers for modeling AC circuits.

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.
 
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Are you referring to the j used in phasor notation?
 
derek181 said:
I am just wondering why or how we introduce the J operator in analyzing ac circuits. I want more of a proof for this.
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.

Quite likely you already knew that, and were hoping for something more elucidatory?
 
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!
 
There go those whacky electrical engineers again! Talking about "j" when they mean "i" and measuring things in degrees that aren't angles!
 
HallsofIvy said:
There go those whacky electrical engineers again! Talking about "j" when they mean "i" and measuring things in degrees that aren't angles!
Yes, I noticed. o_O A mentor should move the thread to the engineering homework forum.

Apologies for any indignation elicited.
 
Last edited:
HallsofIvy said:
There go those whacky electrical engineers again! Talking about "j" when they mean "i" and measuring things in degrees that aren't angles!
What can they do? The notation "i" is occupied for current ... :D

ehild
 

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