# Induction Generator Equations... What the Heck is "j?"

#### thingmaker3

Joined May 16, 2005
5,084
I'm reading Farret & Simoes Integration of Alternative Sources of Energy and I'm having trouble with the big words.

The presented equations often use "j" in conjunction with reactance. Examples: jXr0 where Xr0 is the reactance of the blocked rotor; jX2 where X2 is the stator reactance; jXr where Xr is rotor reactance; jXm where Xm is equivalent reactance of iron in the stator.

Is "j" a calculus notation I might be unaware of? Is it some universal thing I missed somehow? I've read the chapter three times now, and I can't find "j" except in front of the occasional "X."

#### GS3

Joined Sep 21, 2007
408

#### thingmaker3

Joined May 16, 2005
5,084
Thank you! I figured it would be something of that nature - something I missed by not taking higher education. I'll work immediately to fill that gap in my knowledge.

#### nomurphy

Joined Aug 8, 2005
567
That threw me off too, once many a year ago, until someone told me that " j " was just the engineering form of " i " for complex numbers.

I once asked a college instructor if I/we could use j instead of i. He said absolutely NOT, it was a ridiculous question. He had no idea why I asked (he was a mathematician, not an engineer).

As another anecdote, while taking a BASIC language course the instructor was indicating that one MUST read value assignments from right to left as in:

x = a ...or, the value of "a" is assigned to "x" (not "x" to "a").

Okay, but I pointed out that one could read it from left to right by simply changing the semantics, in other words, that "x" is assigned the value of "a".

He never got it. And would tell me it wasn't correct, and "...that's not how it's done."

#### recca02

Joined Apr 2, 2007
1,214
but i wonder why they chose j over i.
i still sometimes use i while solving eqns.

#### Papabravo

Joined Feb 24, 2006
13,933
The use of i for current is so pervasive in electrical engineering that choosing to use j instead of i for those cases where it is required seemed to make sense at the time.

Another thought is that j is also the basis vector for the y-axis, and i is the basis vector for the x-axis. Since the algebra of complex numbers follows the rules of vector algebra using j for the imaginary unit does have a connection to analytic geometry and the Argand diagram.

#### recca02

Joined Apr 2, 2007
1,214

#### thingmaker3

Joined May 16, 2005
5,084
Yes, it does! Reactance is orthogonal to resistance, so if "j" is associated with the y-axis, then it stands to reason one would write "jX" instead of "iX."

So... I've learned today that any complex number z can be expressed as "z = x + iy," or in this instance as "z = x + jy." In my example, the complex number (z) is Zr0, the blocked rotor impedance. x is R0/s = rotor resistance over slip-factor (slip-factor =1 for blocked rotor) - and y is Xr0 = blocked rotor reactance.

z = x + jy

Zr0 = R0/s + jXr0

And the scales fell from my eyes... I love this forum!

#### Dave

Joined Nov 17, 2003
6,970
The use of i for current is so pervasive in electrical engineering that choosing to use j instead of i for those cases where it is required seemed to make sense at the time.

Another thought is that j is also the basis vector for the y-axis, and i is the basis vector for the x-axis. Since the algebra of complex numbers follows the rules of vector algebra using j for the imaginary unit does have a connection to analytic geometry and the Argand diagram.
Indeed Papabravo's explanation is completely correct. In fact the notation is not restricted to electrical engineers any more as electrical engineering merges into other disciplines - mechanical and aerospace engineers now widely use "j" notation because of the increase in use of Control Systems in their work. Mathematicians are now taught both "i" and "j" notation.

Yes, it does! Reactance is orthogonal to resistance, so if "j" is associated with the y-axis, then it stands to reason one would write "jX" instead of "iX."

So... I've learned today that any complex number z can be expressed as "z = x + iy," or in this instance as "z = x + jy." In my example, the complex number (z) is Zr0, the blocked rotor impedance. x is R0/s = rotor resistance over slip-factor (slip-factor =1 for blocked rotor) - and y is Xr0 = blocked rotor reactance.

z = x + jy

Zr0 = R0/s + jXr0

And the scales fell from my eyes... I love this forum!
There is an entire chapter dedicated to complex numbers in the e-book: http://www.allaboutcircuits.com/vol_2/chpt_2/index.html

Discussed in terms of AC waveforms and circuits. Enjoy!

Dave

#### nomurphy

Joined Aug 8, 2005
567
Another thought is that j is also the basis vector for the y-axis, and i is the basis vector for the x-axis. Since the algebra of complex numbers follows the rules of vector algebra using j for the imaginary unit does have a connection to analytic geometry and the Argand diagram.
Bravo, papa.

#### Papabravo

Joined Feb 24, 2006
13,933
I am greatly warmed by the esteem of my colleagues. Thank you, one and all.