L7 1 4Review Nuances of j

>> There are some nuances of j

that deserve a little bit
of additional attention.

We're going to be using
j in complex quantities,

complex exponentials extensively
in electrical engineering.

We need to be comfortable
with a few of these things.

For example, one over j
is equal to negative j.

How do we understand that?

Well, if you've got one over j,

and you multiply numerator and
denominator by j, in other words,

we're going to rationalize the denominator,

by getting rid of the imaginary part
of the denominator,

you come up with,

in the numerator you get j and in

the denominator you have j squared
or j squared is negative one.

So that's equal to j
divided by negative one,

or that's equal to negative j.

So when you move the complex,

the square root of negative one,

term j from the numerator to

the denominator or for
the denominator to the numerator,

you change the sign on it.

Another one that deserves
just a little bit of attention,

e to the j90 is equal to j,

and e to the minus j90
is equal to negative j.

How are we to understand that?

In its polar form,

e to the j90 can be rewritten in

its rectangular form as it
has a magnitude of one,

so it would be one magnitude of it,

times the cosine of 90,

plus j sine of 90.

Of course, the cosine of 90 is zero,

and you're left with j
times the sine of 90.

The sine of 90 is one.

So e to the j90 equals j.

Let's look at it in the complex plane.

If we have a complex number
e to the j90, again,

it is something that has a magnitude
of one and an angle of 90 degrees.

So magnitude of one we're coming up here
along the j-axis, the imaginary axis.

We cover a distance of one,

angle 90, well, that is simply one j.

Finally, when calculating
the magnitude of a complex number,

don't square the j.

The magnitude of a complex
number is the length or

the distance of the point in
the complex plane away from the origin.

By definition and by nature,

the distance or the length of
something is a positive value.

So when we're calculating
the distance or the magnitude of z,

we simply take and calculate the
square root of a squared plus b squared.

You don't put the j in there.

For example, let's let
z equals two plus j3.

The magnitude of z is going to equal

the square root of two
squared plus three squared,

which equals the square root of 13.

What would happen if we did square the j?

Well, this is not how you do it,

but just to show what would
happen if we did it wrong,

we would have then,

we would think that the
square root of z would equal two

squared plus j3 squared.

Well, that would equal the square root
of four plus now jq or j3 squared,

squaring the j would
give you a negative one,

squaring the three would give you nine,

so you'd have a minus nine.

If we were to do it this way,

we would come up with
the length of that thing

being the square root of negative five.

Clearly, that's not what
we're out looking for.

We're looking for the distance there.

The distance is you come over a,

whatever our dimensions are
here and you come up b,

whatever those dimensions are there
and Pythagorean's theorem says that

that distance is a squared plus b
squared square root of it.