L7 4 1 Series and Parallel Impedance

>> Now, let's consider how we combine

impedances that are connected in
either series or in parallel.

We might get a hint from
the fact that because

impedances have the units of ohms,

we would expect them to combine in

exactly the same way that resistances

do and we're going to see
that is in fact the case.

So for example, we've got Z_1 and Z_2
connected in series with each other.

The equivalent impedance Z equivalent
is just equal to Z_1 plus Z_2.

So Z_1 equals 3 plus J_2.

Three being the real part sometimes
referred to as the resistance,

and J_2 the imaginary part also
sometimes referred to as the reactants.

A second impedance Z_2 equaling 5 minus J,

then the equivalent impedance of
those two connected in series

will simply be 3 plus J_2 plus 5,

minus J, which equals 3 plus five is 8.

J_2 minus J_1 is plus J.

That's its rectangular form,

and we can also write
that it is parallel form,

which would give us then
8.06 for the magnitude E to

the J at 7.103.

All right, now let's look
at these two in parallel.

In parallel, it turns out that as
it was with resistance is also

1 over Z_eq is equal to 1 over Z_1,

plus 1 over Z_2.

Now, sometimes we refer
to instead of impedances,

we'll refer to admittances where admittance
is called y is equal to 1 over Z.

So in this case, we could say then
that Y_eq equals Y_1, plus Y_2.

All right. Let's just simplify
this in terms of impedances.

We know then that 1 over Z_eq.

We need to combine these two terms to get

the common denominator or to get
a common denominator to combine them.

So it would be the common denominator
would be Z_1 times Z_2,

and in the numerator
would have Z_2 plus Z_1.

Thus that now we can invert them,

and we get Z_eq is equal
to Z_1 Z_2 over Z_1,

plus Z_2 or the

product of the impedances divided
by the sum of the impedances.

I'll leave it to you to go ahead
and plug in Z_1 and Z_2 on these,

but let me just give you that
for these values of Z_1 and Z_2,

we get Z_eq is equal to

2.2 plus 0.6J in rectangular coordinates,

and in parallel coordinates that turns

out to be around a site
in polar coordinates.

That turns out to be 2.28 e to

the positive J15.3 degrees.