Lecture_note_The_Quarter_Wave_Transformer

Q: Hey wait! If the quarter-wave transformer is a matching network , shouldn’t 0in Γ=??

A: Who says it isn’t!

For a quarter wave transformer , we set 1Z such that:

221001L L Z Z R Z Z R =?=

Inserting this into the scattering parameter 11S of the connector, we find:

2

12

111

10101

1L L Z R L Z R L Z R Z Z Z Z Z R Z Z ???Γ===+++

Look at this result! For the quarter-wave transformer, the connector 11S value (i.e., Γ) is the same as the load reflection coefficient L Γ:

1

1

L L L R Z R Z ?Γ==Γ+

Since the connector is lossless (unitary scattering matrix!), we can conclude (and likewise show) that:

2222

11211S S =+=Γ+Τ

Since 0Z , 1Z , and L R are all real, the values Γ and Τ are also real valued . As a result, 2

2Γ=Γ and 2

2Τ=Τ, and we can likewise conclude:

221Γ+Τ=

Now, using the newly discovered fact that (for a correctly designed transformer) L Γ=Γ:

L

in L

Γ=22

2ΤΓ=Γ?

1?ΓΓΤΓΓ?

1?Γ

And also are recent discovery that 221Τ=?Γ:

in Γ==Τ

=2222

ΤΓ

Γ?

1?ΓΤΓΓ?Γ?Γ

=0

A perfect match ! The quarter-wave transformer does indeed work!