The Smith Chart is a mapping between reflection coefficient and normalized
impedance characterized by a pair of equations:

       z - 1
Gamma = -----
       z + 1

    Gamma + 1
z = --------
    1 - Gamma

The impedance matching problem is mainly finding the

z

that having reflection coefficient

Gamma  = ℜ  + jℑ
    z    z      z

satisfies:

            2    2        2
           ℜ  + ℑ   = Gamma
            z    z
        1   2    2    1
 ( ℜ  - - )   + ℑ   = -
    z   2        z    4

Using some algebra it is easy to find out

         2
ℜ  = Gamma
 z

,

        _____________
     _ |    2       4
ℑ  =  \|Gamma  - Gamma
 z

. Now, we already know that the real component of

z

is

1

. Then the imaginary part of the two matching impedance would be:

            2ℑ
              z
 +/- -----------------
      2          2
     ℜ  - 2ℜ  + ℑ  + 1
      z     z    z