Design of impedance matching transformer (1)

 This year is expected to be the best year of MW DXing for the first time of 11 years. So I am planning to build D-Kaz antenna by myself.  I need to make an impedance matching transformer that converts the antenna impedance around 800 ohms to 50 ohms of the input impedance of SDR effectively.
  To design the transformer, I needed to review a reactance circuit with mutual inductance.  I read some textbook that I used to refer to in my college days a long time ago.

  Figure1 shows an impedance matching circuit using a transformer. The resistance of each coil is ignored.

Figure1 an impedance matching circuit using a transformer 

From this figure, the following formulas can be obtained.

(These formulas are revised on Feb.6 2019）

  From these formulas, I can understand that a primary impedance Z1 is matched by secondary impedance R2 by the ratio of primary and secondary coil turns. However, the following condition is required to let the circuit work properly.

　I am wondering what multiplier α should be applied to obtain reactance of L1.
I am trying to find the clue to decide α in several references and textbooks.

According to the paper of "Impedance Matching Transformers for Receiving Antennas at Medium and Lower Shortwave Frequencies" written by Bill Bowers, John Bryand and Nic Hall-Patch, They say that the reactance of primary winding coil may be found by multiplying the impedance of the antenna to be matched by a factor of 4. They are unaware of any empirical data that has been published to support the value of this particular multiplier being equal to 4. During Bill Bower's study of impedance transformer design for LF use, his data support a K value much more nearly 6.

 I am still struggling for understanding how to design the impedance matching circuit using a transformer.  I would very appreciate if you could give me any suggestion.