This reference offers a good comparison between a number of the formulas for microstrip impedance:

https://qucs.github.io/tech/node75.html

Another very good resource is the Transmission Line Design Handbook by Brian Wadell which also has a good bibliography on the subject of microstrip formulas.

On the microwave radio frequency website
There are such formulas, unfortunately, currently only supports Chinese

Many thanks for your answers.
@Dale xiao: could you please give us the link towards the Chinese website ?

I downloaded the Arditi microstrip paper from 1955, he did not show this formula. See:

https://www.microwaves101.com/encyclopedias/history-of-microstrip

I have some old books gathering mold in a box somewhere, I will find them an get an answer to this question.

Steve

Found it in ITT's reference data for radio engineers, 1983, sixth edition, page 24-25.
The formula came from D.D. Grieg and H.F. Engelmann, "Microstrip - New Transmission Technique for the Kilo-Megacycle Range, proceedings for IRE, Vol. 40, Dec. 1952. There are other reference, let me know if they are needed,

I will download that paper later to check it out.

Cool stuff, I like microwave history.

Steve

II downloaded the Grieg and Engelmann paper, the formula is not in it. I also grabbed a few others from the 1950s, but no luck. Grieg and Engelmann were ITT employees, and microstrip was an ITT technology. The authors of the ITT reference data might have had access to memos that we will never find... I might do a little more digging.