transformer, 4 Times rule for multiple of reactance to resistance being transformed

I have this graph showing the 'Z_Normalized sub i' vs the multiple of 
reactance to resistance needing transforming.
I don't know what the 'Z_Normalized sub i' means. Or, how I could use it 
to calculate input resistance, current, voltage or what I can do with it.
Can someone explain it to me?
> https://www.dropbox.com/s/fwfzwrjxg9e7ayo/4%20Times%20Rule%20Graph.png?dl=0

                         Thanks, Mikek

On 16/09/2022 22:48, amdx wrote:
> I have this graph showing the 'Z_Normalized sub i' vs the multiple of 
> reactance to resistance needing transforming.
> I don't know what the 'Z_Normalized sub i' means. Or, how I could use it 
> to calculate input resistance, current, voltage or what I can do with it.
> Can someone explain it to me?
>> https://www.dropbox.com/s/fwfzwrjxg9e7ayo/4%20Times%20Rule%20Graph.png?dl=0 
>>
> 
>                          Thanks, Mikek
> 

My take is that Zi represents the input impedance of the transformer. So 
for example if you are working in a 100ohm system and the primary 
inductance is such that its reactance is 400ohm (the 4x thick black 
line) then the impedance presented is 97ohms reading from the graph with 
phase shift of 15deg.

The rule of thumb I always remembered was that the primary XL should be 
at least ten times the wanted impedance. That graph is interesting 
showing the effects of using a much lower winding inductance.

piglet

On 9/17/2022 1:17 AM, piglet wrote:
> On 16/09/2022 22:48, amdx wrote:
>> I have this graph showing the 'Z_Normalized sub i' vs the multiple of 
>> reactance to resistance needing transforming.
>> I don't know what the 'Z_Normalized sub i' means. Or, how I could use 
>> it to calculate input resistance, current, voltage or what I can do 
>> with it.
>> Can someone explain it to me?
>>> https://www.dropbox.com/s/fwfzwrjxg9e7ayo/4%20Times%20Rule%20Graph.png?dl=0 
>>>
>>
>>                          Thanks, Mikek
>>
>
> My take is that Zi represents the input impedance of the transformer. 
> So for example if you are working in a 100ohm system and the primary 
> inductance is such that its reactance is 400ohm (the 4x thick black 
> line) then the impedance presented is 97ohms reading from the graph 
> with phase shift of 15deg.
>
> The rule of thumb I always remembered was that the primary XL should 
> be at least ten times the wanted impedance. That graph is interesting 
> showing the effects of using a much lower winding inductance.
>
> piglet
>
  That's the way I tried to use it. A 50 ohm transformation resistance 
with a 50 ohm transformer reactance, would look like
50Ω x .7 (as on the graph) or 35Ω at the 1 multiple point. It just 
didn't agree with a low end roll off of a transformer
someone graphed. All the ham radio books I have seen used a 4 times 
multiple, I always wondered how that worked and finally found that graph.
I worked with a physicist a that designed power amps for ultrasonic 
transducers, his designs used 3.5 times.
They worked, not sure what 4 times would have changed.

                   Thanks, Mikek