[TowerTalk] S21 to Rp & Cp - Equivalent Circuit

jimlux jimlux at earthlink.net
Fri Jan 12 08:38:39 EST 2018

```On 1/11/18 10:48 AM, jimlux wrote:
> On 1/11/18 10:01 AM, Máximo EA1DDO_HK1H wrote:
>> Hi Jim, Steve, all,
>>
>>
>> Playing with those formulas I´ve got an inconsistency.
>>
>>
>> To get R and jX from LogMag and Phase I can use two different
>> formulas, one from Steve and one from Jim, but the results are not
>> same, so I don´t know where the mistake is.
>>
>>
>> Example from Excel;
>>
>>
>> LogMag -29.601925
>>
>> Phase  -77.10851
>>
>>
>> Steve´s formula:
>>
>> Rx = (100*10^(-LogMag/20))/(SQRT(1+(TAN(Phase/57.29578))^2))-100  =
>> 573.916
>>
>> X = -(Rx+100)*TAN(Phase/57.29578)  =  2944.484
>>
>>
>>
>> Jim´s formula:
>>
>>
>> R =
>> COMPLEX(10^((LogMag/20)*(COS((Phase*PI())/180))),10^((LogMag/20)*(SIN((Phase*PI())/180))))
>>
>>
>> then Rx+jX = IMPRODUCT(50,IMDIV(IMSUM(1,R),IMSUB(1,R)))  =  -49.930
>> +3.606
>>
>>
>> As you can see, 573 is not same as -49 OHm
>>
>> And 2944 is not 3.606
>>
>>

working in Octave (free version of Matlab), which is a bit easier than Excel

>> s11mag = -29.6
s11mag = -29.600
>> s11ph = -77.1
s11ph = -77.100
>>
>> gamma = 10^(s11mag/20)
gamma =  0.033113
>> gamma = gamma*complex(cos(s11ph/180*pi),sin(s11ph/180*pi))
gamma =  0.0073925 - 0.0322774i
>> z = 50* ( 1+gamma)/(1-gamma)
z =  50.6383 -  3.2725i

This is the series form, and close to 50 ohms, as expected (a return
loss of 30dB is a good match)
Convert to parallel form

>> rs = real(z)
rs =  50.638
>> xs = imag(z)
xs = -3.2725
>> comterm = (rs/xs+xs/rs)
comterm = -15.538
>> rp=xs*comterm
rp =  50.850
>> xp = rs*comterm
xp = -786.84
>>

Which matches *neither* of the other versions.

And using a different way to convert series to parallel
>> Q = xs/rs
Q = -0.064626

>> xp = xs*(Q^2+1)/Q^2
xp = -786.84
>> rp=rs/(Q^2+1)
rp =  50.428
>>

Slightly different R - maybe it's a numerical precision thing Q^2 is small

```