# [TowerTalk] S21 to Rp & Cp - Equivalent Circuit

Steve Hunt steve at karinya.net
Sun Jan 14 08:19:23 EST 2018

```The general case is simpler than I originally had it, because I'd missed
a simplification using a trig identity. The revised version is here:
http://www.karinya.net/g3txq/chokes/s21.pdf

R = (100.cos(θ).10 ^(- ^|S21| / ^20) ) - 100

X = (R+100).tan (θ)

Steve G3TXQ

On 13/01/2018 20:42, Richard (Rick) Karlquist wrote:
> There is a much simpler way to do these calculations that
> doesn't involve any complex arithmetic.  It applies to
> the two special cases where the impedance is:
>
> 1) Pure resistive or 2) Pure reactive.
>
> For the first case, R = 2Zo[10^(-A/20)-1]
> where A = magnitude of S21 expressed in dB
> Every 2 port VNA displays A = |S21|<dB>.
>
> For example, if |s21| = -20 dB, R=900
> where Zo=50 ohms
>
> For the second case, X = 2Zo sqrt[10^(-A/10)-1]
> where A = magnitude of S21 expressed in dB
>
> For example, if |s21| = -20 dB, X=995
> where Zo=50 ohms
>
> I printed a lookup table made with Excel where
> I can just look up R or X. I do need to verify
> that the other variable (be it X or R is near
> zero).  For example:
>
> |s21|<dB>:  R:      X:
> -5          78       147
> -10        216       300
> -15        462       553
> -20        900       995
> -25       1.68k    1.78k
> -30       3.06k    3.16k
> -35          5k       5k
> -40         10k      10k
>
> If the impedance is complex, |Z| will be between these values.
>
> For large values, both of these asymptotically approach:
>
> |Z| = 2Zo[10^(-A/20)]
>
> For example, if |s21| = -40 dB, |Z| ~= 10,000
>
> Knowing the peak impedance of the choke (which will
> always be pure resistive) and the low frequency
> impedance (often pure inductive) is often enough for
> ham radio work.
>
> A possible plan is to use s11 to measure the impedance
> at frequencies where it is up to around 1k ohm.  The R and X can be
> read directly off the Smith chart marker, and the measurement is
> fairly accurate in that case. Above 1k, s22 can be used and the math
> is very simple.
>
> Rick N6RK
>
> On 1/13/2018 12:26 AM, Jim Brown wrote:
>> On 1/12/2018 7:09 PM, Norm n3ykf wrote:
>>> That software has a conversion utility from S11.
>>>
>> Except that S11 is the WRONG way to measure ferrite chokes. Read the
>> rest of the thread to learn why.
>>
>> 73, Jim K9YC
>>
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