[TowerTalk] S21 to Rp & Cp - Equivalent Circuit

Richard (Rick) Karlquist richard at karlquist.com
Sat Jan 13 15:42:19 EST 2018

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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