S parameters of a real transformer  - Page 1
Forum section: electronics / beginners

author: JonathanTee
author_rank: Newbie
timestamp: November 08, 2022, 04:39:39 pm
content: Hi Everyone,The scattering matrix of an "ideal Transformer" is a textbook example, it represents the impedance step introduced by the turns ration n:.I am trying to figure out the S-parameters of a transformerwith the magnetization branch. I have the VNA on my bench so basically I want to reproduce the result which i see on the screen.So far I have tried the following things:1) First attempt.Assuming that we have two cascaded 4 port networks (one is the ideal transformer, and the other is magnetizing branch) we can cascade them using T parameters. So I converted S parameters of the ideal transformer to T parameters and multiplied them with the T parameters of the magnetization branch. This almost works properly. The values S21, S12, S11 match with the measurement, the exception is S22, which in the calculations is always equal an absolute value to S11. For VNA measurement S11 and S22 are different.2) Second attempt.Described the transformer as Z matrix in a following way:Z11 = 1j*2*pi*freq(i)*L;Z22 = 1j*2*pi*freq(i)*L*n^2;Z21 = 1j*2*pi*freq(i)*L*n;Z12 = 1j*2*pi*freq(i)*L*n;and then converted Z matrix to S parameters. Here again the values S21, S12, S11 match with the measurement, and S22 does not.So here are the plots:Additional thought was to use the formulafrom to get S matrix for different source and load impedances. This does not work at all though. Here is the link to the article.https://empossible.net/wp-content/uploads/2018/03/IEEETransMTT_v42_n2_p205-Parameter-Conversion-Tables.pdfWill be really grateful for any advises, I am not really experienced at S-parameters magic, so maybe some true master can pinpoint what am I doing wrong..

author: mag_therm
author_rank: Frequent Contributor
timestamp: November 09, 2022, 01:09:50 pm
content: I think 3 out of 4 is OK , considering the over simplified lossless low frequency model used while sweeping to above 1 MHz.Measured S22 being different from model might be due to the omission of the mutual inductance term -jWMAdding that to your model, and leaving magnetising current as tee on primary, will increase model to (4) Kirchoff loopsOther additions for a high frequency model would include adding complex permeability which will split your I_mag Tee,and adding shunt and interwinding capacitance.But I suggest to try just adding -jWM first.

author: jonpaul
author_rank: Super Contributor
timestamp: November 09, 2022, 02:43:04 pm
content: errors in model, add series leakage inductanceadd series resistance losses Pcuadd shunt core losses Pfej

author: JonathanTee
author_rank: Newbie
timestamp: November 10, 2022, 10:02:01 am
content: Thanks for the replies friends,mag_therm, well the mutual inductance is what i call a "magnetization branch" in my case. Or have I understood you wrong? Also losses in the transformer are rather low, because the S21 lies close to 0dB in the bandpass.The leakage inductance Ls is indeed quite high for one side of a transformer, and if I add it into the equations, I get:Z11 = 1j*2*pi*freq(i)*(L+Ls);all other Z12, Z22, Z21 are the same. And eventhough the resulting S parameters will be a bit different, the S11 will still be perfectly equal to S22, which is not true in reality.All in all I tend to think that the wrong value of S22 isnotthe result of losses or high frequency or any parasitic things, but just my equations approach is wrong at some point.

author: mag_therm
author_rank: Frequent Contributor
timestamp: November 10, 2022, 01:27:46 pm
content: No,Mutual Inductance is caused by flux leakage between windings and is M = k* sqrt(Lp*Ls)where k is called coefficient of coupling, and is assumed =1 in the simple low frequency models.In actual transformers M causes the secondary voltage to drop ( drop increasing with frequency) when the load Z_out is connected to secondary.You should find a reference to it, showing where jWM fits in the Kirchoff meshes and how it is measured.If this is course related perhaps you should use references.