Reactive load impedance in a transformer (Circuits I don't understand #1)

[M. Hamed]

Having a resonant circuit in the secondary of a transformer has been bothering me for quite some time. Circuit books teach that a transformer will transform the impedance by square of the turn ratio. Wes Hayward in "Introduction to RF Design" says that this holds true even when the impedance is reactive.

This is either misleading or I am confused. Wouldn't that mean that if you have a cap connected in parallel with the secondary then that capacitive impedance will transform as is to the primary multiplied by some factor.

My analysis if correct shows that this is not the case. Resonance between the secondary inductance and the cap play a big role in the result. Am I missing something?

Case in point in the circuit here:

https://www.dropbox.com/s/t6vt868q3xwl8c7/Transformer with a Cap in the Secondary.asc

at 10 MHz a 2.5 nF cap is 6.8 Ohms if that was transformed by the turn ratio it would be .06 Ohm. Simulation shows that the voltage at the primary is actually 800mV with a 1 V supply, showing that the impedance is more like 200 Ohms.

This seems like a direct result of the 100nH inductor resonating with the 2..5n capacitor.

The load impedance didn't transform as dictated by the turns ratio!!

 

[M. Hamed]

But once you take the transformer's own inductance
into account, on a transformer with a coupling constant of unity the
turns-ratio-squared stuff is exactly right.

How do I take the transformer's own inductance into account? Would that be another way of saying, do not operate near resonance of the secondary with the load cap?

 

[Fred Abse]

How do I take the transformer's own inductance into account? Would that be
another way of saying, do not operate near resonance of the secondary with
the load cap?

An .ac analysis, rather than .tran might show you more.

 

[M. Hamed]

An .ac analysis, rather than .tran might show you more.

Thanks, good idea. I did, and the results here showing the relation betweenvoltage and current at each side of the transformer:

https://www.dropbox.com/s/usi2xn20qayq088/V12.PNG
https://www.dropbox.com/s/3xrf8mj1dfz8fue/I12.PNG

It shows that VL1 and VL2 always differ by 20dB (turn ratio of 10) while the current ratio is only 10 at frequencies much higher than the resonant frequency 10 MHz. To me this suggests that impedance ratio is hundred only at a frequency higher than about 20 MHz.

 

[M. Hamed]

Model the transformer as an ideal transformer with parasitics outside.

I'm not sure I fully understand. Would you consider the inductance L1, L2 representing the transformer in LTSpice part of the parasitics?

 

[M. Hamed]

Crack open your 2nd-year circuits book and review.

I did that before posting. The method that was used in the book is to add a JwMI2 term for mutual inductance to the JwLI1 term for self inductance.

I used that method with mesh analysis and came up with a result that agrees with the simulation that I showed. V1/V2 always follows the turn ratio at all frequencies while I1/I2 doesn't around resonance frequency.

I am still not sure where I could be wrong

 

[M. Hamed]

This seems to agree with me:

http://www.qsl.net/va3iul/Impedance_Matching/Impedance_Matching.pdf

"Transformers match only the “real” part of the impedance. If there is a large amount of reactance in the load, a transformer will not eliminate these reactive components. In fact, a transformer may exaggerate the reactive portion of the load impedance. This reactive component results in power that is reflected to the generator."

 

[M. Hamed]

Wow, I finally managed to understand it. It's funny how you can read something over and over without it clicking until you do some hard work on your own and suddenly everything falls into place!