Reactive load impedance in a transformer (Circuits I don't u

[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%20with%20a%20Cap%20in%20the%20Secondary.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!!

 

[Tim Wescott]

On Sun, 04 Aug 2013 07:27:08 -0700, M. Hamed wrote:

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%20with%20a%20Cap%
20in%20the%20Secondary.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!!

The "turns ratio squared" business is accurate as long as two conditions
hold: one, you can ignore, or you take into account, the transformer's
inductance, and two, the coupling between the coils is good enough.

I can't quote numbers on the coupling constant -- I'd have to do some hen-
scratching on paper. 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.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

 

[Fred Abse]

On Sun, 04 Aug 2013 11:33:09 -0700, M. Hamed wrote:

On Sunday, August 4, 2013 9:41:30 AM UTC-7, Tim Wescott wrote:
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?

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

--
"Design is the reverse of analysis"
(R.D. Middlebrook)

 

[John Larkin]

On Sun, 4 Aug 2013 11:33:09 -0700 (PDT), "M. Hamed" <mhdpublic@gmail.com> wrote:

On Sunday, August 4, 2013 9:41:30 AM UTC-7, Tim Wescott wrote:
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?

Model the transformer as an ideal transformer with parasitics outside.

The parasitics are magnetizing inductance and leakage inductance and various
capacitances and ohmic losses.

So do your analysis around that model. The ideal transformer will then transform
the impedances as n^2.

I took two semisters of power electronics (60 Hz, utility type power) and about
all I got out of it was a good mental transformer model.


--

John Larkin Highland Technology Inc
www.highlandtechnology.com jlarkin at highlandtechnology dot com

Precision electronic instrumentation
Picosecond-resolution Digital Delay and Pulse generators
Custom timing and laser controllers
Photonics and fiberoptic TTL data links
VME analog, thermocouple, LVDT, synchro, tachometer
Multichannel arbitrary waveform generators

 

[M. Hamed]

On Sunday, August 4, 2013 9:41:30 AM UTC-7, Tim Wescott wrote:

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?

 

[M. Hamed]

On Sunday, August 4, 2013 11:59:11 AM UTC-7, Fred Abse wrote:

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

Thanks, good idea. I did, and the results here showing the relation between voltage 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]

On Sunday, August 4, 2013 12:09:58 PM UTC-7, John Larkin wrote:

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?

 

[John Larkin]

On Sun, 4 Aug 2013 14:21:46 -0700 (PDT), "M. Hamed" <mhdpublic@gmail.com> wrote:

On Sunday, August 4, 2013 12:09:58 PM UTC-7, John Larkin wrote:
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?

That would allow you to analyze and maybe understand your situation better.

Plop down L1 and L2, two huge inductors, like megaHenries, with K=1 to
approximate an ideal transformer.

Then add the known magnetizing inductance of your real transformer across the
secondary as L3. Add your resonating capacitor. Now the resonant impedance C||L3
is *outside* the ideal transformer and the impedance transformation to the
primary makes more sense.


--

John Larkin Highland Technology Inc
www.highlandtechnology.com jlarkin at highlandtechnology dot com

Precision electronic instrumentation
Picosecond-resolution Digital Delay and Pulse generators
Custom timing and laser controllers
Photonics and fiberoptic TTL data links
VME analog, thermocouple, LVDT, synchro, tachometer
Multichannel arbitrary waveform generators

 

[Tim Wescott]

On Sun, 04 Aug 2013 11:33:09 -0700, M. Hamed wrote:

On Sunday, August 4, 2013 9:41:30 AM UTC-7, Tim Wescott wrote:
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?

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

Would that
be another way of saying, do not operate near resonance of the secondary
with the load cap?

No.

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

 

[M. Hamed]

On Sunday, August 4, 2013 10:26:13 PM UTC-7, Tim Wescott wrote:

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!

 

[John Fields]

On Wed, 14 Aug 2013 23:49:00 -0700 (PDT), "M. Hamed"
<mhdpublic@gmail.com> wrote:

>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!

---
Congratulations!

--
JF