# When to treat transformers as non-ideal?

For an experiment I need to sum a DC and AC Potential. My solution to this was to construct the following circuit:

simulate this circuit – Schematic created using CircuitLab

Testing this circuit I found the following:
1. It works. I recover the AC signal with a DC offset. 2. The transformation ratio was inconsistent 3. The inconsistency comes from the DC offset?

The set-up is:
Test 1:
$V_{ac} = (0.5$V$)$Sin$[2\pi 8 t]$
$V_{dc} = (5$V$)$
$V_{out} = (20.8$V$)$Sin$[2\pi 8 t] + (5$V$)$
$\frac{V_{out}-V_{dc}}{V_{ac}} = 41.6$

Test 2:
$V_{ac} = (1$V$)$Sin$[2\pi 8 t]$
$V_{dc} = (15$V$)$
$V_{out} = (48.8$V$)$Sin$[2\pi 8 t] + (15$V$)$
$\frac{V_{out}-V_{dc}}{V_{ac}} = 48.8$

Approximately a 17% increase.

I reran these experiment at a later date and, found a transformation ratio of ~50 which is the expected result. The second time these we ran, the ratio did not change. Clearly I am neglecting to account for some practical aspect of using a transformer. I will be re-running the tests a third time with an AC source that has a floating ground so I can connect it to the DC supply.

Finally, I noticed that the transformation ratio nearly doubles at a f$\approx$810Hz and then falls off rather quickly. I imagine this must be due to some inductance / resonance effect, but I did not expect resonance anywhere. My AC source has an output impedance of 50Ohms.

I'd love to hear your thoughts on the causes for the inconsistent ratio. If you have any other solutions to adding AC and DC potentials (Max AC swing $\approx$ 6kV, Max f$\approx$1kHz), I'd love to hear those as well.

Details on experiment I am applying this AC+DC potential between to concentric spheres to create a minimum potential in free space. This allows a charged particle with a low enough charge to mass ratio to levitate in open air. I am studying the non-linear dynamics of said particle. What this means for the circuit is that I need to produce the potential with no load attached (no current runs from one sphere to the next). More the physics can be found here

Transformer Details I am running in the forwards direction. That is, I expect ~ $V_{out} = 50 V_{in}$ assuming the signals are sinusoidal. The core material is unknown. The turn ratio is unknown. The transformer is rated as a low power 6kV mains transformer. When plugged in directly to the mains, then, it should produce 6kV. I haven't tested this with a wall outlet, but I wouldn't be surprised to find that it actually only produces about 5.5kV. Various experiments show the transformation ratio to hover around 50, thus this was what I am taking as the expected value. Power is regulated through the use of two 10MOhm resistors in the path of the current, in case of a short. The test I describe above used current limited sources rather than the 10MOhms.

• With a large step up, and using a 'mains transformer' (the type of core here is vital to understanding, as is the number of turns (estimatable from the ratings), not just the turns ratio) I would always expect a secondary resonance at some frequency, maybe not as low as 800Hz. Pure output voltage offset should not cause the change in apparent ratio you've witnessed, so look for some other cause. You've drawn an open circuit load, so negligible current is flowing. If there is a current, and it changes with offset, and it saturates the transformer, that could cause a difference – Neil_UK Nov 21 '16 at 6:24
• Thanks for the thoughts, I think you've led me to some good googling. Indeed there is no load on circuit. I thought that since the ideal gain is 50, there should be no reason for it to nearly double, even at a resonant frequency. I think it might have to to with the impedance of the transformer and my supply. Looking into transformer saturation. – Edgar P Nov 21 '16 at 6:58
• How can 20.8V/1V = 41.6? – Bruce Abbott Nov 21 '16 at 7:39
• Resonance rarely occurs in mains transformers. Generally the number of turns, and hence the self capacitance, is low enough that by the time you've reached a suitable frequency, the iron has got so lossy that any resonance is very low Q. But a very high voltage transformer may. I've driven an iron wire core ignition coil at resonance, about 8kHz IIRC. – Neil_UK Nov 21 '16 at 8:09
• If you want to add a DC and AC component surely a simple op amp circuit (adder) would suffice. A lot easier than messing around with reverse connected transformers. – JIm Dearden Nov 21 '16 at 10:10