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I hope someone here can help.  I have a AC to AC step down Unrectified transformer, i.e., no circuit board whatsoever.  Primary winding has continuity, secondary wire has continuity.  No continuity between primary and secondary wires in any way.  In short, power goes in, but no voltage comes out.  If someone that knows precisely how transformers work can explain to me how a magnetic field can be generated but no electricity generated on the 2ndary wire I would appreciate it.

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A transformer has mainly 2 parts the primiry coil and the secondery coil, it is usualy sometimes found that the copper wire in the primiry coil is thin as compared with the secondry

the working all depends on the number of turns on each side that is at the primiry and the secondary coils and the both of them are not connected because a transformer works on the principle of induction the AC current coming to the primiry or the secondry creates a magnetic flux and induces current on the other coil,

I will also give u the formula to calculate the voltage input or output or to calculate the number of turns needed in the secondry or primiry to attain a certain voltage

number of turns in the secondry coil /number of turns in the primiry coil =voltage in the secondry coil /voltage in the primiry coil

I hope this helped you

Ok, usually not my stuff but anyway to sum it up:
Assuming you used AC to feed the transformer...
Primary is input, secondary is output.
Number of windings, voltages and amperes are all related.
In a perfect world the ration of the number of turns can be used to calculate the outputs.
So if you feed ac into a working transformer then ac will come out if the input is sufficient and the frequency is suitable.
Nothing on the sencondary in this case would mean your input voltage is too low.
For obvious reasons you should confirm by checking if the primary uses any real power.
Some transformers use a temp fuse on the secondary or primary winding, usually hidden under some tape - if this fuse is blown nothing will come out.

The obvious "jiggle the cables" debugging question: the primary is indeed connected to an AC source, right?

check the plug that goes into the wall there is a fuse depending on the plug type, as well a fuse inside the box.

i spent a few months in a country that required them and those were commonly found in the trash because sand and dust will get in them or they dropped and got cracked.

they are basically 70 lbs pieces of junk.

is it hard to replace? What voltage and amperage

What type of transformer, IE (laminated steel or powdered iron)?

Powdered Iron breaks easy making a gap in the magnetic field.

What is the continuity you are getting?

It should only be a couple ohms to almost nothing on the secondary winding and proportionately higher on the primary winding.

If the impedance is disproportionately high on one of the winding's there is a break in the winding that still touches.

Thanks for the reply :D The entire structure is enclosed in an inpenetrable rubberized, weatherproof block and based on the weight I would say steel layers. I am getting a few ohms on both, more on the primary side. I never thought about it being a physical problem, I am guessing that somehow the rubber or metal over time has somehow become a field eliminator. I just have never seen one fail with full-voltage going in and zero going out even though both sides have continuity. Thanks again.

That sounds good.

Rubberized or silicone jell covered most likely powdered iron core cracks when dropped, even though covered in a jell.

In the past black tar was used, you could use solvent to remove the tar and repair the transformer. Not so today.