I have a transformer being driven by a square wave at about 9 kHz with a 10/11 duty cycle. In order to measure the voltage on the secondary I have to set my meter to AC. Why is it so? The primary voltage is not AC so why is the secondary AC? and is it a sinusoidal wave? or alternating square wave? My theory is that it has to do with the way to magnetic field on the primary is constantly increasing and decreasing which would induce an alternating current on the secondary.
The output of a transformer is always AC. In the long run, the average current thru a transformer secondary with resistive load is always 0.
For example, consider driving the primary with a 0-10 V pulse train that is low for 9 µs and high for 1 µs. The DC level is 1 V going into the primary. However, that 1 V will be lost coming out of the secondary. The open-circuit voltage you get out is -1 V for 9 µs and +9 V for 1 µs, multiplied by the turns ratio. If the primary has 100 turns and the secondary 300, then you will get -3 V and +27 V, for example.
A good meter set to DC will always show this as 0 V.
Different AC voltmeters will show you different voltages for the -1 to +9 V example pulse waveform. That is because most meters don't read true RMS. Most will measure the average of the absolute value, then apply the correction factor from that to RMS for a sine wave. That correction factor will be incorrect for something like the pulse train in this example.
To get the voltage of non-sinusoid waveforms, you either have to get a true RMS meter, and make sure your signal is within its frequency range, or use a scope and do the math yourself. Some scopes have RMS measurement built-in as a math function, which makes them in effect true RMS meters. Again, keep the limitations of the RMS meter in mind.
Depending on the transformer, it might be a good or poor bandpass filter by itself. The better the bandpass, the more sine-y the output becomes, but it only does anything when fed that one frequency.
Most likely though, is that you still have a fairly square-ish waveform coming out of it, which like Plasma said, will not play well with either setting. Also, the transformer will remove any DC offset (which you shouldn't give it to start with; they don't like that) so that the DC setting will likely read close to zero on average.
You really need an oscilloscope to see what's going on, which you can sometimes approximate using a PC sound card. At 9kHz, your signal might be on the edge of what a good one can do, so try to find a card that can record at 96kHz or higher, and even then the internal filtering might round off some corners above 20kHz. Also make good and sure that the signal into the card is low enough to not blow it up (done that before), so do some math and aim for about 2V max.