Transformer for Audio

Audio transformers are "wide band" transformers. In essence a "transformer" is two or more windings coupled by a common magnetic field. It is this magnetic field which provides the means to pass voltages and currents from the primary winding to the secondary winding when alternating current flows.
Essentially the main purpose of an interstage audio transformer is to isolate the DC and couple the signal, with minimal loss. The transformer windings look like short circuits to DC, yet are seen as complex impedances to the AC signal.
Much which follows on the topic of audio transformers is of necessity somewhat over simplified to give a general overview. A transformer schematic is depicted in Figure 1 below.

Figure 1. - audio transformers schematic 

  Audio transformers were originally used in valve or tube amplifiers as interstage and output transformers. Early transistor amplifiers similarly used audio transformers for coupling and output stages.
There is nothing particularly mystical about audio transformers although their use is not particularly widespread today. I expect the greatest interest would come from people interested in restoring or building tube audio amplifiers. We will first consider the basic requirements.
We must assume fidelity is foremost in the minds of most restorers. Therefore the first requirement must be a relatively wide bandwidth, this is in the audio sense. My research into long forgotten and dusty, musty papers in my secret collection leads me to freshen my mind that audio transformer coupling is essentially applicable to power stages for impedance matching. Resistance-coupled amplifiers essentially produce undistorted output signal voltages regardless of plate-load-resistance value.
Maximum output power is produced when the impedance of the load matches the plate impedance of the tube. Using triodes, somewhat less output power but significantly less distortion results when the load impedance is between two to three times the plate impedance.
However tetrode or pentode tubes have very high plate impedances and the primary impedance of the coupling transformer is often between 1/10 to 1/5 of the tube plate impedance because it it not entirely feasible to produce audio transformers with primary impedances beyond 20K.

Primary impedance of audio transformers

Let us assume we are going to design a transformer of 20,000 ohms primary impedance. Now there is a direct mathematical relationship between the inductance of the primary, the design impedance and roll off frequencies. For very obvious reasons you would want steer somewhat clear of mains frequencies. One superior advantage of battery powered equipment is that "50 / 60 Hz mains hum" is not a problem. Of necessity because of the voltages involved all tube equipment derives its power from a "mains supply". Battling 50 / 60 Hz mains hum is an absolute pain so our theoretical transformer will begin "rolling off" at 100 Hz. Here is the formula.

Figure 2. - audio transformers formulas

In our fomula 4 * pi may be taken as 12.566, f is 100 Hz and of course R is 20,000 ohms. This formula is for a 3 dB roll off at 100 Hz. Substituting all those numbers into our formula and remembering all units are Henries, Ohms and Hertz we get a primary inductance of 15.92 Henries. That is a huge and possibly impossible value to obtain.