What are transformers?

The name transformers is derived from the fact that when two coils are placed in close inductive proximity to one another the lines of force from one cut across the the turns of the other inducing an ac current, energy is transformed from one winding to another and this is called transformer action.
There are a great variety of transformers for a variety of applications including power transformers, audio transformers and rf transformers among others. All work on the above principle.

Power transformers

As the name implies a power transformer is designed to usually translate voltage from one level to another. Another type called a current transformer will not be discussed here. The schematic of a transformer is depected in figure 1 below. Consider also the topics covered under power supplies where power transformers are used.

Figure 1. - general transformer schematic

Some power transformers have a centre tap on the secondary side. Note in figure 1 above the left hand side is usually denoted the "primary" whilst the right hand side is denoted the power transformers "secondary" side. Most power transformers are designed for frequencies in the region of 50 / 60 Hz which are the principle mains frquencies around the world.
Some examples of power transformers are shown in figure 2 below.

Figure 1. - photograph of different types of power transformers

The transformer on the upper left has "flying leads" for, in the case of all these transformers, the incoming voltage is the Australian standard 240V AC. The secondary side has three "lugs" to connect to and this is a 240V - 6.3V CT transformer.
The transformer on the upper right hand side is a multi-tap type with "flying leads" on both sides. The output allows you to select 6.3V, 9V, 12V and 15V depending upon your requirements, maximum current is 1A.
The transformer on the bottom left is called a "plug pack" in Australia. This one plugs directly into a power point and because a rectifier is included within the plug pack it produces 12DC @ 1A on its output. Note the four way connector shown at the centre, very bottom of the picture. This connector is designed to connect to the four sockets the manufacturer considers most popular.
The power transformer on the bottom right has a bit more "grunt" but only providing "lugs" for connecting leads. It is also a "multi-tap" type but designed to provide 2 amps.
Modern power transformers are wound on a "bobbin" which fits a core manufactured of materials to suit mains frequencies. The power handling capacity of a power transformer is determined by the physical size of the core and its properties. Design information is available from manufacturers. Ultimately the design information will provide the number of turns per volt. It is important to note that "toroid" power transformers are becoming increasingly popular and can handle larger amounts of power for the same physical dimensions and are thought by many "experts" to offer superior performance, particularly in higher power audio amplifiers.
The relationship of turns per volt holds good for both primary and secondary. A transformer designed for a nominal 250V AC input and a nominal 6.3V secondary output has a turns ratio of 250 / 6.3 or about 40:1
"Good design" usually leads to the cross-sectional "copper" areas of both the primary and the secondary being equal. Purely by way of illustration and not necessarily related to the real world, if the primary consisted of 2,400 turns of #34 gauge wire which is 0.16mm dia we would have a total cross sectional area of:
2,400 * [(0.16 ² * pi) / 4]. Where [(pi X D ²) / 4] is the customary formula to determine the area of a circular object.
Here you should get 2400 X 0.0201 = 48.24 mm ²
Therefore if our turns ratio was 20:1 for a 12V secondary, we would get 200 turns secondary still occupying approximately 48.24 mm ². With a little high school algebra we determine this gives us a secondary diameter of 0.55 mm diameter which is around about #24 gauge wire.
CAUTION: Of necessity I have greatly simplified the above to give a broad overview of how power transformers are designed. Always consult manufacturers such as Magnetics for tables and correct design information.
Remember "electricity KILLS!". Home construction of power transformers is a lost cause because the cost of component parts will always greatly exceed the cost of an off-the-shelf transformer.

Audio transformers

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 is contained on the topic of audio transformers is of necessity somewhat over simplified to give a general overview.
Go to: Audio Transformers

RF transformers

RF transformers generally fall into two categories, band pass filters and broad band transformers. Bandpass filters might well fall into the category of those used in IF amplifier filters while broad band transformers are generally used for impedance matching.

Figure 3. - schematic of an rf transformer

The type of broad band transformer depicted in figure 3 is often wound on a ferrite toroid of sufficient permeability to give a reactance of about 5 tomes the highest impedance at the lowest frequency of interest.