Transformer design is a well-developed subject, and the reader is referred to the many textbooks that treat it. It is important not simply to design a transformer that will do, but one that is economical, efficient and makes the best use of available materials. However, it is not difficult to design a serviceable transformer, and by understanding how this is done we will know better why transformers are as they are. As in all design processes, there are numerous trade-offs between competing requirements, and some sort of optimum is sought. On the other hand, the properties of available materials rigidly limit the possibilities.
The size, shape and material of the core must be chosen, and the number of turns of the primary and secondary windings. The size of the wire and the insulation determines if the windings will fit in the space available. The windings must be arranged for minimum leakage flux. The insulation determines the permissible temperature rise, and means of cooling must balance the loss of energy under load. We will treat only the core and the number of turns, the fundamental parameters, in our analysis. A complete design must, of course, include all these features.
Two basic equations are used in transformer design. The first is essentially Faraday's law, E = dΦ/dt x 10-8 V, where Φ is maxwells (gauss times square centimeters). For transformers, this is written Bmax = √2 E 10-8 / 2πfNAK, where E = rms voltage in a winding, N = number of turns, f = frequency (Hz), A = area of core (cm2), K = stacking factor (the proportion of A occupied by iron). A sinusoidal variation in flux is assumed, which is a reasonable assumption when the core does not saturate, but by no means exact. Bmax is generally assumed as a design parameter, as well as a value for E/N in volts per turn, and the necessary A results. A typical value for a small transformer is E/N = 0.1 V/turn.
The second is Ampere's law, H = 0.4πNI/l, where l is the length of the magnetic circuit. From the value of Bmax, the value of Hmax can be found from the magnetization curve of the core material, and this equation used to determine N, when a reasonable value of I is assumed (say, 5% of the full-load current).
Let's try to proportion a transformer for 120 V, 60 Hz supply, with a full-load current of 10 A. All the AC values we use will be effective values. The core material is to be silicon-steel laminations with a maximum operating flux density Bmax = 12,000 gauss. This is comfortably less than the saturation flux density, Bsat. The first requirement is to ensure that we have sufficient ampere-turns to magnetize the core to this level with a permissible magnetizing current I0. Let's choose the magnetizing current to be 1% of the full-load current, or 0.1 A. The exact value is not sacred; this might be thought of as an upper limit. From past experience, we should have some idea of the size of core that will be required. Here, we will assume a simple, uniform magnetic circuit for simplicity. In an actual case, a more complicated magnetic circuit would have to be considered. If l is the length of the magnetic circuit, H is 0.4πN(√2I0)/l, and the magnetization curve for the core iron gives the H required for the chosen Bmax. From this, we can find the number of turns, N, required for the primary.
We could also estimate the ampere-turns required by using an assumed permeability μ. Experience will furnish a satisfactory value. It is not taken from the magnetization curve, but from the hysteresis loop. Let's take μ = 1000. Then, N = Bmaxl / 0.4π√2 μI0. If we estimate l = 20 cm, the number of primary turns required is N = 1350. The rms voltage induced per turn is determined from Faraday's Law: √2 e = (2πf)BmaxA x 10-8. Now, e must be 120 / 1350 = 0.126 V/turn, f is 60, and Bmax = 12,000 gauss. We know everything but A, the cross-sectional area of the core. We find A = 2.8 cm2.
This may, or may not, be an acceptable result. If not, we simply change our assumptions and try again. Design, after all, is an iterative process. By considering the above calculations, we can appreciate the changes that a different frequency, voltage and current rating, permeability or maximum flux density would produce. The number of secondary turns is now easily found from the volts/turn and the desired ratio. A 24 V secondary would have 24 / 0.126 = 190 turns.
The number of primary turns is determined so that the magnetizing current is limited to an acceptable value, and depends on the length of the core. The area of the core is determined by the required volts per turn, now that the total number of turns is known. These are the things that determine the size and weight of a transformer.
Toroidal ferrite or powdered iron cores are now easily available. Ferrite is a high-permeability, high-resistance material that has acceptable losses at high frequency. Powdered iron has granules insulated from each other for high resistivity (low eddy-current losses) and is good at moderate frequencies. Such cores can be used to experiment with transformer design, using a signal generator as a power source. The length, area and permeability of the core is now known at the beginning, making the above calculations somewhat easier. An oscilloscope can be used for measurements.
Powdered iron and ferrite cores have low Bsat and permeability values. A type 43 ferrite has Bsat = 2750 gauss, but a maximum permeability of 3000, and is recommended for frequencies from 10 kHz to 1 MHz. Silicon iron is much better magnetically, but cannot be used at these frequencies. The approximate dimensions of an FT-114 ferrite core (of any desired material) are OD 28 mm, ID 19 mm, thickness 7.5 mm. The magnetic dimensions are l = 74.17 mm, A = 37.49 mm2, and volume 2778 mm3. Similar information is available for a wide range of cores. There are tables showing how much wire can be wound on them, and even the inductance as a function of the number of turns. Consult the ARRL Radio Amateur's Handbook, or specifications from Amidon Associates.
Transformers with air cores are used in radio work. They have no core losses, and are not limited in frequency (which is why they are used). A little calculation will show how hopeless an air-core transformer is at power frequencies. Air core transformers have large leakage fluxes, which cannot be avoided, and therefore poor regulation. Windings must be carefully designed to give the largest mutual flux possible.