I decided to write a Blog on the basics of transformer design as I believe there are a lot of members that may not have been exposed to it before. Some think it's a black art or black magic, but it's actually based on rock solid science. I'm going to stick with the basics and not get too deep in this blog. And I'm planning on writing a companion Blog on Inductor design soon.
Number one rule: a transformer only operates on a changing input voltage. This usually means AC or pulses, but let's stick with AC for this blog. To keep things simple, let's use a sine-wave for an AC input. A transformer has some pretty great properties. It can step-up or step-down voltages. It can do the same with currents. It can even do the same with impedances and hence, can match an input impedance to an output impedance so as to obtain a maximum transfer of power from an input source to a load. The ability to change voltage, current, or impedance is controlled by the turns ratio between primary and secondary. The equation that governs this is: Vsec = (N2/N1)*Vpri, where Vpri is the primary winding voltage, Vsec is the secondary winding voltage, N1 is the primary number of turns and N2 is the secondary number of turns. For impedance, we simply use the ratio of secondary to primary turns squared instead of the ratio of turns itself. In equation form, it is: ZSec/Zpri = (Nsec/Npri)^2, which is N^2sec/N^2pri.
Another amazing property of a transformer is it's ability to isolate the primary circuit from the secondary circuit, yet, it can transfer power from one side to the other. This is true only for an isolation transformer. There is also an Autotransformer which is the equivalent of an isolation transformer with the bottom leads connected together so that isolation is no longer possible, but the step-up and step-down property still works.
Most transformers require a ferromagnetic core made of typically, Silicon Steel for frequencies up to 1 KHz, or Ferrite for frequencies from about 10 KHz up to several MHz. There are other, specialized core materials, but we won't get into them here. When using a magnetic core, if enough voltage is applied to the winding, which in turn creates a magnetic flux in the core, the core will saturate. When this happens, the core no longer functions as a magnetic core and a large current will flow in the primary. In order to avoid core saturation, the transformer must be designed for a certain range of input voltage and frequencies. Actually, it's the product, called volt-seconds that's important. The flux density must be kept below the saturation value for the type of core material used. For magnetic Silicon Steel, this is between 12.5 to 16 KiloGauss in the CGS system of units, which is equivalent to 1.25 to 1.6 Tesla in SI units. For Ferrite, saturation typically happens at 3.0 to 3.5 KG (0.3 to 0.35 Tesla). In order to design a transformer that will not be in saturation, the following formula is used for a sine-wave input: N = E*10^8/(4*1.11*F*A*B), where N= number of primary turns, E= primary voltage in volts, F = sine frequency in Hz, A = core cross sectional area in cm^2, and B= flux density in Gauss. As an example, if we were going to use a standard EI-150 core for a 120 VAC, 60 Hz transformer, the core area for a square stack can be found in the core manufacturer's datasheet as 1.5" x 1.5" x 6.45 cm^2/in^2 = 14.5 cm^2. For a core grade of 29 gauge M6, the maximum flux density is rated at 14.5 KG at a loss of about 6 watts/Lb. The primary turns needed,therefore, will be: N = 125E8/(4*1.11*60*14.5*14.5E3) = 223.17 turns, which can get rounded down to 223 turns in this case. The primary wire size needed depends on the primary current. Note that we used 125 VAC for the primary voltage. This is to give us a little safety marking against core saturation.
In our example, let's say the transformer secondary output power shall be 150 watts, and will be 10 VAC RMS at 15 Amps. For a power line frequency transformer such as this, we can use the current density to determine the wire sizes, starting with the secondary. A good value of current density is 600 C.M./A (circular mils/amp). We need to multiply the secondary current by the current density to get the needed wire cross sectional area. 15 Amps * 600 C.M./Amp = 9,000 circular mils. Next, using a magnet wire chart for round wire, we find that the closest wire size in the American Wire Gauge (AWG) is AWG #10 at 10,384 C.M. Using a current density of 600 C.M./Amp should give an output voltage regulation of approximately 5% from no load to full load.
Next, we determine the secondary turns and must take the 5% load regulation into consideration. We do this by increasing our no load secondary voltage by the 5% load regulation or in our example, 10VAC + 5% = 10.5 VAC. To compute the secondary turns, we use the equation: Nsec/Npri = 10.5 VAC/120 VAC. After rearranging, the equation becomes: Nsec = Npri*10.5/120, or Nsec = 233*10.5/120 = 20.39 turns, which we should round down to 20 turns.
For the primary wire size, we take first compute the primary current at full load of 15 amps in this case. PriCurrent = SecondaryCurrent*10.5/120, so PriCurrent = 15*10.5/120 or 1.313 Amps. For the primary, we can relax the current density a bit since the load regulation won't be affected by the primary wire size. A good rule-of-thumb value to use is 500 C.M./Amp. Therefore, we need: 1.313 Amps * 500 C.M./Amp = 656.5 C.M. Again, we go to the magnet wire chart for round wire and find the closest wire size to 656.5 circular mils, which is: AWG #22 at 640.1 C.M.
That pretty much completes the initial transformer design which I'll summarize here:
Transformer Primary Voltage: 120 VAC, 60 Hz
Transformer Secondary Voltage: 10 VAC at 15 amps
Primary Wire: 223 Turns of AWG #22
Secondary Wire: 20 Turns of AWG #10
Some of you may wonder how the core size is chosen. It's largely done by using the Area Product, which is Aw*Ac*k, where Aw is the window area (the available area the wire must fit into), Ac is the core area, and k is a constant that depends on factors I won't get into here. For those that are interested, a great book on the subject is by the late, Col William T. McLyman of CalTech who came up with this concept of numerically determining the smallest core needed for a given set of transformer parameters. The name of the book is: Transformer_&_Inductor_Design_Handbook.
And if your using a standard transformer lamination, there's no need to always use a square stack. Sometimes, depending on the available space, it's better to use a stack longer than square. Rarely have I seen transformers with a stack shorter than square, but it's sometimes done. The standard laminations have are sized and named after their center leg width in inches. For example, an EI-100 has a 1.00 inch wide center leg. The outer legs are half the width of the center leg, as are the I pieces.
Here's a link to one of the best magnet wire data books available. It's from their Superior Essex, Furukawa Division: https://essexfurukawa.com/wp-content/uploads/2019/09/Essex-Wire-Engineering-Data-Handbook-EN.pdf
I'll be glad to answer any questions as I didn't cover all aspects of transformer design. And although most of this material does apply to high-frequency transformers, the primary turns equation usually needs to be modified, especially for unidirectional waveforms (pulsed DC).
