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Among the first decisions engineers must make when designing a new switchmode power supply are transformer size, working frequency and the selection of turns for the most optimum design. Unfortunately, designers can't just plug their requirements into an electrical formula and get a precise answer.

Designers may opt for minimum copper loss and best regulation (giving a core loss limited design) or minimum core loss (giving a copper loss limited design) or an optimum efficiency design. Normally, we design for optimum efficiency to minimize the temperature rise and permit the maximum transferred power on the chosen core size. In practice, core size depends on many factors that may be hard to define in the early stages of the design.

One starting point for initial core size selection is found using one of the nomograms provided by the core manufactures (**Fig. 1**). Typically, these nomograms assume a temperature rise in free air conditions of 30°C so, at best, they only provide a starting point. Next, you must perform an optimum transformer design on the selected core and check the performance in the actual prototype unit under worst-case working conditions. Use a thermocouple to measure the temperature rise. Even experienced designers may do more than one design before getting an acceptable result.

Apart from other factors, optimum efficiency is desirable to minimize temperature rise. Fortunately, after choosing an initial core size, the design for optimum efficiency is a well-defined process. Optimum efficiency occurs when the total losses are minimized. Copper loss increases as the number of turns increase. (If the available winding space is to be fully utilized, increasing the turns makes the length of the winding greater and means a smaller diameter wire must be used). Core loss depends on the core material, core volume, working frequency and flux density.

Flux density is inversely proportional to turns, so while copper loss increases with turns, core loss decreases. Optimum efficiency is found where the sum of these two losses is at a minimum. Normally, by the time you reach the transformer design stage, core material, initial core size and working frequency have been chosen. The only variable now is the choice of turns.

**Fig. 2** shows actual measured loss values made on an EC41 ferrite core using various numbers of turns. It shows how the copper loss reduces but core loss increases as the turns are decreased; hence, the working flux density swing is increased. Examples are shown for both a 50-kHz and 20-kHz design. Notice how the sum of the copper and core losses has a clear minimum of 2 W near where copper and core losses are equal in both examples. For optimum design, we try to design for this condition.

Also notice that the 20-kHz design has a loss of 2 W at a transferred power of 150 W (98.7% efficiency), while the 50-kHz design has the same loss at a transferred power of 210 W to give 99% efficiency. So, the core can transfer more power at the higher frequency with the same temperature rise. Notice the flux density swing (peak-to-peak flux density change) at minimum loss is near 180 mT for the 50-kHz design and 330 mT for the 20-kHz design. For the 20-kHz case, the core would be near saturation for a forward converter design. Thus, at a lower frequency, the design becomes “saturation limited” and optimum efficiency is no longer possible.

Usually, we optimize a design by choosing the correct working flux density, which defines the number of turns. So, how do we establish the optimum flux density to minimize loss in a given design?

**Fig. 3** shows a nomogram designed by the author for selecting the optimum flux density and core size in the 30-W to 1000-W power range and the 10-kHz to 100-kHz frequency range. To use this nomogram, enter from the top with the chosen working frequency. Where this frequency line intersects with the required power line, you get a working point. From this point, project vertically down to the lower scale to get the optimum flux density, and project left to get an area product (AP) number. (The example shows 20 kHz and 150 W with a flux density of 330 mT and an AP of 2.2.)

AP is a figure of merit indicating core size. Manufacturers often provide the AP for their cores, or it may be obtained by multiplying the center core pole area by the available winding window area in centimeters. Examples of cores meeting the AP number are shown projecting to the right. For an AP of 2.2, near sizes include PQ32/30, ETD 34/17/11 and EC41. The EC41 was used in the example for **Fig. 2**, and the nomogram confirms the optimum flux density of 330 mT for the 20-kHz example.

Use the following formula to calculate optimum primary turns (N):

where V is primary voltage (V), t is transistor “on” period (µs), B is optimum flux density swing (mT), and A_{e} is area of core center pole (mm^{2}).

Other windings may be determined based on the required voltages. For best results, the method of winding and wire sizes must be optimized to reduce skin and proximity effects.

Last, check the design for optimum efficiency by calculating copper and core losses and by verifying they share almost 50% of total losses. **Fig. 3** assumes free air-cooling and a 30°C temperature rise, so a second iteration may be needed in the final application.

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