9.2.4.7 Transformer Selection and Design

After determining the value of current sense resistor R_CS, determine the maximum peak current at maximum demand point on the modulator. Accommodate for the I_PEAK adjustment for frequency dithering. Use this value when calculating the margin for core saturation. In this case, I_PK(sat) calculates to 4.13 A.

Equation 29.

In subsequent calculations of required primary turns etc, the average maximum peak current, I_PK(max) , during the frequency dither period should be used, which calculates to 4.0 A.

Equation 30.

Knowing I_PK(max), L_PRI and the turns ratio, the choice of transformer size and core shape and type dictates the required number of primary, secondary and bias turns, and the size of the air-gap. Various trade-offs, design preferences, and transformer design targets (size, cost, target losses, etc.) influence the specific choice of transformer core in any given design.

In the case of the UCC28630EVM-572 (PWR572 EVM), core area-product geometry was used to choose the minimum core size available to meet the power level. The core geometry factor K_g is a figure-of-merit that reflects the core power capability, in terms of its physical size, shape and design. It combines the core effective cross-sectional area, A_e, winding window area, A_w, and the mean length per turn (MLT) of wire around the core.

Equation 31.

Estimate the required design core geometry, K_G(des), using the required transformer inductance L_PRI, maximum peak current I_PK(max), allowed maximum core flux density B_max and a target copper loss budget, P_CU.

Equation 32.

where

• ρ_cu is the resistivity of Copper (approximately 1.7 × 10^-8 Ωm at room temperature, 2.2 × 10^-8 Ωm at 100°C),
• K_u is a winding window utilization factor that accounts for the percentage of the window that is occupied by Copper

K_u can often be as low as 25%, due to the fill factor (gaps between wires), wire insulation (especially for triple-insulated wire), and the need for insulating tapes and EMC shielding layers. The estimate of the required core geometry needs an estimate of the aggregate total winding current I_TOT. The analysis models the flyback transformer primary and secondary windings as a single lumped non-isolated inductor (such as a single winding buck inductor), only for the purpose of sizing the required core winding window to achieve the target copper loss. In this case, the secondary-side current amplitude reflects to the primary side so that aggregate total primary current. I_TOT can be estimated in Equation 33.

Equation 33.

where

• d is the primary on-time duty cycle
• d_SEC is the secondary-side flyback period duty cycle

At rated power and minimum bulk capacitor voltage, the inductance L_PRI has been chosen to achieve boundary-mode conduction, therefore the duty cycle is given in Equation 34.

Equation 34.

and

Equation 35.

At the boundary conduction point, the primary peak current I_PK is at the level set by the modulator, V_CS(bcm). So from Equation 33, I_TOT becomes Equation 36.

Equation 36.

Equation 36 calculates I_TOT as 2.6 A. Thus the required design K_G(des), assuming K_U of 25%, B_max of 315 mT and a target of 1-W copper loss, is shown in Equation 37.

Equation 37.

Equation 37 indicates that this design requires a core size and shape with a K_G of more than 6.9 × 10^-12. A review of commonly used cores indicated that the RM10/I core set meets this requirement. With A_e of 96.6 mm², A_w of 44.2 mm² and mean length per turn (MLT) of 52 mm, K_G(RM10) is 7.9 × 10^–12, giving some margin over the design target.

Equation 38.

With the chosen core, the actual primary, secondary-side and bias turns can be calculated. The required primary turns depend on the allowed B_max. For most power ferrites, a value in the region of 315 mT is commonly used.

Equation 39.

Round N_P to 34. Now the required secondary-side turns can be calculated, using the previously calculated turns ratio per Equation 23.

Equation 40.

Again, N_S is rounded to 6. Due to the integer rounding of the turns count, ensure that the actual turns ratio is within 5% of original target (if outside this range, secondary-side rectifier or primary MOSFET stress may be too high).

Equation 41.

From Equation 24, the required bias turns can be calculated using Equation 42.

Equation 42.

Again, N_B is rounded to 4. The effect of integer scaling in the turns is verified by calculating the expected bias voltage versus target.

Equation 43.

The V_BIAS target was 12 V, so this is acceptable.

The required core inductance factor, A_L, to achieve the target inductance can be calculated as in Equation 44. The transformer manufacturer uses this factor to gap the core center leg.

Equation 44.

Finally, calculate the required air-gap length l_g, based on the required inductance and the core geometry.

Equation 45.

where

• μ₀ is the permittivity of free-air
• μ_r is the relative permeability of the chosen core ferrite material
• A_CENTRE is the cross-sectional area of the core center leg
• l_m is the core average magnetic path length

For the RM10/I core in 3C95 material (chosen for low core loss over a wide temperature range), the required air-gap length is calaulated using Equation 46.

Equation 46.

Typically, the air-gap calculation in Equation 45 underestimates l_g, due to flux fringing in the air-gap. The fringing causes the affective area of the air-gap A_g to be somewhat larger than the ferrite core center leg A_CENTRE, depending on the gap length. This difference requires an increase in the required air-gap length to get the required inductance, which results in a further increase in fringing. However use Equation 45 to determine an initial value for l_g, which can then be used to estimate A_g. For round centre legs, the increase in effective area within the gap can be estimated empirically from Equation 47

Equation 47.

where

• D_CENTRE is the center leg diameter

(For more information about this subject, download the paper Inductor and Flyback Transformer Design, Lloyd Dixon, TI Power Supply Design Seminar SLUP127).

Because Equation 45 assumes that A_g equals A_CENTRE, it must be modified using Equation 48.

Equation 48.

Re-iterating the air-gap calculation in Equation 49 .

Equation 49.

Typically, after the second iteration above in Equation 48, the estimated air-gap is very close to the required value. Further iterations can be made, but should not be necessary.