L = N² Al — Voltage Regulation

Part 21 · Voltage Regulation · page 2 of 3

\(L = N^2 A_L\)

Key point: on one core, \(L\) grows with turns squared. The secondary shares that core, so \(L_s = L_p(N_s/N_p)^2\). \(\tfrac12 L_p i_p^2\) at turn-off is the packet the secondary dumps.

Orange dot is \(N_p\). Green dot is \(N_s\). Hollow ring is \(2N_p\). The dashed line is the wrong \(L \propto N\) guess.

1
Eight turns on 80 nH/t².
Np 8, Ns 8, AL 80. Lp is 5.1 µH. Ls is 5.12 µH. AL is the core: same ferrite, same gap, same number. The hollow ring is already telling you the next step. 16 turns lands at 20 µH, four times this Lp.
2
Double Np. L goes ×4.
Np 16, same AL. Lp is 20.5 µH. That is 4× the 5.1 µH, not 2×. The hollow ring moved to 32 turns and 82 µH. The dashed line is what L would be if it tracked N. The dot is on the parabola.
3
Ns does not pick Lp.
Ns 4. Np stays 16. Lp stays 20.5 µH. Ls is 1.28 µH. Ls/Lp is 0.063, which is (4/16)². Change secondary turns or change the core. Those are the two knobs. Ls is not a third one.
4
More AL lifts both dots.
AL 200, Np 16, Ns 8. Lp is 51.2 µH. Ls is 12.80 µH. Ls/Lp is still 0.250. 2Np → L reads 205 µH. A smaller gap raises AL and drops Isat. Isat is not on this chart. The flyback page wanted Lp large enough that Δi stays under 2 Iavg. This is that Lp.
5
One packet, two inductances.
AL back to 80. Ns 12. Lp stays 20.5 µH. Ls is 11.52 µH. Ls/Lp is 0.563, (12/16)². ½Lp ip² at the FET edge is what the secondary is allowed to dump. The two L readings are that one energy, written two ways.

Leakage, fringing, and \(A_L\) versus temperature are not in this chart. Gap design is a later magnetics page.

Parabola · same core

Dot climbs L = N² AL · dashed line is the wrong “L ∝ N” guess · hollow ring is 2Np → 4Lp

Np turns Ns turns AL nH/t²

L_p—

L_s—

A_L—

2N_p → L—

L_s/L_p—

Both windings sit on the same parabola because they share \(A_L\). The runner is \(L=N^2 A_L\). The dashed line is what people guess (\(L\propto N\)).