Three design formulas for single-ended flyback transformers

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Three design formulas for single-ended flyback transformers [Copy link]

Three design dormulas of flyback transformer

Abstract: The different expressions of the three design formulas are analyzed and necessary deductions are made to clarify their relationship and whether they are correct or not.
Keywords: Primary (critical) inductance, core air gap, primary winding, design formula

. Designing transformers is inseparable from formulas. Sometimes it is found that the expressions of several books are not exactly the same. This makes some designers who are just getting started at a loss and confused. Different expressions, some of which use different parameters, have the same actual results. However, some are wrong and should not be used.
Now, let's analyze the different expressions and necessary deductions of the three main design formulas for the single-ended flyback transformer of the switching power supply in the three books on hand: finding the primary (critical) inductance, calculating the core air gap and finding the number of primary winding turns.
These three books are: "Electronic Transformer Handbook", "Modern High-Frequency Switching Power Supply Practical Technology" and "Design, Production, Debugging and Maintenance of Integrated Switching Power Supplies". (Hereinafter, they are abbreviated as <Manual>, <Technology> and <Production>.)
The first set of formulas: Calculate the primary (critical) inductance

① <Manual> p.387.

② <Technology> p.76.

③ <Production> p.108.
The second set of formulas: Calculate the core air gap

④ <Manual> p.388.

⑤ <Technology> p.75.

⑥ <Technology> p.75.

⑦ <Production> p.109.
The third set of formulas: Calculate the number of turns of the primary winding

⑧ <Manual> p.389.

⑨ <Technology> p.77.

⑩ <Production> p.109.
Due to the unification of the units of various parameters, the above formulas have been slightly changed compared with the original text.
The units of various parameters are as follows: Up1, Up2, E, Vi, Vo-V, T-μs, PW, B-Gs, RL-Ω, g-cm, Ip-A, AL--nH/N2, Lp-μH.
The relationship between the parameters in the formula is: n=ETon/VoToff, E=Vi=Up1, Vo=Up2. Among them, E, Vi, Up1 refer to the input DC voltage minus the circuit voltage drop.
The transformer efficiency is ignored in the formula calculation.
Now, let's analyze and derive the first set of formulas.
According to

:

(1-1)
According to 2PoT=LpIp2, we can get:

, (1-2)
From (1--1), we know:
LpIp=ViδT=ViTon, (1-3)
From (1--2) and (1--3), we can get:

, (1-4)
Substitute (1--4) into (1--1):

, (1-5)
Replace Vi, δ with E, D, which is the right side of formula ②.
Substitute Po=Vo2/RL into (1--5) So:

Multiply the numerator and denominator by T to get:

(1-6)
Replace Vi with E, which is the middle part of formula ②.
Comparing equations ① and ②, we can find that

the two formulas are consistent as long as they are proved.
Up1 and Up2 are replaced by E and Vo respectively. Substitute n=ETon/VoToff into:

.
In the second set of formulas, it can be clearly seen that one of equations ④ and ⑦ must be wrong. The numerators are the same, but the denominators are ΔBm and Bm2 respectively.
In the brackets of equation ⑥:

and

are different from each other. Otherwise, the difference between the two is equal to 0.
AL refers to the inductance coefficient when the core has no air gap, and Lp is the inductance value when there is an air gap.
The effective length of the magnetic path after the core is gapped

is e=

c+μ r

ɡ (2-1)

c is the magnetic path length of the core, μ r is the relative magnetic permeability.
After moving (2--1), we can get:
μ r

ɡ=

e-

c After moving

(2--3) , we can get: , (

2-4) After substituting (2--4) into (2--2), we can get: . (2-5) μo is replaced by 0.4, and the coefficient is not considered. This formula is exactly the same as formula ⑤. (2--3) is the formula for the inductance coefficient with air gap. So the formula for the inductance coefficient with and without air gap is: , (2-6) After moving (2--6), we can get: , (2-7) Substituting (2--4) and (2--7) into (2--2), we have: , and propose the common factor μo Ae, . (2-8) μo is replaced by 0.4, and the coefficient is not considered. This formula is exactly the same as formula ⑥. Bm=μo μ r H=μo μ r NI/e, (2-9) Substituting into (2--9), we get: , (2-10) Multiplying (2--9) and (2--10), we get , (2-11) Transposing (2--11), we get , (2-12) Substituting (2--12) into (2--2), we get: , (2-13) Substituting μo with 0.4, ignoring the coefficient, this formula differs from formula ⑦ by one . In fact, formula ⑦ omits it. The final result is that formulas ⑤, ⑥, and ⑦ are consistent. Formula ④ is wrong.

In the third group of formulas, it can be seen that formulas 9 and 10 are actually consistent. The question is whether formula 8 is the same as formulas 9 and 10.
After transposing (2--10), we can get:

, (3-1)
(3--1) is consistent with formulas 9 and 10.
After transposing (2--9), we can get:

, (3-2)
After transposing (2--2), we can get

Substitute it into (3--2)

, (3-3)
Replace μo with 0.4, ignoring the coefficient. This formula differs from formula 8 by one

.
Formula 8 omits it.
Since (2--9) = (2--10), formulas 8, 9, and 10 are consistent.
From the three groups of 10 formulas, it is recommended to use formulas 3, 5, and 10.
Design example: Po=100W, E=300V, δ=0.4, fsw=50kHz, T=20μs. Vo=20V, Io=5A.
1. Calculate LpIp2:
LpIp2=2PoT=2×100×20=4000Vμs.
2. Calculate LpIp:
LpIp=ViTon=（300-20）×04× 20=2240 Vμs.
20V is the circuit voltage drop.
3. Calculate Ip:

A.
4. Calculate Lp:

H.
5. Select the core:

This formula applies to the following conditions: Bm-Gs, fsw-Hz, J-400A/cm2, δ-0.4, η-0.8.
Please refer to "International Electronic Transformer" 2003.5.p.70. "Two Methods for Selecting the Core in High-Frequency Transformer Design".
The WaAe of the EI35 core is 1.31cm4, so it is OK to choose it. The core has: Ae=1.01cm2, lc=6.71cm, μr=2100.
6. Calculate the number of turns of the primary winding: 7. Calculate the length of the core air gap: 8. Calculate the number of turns of the secondary winding: References [1] Electronic Transformer Handbook [2] Modern High-Frequency Switching Power Supply Practical Technology [3] Design, Manufacture, Debugging and Maintenance of Integrated Switching Power Supplies [4] Two Methods for Selecting the Core in High-Frequency Transformer Design - International Electronic Transformer 2003.5.p.70.

Article author: Huang Yongfu

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Three design formulas for single-ended flyback transformers [Copy link]

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