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How to design a flyback transformer

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The transformer for a flyback converter is used as the converters inductor as well as an isolation transformer.

1 Variables and acronyms
2 Initial calculations
3 Inductance calculations
4 Core calculations
- 4.1 Core selection
5 Current calculations
6 Power Loss
7 References

Variables and acronymsEdit

Universal constants
- Permittivity of free space μ_o (Wb A^-1 m^-1)
  - $μ o = 4π10 - 7$ (Wb A^-1 m^-1)

Wire variables:
- $ρ$ , Wire resistivity (Ω-cm)
- $I t o t$ , Total RMS winding currents (A)
- $I m, m a x$ , Peak magnetizing current (A)
- $I R M S$ , Max RMS current, worst case (A)
- $P c m$ , Allowed copper loss (W)
- $A c$ , Cross sectional area of wire (cm²)

Xformer/inductor design parameters
- $n 1, n 2$ , turns (turns)
- $L m$ , Magnetizing inductance (for an xformer) (H)
- $L$ , Inductance (H)
- $K u$ , Winding fill factor (unitless)
- $B m a x$ , Core maximum flux density (T)

Core parameters
- EC35, PQ 20/16, 704, etc, Core type (mm)
- $K g$ , Geometrical constant (cm⁵)
- $K g f e$ , Geometrical constant (cm^x)
- $A c$ , Cross-sectional area (cm²)
- $W A$ , Window area (cm²)
- $M L T$ , Mean length per turn (cm)
- $l m$ , Magnetic path length (cm)
- $l$ , or $l g$ , Air gap length (cm)
- $μ$ , Permittivity (Wb A^-1 m^-1)
- μ_r, Relative Permittivity (unitless)
  - $μ = μ o μ r$

Acronyms

RMS: root-mean-squared - $x_\text{rms} = \sqrt{ \langle x^2 \rangle} \,\!$ (where $\langle \ldots \rangle$ denotes the arithmetic mean)
MLT: mean length turn
AWG: American wire gauge

Initial calculationsEdit

Variables

$V o$ - output voltage [V]
$V i n$ - input voltage [V]
$V D$ - diode voltage drop [V]
$V R d s$ - transistor on voltage [V]
$N$ - turns ratio [unitless]
$D$ - duty cycle [unitless]

Calculate turns ratio

$\frac{ V_o + V_D }{ V_{in} - V_{Rds} } = \frac{ 1 }{ N } * \left ( \frac{ D_{max} }{ 1 - D_{max} } \right )$

Diode
- Rectifier: $V D = 0.8 V$
- Schottky diode: $V D = ?$

Inductance calculationsEdit

The inductance of the transformer, $L m$ , controls the current ripple.

Say you want a current ripple 50% of average current.
$Δ i = 0.5 * I$

Solve for $L m$

let $n = \frac{n_2}{n_1}$

$I=\frac{n}{D'}I_{load}$

$\Delta i = \frac{nI_{load}}{2D'}$

$L_m = \frac{V_g D T_s}{2 \Delta i}$

$L_m=\frac{\mu A_c n_1^2}{l}$

The permittivity of free-space is so much larger than the permittivity the transformer material, that the magnetic path length, $l$ , can be estimated to be the air gap lenght, $l g$ . so $l = l g$ and
$L_m=\frac{\mu_o A_c n_1^2}{l_g}$

Solve for $n$

Minimize total power loss: $P t o t = P f e + P c u$
Core loss: $P f e = K f e Δ B β A c l m$

$B_{ac} = \frac{L_m \Delta i}{n_1 A_c}$
The $β$ and $K f e$ are in the core material's datasheets

Core calculationsEdit

Core selectionEdit

Variables

$P F e$ - power loss in the core [ $W$ ]
$B s a t$ - saturation flux density [ $T$ ]
$B m a x$ - max flux density [ $T$ ]
$Δ B$ - change in flux density [ $T$ ], aka $B a c$
$A w$ - winding area [ $c m 2$ ]
$A e$ - effective cross-setional area of the core [ $c m 2$ ]
$A P$ - Area Product [ $c m 4$ ]
$K u$ - window utilization factor, or fill factor [unitless]
$N P$ - number of turns on the primary [unitless]
$N S$ - number of turns on the secondary [unitless]
$N B$ - number of turns on the bias [unitless]
$μ o$ - permittivity of free space (air) $μ o = 2π10 - 7$ [H/m]

Material specifications

Grade	$B s a t$ [T]	Specific Power Losses @100 °C [W/cm3]	Manufacturer
B2	0.36	$P_{Fe} = 1.15 * 10^{-5} * \Delta B^{2.26} * f_{sw}^{1.11}$	THOMSON
3C85	0.33	$P_{Fe} = 1.54 * 10^{-7} * \Delta B^{2.62} * f_{sw}^{1.54}$	PHILIPS
N67	0.38	$P_{Fe} = 8.53 * 10^{-7} * \Delta B^{2.54} * f_{sw}^{1.36}$	EPCOS (ex S+M)
PC30	0.39	$P_{Fe} = 1.59 * 10^{-6} * \Delta B^{2.58} * f_{sw}^{1.32}$	TDK
F44	0.4	$P_{Fe} = 2.39 * 10^{-6} * \Delta B^{2.23} * f_{sw}^{1.26}$	MMG

Calculate minimal AP needed

$AP_{min} = 10^3 * \left ( \frac{ L_p * I_{Prms} }{ \Delta T^{ \frac{1}{2} } * K_u * B_{max} } \right )^{1.316}$ [ $c m 4$ ]

$B m a x$ should be less than $B s a t$ , to avoid core saturation. for example $B s a t > 0.3 T$ , then for a conservative calculation use $B m a x = 0.25 T$