# 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.

# Variables and acronymsEdit

• Universal constants
• Permittivity of free space $\mu_o$ (Wb A−1 m−1)
• $\mu_o = 4\pi 10^{-7}$ (Wb A−1 m−1)

• Wire variables:
• $\rho$, Wire resistivity (Ω-cm)
• $I_{tot}$, Total RMS winding currents (A)
• $I_{m,max}$, Peak magnetizing current (A)
• $I_{RMS}$, Max RMS current, worst case (A)
• $P_{cm}$, 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)
• $B_{max}$, Core maximum flux density (T)

• Core parameters
• EC35, PQ 20/16, 704, etc, Core type (mm)
• $K_g$, Geometrical constant (cm5)
• $K_{gfe}$, Geometrical constant (cmx)
• $A_c$, Cross-sectional area (cm2)
• $W_A$, Window area (cm2)
• $MLT$, Mean length per turn (cm)
• $l_m$, Magnetic path length (cm)
• $l$, or $l_g$, Air gap length (cm)
• $\mu$, Permittivity (Wb A−1 m−1)
• $\mu_r$, Relative Permittivity (unitless)
• $\mu = \mu_o \mu_r$
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]
• $V_{in}$ - input voltage [V]
• $V_D$ - diode voltage drop [V]
• $V_{Rds}$ - 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: $V_D = 0.8V$
• 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.
$\Delta i = 0.5 * I$

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 length, $l_g$. so $l = l_g$ and
$L_m=\frac{\mu_o A_c n_1^2}{l_g}$

Solve for $n$

Minimize total power loss: $P_{tot} = P_{fe} + P_{cu}$
Core loss: $P_{fe} = K_{fe} \Delta B^\beta A_c l_m$

$B_{ac} = \frac{L_m \Delta i}{n_1 A_c}$
The $\beta$ and $K_{fe}$ are in the core material's datasheets

# Core calculationsEdit

## Core selectionEdit

Variables
• $P_{Fe}$ - power loss in the core [$W$]
• $B_{sat}$ - saturation flux density [$T$]
• $B_{max}$ - max flux density [$T$]
• $\Delta B$ - change in flux density [$T$], aka $B_{ac}$
• $A_w$ - winding area [$cm^2$]
• $A_e$ - effective cross-setional area of the core [$cm^2$]
• $AP$ - Area Product [$cm^4$]
• $K_u$ - 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]
• $\mu_o$ - permittivity of free space (air) $\mu_o = 2 \pi 10^{-7}$ [H/m]

Material specifications
Grade $B_{sat}$ [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

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

• $B_{max}$ should be less than $B_{sat}$, to avoid core saturation. for example $B_{sat} > 0.3T$, then for a conservative calculation use $B_{max} = 0.25T$
• $\Delta T = T_{max} - T_{amb}$
Generally $T_{max} = 100C$ and $T_{amb}=30C$
• Using $K_u=0.3$ for off-line power supplies is a good estimate
Calculate minimum number of primary and secondary turns
• $N_{P-min} = \frac{ L_p * I_{pk} * 10^4 }{ B_{max} * A_e }$
• $N_{S-min} = \frac{ N_{P-min} }{ N }$
Calculate actual number of turn on the primary and secondary to be used.
• $N_S$: Round up $N_{S-min}$ to the nearest integer
• $N_P = N * N_S$
Calculate air gap

$l_g = \frac{ \mu_o * N_P^2 * A_e * 10^{-2} }{ L_p }$

# Current calculationsEdit

Variables
• $I_{pk}$ - Ripple current max peak
• $I_{min}$ - Ripple current min peak
• $\Delta I_{pp}$ - pk-pk ripple current $I_{pk} - I_{min}$
Peak current

$I_{pk} = \left ( \frac{ I_{out-max} }{ N } \right ) * \left ( \frac{ 1 }{ 1 - D_{max} } \right ) + \frac{ \Delta I_L }{ 2 }$

DC current

$I_{dc}=D \frac{I_{pk}+I_{min}}{2}$

RMS current

$I_{rms}=\sqrt{ D \left ((I_{pk}+I_{min}) + \frac{1}{3} (I_{pk}+I_{min})^2 \right )}$

AC current

$I_{rms}=\sqrt{ I_{rms}^2 - I_{dc}^2 }$

# Power LossEdit

$P_{tot}=P_{fe}+P_{cu}$