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 acronyms

• 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 (cm^x)
  • $A_{c}$, 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 calculations

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 calculations

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 $n={\frac {n_{2}}{n_{1}}}$

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

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

$L_{m}={\frac {V_{g}DT_{s}}{2\Delta i}}$

$L_{m}={\frac {\mu A_{c}n_{1}^{2}}{l}}$

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 calculations

Core selection

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 calculations

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 Loss

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

References

• U of Colorado - Flyback transformer design
• TI - "Magnetics Design 4 - Power Transformer Design" - very good, long, description of transformers and design
• TDK ferrite materials
• IRF - Flyback Transformer Design - nice description of howto wind the transformer
• TI - Magnetics Design 5 - Inductor and Flyback Transformer Design - describes various converters DCM and CCM
• OFFLINE FLYBACK CONVERTERS DESIGN METHODOLOGY WITH THE L6590 FAMILY - very good, full description of designing an offline flyback converter
• Isolated 50 Watt Flyback Converter Using the UCC3809
• TOPSwitch Flyback Transformer Construction Guide