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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.

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}

ReferencesEdit

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