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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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