902 - Transformer Design¶

High-frequency transformer modeling with leakage and parasitics.

Overview¶

Transformer modeling in power electronics requires: - Magnetizing inductance - Leakage inductance - Winding capacitance - Core losses

Transformer Equivalent Circuits¶

Ideal Transformer¶

\[\frac{V_1}{V_2} = \frac{N_1}{N_2} = n$$ $$\frac{I_1}{I_2} = \frac{N_2}{N_1} = \frac{1}{n}\]

Practical Model¶

  Primary Side          Ideal TX         Secondary Side
      │                   │ │                  │
     Rp    Llk,p         ┌┴─┴┐        Llk,s   Rs
  ●──/\/\──⊏⊐──┬────────┤ n:1├────────⊏⊐──/\/\──●
               │        └┬─┬┘
              Lm         │ │
               │
              ═╧═

Where: - Lm = magnetizing inductance - Llk,p, Llk,s = leakage inductances - Rp, Rs = winding resistances

Magnetic Circuit Model¶

Permeance Network¶

         P_leak,p      P_leak,s
   MMF1 ────┬──────┬──────┬──── MMF2
            │      │      │
         P_gap   P_core   │
            │      │      │
   ─────────┴──────┴──────┴─────

Parameter Extraction¶

Magnetizing Inductance: $$L_m = N_1^2 \cdot P_{core}$$

Leakage Inductance: $$L_{lk} = N^2 \cdot P_{leak}$$

Design Procedure¶

Step 1: Core Selection¶

Area product method: $$A_p = A_e \cdot A_w = \frac{P_{out}}{K_f \cdot K_u \cdot J \cdot B_{max} \cdot f}$$

Where: - Ae = core cross-section - Aw = window area - Kf = waveform factor - Ku = window utilization - J = current density - Bmax = peak flux density

Step 2: Turns Calculation¶

Primary Turns: $$N_1 = \frac{V_1}{4 \cdot f \cdot B_{max} \cdot A_e}$$

Secondary Turns: $$N_2 = \frac{N_1}{n}$$

Step 3: Wire Selection¶

Current density (typical 3-5 A/mm² for natural cooling): $$A_{wire} = \frac{I_{rms}}{J}$$

Skin depth at frequency f: $$\delta = \sqrt{\frac{\rho}{\pi f \mu}}$$

Step 4: Leakage Inductance¶

Depends on winding arrangement: - Interleaved: lower leakage - Separated: higher leakage

\[L_{lk} \approx \frac{\mu_0 N^2 MLT}{3 h_w} \cdot \left(n_p \cdot b_p + \frac{n_i \cdot b_i}{3}\right)\]

Winding Arrangements¶

Primary-Secondary (P-S)¶

Simple construction
High leakage inductance
Suitable for flyback

Interleaved (P-S-P-S)¶

Reduced leakage
Lower proximity effect
More complex

Sandwich (P-S-P)¶

Moderate leakage
Good balance

Parasitic Capacitances¶

Inter-winding Capacitance¶

\[C_{ps} = \frac{\epsilon_0 \epsilon_r A_{overlap}}{d_{insulation}}\]

Important for: - Common-mode noise - High dv/dt applications

Intra-winding Capacitance¶

Affects self-resonance frequency: $$f_{res} = \frac{1}{2\pi\sqrt{L_{lk} C_{self}}}$$

GeckoCIRCUITS Transformer Model¶

Setting Up¶

Use coupled inductor component
Set:
Turns ratio
Magnetizing inductance
Leakage inductance (referred to primary)
Winding resistances
For detailed analysis:
Use permeance network
Add saturation to core permeance

Simulation Exercises¶

Compare ideal vs practical transformer
Measure leakage inductance effect on regulation
Model flyback transformer with gap
Analyze resonance with capacitance