1 Introduction
1.1 Research contributions
2 Review of the high-frequency inductive components design and optimization methods
2.1 Challenges in the design of HF transformer
2.2 Single-objective methods
2.3 Multi-objective methods
2.4 200-W flyback converter design complexity
3 Design approach
3.1 Problem description
Output voltage | 120 V |
Input voltage | 48 V |
Output power | 200 W |
Maximum temperature rise | 60 °C |
Frequency | 10–20–30–40–50–60 kHz |
3.2 Core loss prediction
3.3 Winding loss
3.4 Volume modeling
3.5 Transformer cost modeling
3.6 Problem formulation
Variable | Unit | Type | Interval |
---|---|---|---|
Magnetic material | U | Integer | [1–24] |
Duty cycle | – | Integer | [1–72] |
Core reference | U | Integer | [1–n] |
W
c
| U | Integer | [1–2] |
N
p
| U | Integer | [1–100] |
N
s
| U | Integer | [1–100] |
d
p
| U | Integer | [1–60] |
d
s
| U | Integer | [1–60] |
l
g
| M | Real | [1e−4, 4e−3] |
η
i
| – | Real | [0.95–0.99] |
3.6.1 Design equations of the transformer
3.6.2 Design constraints
-
The core saturation, the gain and the air gap are as follows:$$ B_{\rm{max} } \le 0.7 B_{\text{s}} $$(14)$$ {\text{Gain}} = \frac{{N_{\text{s}} D}}{{N_{\text{p}} \left( {1 - D} \right)}} $$(15)$$ l_{\text{g}} = \frac{{\mu_{0} I_{\text{pmax}}^{2} L_{\text{p}} }}{{A_{\text{s}} B_{\rm{max} }^{2} }} - \frac{{l_{\text{c}} }}{{\mu_{\text{r}} }} $$(16)
-
Conductor diameters: They are determined by [31].$$ d_{\text{p}} = \sqrt {\frac{{ I_{\text{pmax}} }}{7.2}} ;\quad d_{\text{s}} = \sqrt {\frac{{I_{\text{smax}} }}{7.2}} $$(17)
-
The winding area Aw should be capable of allocating the two windings. fu is equal to 0.4.$$ A_{\text{w}} \ge \frac{{\pi \left( {N_{\text{p}} d_{\text{p}}^{2} + N_{\text{s}} d_{\text{s}}^{2} } \right)}}{{f_{\text{u}} }} $$(18)
-
Temperature rise Tr:$$ T_{\text{r}} = \frac{{53 \left( {P_{\text{c}} + P_{\text{w}} } \right)}}{{V_{\text{c}}^{0.53} }} \le 60 \,^\circ {\text{C}} $$(19)
-
Currents ripple factor: it should be lower than rmax.$$ I_{\text{pr}} \le r_{\rm{max} } ;\quad I_{\text{sr}} \le r_{\rm{max} } I_{\text{savg}} $$(20)
-
Peak MOSFET voltage: the maximum allowable voltage across the MOSFET time must be lower than 400 V.
-
Leakage inductance: it is kept lower than 2% of the magnetizing inductance. Model presented in [42] is used.
3.6.3 Optimization variables
4 Multi-objective optimization in discrete research space
4.1 Bi-objective optimization
30 kHz | 40 kHz | 50 kHz | 60 kHz | |
---|---|---|---|---|
Losses (W) | 4.88 | 4.37 | 4.02 | 3.66 |
Cost ($) | 5.68 | 3.9764 | 3.96 | 3.32 |
Volume (cm3) | 63.41 | 44.11 | 44.12 | 42.75 |
Material/Wc | F/2 | F/2 | F/2 | F/2 |
Shape/Ref | ETD/7 | ETD/6 | ETD/6 | ETD/6 |
dp (mm) | 1.82 | 1.8 | 1.82 | 2.05 |
ds (mm) | 0.91 | 1.02 | 1.02 | 0.91 |
Np/Ns | 23/47 | 24/27 | 17/32 | 9/37 |
lg (mm) | 1 | 0.7 | 0.6 | 0.5 |
5 Comparison with classical area product method and experimental verification
5.1 Comparison with the area product methods
Proposed approach | Lloyd approaches [36] | Sanjaya approach [34] | ||
---|---|---|---|---|
1st approach | 2nd approach | |||
Primary inductance (μH) | 233 | 216.45 | 216.45 | |
Maximum swing flux (T) | 0.1 | 0.1 | 0.1 | |
Area product (cm4) | – | 17.78 | 10.24 | 2.09 |
Selected core | ETD59 | EE 65 | ETD59 | ETD49 |
Selected material | F | F | F | |
Turns ratio | 0.48 | 0.4 | 0.4 | |
Primary/secondary number of turns | 23/47 | 15/38 | 22/55 | 38/95 |
Primary/secondary diameter (mm) | 1.82/1.45 | 1.28/0.77 | 1.28/0.77 | |
Primary/secondary number of layers | 1/1 | 1/1 | 2/2 | |
Primary/secondary resistance (mΩ) | 81.12/327.52 | 76.54/396 | 112.25/574 | 663/2100 |
Winding loss (W) | 4.72 | 5.44 | 7.94 | 39 |
Core loss (W) | 0.17 | 0.27 | 0.17 | 0.08 |
Transformer loss (W) | 4.88 | 5.71 | 8.11 | 39.08 |
5.2 Experimental results
6 Conclusion
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The approach solves the limitations of the existing works by considering the effect of the duty cycle and the efficiency as main optimization variables. The approach takes also into account the nonlinear relationships between variables, objective functions and design constraints.
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The proposed approach allows better minimization of the loss and volume in comparison with existing classical methods.
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The approach is very effective as it improves the accuracy of the results compared to other techniques, and it reduces the design time. The optimization needs 30 min to get the optimum Pareto fronts which can take longer time using classical techniques.
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Results show that the optimal magnetic material and optimal core shape depend on the switching frequency.
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Pareto fronts of the objective functions (loss, cost and volume) are inversely proportional to the switching frequency.