SLUAAL2 Application note

SLUAAL2 june 2023 UCC256402 , UCC256403 , UCC256404

1.18.1 LLC Resonant Inductor Design

Figure 1-36 LLC Resonant Inductor Design Procedure

Here for designing the resonant inductor, the same method given in Reference [14] is followed. PFC LLC EVM [15] is considered as an design example.

Step 1: Specifications

Equation 65.

Resonant Inductor value L_{r} = 75 u H

Equation 66.

\begin{array}{l} At minimum Input voltage and at maximum output power: \\ Peak current of the resonant inductor I_{p} = 1.9 A \\ RMS current of the resonant inductor I_{r m s} = 1.27 A \\ Switching frequency f_{s w} = 77 k H z \end{array}

Equation 67.

\begin{array}{l} At Rated Input voltage and at maximum output power: \\ Peak current of the resonant inductor I_{p} = 1.78 A \\ RMS current of the resonant inductor I_{r m s} = 1.22 A \\ Switching frequency f_{s w} = 88 k H z \end{array}

Step 2: Pick a core based on the maximum energy to be stored

Equation 68.

Area Product of a core A_{p} = W_{a} A_{c} = \frac{L I_{p}^{2}}{K_{u} J_{m} B_{m}} (m^{4})

[Equation 10.100 in Reference 16]

Equation 69.

where W_{a} is window area and A_{c} is core area

Equation 70.

K_{u} is window utilization factor (for Litz wire it is : 0 .3 to 0 .4)

Equation 71.

J_{m} is peak current density: 4 t o 6 A / m m^{2} (under natural cooling condition)

Equation 72.

B_{m} is peak flux density of the core

B_m is chosen such that at the operating frequency, core loss power density should be less than 150 mW/cm³ for natural convection cooling.

In general, suggested magnetic materials for reducing core losses are 3C95, 3F4 from Ferroxcube (Ferroxcube Cores and Accessories) and PC47, PC90, PC95 from TDK ( TDK Cores and Accessories).

Equation 73.

For K_{u} = 0.3, J_{m} = 4 A / m m^{2}, B_{m} = 0.15 T, A_{p} \geq 1320 m m^{4}

For this design, RM8 core with 3C95 material is selected. This core's effective cross section area $A_{c} = 63 m m^{2}$ (Ferroxcube RM8 core data sheet) and minimum winding area $W_{a} = 31 m m^{2}$ (Ferroxcube RM8 bobbin data sheet). Cores, Bobbin, Clamp can be obtained from following links: RM8 with 3C95, RM8 Bobbin, Clamp for RM8 Core.

Step 3: Wire Selection and Airgap calculation

Equation 74.

The effective cross-sectional area of the bare winding is A_{w} = \frac{I_{p}}{J_{m}} = \frac{1.78}{4} = 0.4450 m m^{2}

Equation 75.

Let's initially
                            select AWG21 which has a copper area about 0 .4116 m m^{2} which is closest to required copper area .

Equation 76.

So the actual current density would be J_{m_a c t} = \frac{1.78}{0.4116} = 4.32 \frac{A}{m m^{2}}

In order to reduce both skin and proximity losses, Litz wire is considered for this design .

Equation 77.

In general, for high frequency (around
                            100kHz) designs, AWG38-42 should be selected for each strand .

The skin depth of copper at 88 k H z is δ = \frac{66.2}{\sqrt{f}} (m m) = \frac{66.2}{\sqrt{88, 000}} = 0.2232 m m

[Equation 10.148 in Reference 16]

Equation 78.

Here AWG 38 is chosen with 50 strands for following reasons:

Equation 79.

1 . Its over all copper area is equivalent to AWG21 copper area

Equation 80.

2 . Each strand diameter is much less than skin depth so that current through the each strand will be uniform

Equation 81.

3 . And its readily available

(Litz Wire Data from MWS Wire Industries)

Equation 82.

With insulation, the overall diameter (d_{o}) of the U N S E R V E D L I T Z W I R E of 38AWG ​ with 50 strands is 0 .9398mm

(Litz Wire Data from MWS Wire Industries)

Equation 83.

The cross sectional area of the insulated wire is A_{w o} = \frac{π d_{o}^{2}}{4} = \frac{π \times {0.9398}^{2}}{4} = 0.6937 m m^{2}

Equation 84.

The minimum number of turns is N = \frac{K_{u} W_{a}}{A_{w o}} = \frac{0.3 \times 31}{0.6937} = 13.4

Equation 85.

Pick N = 14. The air-gap length is l_{g} = \frac{μ_{o} A_{c} N^{2}}{L_{r}} = \frac{4 π \times 10^{- 7} \times 63 \times 10^{- 3} \times 14^{2}}{75 \times 10^{- 6}} = 0.207 m m

Step 4: Copper Loss Calculation

Equation 86.

The winding width of the bobbin is 8 .9mm

(Ferroxcube RM8 bobbin data sheet)

Equation 87.

Since there are 14 bundle turns each with 0 .9398mm over all diameter, total number of bundled winding layers (N_{l}) would be 2 .

Equation 88.

Each bundle has a total number of strands k = 50

(represented in Figure 1-37 as a $7 \times 7 matrix$ )

Equation 89.

The strands in each bundle are modelled as a square with \sqrt{k} strands on each side of the bundle as shown in below figure

Figure 1-37 Model of the Litz Wire Winding Around a Central Core

Equation 90.

\begin{array}{l} The effective number of layers of the Litz-wire winding with the square arrangement of the strands in a bundle \\ is given by N_{l l} = N_{l} \times \sqrt{k} = 2 \times \sqrt{50} = 14.14 ≃ 14 \end{array}

Equation 91.

\begin{array}{l} The total number of strands in each layer is given as N_{s l} = \\ the number of bundle turns in a layer \times number of strands of each side of the square bundle \\ = 7 \times \sqrt{50} = 49.49 ≃ 50 \end{array}

Equation 92.

The mean length of the turn is l_{T} = 42 m m

(Ferroxcube RM8 bobbin data sheet)

Equation 93.

\begin{array}{l} The DC resistance of the single AWG38 strand is R_{w D C s} = l_{T} \times (A W G 38 DC resistance per m) \\ = 42 \times 2.1266 = 89.32 m Ω \end{array}

Equation 94.

The AC power loss of a single layer is given by P_{a c} = DC power loss of a single layer \times φ \times Q^{'} (φ, m)

[Table 10.1 in Reference 16]

[Equation 10.80 in Reference 9]

Equation 95.

\begin{array}{l} Here φ = \sqrt{η} \sqrt{\frac{π}{4}} \frac{d_{s}}{δ} \\ where η is porosity factor, d_{s} is strand bare wire diameter, δ is skin depth for a given frequency \end{array}

[Equation 10.74 in Reference 9]

Equation 96.

Q^{'} (φ, m) = (2 m^{2} - 2 m + 1) G_{1} (φ) - 4 m (m - 1) G_{2} (φ)

[Equation 10.81 in Reference 9]

Equation 97.

\begin{array}{l} G_{1} (φ) = \frac{\sinh (2 φ) + \sin (2 φ)}{\cosh (2 φ) - \cos (2 φ)} \\ G_{2} (φ) = \frac{\sinh (φ) \cos (φ) + \cosh (φ) \sin (φ)}{\cosh (2 φ) - \cos (2 φ)} \end{array}

[Equation 10.76 in Reference 9]

Equation 98.

\begin{array}{l} m for each layer can be found by m = \frac{m m f (h)}{m m f (h) - m m f (0)} \\ where h is thickness of each layer \end{array}

[Equation 10.81 in Reference 9]

Equation 99.

\begin{array}{l} In this design example, the DC power loss of a single layer is given as = \\ square of the RMS current through each strand \times DC resistance of a single strand \times total number of strands in a single layer \\ = {(\frac{I_{r m s}}{k})}^{2} \times R_{w D C s} \times N_{s l} = {(\frac{1.22}{50})}^{2} \times 89.32 m Ω \times 50 = 2.66 m W \end{array}

Equation 100.

Porosity factor η = \frac{Number of strands per layer (N_{s l}) \times S trand diameter with insulation}{width of the bobbin} = \frac{50 \times 0.124 m m}{8 .9mm} = 0.7

[Equation 10.73 in Reference 9]

Equation 101.

In this design, mmf due to each layer = current through each strand \times number of strands in a layer = \frac{I_{r m s}}{k} \times N_{s l}

Equation 102.

S o, layer 1's m value can be determined by m = \frac{(\frac{I_{r m s}}{k} \times N_{s l})}{(\frac{I_{r m s}}{k} \times N_{s l}) - 0} = 1

[Equation 10.90 in Reference 9]

Equation 103.

Layer 2's m value can be determined by m = \frac{2 (\frac{I_{r m s}}{k} \times N_{s l})}{2 (\frac{I_{r m s}}{k} \times N_{s l}) - (\frac{I_{r m s}}{k} \times N_{s l})} = 2

Equation 104.

Similarly, m values for other adjacent layers increase by 1 . Since there are 14 layers, m value increases up to 14 .

Equation 105.

φ = \sqrt{η} \sqrt{\frac{π}{4}} \frac{d_{s}}{δ} = \sqrt{0.7} \sqrt{\frac{π}{4}} \frac{0.1007}{0.2232} = 0.335

where 0.1007 is AWG38 bare wire diameter in mm [Table 10.1 in Reference 16].

$The copper loss for all the layers is given by P_{w} = DC power loss of a single layer \times \sum_{m = 1}^{14} φ Q^{'} (φ, m) = 47 m W$

Step 5: Flux Density and Core-Loss Calculation

Equation 106.

The amplitude of the core magnetic flux density at rated input
                            voltage is B_{m} = \frac{μ_{o} N I_{p}}{l_{g}} = \frac{4 π \times 10^{- 7} \times 14 \times 1.78}{0.2 \times 10^{- 3}} = 0.157 T

Equation 107.

The core loss per unit volume (P_{v}) at 0.157 T, 88 k H z is 150 m W / c m^{3}

(Use power loss calculator in the Ferroxcube design tool for finding core loss of the material Core Loss Calculator)

Equation 108.

The total core loss is P_{C} = V_{C} P_{v} = 150 m W / c m^{3} \times 2440 m m^{3} = 366 m W

Equation 109.

The amplitude of the core magnetic flux density at minimum input
                            voltage is B_{m_\max} = \frac{μ_{o} N I_{p_\max}}{l_{g}} = \frac{4 π \times 10^{- 7} \times 14 \times 1.9}{0.2 \times 10^{- 3}} = 0.167 T

Equation 110.

At worst case current, B_{m_\max} is less than saturation flux density of the ferrite material .

Step 6: Temperature Rise and Bobbin Fit Calculations

Equation 111.

Total power loss of the inductor is P_{w c} = P_{w} + P_{C} = 0.4 W

Equation 112.

The surface area of RM-8 core is A_{t} = 20.2 c m^{2}

Table 3-43

The surface power loss density is ψ = \frac{P_{w c}}{A_{t}} = \frac{0.4 W}{20.2 c m^{2}} = 0.0206 W / c m^{2}

Equation 113.

The temperature rise of the inductor is Δ T = 450 ψ^{0.826} = 450 \times {(0.0206 W / c m^{2})}^{0.826} = {18.25}^{o} C

[Equation 10.193 in Reference 16]

Equation 114.

The actual core window utilization factor is K_{u} = \frac{N A_{w o}}{W_{a}} = \frac{14 \times 0.6937}{31} = 0.3133