Power Supply Design Seminar Topic 4 Presentation: Practical Magnetic Design: Inductors and Coupled Inductors Reproduced from 2012 Texas Instruments Power Supply Design Seminar SEM2000, Topic 4 TI Literature Number: SLUP305 © 2012, 2013 Texas Instruments Incorporated Power Seminar topics and online power training modules are available at: ti.com/psds Topic 4 Practical Magnetic Design: Inductors and Coupled Inductors Louis Diana SLUP305 Agenda • Theory of operation and design equations – Design flow diagram discussion – Inductance calculations – Ampere’s law for magnetizing force – Faraday’s law for flux density – BH curves and magnetic saturation – Core loss – AC and DC wire losses – Effective permeability and inductance rolloff – Coupled-inductor winding layers and considerations • Example – flyback-coupled inductor design using EFD core Texas Instruments – 2012 Power Supply Design Seminar 4-2 SLUP305 Design Flow Specifications required: 1. Inductance 2. Turns 3. Peak current 4. RMS current 5. Output power 6. Frequency Specification Pick a Core Based on Output Power Inductance Calculation Yes Wire-Loss Calculations Yes Change Core Size No Flux Density and CoreLoss Calculations Yes No Bobbin-Fit Calculations No Change Wire Size or Filar Yes Build and Test Magnetic Texas Instruments – 2012 Power Supply Design Seminar 4-3 SLUP305 Magnetic Parameters and Conversion Factors SI CGS CGS to SI Flux Density B Tesla Gauss 10-4 Field Intensity H A-T/m Oersted 1,000/4π Permeability µ 4 π x10-7 1 4 π x10-7 Area Ae m2 cm2 10-4 Length lg, le m cm 10-2 Texas Instruments – 2012 Power Supply Design Seminar 4-4 SLUP305 Inductance The inductance, L, of a wound core can be calculated from the core geometry using the following equation: 0.4×π×N 2 ×Ae×10−8 L= le lg + µ µ = Core permeablity N = Number of turns Ae = Core cross - section (mm2 ) le = Core magnetic path length (mm) lg = Gap (mm) Manufacturers list inductance for a given core and gap as inductance per turn squared, referred to as Al . L = Al×N 2 Texas Instruments – 2012 Power Supply Design Seminar For Al in nH/per turn 2 4-5 SLUP305 Core Geometry Mean magnetic path length (le) and effective area (Ae) are given in most data sheets, but le can also be calculated using the following equation: OD ID le = π(OD-ID) OD ) ln( ID Ae le Texas Instruments – 2012 Power Supply Design Seminar 4-6 SLUP305 Ampere’s Law Ampere’s law states that the total magnetic force along a closed path is proportional to the ampere-turns in the winding that the path passes through. SI: F = ∫ H×dl = N×I ≈ H×le H≈ CGS: F = N×I ampere-turns/meter(A-T/m) le ∫ H×dl = 0.4×π×N×I ≈ H×le 0.4×π×N×I H≈ (Oersteds) le H = Magnetizing force N = Number of turns le = Core magnetic path length (m for SI, cm for CGS) I = Peak magnetizing current (amperes) Texas Instruments – 2012 Power Supply Design Seminar 4-7 SLUP305 Faraday’s Law The total magnetic flux Φ passing through a surface of area Ae is related to the flux density β. ΔΦ = SI: 1 E×dt ∫ N E×dt dΦ dβ ∫ E = N× ≈ N×Ae× ⇒ β = dt dt N×Ae β = Flux density (Tesla) CGS: 10 8 ΔΦ = E×dt ∫ N E = N dΦ N×Ae×dβ × ≈ ⇒β = 8 8 dt 10 10 ×dt β = Flux density (Gauss) E = Voltage across coil (V) 8 ×10 E×dt ∫ N×Ae Ae = Effective core area (m 2 ) Texas Instruments – 2012 Power Supply Design Seminar 4-8 SLUP305 BH Curves β βmax βmax βac Ηmin β = Flux density (Tesla) Ηmax Np = Number of primary turns 2 Ae = Effective core area (m ) Vin = Voltage (V) Bac is the AC flux density and Bmax is the peak flux density. Vin×ton βac = Ae×Np Tesla Lp×Ip βmax = Ae×Np Tesla ton = On time (s) Lp = Primary inductance (H) Ip = Peak primary current (A) Texas Instruments – 2012 Power Supply Design Seminar 4-9 SLUP305 Core Loss For a Ferroxcube core: βacpeak = βac 2 Convert Ve from mm 3 to m 3 Ve(m 3 ) = Ve 1,000 3 Pcoreloss = Pv(KW m 3 )×Ve(m 3 ) Specific power loss as a function of one-half of peak-to-peak flux density with frequency as a parameter. Courtesy of Ferroxcube Texas Instruments – 2012 Power Supply Design Seminar 4-10 SLUP305 Permeability and Inductance Rolloff As current is increased in a magnetic and the flux density gets closer to core saturation, the permeability of the core starts to roll off. CBW477 10 4 handbook, halfpage 0.4×π×N 2 ×Ae×10-8 L= le lg+ µ 3F3 µ µ rev 10 3 10 2 10 1 10 10 2 3 H (A/m) 10 Permeability as a function of magnetizing force (H) (courtesy of Ferroxcube). Courtesy of Ferroxcube Texas Instruments – 2012 Power Supply Design Seminar 4-11 SLUP305 AC–DC Wire Loss Determine wire area and AC losses: Open J Wire area required = Irms / J J = 400 A/cm 2 Actual δ(skin depth cm) = Equivalent ρ 7.6 ≈ at 100°C π×µ×f f Number of wires required = Wire area required Area of wire chosen Ratio of AC to DC losses Total winding resistance = AC–DC wire resistance of selected wire×Number of turns× Texas Instruments – 2012 Power Supply Design Seminar Length per turn Number of wires used 4-12 SLUP305 AC–DC Wire Loss Using look-up tables from the “Magnetics Design Handbook (MAG100A)” by Llyod H. Dixon: Determine the wire area in cm 2 and the radius in cm { Annular inner ring (cm 2 ) = π× radius of wire 2 −[wire radius(cm)−skin depth(cm)]2 } Area of wire in (cm 2 ) Ratio Rac to Rdc (Rskin) = Annular inner ring (cm 2 ) AC–DC wire resistance = DC resistance of chosen wire×Ratio Rac to Rdc (Rskin) Texas Instruments – 2012 Power Supply Design Seminar 4-13 SLUP305 Proximity Effects AC current in a conductor induces eddy currents in adjacent conductors by a process called the proximity effect. P1 P2 S S S 3 2 1 F=Nl w Pa Sa Sb Pb F=Nl w Power loss Pm in layer m is: ' wire_dia * 2 × Rdc , Pm = I 2 "#( m −1) + m 2 $% × ) ( + δ Texas Instruments – 2012 Power Supply Design Seminar 4-14 SLUP305 Winding Factor and Bobbin Fit Calculations 0 14.8 í 0.2 Winding data and area product for EFD20/10/7 coil former with 10-solder pads Number of Sec,ons 1 Winding Area (mm2) Minimum Winding Width (mm) Average Length of Turn (mm) 13.5 34.1 27.7 Area Product Ae x Aw (mm4) Type Number 859 CPHS-­‐EFD20-­‐1S-­‐10P 0 í 0.15 10.3 9.2 + 0.15 0 9.5 max. 3.9 +0.1 0 5 í 0.1 0 Courtesy of Ferroxcube Winding factor for a bobbin should be in the .3 to .7 range depending on isolation requirements Turns per layer = Build up = 1 0.3 7.5 ± 0.05 15 ± 0.05 21.7 max. Bobbin width (mm) -2 Diameter of wire in (mm) 0 14.8 í0.2 Winding area in (mm) Winding width in (mm) Number of avaliable layers = Buildup Diameter of chosen wire Total bobbin turns = Turns per layer×Number of layers Winding fill factor = (13.5 min.) Turns needed Total bobbin turns avaliable 0.3 1.8 21 23.7 max. 2 Dimmensions in mm Texas Instruments – 2012 Power Supply Design Seminar 4-15 SLUP305 Transformer Layers Transformer layout is very important because it affects primary to secondary coupling and leakage inductance. Fourth layer: 32awg single filar half of primary. Third layer: 32awg trifilar. Second layer: 32awg trifilar. First layer: 32awg single filar half of primary. Texas Instruments – 2012 Power Supply Design Seminar 4-16 SLUP305 Design Flow Summary • Complete specification • Pick a core (see section 4, “Power Transformer Design” by Lloyd H. Dixon) • Calculate inductance and turns-based Al • Calculate copper loss • Calculate flux density and core loss • Calculate temperature rise (see “Constructing Your Power Supply – Layout Considerations” by Robert Kollman) • Iterate as required to get the turns to layer nicely on the core for good coupling and low leakage inductance Texas Instruments – 2012 Power Supply Design Seminar 4-17 SLUP305 Flyback-Coupled Inductor Example Specifications required: 1. Inductance 2. Turns 3. Peak current 4. RMS current 5. Output power 6. Frequency Specification Pick a Core Based on Output Power Inductance Calculation Yes Wire-Loss Calculations Change Core Size Yes No Flux Density and CoreLoss Calculations Yes No Bobbin-Fit Calculations No Change Wire Size or Filar Yes Build and Test Magnetic Texas Instruments – 2012 Power Supply Design Seminar 4-18 SLUP305 Flyback-Coupled Inductor Example Specifications required: General Topology Main Output Power Maximum Switching Frequency at Full Load Input Minimum Input Voltage Maximum Input Voltage Primary Peak Current Primary RMS Current Outputs Secondary Output Voltage Secondary Peak Current Secondary RMS Current Bias Voltage Bias Current Inductance and Turns Ratio Primary Inductance Leakage Inductance Primary-to-Secondary Turns Ratio Texas Instruments – 2012 Power Supply Design Seminar Quasi-Resonant Flyback 10 W 140 kHz 85 VAC, 76 VDC 265 VAC, 375 VDC 1.155 A 0.425 A 5V 13.861 A 5.382 A 16 V 50 mA 190.918 µH 3.818 µH 12 4-19 SLUP305 Flyback-Coupled Inductor Example • Core selection is based on − Core size − Core material − Core gap See section 5 in “Magnetics Design Handbook” by Lloyd H. Dixon (MAG100A) • An EFD20 core with 3F3 material was chosen − Al was set to 82 nH − Key core and bobbin data Winding Area Min. Winding (mm2) Width (mm) 27.7 13.5 Length per Turn (mm) Area Product (mm4) Buildup (mm) Effective Volume (mm3) 34.1 859 2.052 1460 Texas Instruments – 2012 Power Supply Design Seminar 4-20 SLUP305 Flyback-Coupled Inductor Example • Calculate primary turns, Np, based on required inductance and Al value ⎛ Lp ⎞⎟1/2 Np = ⎜⎜⎜ ⎟⎟ ⎝ Al ⎠ Lp = 190.918 µH Al = 82 nH Np = 48 • Calculate secondary turns-based turns ratio Ns = Np Turns ratio Turns ratio = 12 Ns = 4 Texas Instruments – 2012 Power Supply Design Seminar 4-21 SLUP305 Flyback-Coupled Inductor Example • Determine required primary wire area based on current density, J Primary wire area = Ipri_max J J = 400 A/cm 2 Ipri_max = 0.425 A Primary wire area = 1.0625 × 10-3 cm 2 • Determine skin depth at 100°C δ(skin depth cm) ≈ 7.6 f f = 140 kHz Skin depth = 0.0203 cm Texas Instruments – 2012 Power Supply Design Seminar 4-22 SLUP305 Flyback-Coupled Inductor Example • Calculate annular ring area ( ) { Annular inner ring cm 2 = π× radius of wire 2 – [wire radius (cm) – skin depth (cm)] 2 } – If skin depth is greater than wire radius annular ring area = wire area • Calculate AC to DC resistance, Rac/Rdc ratio Ratio Rac to Rdc (Rskin) = Area of wire in (cm 2 ) Annular inner ring (cm 2 ) AWG Cu Radius (cm) Cu Area (cm2) Annular Ring Area (cm2) Rac/Rdc Rskin 24 0.0255 0.002047 0.001958 1.0453 26 0.02 0.001287 0.001287 1.0000 28 0.016 0.00081 0.000810 1.0000 30 0.0125 0.000509 0.000509 1.0000 Texas Instruments – 2012 Power Supply Design Seminar 4-23 SLUP305 Flyback-Coupled Inductor Example • Calculate number of wires required based on wire area, current density and Rac/Rdc ratio Numbers of wires required = Wire area required Area of wire chosen Ratio of AC to DC losses AWG Wire Area Required (cm2) Cu Area (cm2) Rac/Rdc Rskin Number of Wires Required 24 0.0010625 0.002047 1.0453 1 26 0.0010625 0.001287 1.0000 1 28 0.0010625 0.00081 1.0000 2 30 0.0010625 0.000509 1.0000 3 32 0.0010625 0.00032 1.0000 4 Texas Instruments – 2012 Power Supply Design Seminar 4-24 SLUP305 Flyback-Coupled Inductor Example • Look at how the windings might fit in the bobbin window – Initially estimate that half the area is used by the primary Turns per layer = Bobbin width (mm) Diameter of wire (mm) Number of available layers = Buildup Diameter of chosen wire Buildup = −2 Winding area in (mm) Winding width in (mm) AWG Insulated Diameter (cm) Min. Winding Width (mm) Turns per Layer Available Layers Number of Wires Required 24 26 28 30 32 0.057 0.046 0.037 0.03 0.024 13.5 13.5 13.5 13.5 13.5 21 27 34 43 54 3 4 5 6 8 1 1 2 3 4 Texas Instruments – 2012 Power Supply Design Seminar 4-25 SLUP305 Flyback-Coupled Inductor Example Estimate primary winding copper losses • One wire of wire gauge 26 was chosen Effective core parameters (courtesy of Ferroxcube). Length_per_turn_cm = 34.1 mm 10 Rcu_pri_26awg = Rcu26_100°C×Np× Length_per_turn_cm = 3.41 cm Length per turn Number of primary wires Rcu_pri_26awg = 0.293 Ω Pcu_pri_loss_26awg = Ipri_rms 2 ×Rcu_pri_26awg Pcu_pri_loss_26awg = 0.053 W Texas Instruments – 2012 Power Supply Design Seminar 4-26 SLUP305 Flyback-Coupled Inductor Example Ns = 4 Determine secondary wire AWG and losses: Secondary wire area = Isecondary J Number_of_wires_sec_28awg = = 14×10-3cm 2 where J = 400 A/cm 2 Area_req_sec_cm 2 Area_28awg_cm 2 Rskin_28awg_100°C Number_of_wires_sec_28awg = 16.667 Number_of_wires_sec_28awg_used = 5 Rcu_sec_28awg = Rcu28_100°C×Ns× Length per turn Number of secondary wires Rcu_sec_28awg = 7.761×10 -3 Pcu_sec_loss_28awg = Isec_rms 2 ×Rcu_sec_28awg Pcu_sec_loss_28awg = 0.226 W Texas Instruments – 2012 Power Supply Design Seminar 4-27 SLUP305 Flyback-Coupled Inductor Example Determine bias wire AWG and losses: Number of bias wires = 1 Nbs = 13 Rcu_bias_32awg = Rcu32_100°C×Nsb× Length per turn Number of bias wires Rcu_bias_32awg = 225×10-3 Pcu_bias_loss_32awg = Ibias_rms 2 ×Rcu_bias_32awg Pcu_bias_loss_32awg = 4.197×10-4 W Texas Instruments – 2012 Power Supply Design Seminar 4-28 SLUP305 Flyback-Coupled Inductor Example Determine core flux density: Ae = 31 mm 2 Np = 48 Ipri_p = 1.155 A tonmax = 2.9 µs MBW015 500 -25ºC -100ºC B (mT) Vinmin×tonmax Bac = Ae×Np 400 Bac = 148.17 mT 200 3F3 300 100 Bmax = Lp×Ipri_p Ae×Np Bmax = 148.19 mT Texas Instruments – 2012 Power Supply Design Seminar 0 -25 0 25 50 150 H (A/m) 250 Courtesy of Ferroxcube 4-29 SLUP305 Flyback-Coupled Inductor Example Determine core loss: 10 MBW048 4 3F3 T=100 oC 10 3 10 2 25 kHz Bac Bunipolar = 2 700 kH z 400 kH z 200 kHz 100 kHz Pv (kW/m 3) Bunipolar_mt = 74.1 mT From graph Pcore = 60 kW/m 3 Ve = 1460 mm 3 Ve_per_m 3 = Pcore_loss = Pcore×Ve_per_m 3 10 1 Ve 1,000 3 Pcore_loss = 0.088 W Texas Instruments – 2012 Power Supply Design Seminar 10 10 2 B (mT) 10 3 Specific power loss as a function of Peak flux density with frequency as a parameter. Courtesy of Ferroxcube 4-30 SLUP305 Flyback-Coupled Inductor Example Magnetic power dissipation: Pcu_loss_tot = Pcu_pri_loss + Pcu_sec_loss + Pcu_bias_loss Ptot_mag = Pcu_loss_tot + Pcore_loss Ptot_mag = 0.367 W Determine bobbin fit factor: AWG Copper Diameter in cm with Insulation Copper Area in cm2 with Insulation 26 0.046 0.001671 28 0.037 0.001083 30 0.03 0.000704 32 0.024 0.000459 bobbin_width_mm = 13.5 mm bobbin_wdth_cm = bobbin_width_mm×.1 bobbin_width_cm = 1.35 cm Texas Instruments – 2012 Power Supply Design Seminar 4-31 SLUP305 Flyback-Coupled Inductor Example Determine bobbin fit factor continued: Turns_per_layer26 = bobbin_width_cm −2 dia_26awg_Inso_cm Turns_per_layer28 ≈ 34 Turns_per_layer26 ≈ 27 Turns_per_layer30 ≈ 43 Winding_area_mm = 27.7 mm 2 Winding_area_cm 2 = .277 winding_area_cm 2 Build_up_cm = bobbin_width_cm Build_up = .205 cm Layers = Build_up_cm dia_26awg_Inso_cm Layers ≈ 4 Total_bobbin_turns_26awg = Turns_per_layer×Layers Total_bobbin_turns_26awg =108 Turns_needed = Np×number_of_wires_pri+ Ns×number_of_wires_sec + Nsb×number_of_wires_bias Turns_needed = 81 Winding_factor = Turns_needed Total_bobbin_turns_26awg Texas Instruments – 2012 Power Supply Design Seminar Winding_factor = .75 4-32 SLUP305 Flyback-Coupled Inductor Example Coupled inductor layout: First layer: 24 turns of 26awg single filar half of primary Second layer: 20 turns of 28awg five filar secondary plus 13 turns of 30awg bias winding Third layer: 24 turns of 26awg single filar half of primary Black lines are tape. Texas Instruments – 2012 Power Supply Design Seminar 4-33 SLUP305 Summary • Discussed theory of operation and design flow • Determined inductance, flux density, core loss, AC and DC wire losses • Discussed BH curves and magnetic saturation, effective permeability and inductance rolloff • Discussed transformer winding layers and how to interleave the winding for good coupling and low leakage • Went through an example of a flyback-coupled inductor design using EFD core Texas Instruments – 2012 Power Supply Design Seminar 4-34 SLUP305 TI Worldwide Technical Support Internet Asia TI Semiconductor Product Information Center Home Page support.ti.com TI E2E™ Community Home Page e2e.ti.com Product Information Centers Americas Brazil Mexico Phone +1(512) 434-1560 Phone 0800-891-2616 Phone 0800-670-7544 Fax Internet/Email +1(972) 927-6377 support.ti.com/sc/pic/americas.htm Europe, Middle East, and Africa Phone European Free Call 00800-ASK-TEXAS (00800 275 83927) International +49 (0) 8161 80 2121 Russian Support +7 (4) 95 98 10 701 Note: The European Free Call (Toll Free) number is not active in all countries. 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