## Analysis of Specific Transformer Design

Wound on double-E core with a 1 cm using 3F3 ferrite. Ipri 4 A rms, sinusoidal waveform Vpri 300 V rms. Winding window split evenly between primary and secondary and wound with Litz wire. Transformer surface black (E 0.9) and Ta < 40 C. Find core flux density, leakage inductance, and maximum surface temperature Ts, and Areas of primary and secondary conductors, N A N A pri cu,pri sec cu,sec where kcu,pri kcu,sec kcu since we assume primary and secondary are wound with same type of conductor....

## Basis of Transformer Design

Design proceedure starting point - transformer V-A rating S S v I + v I 2 V -I pn pri v sec sec pn pri d( ) pri Acore 00 ac pri pri dt ' P Wis Acu,pri pri 'pri 4.4 kcu f Acore Aw Jrms Bac Equation relates input specifications (left-hand side) to core and winding parameters (right-hand side). Desired design procedure will consist of a systematic, single-pass method of selecting kcn, Acore, Aw, Jrms, and B

## Core Database Basic Transformer Design Tool

Interactive core database (spreadsheet-based) key to a single pass tramsformer design procedure. User enters input specifications from converter design requirements. Type of conductor for windings (round wire, Leitz wire, or rectangular wire or foil) must be made so that copper fill factor kcu is known. Spreadsheet calculates capability of all cores in database and displays smallest size core of each type that meets V- I specification. Also can be designed to calculate (and display as desired)...

## Core Material Performance Factor

Volt-amp (V-A) rating of transformers proportional to f Bac Core materials have different allowable values of Bac at a specific frequency. Bac limted by allowable Pmsp. Most desirable material is one with largest Bac. Choosing best material aided by defining an emperical performance factor PF f Bac. Plots of PF versus frequency for a specified value of Pmsp permit rapid selection of best material for an application. Plot of PF versus frequency at Pmsp 100 mW cm3 for several different ferrites...

## Core Shapes and Sizes

Magnetic cores available in a wide variety of sizes and shapes. Ferrite cores available as U, E, and I shapes as well as pot cores and toroids. I Laminated (conducting) materials available in E, U, and I shapes as well as tape wound toroidl and C-shapes. Open geometries such as E-core make for easier fabrication but more stray flux and hence potentially more severe EMI problems. Closed geometries such as pot cores make for more difficult fabrication but much less stray flux and hence EMI...

## Current Flux Density Versus Core Size

Larger electrical ratings require larger current I and larger flux density B. Core losses (hysteresis, eddy currents) increase as B2 (or greater) Winding (ohmic) losses increase as I2 and are accentuated at high frequencies (skin effect, proximity effect) To control component temperature, surface area of component and thus size of component must be increased to reject increased heat to ambient. At constant winding current density J and core flux density B, heat generation increases with volume...

## Details of Interactive Core Database Calculations cont

Calculate number turns of N in winding N k Aw A 4. Calculate air-gap length Lg. Air-gap length determined on basis that when inductor current equals peak value I, flux density equals peak value B. Formulas for air-gap length different for different core types. Example for double-E core given in next slide. 5. Calculate maximum inductance Lmax that core can support. Lmax N Acore Bpeak Ipeak . If Lmax > required L value, reduce Lmax by removing winding turns. Save on copper costs, weight, and...

## Details of Interactive Inductor Core Database Calculations

User inputs L, I, Irms, Idc, f, Ts, Ta, and kcu Stored information (static, independent of converter requirements) Core dimensions, Aw, Acore, Vc, Vw, surface area, mean turn length, mean magnetic path length, etc. Quantitative core loss formulas for all materials of interest including approximate temperature dependence. Calculation of core capabilities (stored energy value) 1. Compute converter-required stored energy value L I Irms. 2. Compute allowable specific power dissipation Psp Ts - Ta...

## Details of Interactive Transformer Core Database Calculations

User inputs Vpri, Ipri, turns ratio Ndc Nsec, f, Ts, Ta, and kcu Stored information (static, independent of converter requirements) Core dimensions, Aw, Acore, Vc, Vw, surface area, mean turn length, mean magnetic path length, etc. Quantitative core loss formulas for all materials of interest including approximate temperature dependence. 1. Compute converter-required stored energy value S 2 Vpri Ipri 2. Compute allowable specific power dissipation Psp Ts - Ta R0sa Vc + Vw . R0sa h As or...

## Eddy Currents Increase Winding Losses

AC currents in conductors generate ac magnetic fields which in turn generate eddy currents that cause a nonuniform current density in the conductor . Effective resistance of conductor increased over dc value. dimensions greater than a skin depth. w 2p f, f frequency of ac current m magnetic permeability of conductor m mo for nonmagnetic conductors. s conductivity of conductor material. Numerical example using copper at 100 C Mnimize eddy currents using Leitz wire bundle. Each conductor in...

## Hysteresis Loss in Magnetic Materials

Area encompassed by hysteresis loop equals work done on material during one cycle of applied ac magnetic field. Area times frequency equals power dissipated per unit volume. Typical waveforms of flux density, B(t) versus time, in an inductor. Only Bac contributes to hysteresis loss.

## Inductor Design Example

Peak current 5.6 A, sinewave current, Irms 4 A Stored energy L I Irms (3x1Q-4)(5.6)(4) High frequency operation dictates ferrite material. 3F3 material has highest performance factor PF at 100 kHz. Double-E core chosen for core shape. Double-E core with a 1 cm meets requirements. kcu Jrms B Aw Acore * 0 0125 0 0068 for kcu > 0.3 Database output R0 9.8 C W and Psp 237 mW cm3 Core flux density B 170 mT from database. No Idc, Bpeak 170 mT. Litz wire used, so 0.3. 6 A mm Acu (4 A) (6 A mm2) Q.67...

## Iterative Inductor Design Procedure

Iterative design procedure essentially consists of constructing the core database until a suitable core is found. Choose core material and shape and conductor type as usual. Use stored energy relation to find an initial area product AwAc and thus an Use initial values of Jrms 2-4 A mm2 Use initial core size estimate (value of a in double-E core example) to find corrected values of Jrms and Bac and thus corrected value Compare kcu Jrms B Aw Acore with L I Irms and iterate as needed into proper

## Iterative Transformer Design Procedure

Iterative design procedure essentially consists of constructing the core database until a suitable core is found. Choose core material and shape and conductor type as usual. Use V - I rating to find an initial area product AwAc and thus an initial core size. Use initial values of Jrms 2-4 A mmz Use initial core size estimate (value of a in double-E core example) to find corrected values of Jrms and Bac and thus corrected 2 Vpri Ipri and iterate as needed into proper size is found.

## Magnetic Component Design Problem

Challenge - conversion of component operating specs in converter circuit into component design parameters. Goal - simple, easy-to-use procedure that produces component design specs that result in an acceptable design having a minimum size, weight, and cost. Inductor electrical (e.g.converter circuit) Inductor currents rated peak current I, rated rms current Irms , and rated dc current (if any) Idc Allowable power dissipation in inductor or equivalently maximum surface temperature of the...

## Magnetic Component Design Responsibility of Circuit Designer

Ratings for inductors and transformers in power electronic circuits vary too much for commercial vendors to stock full range of standard parts. Instead only magnetic cores are available in a wide range of sizes, geometries, and materials as standard parts. Circuit designer must design the inductor transformer for the particular application. 1. Selecting appropriate core material, geometry, and size 2. Selecting appropriate copper winding parameters wire type, size, and number of turns.

## Optimization of Solid Conductor Windings

Locus of minimum total loss 1.5 dc loss Locus of minimum total loss 1.5 dc loss Nomalized power dissipation p' FRRdc Conductor height diameter F copper layer factor F b bo for rectangular conductors F d do for round conductors h effective conductor height Transformer leakage inductance causes overvoltages across power switches at turn-off. Leakage inductance caused by magnetic flux which does not completely link primary and secondary windings. Direction and relative magnitude of leakage...

## Power Dissipation in Windings

Average power per unit volume of copper dissipated in c pper winding Pcu,sp Pcu (Jrms)2 where Jrms Irms Acu and pcu copper resistivity. Average power dissipated per unit volume of winding pw,sp kcu Pcu (Jrms)2 Vcu k Vw where Vcu total volume of copper in the winding and Vw total volume of the winding. N number of turns Acu cross-sectional area of copper conductor from which winding is made Lw bw lw Aw b , lw area of winding window. kcu 0.3 for Leitz wire kcu 0.6 for round conductors kcu 0.7-0.8...

## Quantitative Description of Core Losses

Eddy current loss plus hysteresis loss core loss. Empirical equation - Pmsp k fa Bac f frequency of applied field. Bac base-to-peak value of applied ac field. k, a, and d are constants which vary from material to material Pm,sp 15x10-6 f13 Bacl2'5 mW cm3 for 3F3 ferrite. (f in kHz and B in mT) Pm,sp 3.2x10-6 f18 Bacl2 mW cm3 METGLAS 2705M (f in kHz and B in mT) Example 3F3 ferrite with f 100 kHz and Bac 100 mT, Pm,sp 60

## Review of Inductor Fundamentals

No core losses or copper winding losses Linearized B-H curve for core with mm > > mo lm > > g and A > > g2 Magnetic circuit approximations (flux uniform over core cross-section, no fringing flux) Hm lm + Hg g N I (Ampere's Law) Bm A Bg A f (Continuity of flux mm Hm Bm (linearized B-H curve)

## Scaling of Core Flux Density and Winding Current Density

Power per unit volume, Psp, dissipated in magnetic component is Psp k a k1 constant and a core scaling dimension. surface-to-ambient thermal resistance of component. For optimal design Pw sp Pc sp Psp Ts - Ta ' R0sa proportional to a2 and (Vw + Vc) Plots of Jrms , Bac , and Psp versus core size (scale factor a) for a specific core material, geometry, frequency, and Ts - Ta value very useful for picking appropriate core size and winding conductor size.

## Setting DoubleE Core Airgap Length

Set total airgap length Lg so that Bpeak generated at the peak current Ipeak- Lg Ng g Ng number of distributed gaps each of length g. Distributed gaps used to minimize amount of flux fringing into winding and thus causing additional eddy current losses. m,core ' ,xm,gap m,gap - Ag For a double-E core, Ag (a + Tp ) (d + rp ) Ng Ng Ag ad + (a + d) Ng Ng < < a Insertion of expression for Ag(Lg) into expression for Lg(Ag) and solving for Lg yields Above expression for Lg only valid for double-E...

## Simple Nonoptimal Inductor Design Method

Assemble design inputs and compute required LI Ir No Check power dissipation yes and surface temperature. Excessive . No Check power dissipation yes and surface temperature. Excessive . Choose core geometry and core material based on considerations discussed previously. Assume Jrms 2-4 A mm2 and Bac 50-100 mT and use LI Us kcu Jrms Bac Aw Acore to find the required area product Aw Acore and thus the core size. Assumed values of Jrmsand B based on experience. Complete design of inductor as...

## Stored Energy Relation Basis of Inductor Design

Input specifications for inductor design Rated dc current (if any) Idc. Maximum inductor surface temperature Ts and maximum ambient temperature T Design procedure starting point - stored energy relation Selection of core geometric shape and size Winding conductor geometric shape and size Equation relates input specifications (left-hand side) to needed core and winding parameters (right-hand side) A good design procedure will consists of a systematic, single-pass method of selecting kcu, Jrms,...

## Thermal Considerations in Magnetic Components

Losses (winding and core) raise core temperature. Common design practice to limit maximum interior temperature to 100-125 C. Core losses (at constant flux density) increase with temperature increases above 100 C Saturation flux density Bs decreases with temp. Increases Nearby components such as power semiconductor devices, integrated circuits, capacitors have similar limits. Temperature limitations in copper windings Copper resistivity increases with temperature increases. Thus losses, at...

## Types of Core Materials

METGLASS (Fe-B, Fe-B-Si, plus many other compositions) Resistivity _ (10 - 100) Pcu Bs 1 - 1.8 T (T tesla 104 oe) METGLASS materials available only as tapes of various widths and thickness. Resistivity p very large (insulator) no ohmic losses and hence skin effect problems at high frequencies. Other iron alloys available as laminations of various shapes. Powdered iron can be sintered into various core shapes. Powdered iron cores have larger effective resistivities.

## Analysis of Specific Transformer Design cont

Power dissipation in winding Pw kcu Pcu(Jrms) Vw Jrms (4 A) (0.64 mm ) (16 A) (2.6 mm ) 6.2 A mm Pw (0.3)(2.2x10 ohm-m) (6.2x10 A m (1.23x10 m ) Vpri,max Npri Ac Bac (1.414)(300) 425 V Pcore (13.5 cm )(1.5x10 )(100 kHz) '- (140 mT) ' 1.9 W Leakage inductance L eak lw 8 a 8 cm l w (2)(1.4a) + (2)(1.9a) + 2p (0.35b w) 8 a Ts (9.8)(3.1 + 1.9) + 40 89 C No change in core flux density. Constant voltage applied to primary keeps flux density constant. Pw (3.1)(1.25)2 4.8 watts Ts (9.8)(4.8 + 1.9) + 40...

## Analysis of a Specific Inductor Design

Maximum current 4 ams rms at 100 kHz Double-E core with a 1 cm using 3F3 ferrite. Distributed air-gap with four gaps, two in series in each leg total gap length Sg 3 mm. Winding - 66 turns of Leitz wire with Acu 0.64 mm2 Inductor surface black with emissivity 0.9 Find inductance L, Ts max effect of a 25 overcurrent on Ts Power dissipation in winding, Pw Vw kcu pcu (Jrms)2 3.2 Watts Vw 12.3 cm3 (table of core characteristics) pcu at 100 C (approx. max. Ts) 2.2x10-8 ohm-m Jrms 4 (.64) 6.25 A mm2...

## Review of Transformer Fundamentals

Assumptions same as for inductor Starting equations H1Lm Nil H2Lm N2 Ampere's Law HmLm H1 - H2 Lm N1I1- N2I2 MmHm Bm linearized B-H curve Faraday's Law Net flux f f 1 - f2 MmHmA Results assuming pm i.e. ideal core Cross-sectional l m mean path ng area of core A Cross-sectional l m mean path ng area of core A

## Proximity Effect Further Increases Winding Losses

Proximity effect - losses due to eddy current generated by the magnetic field experienced by a particular conductor section but generated by the current flowing in the rest of the winding. Design methods for minimizing proximity effect losses discussed later. Pw Pdc Pec Pec eddy current loss. Pw Rdc Rec Irms Rac Irms Rac FR Rdc 1 Rec Rdc Rdc Minimum winding loss at optimum conductor size. High frequencies require small conductor sizes minimize loss. Pdc kept small by putting may small-size...