(* * LANGUAGE : ANS Forth * PROJECT : Forth Environments * DESCRIPTION : Compute the proximity losses of a layered winding * CATEGORY : Utility * AUTHOR : Marcel Hendrix * LAST CHANGE : November 15, 2000, Marcel Hendrix *) NEEDS -miscutil NEEDS -fsl_util NEEDS -sketcher REVISION -layers "ÄÄÄ Proximity losses Version 0.01 ÄÄÄ" PRIVATES DOC (* Computes the losses in a layer of solid round wire. The losses are a balance between DC losses (increase for thinner wire) and proximity losses (increase for thicker wire). A general formula is found for a single layer of foil, located in the H-field at a position where the MMF ratio is m. The MMF ratio is easily sketched approximately, or could come from a FEM calculation. As an example, we calculate a simple two winding transformer with N1 is N2. For this special case the Fr can be calculated in closed form. Note0: The calculation is of course also valid for an inductor. However, for inductors we can use the Snelling/Bracke/Jongsma equations that give the optimal wire diameter directly. Note1: In a litze wire the current can't crowd on the surface, so there are "no" proximity and skineffect losses. Note2: Proximity loss goes down dramatically when layers are interleaved. This can be accurately predicted when MMF diagrams are sketched and then the losses per layer are computed and summed (using the general formula). See: 'Fundamentals of Power Electronics', Chapter 12, 'Basic Magnetics Theory', Robert W. Erickson. *) ENDDOC -- -- Wire and layer characteristics (here .3 wire on ETD34) --- -- 100e3 FVALUE Fswitch -- switching or test frequency (sinusoidal) 0.351e-3 FVALUE d_out -- outer diameter, includes isolation 0.300e-3 FVALUE d_nom -- copper diameter 42.1e-3 FVALUE MLT -- mean length of turn in two successive layers 20e-3 FVALUE bw -- breadth of winding #61 VALUE t/l -- turns/layer ( for ETD34 ) 10e FVALUE m -- MMF ratio 25e FVALUE Tambient -- ambient temperature in K -- -------------------------------------------------------------- -- : .HOW? ( -- ) CR ." The inductor is wound as follows:" CR CR ." winding breadth = " bw F.N1 ." m" CR ." mean length of turn = " MLT F.N1 ." m" CR ." turns/layer = " t/l DEC. CR ." MMF ratio (m) = " m F.N1 CR ." wire d_out = " d_out F.N1 ." m" CR ." Fswitch = " Fswitch F.N1 ." Hz" CR ." Ambient temperature = " Tambient F.N1 ." deg C" ; 1.724e-4 FCONSTANT rho PRIVATE -- resistivity 4e-3 FCONSTANT Drhocu_dt PRIVATE -- temperature sensitivity of copper resistance 4e-7 PI F* FCONSTANT mu0 PRIVATE -- relative permeability of air -- The resistivity of copper, taking temperature into account : RESISTIVITY ( F: -- r ) rho Tambient 25e F- Drhocu_dt F* F1+ F/ ;P -- Skindepth in copper (Mur = 1) : SKINDEPTH ( F: -- r ) resistivity PI mu0 F* Fswitch F* F/ FSQRT ;P -- Conductor spacing factor or winding porosity. This factor is introduced -- because round wires are transformed into layers of foil conductor. : POROSITY ( F: -- r ) [ PI/4 FSQRT ] FLITERAL d_out F* t/l S>F F* bw F/ ;P -- Effective ratio of round wire conductor height to skin depth. : PHI ( F: -- r ) POROSITY FSQRT d_out F* skindepth F/ ;P -- The DC resistance is rho*lb/Aw, which can be transformed to ... : Rdc ( F: -- r ) resistivity MLT F* t/l DUP * * S>F F* bw FSQR porosity F* F/ ; -- ----------------------------------------------------------------------------------- -- Function G1(phi) : G1 ( F: phi -- r ) F2* FLOCAL 2*phi 2*phi FSINH 2*phi FSIN F+ 2*phi FCOSH 2*phi FCOS F- F/ ;P -- Function G2(phi) : G2 ( F: phi -- r ) FLOCAL phi phi FSINH phi FCOS F* phi FCOSH phi FSIN F* F+ phi F2* FDUP FCOSH FSWAP FCOS F- F/ ;P : Q' ( F: phi m -- r ) FLOCALS| m phi | m FSQR F2* m F2* F- F1+ phi G1 F* m 4e F* m F1- F* phi G2 F* F- ;P -- The relative powerloss in a layer with thickness defined through PHI, -- located in the H-field at a position where the MMF ratio is m. : P/Pdc ( F: -- r ) PHI FDUP m Q' F* ; -- Plotting 0e FVALUE mm PRIVATE : log_Q'[m] ( F: phi -- r ) FALOG mm Q' FLOG ;P : log_phi*Q'[m] ( F: phi -- r ) FALOG FDUP mm Q' F* FLOG ;P : DO_MPLOT ( xt -- ) ( F: ini -- ) LOCAL xt PSINGLE ( F: ini ) xt EXECUTE PMULTI 0.5e xt EXECUTE 1.0e xt EXECUTE 1.5e xt EXECUTE 2.0e xt EXECUTE 4.0e xt EXECUTE 5.0e xt EXECUTE 6.0e xt EXECUTE 8.0e xt EXECUTE 10e xt EXECUTE 12e xt EXECUTE 15e xt EXECUTE PSINGLE ;P : *P/Pdc ( F: m -- ) TO mm -1e 1e ['] log_phi*Q'[m] 'SKETCH ;P : *P/Pdc_skin ( F: m -- ) TO mm -1e 1e ['] log_Q'[m] 'SKETCH ;P \ This is a familiar picture that shows that the smaller PHI, the closer the wire loss \ will be to the loss caused by the wire DC resistance. The problem is, of course, that \ thin wires have very high DC losses... : P/Pdc(phi,m) ( -- ) ['] *P/Pdc 3e DO_MPLOT ; \ That's why we need this picture: it shows the optimum wire diameter. Thinner diameter \ increases DC loss, thicker diameter increases proximity loss. : P/Pdc_d(phi,m) ( -- ) ['] *P/Pdc_skin 0.5e DO_MPLOT ; -- -- Example: power loss in a transformer winding --------------------------------------- \ Assume a transformer where primary and secondary each have M layers. The transformer \ is coaxially wound, primary first. The normalized MMF is F / t/l*Ip. \ The proximity effect increases the copper loss in each layer by the factor phi*Q'(phi,m). \ The total increase is found by summing over the primary layers. \ Fr = Ppri/Ppri,dc = M^-1 * SUM(m=1, m=M) phi*Q'(pi,m) \ Because of the symmetry in this special case (equal number of primary and secondary layers), \ the factor for the secondary is the same. It follows 10e FVALUE M1:1 : Fr/phi ( F: phi -- r ) FALOG FLOCAL phi phi G1 FDUP phi G2 F2* F- M1:1 FSQR F1- 0.6666e F* F* F+ FLOG ;P : Fr ( F: phi -- r ) FALOG FLOCAL phi phi G1 FDUP phi G2 F2* F- M1:1 FSQR F1- 0.6666e F* F* F+ phi F* FLOG ;P : *Fr/phi ( F: M -- ) TO M1:1 -1e 1e ['] Fr/phi 'SKETCH ;P : *Fr ( F: M -- ) TO M1:1 -1e 1e ['] Fr 'SKETCH ;P : Fr(phi,m) ( -- ) ['] *Fr 3e DO_MPLOT ; : Fr/phi(phi,m) ( -- ) ['] *Fr/phi 0.5e DO_MPLOT ; :ABOUT CR ." **** Proximity loss in inductor wound with solid round wire ****" CR CR ." Try: P/Pdc ( F: -- r ) -- copper loss factor increase" CR ." P/Pdc(phi,m) -- show copper loss factor increase" CR ." P/Pdc_d(phi,m) -- show copper loss factor increase when d=skindepth" CR ." .HOW? -- Construction details of the inductor" CR ." TO bw -- (float) width of winding area" CR ." TO m -- (float) MMF ratio" CR ." TO MLT -- (float) mean length of turn [m]" CR ." TO t/l -- (integer) turns per layer" CR ." TO d_out -- (float) outer diameter of [round] wire [m]" CR ." TO Tambient -- (float) ambient temperature [deg C]" CR ." TO Fswitch -- (float) switching frequency [Hz]" CR CR ." ** Example transformer 1:1 ** " CR ." TO M1:1 -- (float) number of primary and secondary windings" CR ." Fr(phi,m) -- plots Fr or Ppri/Ppri,dc" CR ." Fr/phi(phi,m) -- plots Ppri/Pri,dc|phi=1" ; DEPRIVE .ABOUT -layers CR (* End of Source *)