PHYSICAL CONSTANTS USED IN TRANSMISSION LINE WORK. E0_meters E0_inches U0_meters U0_inches

Transcription

1 PHYSICAL CONSTANTS USED IN TRANSMISSION LINE WORK file: consan.mcd Elecric permiiviy of free space (meric) E0_meers F/m Recalculae in in. E0_inces E0_meers.0254 Display calculaed value E0_inces Magneic permeabiliy of free space (meric) U0_meers H/m Recalculae in in. U0_inces U0_meers.0254 Display calculaed value We ofen need is number U0_inces U0_inces Speed of lig (meric) C_meers m/s Recalculaed in in. C_inces C_meers.0254 Display calculaed value C_inces Propagaion delay a lig speed (ps/in.) C_inces CONSTANT.MCD -- 9/7/ p.1

2 DC RESISTANCE OF COPPER WIRES AND TRACES file: resis.mcd Conversion formulas included in is spreadsee: Diameer o AWG AWG o diameer Tickness o copper plaing weig Copper plaing weig o ickness AWG() DIAMETER() CPW() THICKNESS() Resisance formulas included in is spreadsee: DC resisance of round wires From diameer From AWG wire size A room emperaure only RROUND() RROUND_AWG() RROUND_RT() DC resisance of prined circui board races From race ickness and wid RTRACE() Using copper plaing weig RTRACE_CPW() A room emperaure only RTRACE_RT() DC resisance of power or ground planes Using ickness and via diameer RPLANE() Using copper plaing weig RPLANE_CPW() Variables used: Bulk resisiviy of copper om-in. Tis coefficien is sligly differen from e bulk resisiviy of pure copper (6.58E-07) owing o e annealing process used in making wire, and cemical imperfecions in e copper used for making pracical wires. In pracice, e resisance of wo wires making up a wised pair may ofen be maced as well as 10%, bu almos never as well as 1%. RESIST.MCD -- 9/7/ p.1

3 Termal coefficien of resisance.0039 per deg. C If e resisance of a copper wire is R a room emperaure, en a a emperaure 1 o C iger i will be R(1 + ). Tis coefficien applies o sandard annealed copper wires. Te coefficien for pure copper in is bulk sae varies sligly. Over a emperaure range 0-70 o C e resisance of copper wires varies 28%. x d Leng of wire (in.) (or separaion beween conac poins on ground plane) Diameer of wire (in.) (or diameer of conac poin on ground plane) AWG American wire gauge (Englis unis) emp Temperaure ( o C) w Wid of prined circui board race (in.) Tickness of prined circui board race (in.) cpw Tickness of prined circui board races, in unis of copper plaing weig (oz/f 2 ) Conversions beween American Wire Gauge (AWG) and diameer (in.): DIAMETER( awg) 10 awg10 20 AWG( d) 10 20log( d) General formula for resisance of a round wire (): RROUND( d x emp) 4 x 1 ( emp 20) d 2 RESIST.MCD -- 9/7/ p.2

4 Resisance of a round wire specified by AWG size insead of diameer (): RROUND_AWG( awgx emp) RROUND( DIAMETER( awg) x emp) Resisance of a round wire a room emperaure (): RROUND_RT( dx) RROUND( d x 20) Conversion beween ickness, (in.) and copper plaing weig, cpw (oz): CPW() THICKNESS( cpw).00137cpw Resisance of a circui race (): RTRACE( w x emp) x w 1 ( emp 20) Resisance of a race specified by plaing weig insead of ickness ( ): RTRACE_CPW( wcpw x emp) RTRACE( wthickness( cpw) x emp) Resisance of a circui race a room emperaure (): RTRACE_RT( w x) RTRACE( w x 20) RESIST.MCD -- 9/7/ p.3

5 Resisance of a power or ground plane ( ): Wen using long, skinny races or wires, e approximaions above work exremely well. Eac formula assumes a uniform disribuion of curren rougou e conducing body, for wic resisance is direcly proporional o leng. Currens circulaing in a large ground or power plane are no uniform. Consequenly, e resisance measured beween wo poins on a ground or power plane is no direcly proporional o e separaion beween measuremen poins. Te following equaion models e resisance beween wo conac poins on a ground plane. Tis model assumes eac conac poin ouces e ground plane over some finie area. Te approximae diameer of e conac poin deermines e overall resisance. If e conac poins lie near any edge of e plane, e resisance beween em may go up by a facor of 2. Te resisance near corners may rise even iger. d1 Diameer of 1s conac poin (in.) d2 Diameer of 2nd conac poin (in.) Tickness of plane (in.) cpw Tickness of plane, copper plaing weig (oz) x Separaion beween conac poins (in.) emp Temperaure ( o C) Resisance of a power or ground plane ( ): RPLANE( d1 d2 x emp) ln 2x ln 2x 1 ( emp 20) 2 d1 d2 Resisance of a power or ground plane specified by plaing weig insead of ickness ( ): RPLANE_CPW( d1d2 cpw x emp) RPLANE( d1d2 THICKNESS( cpw) x emp) RESIST.MCD -- 9/7/ p.4

6 CAPACITANCE OF TWO PARALLEL PLATES file: capac.mcd Formulas included in is spreadsee: Capaciance of wo plaes Impedance magniude of capacior a one frequency Impedance magniude of capacior as seen by rising edge CPLATE() XCF() XCR() Variables used: w Wid of plae overlap (in.) Capaciance of wo parallel plaes x Leng of plae overlap (in.) Heig of one plae above e oer (in.) x w Relaive elecric permiiviy e r beween plaes er Relaive dielecric consan of maerial beween plaes capac Capaciance of wo plaes (F): CPLATE( w x erxw A power and ground plane separaed by in. of FR-4 dielecric (er = 4.5) sare a capaciance of 100 pf/in. 2 Halving e separaion doubles e capaciance. Impedance magniude of capacior a frequency f ( ): c Capaciance (F) f Frequency (Hz) XCF( c f) 1 2fc CAPAC.xmcd -- 1/23/ p.1

7 Te impedance, a 100 MHz, of a 100-pF capacior is XCF Impedance magniude of capacior as seen by rising edge ( ): c r Capaciance (F) 10-90% rise ime (s) XCR( c r) r c Te impedance, as seen by a 5-ns rising edge of a 100-pF capacior is XCR CAPAC.xmcd -- 1/23/ p.2

8 INDUCTANCE OF CIRCULAR LOOP file: circular.mcd Formulas included in is spreadsee: Inducance of circular wire loop Impedance magniude of inducor a one frequency Impedance magniude of inducor as seen by rising edge LCIRC() XLF() XLR() Variables used: d Diameer of wire (in.) Inducance of a circular wire loop x Diameer of wire loop (in.) x d circular Inducance of wire loop (H): LCIRC( dx) xln 8x d 2 A loop of 24-gauge wire e size of e loop beween your umb and forefinger as abou 100 nh of inducance. Canging e wire diameer from AWG 24 o AWG 14 makes lile difference. Te log funcion is raer insensiive o wire size. LCIRC (.011.3) LCIRC (.1 1.3) CIRCULAR.MCD -- 9/7/ p.1

9 Impedance magniude of inducor a frequency f (): l f Inducance (H) Frequency (Hz) XLF( l f) 2fl Te impedance, a 100 MHz, of a 100-nH inducor is XLF Impedance magniude of inducor as seen by rising edge ( ): l r Inducance (H) 10-90% rise ime (s) XLT( l r) l r Te impedance, as seen by a 5-ns rising edge, of a 100-nH inducor is XLT CIRCULAR.MCD -- 9/7/ p.2

10 INDUCTANCE OF RECTANGULAR LOOPS file: recangl.mcd Formulas included in is spreadsee: Inducance of recangular wire loop Impedance magniude of inducor a one frequency Impedance magniude of inducor o rising edge Variables used: LRECT() XLF() XLR() Inducance of a recangular wire loop d x Diameer of wire (in.) Leng of wire loop (in.) x y Bread of wire loop (in.) y d recangle Inducance of wire loop (H): LRECT( d x y) xln 2y d yln 2x d A loop of 24-gauge wire 1 in. 2 as abou 100 nh of inducance. Canging e wire diameer from AWG 30 o AWG 10 makes lile difference. Te log funcion is very insensiive o wire size. If your loop consiss of differen-sized conducors, use e diameer of e smalles one. Impedance magniude of inducor a frequency f (): l f Inducance (H) Frequency (Hz) XLF( l f) 2fl Te impedance, a 100 MHz, of a 100-nH inducor is 62. RECTANGL.MCD -- 9/7/ p.1

11 Impedance magniude of inducor as seen by rising edge ( ): l r Inducance (H) 10-90% rise ime (s) XLT( l r) l r Te impedance, as seen by a 5-ns rising edge, of a 100-nH inducor is 62. RECTANGL.MCD -- 9/7/ p.2

12 MUTUAL INDUCTANCE OF TWO LOOPS file: mloop.mcd Formulas included in is spreadsee: Muual inducance of wo loops MLOOP() Variables used: Muual inducance of wo wire loops r Separaion beween loop ceners (in.) A1 Surface area of loop 1 (in. 2 ) A1 r A2 A2 Surface area of loop 2 (in. 2 ) I 1 I 2 (We assume e loops are fla, and a eir faces are oriened parallel o eac oer for maximum coupling) mloop Te loops mus be well separaed for e MLOOP() approximaion o work: r A1 r A2 and Muual inducance of wo well-separaed loops (nh): MLOOP( r A1 A2) 5.08 A1 A2 r 3 MLOOP.MCD -- 9/7/ p.1

13 MUTUAL INDUCTANCE OF PARALLEL TRANSMISSION LINES file: mline.mcd Formulas included in is spreadsee: Muual inducance of wo lines MLINE() Variables used: s Separaion beween wire ceners (in.) Heig of wires above ground (in.) Muual inducance of parallel wires suspended above ground plane Round or square ransmission line s x Leng of parallel span (in.) (We assume a wo idenical ransmission lines sare a parallel run of leng x, wi a orizonal separaion s.) mline Le L equal e inducance (H) of e firs ransmission line of leng x (use formula for round, microsrip, or sripline geomery as appropriae): MLINE( L s ) L 1 1 s 2 MLINE.MCD -- 9/7/ p.1

14 COAXIAL TRANSMISSION LINE file: coax.mcd Formulas included in is spreadsee: Coaxial cable caracerisic impedance Coaxial cable propagaion delay Coaxial cable inducance Coaxial cable capaciance Variables used: d1 Diameer of inner wire (in) ZCOAX() PCOAX() LCOAX() CCOAX() Coaxial cable (cross-secion view) d2 Diameer of ouer sield (in) x Leng of cable (in) er Relaive dielecric consan of maerial surrounding e inner wire d 1 d 2 Relaive elecric permiiviy e r of maerial surrounding inner wire coax Caracerisic impedance of coaxial cable (): ZCOAX( d1 d2 60 ln d2 er d1 Propagaion delay per in. for coaxial cable (s/in.): PCOAX( er Inducance of coaxial cable (H): LCOAX( d1 d2 x) x ln d2 d1 Capaciance of coaxial cable (F): CCOAX( d1 d2 er x) x ln d2 d1 er COAX.MCD -- 9/7/ p.1

15 Example coaxial cable calculaions Diameer of AWG 30 inner wire (in.) D1.01 Inside diameer of sield (in.) D2.1 Leng of cable (in.) X Relaive dielecric consan er 2.2 Caracerisic impedance (): ZCOAX( D1 D Toal inducance (H): LCOAX( D1 D2 X) Same resul in nh: LCOAX( D1 D2 X) Inducance per in. (H): LCOAX( D1 D2 1) Toal capaciance (F): CCOAX( D1 D2 er X) Same resul in pf: CCOAX( D1 D2 er X) Capaciance per in. (F): CCOAX( D1 D2 er 1) COAX.MCD -- 9/7/ p.2

16 TRANSMISSION LINE MADE FROM ROUND WIRE (WIRE-WRAP) file: round.mcd Formulas included in is spreadsee: Round wire caracerisic impedance Round wire propagaion delay Round wire inducance Round wire capaciance ZROUND() PROUND() LROUND() CROUND() Variables used: d Diameer of wire (in.) Round wire suspended above ground plane (wire wrap) x Heig of wire above ground (in.) Leng of wire (in.) (Assume air dielecric) d (We assume e wire is suspended in air, for wic e relaive dielecric consan is 1.00.) Caracerisic impedance of round wire above ground plane (): round ZROUND( d) 60ln 4 d Propagaion delay per in. of round wire above ground plane (s/in): PROUND( d) (assume air dielecric) Inducance of round wire above ground plane (H): LROUND( d x) x ln 4 d Capaciance of round wire above ground plane (F): CROUND( d x) x ln 4 d ROUND.xmcd -- 1/23/ p.1

17 Example round wire calculaions Diameer of AWG 30 wire (in.) D.01 Leng of wire (in.) X Heig above ground (in.) H.100 Caracerisic impedance (): ZROUND( D H) Toal inducance (H): Same resul in nh: LROUND( D H X) LROUND( D H X) Inducance per in. (H): LROUND( D H 1) Toal capaciance (F): CROUND( D H X) Same resul in unis pf: CROUND( D H X) Capaciance per in. (F): CROUND( D H 1) ROUND.xmcd -- 1/23/ p.2

18 TRANSMISSION LINE MADE FROM TWISTED PAIR WIRE file: wis.mcd Formulas included in is spreadsee: Twised-pair caracerisic impedance Twised-pair propagaion delay Twised-pair inducance Twised-pair capaciance Variables used: ZTWIST() PTWIST() LTWIST() CTWIST() Twised-pair ransmission line d Diameer of wire (in.) s s x Separaion beween wires (in.) Leng of wire (in.) er Effecive relaive dielecric consan of medium beween wires d Effecive relaive elecric permiiviy e r lies beween e permiiviy of e wire's insulaor and e permiiviy of air (1.00). wis Caracerisic impedance of wised pair (): 120 ZTWIST( ds ln 2s er d Propagaion delay per in. wised pair (s/in.): PTWIST( er Inducance of wised pair (H): LTWIST( ds x) x ln 2s d Capaciance of wised pair (F): CTWIST( d s er x) x ln 2s d er TWIST.MCD -- 9/7/ p.1

19 Example wised-pair calculaions Diameer of AWG 24 wire (in.) D.02 Leng of wire (in.) X Separaion beween wire ceners (in.) S.038 Relaive dielecric consan er 2.5 Caracerisic impedance (): ZTWIST( D S Toal inducance (H): LTWIST( D S X) Same resul in nh: LTWIST( D S X) Inducance per in. (H): LTWIST( D S 1) Toal capaciance (F): CTWIST( D S er X) Same resul in pf: CTWIST( D S er X) Capaciance per in. (F): CTWIST( D S er 1) TWIST.MCD -- 9/7/ p.2

20 MICROSTRIP TRANSMISSION LINES file: msrip.mcd Formulas included in is spreadsee: Effecive relaive permiiviy EEFF() (used inernally) Effecive elecrical race wid WE() (used inernally) Microsrip caracerisic impedance Microsrip propagaion delay Microsrip race inducance Microsrip race capaciance ZMSTRIP() PMSTRIP() LMSTRIP() CMSTRIP() Formulas from: I. J. Bal and Rames Garg, "Simple and accurae formulas for microsrip wi finie srip ickness", Proc. IEEE, 65, 1977, pp Microsrip ransmission line w Tis maerial is nicely summarized in T. C. Edwards, "Foundaions of Microsrip Circui Design," Jon Wiley, New York, 1981, reprined (Wac ou for Edward's error in Equaion 3.52b, were e omis a ln() funcion.) Relaive elecric permiiviy of e subsrae, e r msrip Variables used: w er x Trace eig above ground (in.) Trace wid (in.) Trace ickness (in.) Relaive permiiviy of maerial beween race and ground plane (dimensionless) Trace leng (in.) MSTRIP.MCD -- 9/7/ p.1

21 Effecive relaive permiiviy as a funcion of microsrip race geomery: For skinny races (w < ) E_skny( w er 1 2 er w w 2 For wide races (w > ) E_wide( w er 1 2 er w.500 Composie formula picks skinny or wide model depending on w/ raio: E_emp( w if( w E_wide( w E_skny( w ) Special adjusmen o accoun for race ickness: EEFF( w E_emp( w ( er 1) 4.6 w Wen w/ is skinny, you ge e average of e PCB permiiviy, er, and e permiiviy of air. Wen w/ is wide, (e race is very close o e ground plane) you ge er. MSTRIP.MCD -- 9/7/ p.2

22 Effecive race wid as a funcion of oer parameers (in.): For skinny races (2w < ) WE_skny( w ) w ln 4w For wide races (2w > ) WE_wide( w ) w ln 2 Composie formula picks skinny or wide model depending on w/ raio: WE( w ) if w 2 WE_wide( w ) WE_skny( w ) MSTRIP.MCD -- 9/7/ p.3

23 Caracerisic impedance as a funcion of race geomery (): Accuracy of beer an 2 percen is obained under e following condiions: 0 < / < < w/ < 20 0 < er < 16 For skinny races (w < ) ZMS_skny( w ) 60ln 8 WE( w ) WE( w ) 4 For wide races (w > ) ZMS_wide( w ) WE( w ) ln WE( w ) Composie formula picks skinny or wide model depending on w/ raio: ZMSTRIP( w if ( w ZMS_wide( w ) ZMS_skny( w ) ) EEFF( w MSTRIP.MCD -- 9/7/ p.4

24 Microsrip propagaion delay (s/in.): PMSTRIP( w EEFF( w Inducance of microsrip (H): LMSTRIP( w x) PMSTRIP( w 1. ) ZMSTRIP( w 1. ) x (Use a dummy er value of 1. I doesn' maer for inducance calculaions.) Capaciance of microsrip (F): CMSTRIP( w er x) PMSTRIP( w ZMSTRIP( w x MSTRIP.MCD -- 9/7/ p.5

25 Example microsrip wire calculaions Heig above ground (in.) H.006 Wid of race (in.) W.008 Tickness of race (in.) T Leng of wire (in.) X (1-oz copper plaing weig) Relaive elecric permeabiliy (affecs capaciance, bu no inducance) er 4.5 Impedance (): ZMSTRIP( H W T Toal inducance (H): LMSTRIP( H W T X) Same resul in nh: LMSTRIP( H W T X) Inducance per in. (H): LMSTRIP( H W T 1) Toal capaciance (F): CMSTRIP( H W T er X) Same resul in pf: CMSTRIP( H W T er X) Capaciance per in. (F): CMSTRIP( H W T er 1) MSTRIP.MCD -- 9/7/ p.6

26 Tolerance effecs ZMSTRIP_TOL( d w dw er d ZMSTRIP( d w dw er d ZMSTRIP( w ZMSTRIP( d w dw er d ZMSTRIP_TOL ( ) REFL( xz) z x 0 z x 0 z x 1 z x 1 z x 2 z x 2 REFL MSTRIP.MCD -- 9/7/ p.7

27 STRIPLINE TRANSMISSION LINES file: sline.mcd Formulas included in is spreadsee: Sripline caracerisic impedance Offse sripline caracerisic impedance Sripline propagaion delay Sripline race inducance Offse sripline inducance Sripline race capaciance Offse sripline capaciance ZSTRIP() ZOFFSET() PSTRIP() LSTRIP() LOSTRIP() CSTRIP() COSTRIP() Formulas are from Seymour Con, "Problems in Srip Transmission Lines," MTT-3, No. 2, Marc 1955, pp Sripline ransmission line w Tis maerial is summarized in Harlan Howe, Sripline Circui Design, Arec House, Norwood, MA, b 1 2 Relaive elecric permiiviy, e r sline Variables used: 1 2 b w er x Trace eig above lower ground plane (in.) Trace eadroom below upper ground plane (in.) Separaion beween ground planes, b = (in.) Trace wid (in.) Trace ickness (in.) Trace ickness (in.) Trace leng (in.) SLINE2.MCD -- 9/7/ p.1

28 Sripline caracerisic impedance (:) Accuracy of beer an 1.3% is obained under e following condiions: /b < 0.25 /w < 0.11 er unresriced NOTE: formula ZSTR_K1() correced per insrucions from Rober Canrig of Ricardson, TX. Tanks, Rober. For skinny races (w/b < 0.35) ZSTR_K1( w ) w ln 4w w w 2 ZSTR_skny( bw 60 ln er 4b ZSTR_K1( w ) For wide races (w/b > 0.35) ZSTR_K2( b ) 2 1 ln b 1 1 b b 1 ln 1 1 b 2 1 ZSTR_wide( bw w b 1 b ZSTR_K2( b) 1 er Composie formula picks skinny or wide model depending on w/b raio: ZSTRIP( bw if ( w.35b ZSTR_wide( b w ZSTR_skny( b w ) SLINE2.MCD -- 9/7/ p.2

29 Rarely are e wo parameers 1 and 2 equal in pracice. Te more common case is an assymeric sripline aving e conducing race offse o one side. Offse, or asymmeric, sripline caracerisic impedance ( ) (no accuracy guaraneed): ZOFFSET( 12 w 2ZSTRIP( 21 w ZSTRIP( 22 w ZSTRIP( 21 w ZSTRIP( 22 w Propagaion delay of sripline (s/in.): PSTRIP( Inducance of sripline (H): er (same formula for cenered or offse sripline) LSTRIP( bw x) PSTRIP( 1. ) ZSTRIP( b w 1. ) x In e equaion above, we can assume a relaive permiiviy of 1.; i doesn' affec e answer. Inducance of offse sripline (H): LOSTRIP( 12 w x) PSTRIP( 1. ) ZOFFSET( 1 2 w 1. ) x Capaciance of sripline (F): CSTRIP( bw er x) PSTRIP( ZSTRIP( bw x In e equaions above and below, we mus use e relaive permiiviy. Capaciance of offse sripline (F): COSTRIP( 12 w er x) PSTRIP( ZOFFSET( 12 w x SLINE2.MCD -- 9/7/ p.3

30 Example sripline calculaions Ground plane separaion (in.) B.020 Wid of race (in.) W.006 Tickness of race (in.) T Leng of wire (in.) X (1-oz copper plaing weig) Relaive elecric permeabiliy (affecs capaciance, bu no inducance) er 4.5 Impedance (): ZSTRIP( B W T Toal inducance (H): LSTRIP( B W T X) Same resul in nh: LSTRIP( B W T X) Inducance per in. (H): LSTRIP( B W T 1) Toal capaciance (F): CSTRIP( B W T er X) Same resul in pf: CSTRIP( B W T er X) Capaciance per in. (F): CSTRIP( B W T er 1) SLINE2.MCD -- 9/7/ p.4

31 Tolerance effecs ZOFF_TOL( 1d1 2 d2 w dw er d ZOFFSET( 1 d1 2 d2 w dw er d ZOFFSET( 12 w ZOFFSET( 1 d1 2 d2 w dw er d ZOFF_TOL ( ) REFL( xz) REFL 50 z x 0 z x 0 z x 1 z x 1 z x 2 z x SLINE2.MCD -- 9/7/ p.5