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Below is the mathematics behind this choice. It sho uld be noted there phases can't be plotted on log-log plots since they can be negative. A semi-log plot is usually the best choice for phase plots." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{XPPEDIT 0 1 "with(plots);" "-I%mrowG6#/I+m odulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6&Q%withF'/%%boldGQ&f alseF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I(mfencedGF$6%-F#6#- F,6&Q&plotsF'F/F2F5/F6Q%boldF'/%+fontweightGFA-I#moGF$6.Q\";F'F//F6Q'n ormalF'/%&fenceGF1/%*separatorGF4/%)stretchyGF1/%*symmetricGF1/%(large opGF1/%.movablelimitsGF1/%'accentGF1/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.2 777778emF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7enI(animateG6\"I*animate3 dGF$I-animatecurveGF$I&arrowGF$I-changecoordsGF$I,complexplotGF$I.comp lexplot3dGF$I*conformalGF$I,conformal3dGF$I,contourplotGF$I.contourplo t3dGF$I*coordplotGF$I,coordplot3dGF$I,densityplotGF$I(displayGF$I-dual axisplotGF$I*fieldplotGF$I,fieldplot3dGF$I)gradplotGF$I+gradplot3dGF$I ,graphplot3dGF$I-implicitplotGF$I/implicitplot3dGF$I(inequalGF$I,inter activeGF$I2interactiveparamsGF$I.intersectplotGF$I-listcontplotGF$I/li stcontplot3dGF$I0listdensityplotGF$I)listplotGF$I+listplot3dGF$I+loglo gplotGF$I(logplotGF$I+matrixplotGF$I)multipleGF$I(odeplotGF$I'paretoGF $I,plotcompareGF$I*pointplotGF$I,pointplot3dGF$I*polarplotGF$I,polygon plotGF$I.polygonplot3dGF$I4polyhedra_supportedGF$I.polyhedraplotGF$I*r ootlocusGF$I,semilogplotGF$I*setcolorsGF$I+setoptionsGF$I-setoptions3d GF$I+spacecurveGF$I1sparsematrixplotGF$I)surfdataGF$I)textplotGF$I+tex tplot3dGF$I)tubeplotGF$" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`: =`(rc, 'rc');" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 .-I#miGF$6(Q#rcF'/%'familyGQ+MonospacedF'/%'italicGQ%trueF'/%+foregrou ndGQ*[255,0,0]F'/%0font_style_nameGQ,Maple~InputF'/%,mathvariantGQ'ita licF'-I#moGF$60Q*≔F'F/F5F8/FFXF^oFXFin" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#rcG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#wbG6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 31 "Let's start with an RC circuit:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } {XPPEDIT 0 1 "`:=`(frc, `/`(1, `*`(sqrt(`+`(1, `*`(`^`(w, 2), `*`(`^`( rc, 2))))))));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF' 6&-I#miGF$6&Q$frcF'/%%boldGQ&falseF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'-I#moGF$6.Q*≔F'F//F6Q'normalF'/%&fenceGF1/%*separator GF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%'acc entGF1/%'lspaceGQ,0.2777778emF'/%'rspaceGFN-I&mfracGF$6(-I#mnGF$6%Q\"1 F'F/F<-F#6$-F,6&Q%sqrtF'F//F3F1F<-I(mfencedGF$6%-F#6-FT-F96.Q\"+F'F/F< F>F@FBFDFFFHFJ/FMQ,0.2222222emF'/FPFao-F,6&Q\"wF'F/F2F5-F96.Q'⋅F' F/FF@FBFDFFFHFJ/FMQ&0.0emF'/FPFjoFfo-FU6%Q\"2F'F/F/FAF4FBFDFFFHFJFioFO" }} {PARA 11 "" 1 "" {XPPMATH 20 "*$,&\"\"\"F$*&I\"wG6\"\"\"#I#rcGF'F(F$#! \"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 38 "Define the break frequency (3dB point)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{XPPEDIT 0 1 "`:=`(rc, `/`(1, `*`(wb)));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysli bGF'6&-I#miGF$6&Q#rcF'/%%boldGQ&falseF'/%'italicGQ%trueF'/%,mathvarian tGQ'italicF'-I#moGF$6.Q*≔F'F//F6Q'normalF'/%&fenceGF1/%*separa torGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%' accentGF1/%'lspaceGQ,0.2777778emF'/%'rspaceGFN-I&mfracGF$6(-I#mnGF$6%Q \"1F'F/F<-F#6#-F,6&Q#wbF'F/F2F5/%.linethicknessGQ\"1F'/%+denomalignGQ' centerF'/%)numalignGF\\o/%)bevelledGF1-F96.Q\";F'F/F/FAF4FBFDFFFHFJ /FMQ&0.0emF'FO" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$I#wbG6\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "frc;" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6$-I#miGF$6(Q$frcF'/%'familyGQ+Monospace dF'/%'italicGQ%trueF'/%+foregroundGQ*[255,0,0]F'/%0font_style_nameGQ,M aple~InputF'/%,mathvariantGQ'italicF'-I#moGF$60Q\";F'F/F5F8/F " 0 "" {XPPEDIT 0 1 "`: =`(wb, `*`(`+`(`*`(2.0, `*`(evalf(Pi)))), 600));" "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6+-I#miGF$6&Q#wbF'/%%boldGQ&falseF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6.Q*≔F'F//F6 Q'normalF'/%&fenceGF1/%*separatorGF1/%)stretchyGF1/%*symmetricGF1/%(la rgeopGF1/%.movablelimitsGF1/%'accentGF1/%'lspaceGQ,0.2777778emF'/%'rsp aceGFN-I#mnGF$6%Q$2.0F'F/F<-F96.Q'⋅F'F/FF@FBFDFFFHFJ/FMQ&0.0em F'/FPFY-F,6&Q&evalfF'F/F2F5-I(mfencedGF$6%-F#6#-F,6&Q%πF'F//F3F1F/FAF4FBFDFFFHFJFXFO" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+&=6*pP!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 214 261 "The frequency behaviour is much more obvious on a log-log pl ot. Here we see that at high frequencies the amplitude drops of by a f actor of 10 for each decade. This comes from the fast that as w->infin ity, frc~wb/w. log(f) vs log(wb/w) is indeed a straight line." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 19 0 "" }{XPPEDIT 19 1 "plot(frc, w = 0 .. 100000);" "-I%mr owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6(Q%plotF'/% 'familyGQ+MonospacedF'/%'italicGQ%trueF'/%+foregroundGQ*[255,0,0]F'/%0 font_style_nameGQ,Maple~InputF'/%,mathvariantGQ'italicF'-I(mfencedGF$6 )-F#6)-F,6(Q$frcF'F/F2F5F8F;-I#moGF$60Q\",F'F/F5F8/FD_D7I s!#;7$$\"1ML$e9ui2%!#7$\"172%eG'z*y'!#;7$$\"-voMrU^!\")$\"1DbjGl>7f!#; 7$$\"1nmm\"z_\"4i!#7$\"1w`si,&)*=&!#;7$$\"1ommm6m#G(!#7$\"1`Ac))=8(f%! #;7$$\"1ommT&phN)!#7$\"2NL)y\\^Q7T!#<7$$\"1-+v=ddC%*!#7$\"2%y*[$oa(Rr$ !#<7$$\"2ML$e*=)H\\5!#7$\"29g3Hi*=\"Q$!#<7$$\"2nm;z/3uC\"!#7$\"1,!GJ;l H*G!#;7$$\"-DJ$RDX\"!\"($\"1M26m=;7D!#;7$$\"1nm\"zR'ok;!#6$\"2-om:*yq3 A!#<7$$\"2-+]i5`h(=!#7$\"2a$pP#Q1+(>!#<7$$\"2NLL$3En$4#!#7$\"2Fy\"H;@7 s,7!#;7$$\"2NLL$347TL!#7$\"21I5GVA77\" !#<7$$\"2NLLLjM?`$!#7$\"11N!o(*>81\"!#;7$$\"-D\"o7Tv$!\"($\"1$GY00H=** *!#<7$$\"2NLLLQ*o]R!#7$\"1&[pp^i#*\\*!#<7$$\"1,+D\"=lj;%!#6$\"0;rotD;, *!#;7$$\"-vV&R) !#<7$$\"1MLeR\"3Gy%!#6$\"1#Gr[cSy&y!#<7$$\"1nm;/T1&*\\!#6$\"1RT\"H9pe_ (!#<7$$\"1nm\"zRQb@&!#6$\"0i&3N5U4s!#;7$$\"-v=>Y2a!\"($\"1y&z83-[&p!#< 7$$\"1nm;zXu9c!#6$\"1z\"\\\\0A#*p'!#<7$$\"*&y))Ge!\"%$\"1$*>b/*[TX'!#< 7$$\",DE&QQg!\"'$\"1$eh1&=6Ji!#<7$$\"2/+]7y%3Ti!#7$\"2Wp4wE%[Hg!#=7$$ \"1,+]P![hY'!#6$\"1QKL>MM?e!#<7$$\"1LLL$Qx$om!#6$\"2D^4Y>.Wk&!#=7$$\"+ v.I%)o!\"&$\"1/+Kho!zY&!#<7$$\"1mm\"zpe*zq!#6$\"1&QroqJsJ&!#<7$$\"1-++ D\\'QH(!#6$\"1[Qotcrh^!#<7$$\"1MLe9S8&\\(!#6$\"1ir#\\;hM-&!#<7$$\"1,+D 1#=bq(!#6$\"2Cq6&Guj')[!#=7$$\"1LLL3s?6z!#6$\"11CH9!y)fZ!#<7$$\"-DJXaE \")!\"($\"2bwPu?DSj%!#=7$$\"1ommm*RRL)!#6$\"2w`[X,V*=X!#=7$$\"1om;a<.Y &)!#6$\"1ihE&Q9qS%!#<7$$\"1NLe9tOc()!#6$\"1bNVIBN,V!#<7$$\"1-++]Qk\\*) !#6$\"1(HL0mE'3U!#<7$$\"1ML$3dg6<*!#6$\"2c58))oYr5%!#=7$$\"1nmmmxGp$*! #6$\"1*pax+P/-%!#<7$$\"-D\"oK0e*!\"($\"2DxW=TF>$R!#=7$$\"1,+v=5s#y*!#6 $\"10,W([%y]Q!#<7$$\"(+++\"!\"\"$\"10vN&4Nsw$!#<-%&COLORG6&%$RGBG$\"#5 !\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"\"!!\"\"$\"(+++\"!\"\"%(DEFAU LTG-%+AXESLABELSG6'-I#miG6#/I+modulenameG6\"I,TypesettingGI(_syslibG6 \"65Q\"w6\"/%'familyGQ!6\"/%%sizeGQ#106\"/%%boldGQ&false6\"/%'italicGQ %true6\"/%*underlineGQ&false6\"/%*subscriptGQ&false6\"/%,superscriptGQ &false6\"/%+foregroundGQ([0,0,0]6\"/%+backgroundGQ.[255,255,255]6\"/%' opaqueGQ&false6\"/%+executableGQ&false6\"/%)readonlyGQ&false6\"/%)comp osedGQ&false6\"/%*convertedGQ&false6\"/%+imselectedGQ&false6\"/%,place holderGQ&false6\"/%6selection-placeholderGQ&false6\"/%,mathvariantGQ'i talic6\"Q!6\"-%%FONTG6%%(DEFAULTG%(DEFAULTG\"#5%+HORIZONTALG%+HORIZONT ALG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 235 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 214 261 "The frequency behaviour is much more obvious on \+ a log-log plot. Here we see that at high frequencies the amplitude dro ps of by a factor of 10 for each decade. This comes from the fast that as w->infinity, frc~wb/w. log(f) vs log(wb/w) is indeed a straight li ne." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 0 "" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 19 0 "" }{XPPEDIT 19 1 "loglogplot (frc, w = 10 .. 200000);" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI( _syslibGF'6%-I#miGF$6(Q+loglogplotF'/%'familyGQ+MonospacedF'/%'italicG Q%trueF'/%+foregroundGQ*[255,0,0]F'/%0font_style_nameGQ,Maple~InputF'/ %,mathvariantGQ'italicF'-I(mfencedGF$6)-F#6)-F,6(Q$frcF'F/F2F5F8F;-I#m oGF$60Q\",F'F/F5F8/F#>['*****!#;7$$\"+M(\\\"H7!\")$\"1-0q[o%*****!#;7$$\"*243^\"!\"($ \"1*eh))p>*****!#;7$$\"+m0,d=!\")$\"1%Qe5oy)****!#;7$$\"+XSa#G#!\")$\" 1*HY>r;)****!#;7$$\"+0%)e0G!\")$\"1x_j!4B(****!#;7$$\"+T#)[[M!\")$\"1P op^;e****!#;7$$\"+3%3(QU!\")$\"12u%e(zO****!#;7$$\"+6t+5_!\")$\"1EmTy^ /****!#;7$$\"+1\"zQS'!\")$\"13`T_vb)***!#;7$$\"+)HE8(y!\")$\"17]#z(4#y ***!#;7$$\"+C'Q]n*!\")$\"1'3$>j%3n***!#;7$$\"+;r?*=\"!\"($\"1tv,q$G]** *!#;7$$\"*h8'!\"($\"0Y'prTin)*!#:7$$\" +&fQbh(!\"($\"14;Q%e-?!)*!#;7$$\"+Ysjg$*!\"($\"/1BOyH0(*!#97$$\"+yCc]6 !\"'$\"1k7e(yvWc*!#;7$$\"+lN@99!\"'$\"118Q3g)GO*!#;7$$\"+F-GQL4X,**p)!#;7$$\"+K&4ii#!\"'$\"1'fd` ;8`?)!#;7$$\"+ZZ+GK!\"'$\"1AIxXz*ef(!#;7$$\"*=,x'R!\"&$\"1$o\"!#<7$$\"*Am(QQ!\"%$\"0;t7v9 Ox*!#;7$$\"2'*****4\\=%=Z!#7$\"1HhWLlRkz!#<7$$\"+#zU'*z&!\"&$\"18:)evd l['!#<7$$\"+:%H'Gr!\"&$\"1'Hl@!*H5G&!#<7$$\"+uH:i()!\"&$\"2M+#H#R<&)H% !#=7$$\"+.)**p2\"!\"%$\"2w=qR%*R#)\\$!#=7$$\"+7SzB8!\"%$\"2FIv,#RlYG!# =7$$\"+i59F;!\"%$\"2$)fn&e5F;B!#=7$$\"+4+++?!\"%$\"2it)p03i%)=!#=-%&CO LORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"$+\"!\"\"$\" (+++#!\"\"%(DEFAULTG-%+AXESLABELSG6'-I#miG6#/I+modulenameG6\"I,Typeset tingGI(_syslibG6\"65Q\"w6\"/%'familyGQ!6\"/%%sizeGQ#106\"/%%boldGQ&fal se6\"/%'italicGQ%true6\"/%*underlineGQ&false6\"/%*subscriptGQ&false6\" /%,superscriptGQ&false6\"/%+foregroundGQ([0,0,0]6\"/%+backgroundGQ.[25 5,255,255]6\"/%'opaqueGQ&false6\"/%+executableGQ&false6\"/%)readonlyGQ &false6\"/%)composedGQ&false6\"/%*convertedGQ&false6\"/%+imselectedGQ& false6\"/%,placeholderGQ&false6\"/%6selection-placeholderGQ&false6\"/% ,mathvariantGQ'italic6\"Q!6\"-%%FONTG6%%(DEFAULTG%(DEFAULTG\"#5%+HORIZ ONTALG%+HORIZONTALG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1 " }}{TEXT 235 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 0 1 "`:=`(fhli m, `/`(`*`(wb, `*`(Heaviside(`+`(w, `-`(wb))))), `*`(w)));" "-I%mrowG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6)-I#miGF$6&Q&fhlimF'/%%bo ldGQ&falseF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6.Q*&co loneq;F'F//F6Q'normalF'/%&fenceGF1/%*separatorGF1/%)stretchyGF1/%*symm etricGF1/%(largeopGF1/%.movablelimitsGF1/%'accentGF1/%'lspaceGQ,0.2777 778emF'/%'rspaceGFN-I&mfracGF$6(-F#6#-F,6&Q#wbF'F/F2F5-F#6#-F,6&Q\"wF' F/F2F5/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF]o/% )bevelledGF1-F96.Q'⋅F'F/FF@FBFDFFFHFJ/FMQ&0.0emF'/FPFfo-F,6&Q* HeavisideF'F//F3F1F<-I(mfencedGF$6%-F#6%Fen-F96.Q(−F'F/FF@FBF DFFFHFJ/FMQ,0.2222222emF'/FPFepFVF/F<-F96.Q\";F'F/F/FAF4FBFDFFFHFJF eoFO" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I\"wG6\"!\"\"-I*HeavisideG6$ %*protectedGI(_syslibGF%6#,&F$\"\"\"$!+&=6*pP!\"'F.F.$\"+&=6*pPF1" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 0 1 "`:=`(fllim, Heaviside(`+`(wb, ` -`(w))));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I# miGF$6&Q&fllimF'/%%boldGQ&falseF'/%'italicGQ%trueF'/%,mathvariantGQ'it alicF'-I#moGF$6.Q*≔F'F//F6Q'normalF'/%&fenceGF1/%*separatorGF1 /%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%'accent GF1/%'lspaceGQ,0.2777778emF'/%'rspaceGFN-F,6&Q*HeavisideF'F//F3F1F<-I( mfencedGF$6%-F#6%-F,6&Q#wbF'F/F2F5-F96.Q(−F'F/FF@FBFDFFFHFJ/F MQ,0.2222222emF'/FPF[o-F,6&Q\"wF'F/F2F5F/F<-F96.Q\";F'F/F/FAF4FBFDF FFHFJ/FMQ&0.0emF'FO" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*HeavisideG6$%* protectedGI(_syslibG6\"6#,&$\"+&=6*pP!\"'\"\"\"I\"wGF'!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 214 392 "Here we see another obvious advantage. On a log-log pl ot when we extrapolate the two limiting lines they intersect at the br eak point (3dB point). The is just a graphical solution of 1=w*rc. Thi s construction is an independent method of determining wb. With modern computers though it is best just to fit the function 1/sqrt(1+(w/wb)* *2) to your data. 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O ne can see the limiting behavior at low and high frequency, w/(Qw0) an d w0/(Qw) respectively. 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