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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 113 " Solution to Phase-Shift \+
Oscillator \"Introduction to Electronics for Scientists and Engineers
\" ,Simpson, pg 460. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "as
sume(i1,complex);assume(i2,complex);assume(i3,complex);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "assume(v0,real);assume(v0>0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "assume(Zc,complex);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "assume(Zr,complex);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "assume(R,real);assume(R>0);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "assume(C,real);assume(C
>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "assume(omega,real);
assume(omega>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Zr:=R;Z
c:=1/(I*omega*C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZrG%#R|irG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZcG*&^#!\"\"\"\"\"*&%'omega|irGF(%#
C|irGF(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eq1:=-v0+Zc*i1
+Zr*(i1-i2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G,(%$v0|irG!\"\"
*&*&^#F'\"\"\"%$i1|irGF+F+*&%'omega|irGF+%#C|irGF+F'F+*&%#R|irGF+,&F,F
+%$i2|irGF'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eq2:=Zr*
(i2-i1)+i2*Zc+Zr*(i2-i3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G,(
*&%#R|irG\"\"\",&%$i2|irGF(%$i1|irG!\"\"F(F(*&*&^#F,F(F*F(F(*&%'omega|
irGF(%#C|irGF(F,F(*&F'F(,&F*F(*&**)F'\"\"#F(%$v0|irGF()F1\"\"$F()F2F;F
(F(,**()F'F;F(F:F(F<F(F(**^#!\"'F(F7F()F1F8F()F2F8F(F(**\"\"&F(F'F(F1F
(F2F(F,^#F(F(F,F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eq
3:=Zr*(i3-i2)+Zc*i3+Zr*i3;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eq3G,
(*&%#R|irG\"\"\",&*&**)F'\"\"#F(%$v0|irGF()%'omega|irG\"\"$F()%#C|irGF
1F(F(,**()F'F1F(F/F(F2F(F(**^#!\"'F(F,F()F0F-F()F3F-F(F(**\"\"&F(F'F(F
0F(F3F(!\"\"^#F(F(F>F(%$i2|irGF>F(F(*&*,^#F>F(F,F(F.F(F/F(F2F(F(*(F0F(
F3F(F4F(F>F(*&*,F'F(F,F(F.F(F/F(F2F(F(F4F>F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "solve(\{eq1,eq2,eq3\},\{i1,i2,i3\});" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#<%/%$i1|irG*&*(%#C|irG\"\"\"%'omega|irGF),@*
*F(F)F*F)%$v0|irGF)%#R|irGF)!\"#*0)F.\"\"$F))F*\"\"#F))F(F4F))F.F4F)F-
F))F*F2F))F(F2F)F)*2\"\"&F))F.F4F)F3F)F5F)F.F)F*F)F(F)F-F)F)*0F;F)F3F)
F5F))F.F2F)F7F)F8F)F-F)!\"\"*2^#\"\"'F)F;F)F3F)F5F)F6F))F*F4F))F(F4F)F
-F)F)*,^#F>F)F;F)F3F)F5F)F-F)F)*2\"#7F)F-F)F(F)F*F)F.F)F6F)FBF)FCF)F)*
,F6F)F-F)F7F)F8F)F.F)F>*2^#F/F)F;F)F*F)F(F)F6F)F-F)F7F)F8F)F)*2^#!#5F)
F.F)F*F)F(F)F.F)F*F)F(F)F-F)F)*2^#F4F)F.F)F*F)F(F)F=F)F7F)F8F)F-F)F)*,
F:F)F.F)F*F)F(F)F-F)F>**F=F)F7F)F8F)F-F)F)*,^#!\"'F)F6F)FBF)FCF)F-F)F)
*&^#F)F)F-F)F)F)F)*&,2*(F.F)F*F)F(F)F/*0FOF)F=F)F7F)F8F)F.F)F*F)F(F)F)
*0FGF)F.F)F*F)F(F)F6F)FBF)FCF)F)*0FLF)F.F)F*F)F(F)F.F)F*F)F(F)F)*(F=F)
F7F)F8F)F)**F:F)F.F)F*F)F(F)F>FVF)**FSF)F6F)FBF)FCF)F)F),&FEF)FYF)F)F>
/*&**F6F)F-F)F7F)F8F)F),*FgnF)FinF)**F:F)F.F)F*F)F(F)F>FVF)F>*&,<**F-F
))F(F2F))F*F2F)F;F)F>*0)F.\"\"%F))F*FgoF))F(FgoF)F6F)F-F)F7F)F8F)F)*2F
:F)F1F)FhoF)FioF)F.F)F*F)F(F)F-F)F)*0F1F)FhoF)FioF)F=F)F7F)F8F)F-F)F>*
2F@F)F1F)FhoF)FioF)F6F)FBF)FCF)F-F)F)*,FEF)F1F)FhoF)FioF)F-F)F)*2FAF)F
-F)FcoF)FdoF)F;F)F6F)FBF)FCF)F)*2\"\")F)F;F)F6F)F-F)F7F)F8F)F3F)F5F)F>
*2^#!\"&F)F1F)FdoF)FcoF)F6F)F-F)F7F)F8F)F)*2FbpF)F;F)FdoF)FcoF)F.F)F*F
)F(F)F-F)F)*2FVF)F;F)FdoF)FcoF)F=F)F7F)F8F)F-F)F)*2^#F:F)F6F)F-F)F7F)F
8F)F.F)F*F)F(F)F)**F6F)F-F)F7F)F8F)F)F)*,FXF)F.F)F*F)F(F)FjnF)F>/%$i2|
irG*&,0*2FJF)F6F)F-F)F7F)F8F)F.F)F*F)F(F)F)*0F;F)F6F)F-F)F7F)F8F)F3F)F
5F)F)FhpF>*2F:F)F.F)F*F)F(F)F-F)F5F)F3F)F.F)F)*0F=F)F7F)F8F)F-F)F5F)F3
F)F.F)F>*2F@F)F6F)FBF)FCF)F-F)F5F)F3F)F.F)F)*,FEF)F-F)F5F)F3F)F.F)F)F)
FXF>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "i3:=simplify(evalc(
R^2*v0/(R^3+6*Zc*R^2+5*R*Zc^2+Zc^3)));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#i3G*&*,)%#R|irG\"\"#\"\"\"%$v0|irGF*)%'omega|irG\"\"$F*)%#C|i
rGF.F*,**()F(F.F*F,F*F/F*F***\"\"&F*F(F*F-F*F0F*!\"\"**^#\"\"'F*F'F*)F
-F)F*)F0F)F*F*^#F6F*F*F*,**()F(F9F*)F-F9F*)F0F9F*F***\"#EF*)F(\"\"%F*)
F-FEF*)F0FEF*F***\"#8F*F'F*F:F*F;F*F*F*F*F6" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 53 "assume(X,real);assume(Xreal,real);assume(Ximag,rea
l);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Xreal:=Re(i3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&XrealG*&*,%$v0|irG\"\"\")%#C|irG\"
\"$F()%'omega|irGF+F()%#R|irG\"\"#F(,&*()F/F+F(F,F(F)F(F(**\"\"&F(F/F(
F-F(F*F(!\"\"F(F(,**()F/\"\"'F()F-F:F()F*F:F(F(**\"#EF()F/\"\"%F()F-F@
F()F*F@F(F(**\"#8F(F.F()F-F0F()F*F0F(F(F(F(F6" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "Ximag:=Im(i3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%&XimagG*&*,%$v0|irG\"\"\")%#C|irG\"\"$F()%'omega|irGF+F()%#R|irG\"
\"#F(,&*(F.F()F-F0F()F*F0F(\"\"'F(!\"\"F(F(,**()F/F5F()F-F5F()F*F5F(F(
**\"#EF()F/\"\"%F()F-F?F()F*F?F(F(**\"#8F(F.F(F3F(F4F(F(F(F(F6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "X:=simplify(sqrt(Xreal**2+Xi
mag**2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG*&**%$v0|irG\"\"\")
%#C|irG\"\"$F()%'omega|irGF+F()%#R|irG\"\"#F(F(*$-%%sqrtG6#,**()F/\"\"
'F()F-F8F()F*F8F(F(**\"#EF()F/\"\"%F()F-F>F()F*F>F(F(**\"#8F(F.F()F-F0
F()F*F0F(F(F(F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s
olve(Ximag,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"!F#F#,$*&*$-
%%sqrtG6#\"\"'\"\"\"F+*&%#C|irGF+%#R|irGF+!\"\"#F+F*,$F%#F/F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "X;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&**%$v0|irG\"\"\")%#C|irG\"\"$F&)%'omega|irGF)F&)%#R|i
rG\"\"#F&F&*$-%%sqrtG6#,**()F-\"\"'F&)F+F6F&)F(F6F&F&**\"#EF&)F-\"\"%F
&)F+F<F&)F(F<F&F&**\"#8F&F,F&)F+F.F&)F(F.F&F&F&F&F&!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Gain:=w**3*c**3*r**3/sqrt(r**6*w**6
*c**6+26*r**4*w**4*c**4+13*r**2*w**2*c**2+1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%GainG#\"\"\"\"#H" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "w:=1/(sqrt(6)*r*c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"wG,$*&*$-%%sqrtG6#\"\"'\"\"\"F,*&%\"rGF,%\"cGF,!\"\"#F,F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Gain;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"\"\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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