InputMode:=Word

PrecisionDigits:=64

NotationDigits:=64

"We want to find the resistor network values to make a linear temperature sen~
sor."

"This is a table of the resistance of the thermistor as a function of tempert~
ure Celsius."

m:=[[-55,963749],[-54,895245],[-53,832055],[-52,773734],[-51,719877],[-50,670~
115],[-49,624114],[-48,581565],[-47,542189],[-46,505731],[-45,471956],[-44,44~
0652],[-43,411622],[-42,384689],[-41,359688],[-40,336470],[-39,314896],[-38,2~
94841],[-37,276188],[-36,258833],[-35,242675],[-34,227627],[-33,213606],[-32,~
200535],[-31,188337],[-30,176970],[-29,166333],[-28,156431],[-27,147173],[-26~
,138514],[-25,130418],[-24,122824],[-23,115723],[-22,109119],[-21,102903],[-2~
0,97078.5],[-19,91618.8],[-18,86498.8],[-17,81695.6],[-16,77187.7],[-15,72955~
.4],[-14,68980.3],[-13,65245.3],[-12,61734.6],[-11,58433.6],[-10,55328.4],[-9~
,52406.5],[-8,49656],[-7,47065.9],[-6,44626.1],[-5,42326.8],[-4,40159.4],[-3,~
38115.5],[-2,36187.5],[-1,34368],[0,32650.5],[1,31032.1],[2,29499.9],[3,28052~
.4],[4,26684.6],[5,25391.2],[6,24168.2],[7,23011.2],[8,21916.3],[9,20879.8],[~
10,19898.3],[11,18968.6],[12,18087.6],[13,17252.6],[14,16460.9],[15,15710],[1~
6,14997.7],[17,14321.6],[18,13679.8],[19,13070.4],[20,12491.6],[21,11941.6],[~
22,11418.9],[23,10922],[24,10449.5],[25,10000],[26,9572.32],[27,9165.29],[28,~
8777.79],[29,8408.68],[30,8057.31],[31,7722.43],[32,7403.29],[33,7098.42],[34~
,6808.36],[35,6531.31],[36,6265.75],[37,6016.47],[38,5776.05],[39,5546.53],[4~
0,5327.34],[41,5117.97],[42,4917.94],[43,4726.77],[44,4543.91],[45,4369.33],[~
46,4200.84],[47,4040.81],[48,3889.51],[49,3743.17],[50,3603.1],[51,3467.69],[~
52,3340.55],[53,3217.54],[54,3099.62],[55,2986.6],[56,2878.49],[57,2774.55],[~
58,2675.2],[59,2579.52],[60,2488.2],[61,2400.36],[62,2316.05],[63,2235.13],[6~
4,2157.43],[65,2082.84],[66,2011.18],[67,1942.35],[68,1876.21],[69,1812.65],[~
70,1751.65],[71,1692.82],[72,1636.33],[73,1582.02],[74,1529.77],[75,1479.3],[~
76,1430.9],[77,1384.28],[78,1339.73],[79,1296.74],[80,1255.08],[81,1215.15],[~
82,1176.76],[83,1139.77],[84,1103.96],[85,1069.79],[86,1036.21],[87,1004.2],[~
88,973.67],[89,943.97],[90,915.32],[91,887.69],[92,861.02],[93,835.29],[94,81~
0.46],[95,786.54],[96,763.35],[97,741],[98,719.41],[99,698.57],[100,678.42],[~
101,658.96],[102,640.17],[103,621.97],[104,604.39],[105,587.39]]

"A good model of the resistance of a thermistor is:"

r SUB t=EXP(a SUB 0+a SUB 1/t+a SUB 2/t^2+a SUB 3/t^3)

"t is temperature Kelvin."

"The data in the table needs to be transformed so we can fit it to the model.~
"

"Transform resistance to y and temperature to x."

rt=EXP(a0+a1/t+a2/t^2+a3/t^3)

;Simp(User)
LN(rt)=a0+a1/t+a2/t^2+a3/t^3

t=c+273.15

x=1/t

y=LN(rt)

y=a0+a1*x+a2*x^2+a3*x^3

"Transform m to m1."

m1:=VECTOR([1/(x SUB 1+273.15),LN(x SUB 2)],x,m)

"Fit the transformed data to the model."

f(x):=FIT([x,a0+a1*x+a2*x^2+a3*x^3],m1)

;Approx(#21)
f(x):=-6.976689289502469431223611098003650569587269516127177141268114716*10^6~
*x^3-7.202847605003461292517331272822515126389694064366024764873334414*10^4*x~
^2+4617.915302731585665587490621743690648498587109392037552533546245*x-5.2046~
93178344775286271963872038032912712164209568225962458452448

"Transform back to the original data."

f1(t):=EXP(f(1/(t+273.15)))

"Check to error of the fit multiplied by 10000 so it is easy to see."

VECTOR([x SUB 1,10000*(x SUB 2/f1(x SUB 1)-1)],x,m)

"Pick four equally spaced points so the TI-83 can fit this same model."

VECTOR(t,t,-55,105,(105+55)/8)

;Approx(#28)
[-55,-35,-15,5,25,45,65,85,105]

[-35,5,45,85]

VECTOR([t,f1(t)],t,[-35,5,45,85])

"This is the table of values."

;Approx(#31)
[[-35,2.426949769066949173939958476578048893167428568242522979509575514*10^5]~
,[5,2.539641226908252205782705840511663329021866186316892670993351458*10^4],[~
45,4367.52464901177432161355724344092364985809611068873493492617636],[85,1070~
.320602157991357022319618388484426066797519486255114581700935]]

"Fit these values to see if they will give the same model when rounded to 8 p~
laces."

m2:=APPROX(VECTOR([1/(t+273.15),LN(f1(t))],t,[-35,5,45,85]),8)

;Simp(#35')
m2:=[[0.0041990342,12.39956],[0.0035951824,10.142363],[0.0031431714,8.3819517~
],[0.0027921262,6.9757135]]

f2(x):=FIT([x,a0+a1*x+a2*x^2+a3*x^3],m2)

;Approx(#37)
f2(x):=-6.975971637427425099799276082395907998654540859869827626540463013*10^~
6*x^3-7.203682981032733062062530061457705336739435223637103415285912215*10^4*~
x^2+4617.946602239741789891179733155987294239692031762648048908951392*x-5.204~
731060364272349755352053758814589203954962567461791503193492

f3(t):=EXP(f2(1/(t+273.15)))

"Check to error of the fit multiplied by 10000000 so it is easy to see."

VECTOR([x SUB 1,10000000*(f1(x SUB 1)/f3(x SUB 1)-1)],x,m)

"Construct a matrix to solve for the fit."

m3:=VECTOR(APPEND(VECTOR(x SUB 1^i,i,0,3),[x SUB 2]),x,m2)

;Approx(#43)
m3:=[[1,0.0041990342,1.763188821276964*10^(-5),7.4036901615996595081688*10^(-~
8),12.39956],[1,0.0035951824,1.292533648926976*10^(-5),4.64689422603004300042~
24*10^(-8),10.142363],[1,0.0031431714,9.87952644977796*10^(-6),3.105304498248~
5620222344*10^(-8),8.3819517],[1,0.0027921262,7.79596871672644*10^(-6),2.1767~
328508352271356728*10^(-8),6.9757135]]

"Solve for the model."

ROW_REDUCE(m3)

;Approx(#46)
[[1,0,0,0,-5.204731060364272349755352053758814589203954962567461791503193492]~
,[0,1,0,0,4617.946602239741789891179733155987294239692031762648048908951392],~
[0,0,1,0,-7.203682981032733062062530061457705336739435223637103415285912215*1~
0^4],[0,0,0,1,-6.975971637427425099799276082395907998654540859869827626540463~
013*10^6]]

"Find three equally spaced points for the linear temperature sensor."

VECTOR(t,t,0,100,(100+0)/6)

;Simp(#49)
[0,50/3,100/3,50,200/3,250/3,100]

;Approx(#50)
[0,16.66666666666666666666666666666666666666666666666666666666666666,33.33333~
333333333333333333333333333333333333333333333333333333333,50,66.6666666666666~
6666666666666666666666666666666666666666666666666,83.333333333333333333333333~
33333333333333333333333333333333333333,100]

"These are the points."

[17,50,83]

"Pick two points on the desired output of the linear sensor."

[[0,0.3],[100,0.7]]

"Solve for the line that passes through these points."

f4(t):=FIT([t,a0+a1*t],[[0,0.3],[100,0.7]])

;Simp(#57)
f4(t):=t/250+3/10

"Find the voltage output and thermistor resistance at the three tempreatures.~
"

APPROX(VECTOR([t,f4(t),f3(t)],t,[17,50,83]),8)

;Simp(#60)
[[17,0.368,1.4324521*10^4],[50,0.5,3601.5491],[83,0.632,1140.5009]]

"This is the resistor network which is the temperature sensor."

"Solve for the voltage."

v/rc+(v-va)/(rb+1/(1/ra+1/r))=0

;Simp(User)
v/va=rc*(r+ra)/(r*(ra+rb+rc)+ra*(rb+rc))

v=rc*(rt+ra)/(rt*(ra+rb+rc)+ra*(rb+rc))

;Simp(User)
1/v=(ra*(rb+rc+rt)+rt*(rb+rc))/(rc*(ra+rt))

;Expd(#67)
1/v=-rt^2/(rc*(ra+rt))+rb/rc+rt/rc+1

"Substitute the three voltages and the three resistances."

;Sub(#68)
1/v1=-r1^2/(rc*(ra+r1))+rb/rc+r1/rc+1

;Sub(#70)
1/v2=-r2^2/(rc*(ra+r2))+rb/rc+r2/rc+1

;Sub(#71)
1/v3=-r3^2/(rc*(ra+r3))+rb/rc+r3/rc+1

"Eliminate rb by subtraction."

;Simp(User)
1/v2-1/v1=ra^2/(rc*(r1+ra))-ra^2/(rc*(r2+ra))

;Simp(User)
1/v3-1/v2=ra^2/(rc*(r2+ra))-ra^2/(rc*(r3+ra))

;Fctr(#74)
(v2-v1)/(v1*v2)=ra^2*(r1-r2)/(rc*(r1+ra)*(r2+ra))

;Fctr(#75)
(v3-v2)/(v2*v3)=ra^2*(r2-r3)/(rc*(r2+ra)*(r3+ra))

"Eliminate rc by division."

;Simp(User)
v1*(v2-v3)/(v3*(v1-v2))=(r1+ra)*(r2-r3)/((r1-r2)*(r3+ra))

"Solve for ra."

;Expd(User')
ra=(r1*r2*v3*(v1-v2)+r1*r3*v2*(v3-v1)+r2*r3*v1*(v2-v3))/(r1*v1*(v2-v3)+r2*v2*~
(v3-v1)+r3*v3*(v1-v2))

"Solve for rc."

;Simp(Solve(#76,rc))
rc=ra^2*v1*v2*(r1-r2)/((r1+ra)*(r2+ra)*(v2-v1))

"Solve for rb."

;Simp(Solve(#70,rb))
rb=-(r1*(ra*v1+rc*(v1-1))+ra*rc*(v1-1))/(v1*(r1+ra))

;Expd(#85)
rb=ra^2/(r1+ra)-ra+rc/v1-rc

;Simp(#86)
rb=ra^2/(r1+ra)-ra+rc*(1/v1-1)

"Substitute the three values of voltage and resistance."

[[17,0.368,1.4324521*10^4],[50,0.5,3601.5491],[83,0.632,1140.5009]]

"Find ra."

[ra=(r1*r2*v3*(v1-v2)+r1*r3*v2*(v3-v1)+r2*r3*v1*(v2-v3))/(r1*v1*(v2-v3)+r2*v2~
*(v3-v1)+r3*v3*(v1-v2)),rc=ra^2*v1*v2*(r1-r2)/((r1+ra)*(r2+ra)*(v2-v1)),rb=ra~
^2/(r1+ra)-ra+rc*(1/v1-1)]

;Simp(User)
[ra=111122693724466357/14941694980000,rc=4932567074000*ra^2/(33*(1000*ra+1432~
4521)*(10000*ra+36015491)),rb=205191901879441/(1000*(1000*ra+14324521))+79*rc~
/46-14324521/1000]

"Find rc."

;Simp(User)
[true,rc=12348253060581554680330304424851449/3587955974340583920382254375000,~
rb=79*rc/46-1591779359832686544639997/325155317241070937000]

"Find rb."

;Simp(Sub(#94))
[true,true,rb=7284296100926393733236159562471227/7175911948681167840764508750~
000]

APPROX([ra=111122693724466357/14941694980000,rc=12348253060581554680330304424~
851449/3587955974340583920382254375000,rb=7284296100926393733236159562471227/~
7175911948681167840764508750000],8)

[ra=7437.0875,rc=3441.5843,rb=1015.1038,rt=f3(t)]

"Calculate the voltage v."

v=rc*(rt+ra)/(rt*(ra+rb+rc)+ra*(rb+rc))

;Sub(#100)
v=3441.5843*(f3(t)+7437.0875)/(f3(t)*(7437.0875+1015.1038+3441.5843)+7437.087~
5*(1015.1038+3441.5843))

"Compare v to the desired voltage."

f4(t)

;Simp(#103)
t/250+3/10