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"#D$<Copyright Charles DH Williams (1995), all rights reserved.%& #h'`Please let me know if you find this workbook interesting or useful, or have any suggestions that'( #8'0would improve it. )( *+License and Warranty,-! "#)dqThis workbook "~ C0!A" was written as an educational tool and is free for educational caThis workbook "Oven Control v1.1" was written as an educational tool and is free for educational ). #i/a or individual use but it is not "public domain" -- it is copyright, and all rights are reserved.). #i)aThis workbook or its contents cannot be sold, either by itself or with any other product, without).   #e )]the prior agreement of the author (CDHW). This workbook is provided without warranty, express ) .    #l )dor implied, including but not limited to any implied warranties of merchantability and fitness for a ) .    #j 0bparticular purpose. 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#\ ??/?? d/m/yy  d\-mmm\-yy d\-mmm mmm\-yy h:mm\ am/pmh:mm:ss\ am/pmh:mm h:mm:ss d/m/yy\ h:mm0.000 0.000E+00000.0 0.00E+0 0.0E+0 ##0.0E+0mm:ss@63_-" "* #,##0_-;\-" "* #,##0_-;_-" "* "-"_-;_-@_-*'_-* #,##0_-;\-* #,##0_-;_-* "-"_-;_-@_->;_-" "* #,##0.00_-;\-" "* #,##0.00_-;_-" "* "-"??_-;_-@_-2/_-* #,##0.00_-;\-* #,##0.00_-;_-* "-"??_-;_-@_- / / / =/ +Power_values)'}#0## / / /*/ SOLUTIONS#0</ +Time_values)'#0##/Title,cC C  C  C  C  C C C C C C C C C C C  C  C  C  C  C  C  C  C  C  C  C  C !C  C !C  C  C #C !C C C &C &C !C &C #C !C &C #C !C  C  Normal_Misc Macros Normal_ModelsU } M} UM} UPM} M}  -M b  ;l@  E@ (X@ 2 @ @ @ @ @ @ } < @J @]@ @\ @`@ @\ @@  @F @V @t@ @a s @D @Y @` @h @q@ @R Z !"Oven Control v1.1"# !,$$ Charles D.H. Williams, Exeter, UK.%# ! !& Introduction'()c[This workbook contains a model of the oven-controller system described eslewhere [1,2]. The !h`parameters and results of the simulation are displayed on the "Settings" panel of the Simulation !nfsheet. Values displayed red can be adjusted by the user and are explained briefly in the next section; !methe initial values are typical of those for a domestic electric oven with a simple thermostat (onoff !k ccontrol). The Calculations area (down and to the right of the settings) lists various intermediate     !Q Iresults and determines appropriate initial conditions for the controller.     ! &Settings  ' ( ) Controller Parameters:  * + , -Type+ -#Type of controller; 0=On-Off, 1=PID    ! Proportional"Gain of the PID controller !- Derivative( Damping level for PID controller !-Integral&Reset level for PID controller !-Integral limitQIInhibits PID integral action when Power is limited by P_max; 0=off, 1=on. !-Hysteresis BandRJFor On-Off controller; the difference between turn-on and turn-off levels !Ts Ramp programme:*+,-Start Ts Initial oven temperature !- Start Time<4Time when Ts (set-point temperature) starts changing !-Stop Ts,$Final value of set-point temperature !- Stop Time<4Time when Ts has finished ramping to its final value !Oven Parameters:*+, -P_max&The maximum heater power limit ! -T_e/'The environment (i.e. room) temperature ! -R_o7/Thermal resistance between oven and environment ! -R_ho?7Thermal resistance between the heating element and oven ! -C_o!Heat capacity of the oven ! -C_h)!Heat capacity of heating elements !Calculation Parameters:*+, @! @g" # E$ @% @& @' @( @:) @2* @M+ @e, @2- @. @/ Y0 >1 @t2 @@3 @z 4 @5 @; @2@ - RMS noise0 (Approximately gaussian thermometer noise    !!- Step_DensityQ!IThe relative frequency of time-steps used to solve differential equations!! !!"&Technical Details""'"(")g#_The differential equations are solved by brute force based on a second order Runge-Kutta method### #!i$a[3], with the integral of the temperature error Intgl(T_err) found using a trapesoidal rule [3]. $$$ $!h%`Various possible optimisations, from the point of view of both speed and accuracy, have not been%%% %!d&\made to the sheet to make it easy to modify the models which are defined on the Models macro&&& &!'-sheet:''' '!(((( (!)!)T_o"=Oven(Power,T_o',T_o))) )!*9*1Power=Controller(Power,T'_err,T_err,Intgl(T_err))** *!++++ +!],UTo change the number of grid-points add or delete rows in range named SOLUTIONS which,,, ,!h-`excludes the initial and final rows of values. It may be necessary to refill SOLUTIONS from its--- -!-.%first row downwards after doing this.... .!/& References//'/(/)H0@1. CDH Williams "Temperature Controllers", Handout 19950730/1. 000 0!<142. N2F3. FB Hildebrand "Introduction to Numerical Analysis" 2nd edn, (1974)222 2!3334445.5.5.5.5.;.;.;.;.;.@@PG@2P@2W@2Z@2@.@.@.@.@.G.G.G.G.G.P.P.P.P.P.W.W.W.W.W.Z.Z.Z.Z.Z.a@a.a.a.a.a.]4% R\  ,  Helvetica1E0EPhxPo hF ~1>fN0>fN FF FFf1`0`  ` qp yy mm mF y y=y[ m=m[ mLFL LyL1`0` `U  s s s   q    p       UU QPQPQIOPIOQCIIOPCIIOQCIPCIQCIPCIqqususqspqususqs ss ss ss ss ss ss,Times (r' CONTROLLER(xR (|o (6qR (:{ho$ (xC (|o (x^C (|hhqD&H.F.H&F&D&F.pD&H.F.H&F&D&F. FF& (="Q(;ET (?Nh (HT (Lo (T (eQ-%1)P-%1)= 00 > " A Simulation  (8?[$o>^B d h㈵>"*+_%,1Tms Rmn1 Geneva1 Geneva1 Geneva1 Geneva1h Tms Rmn1Tms Rmn1 Tms Rmn1 Tms Rmn1 Symbol1 Tms Rmn&FPage &P&?'?(?)? d@  Models# Controller,# Oven,#Update,YZOven(Power,T_o,T_o_prime)Z5/Controller(Power,T_prime_err,T_err,Intgl_T_err)Z Update(Dummy)  Misc Macros#Noise,YZ Noise(RMS)V General00.00#,##0 #,##0.00 #,##0;\-#,##0#,##0;[Red]\-#,##0#,##0.00;\-#,##0.00#,##0.00;[Red]\-#,##0.00""\ #,##0;\-""\ #,##0""\ #,##0;[Red]\-""\ #,##0 ""\ #,##0.00;\-""\ #,##0.00%"""\ #,##0.00;[Red]\-""\ #,##0.000%0.00% 0.00E+00 #\ ?/? #\ ??/?? d/m/yy  d\-mmm\-yy d\-mmm mmm\-yy h:mm\ am/pmh:mm:ss\ am/pmh:mm h:mm:ss d/m/yy\ h:mm0.000 0.000E+00000.0 0.00E+0 0.0E+0 ##0.0E+0mm:ss@63_-" "* #,##0_-;\-" "* #,##0_-;_-" "* "-"_-;_-@_-*'_-* #,##0_-;\-* #,##0_-;_-* "-"_-;_-@_->;_-" "* #,##0.00_-;\-" "* #,##0.00_-;_-" "* "-"??_-;_-@_-2/_-* #,##0.00_-;\-* #,##0.00_-;_-* "-"??_-;_-@_-+/ Controller*# / /C_h, /C_o, / / /delta,# /delta_p,* / Derivative, /eta,  /h,/ Hysteresis,! /Hysteresis_Band,/Integral,/Integral_Limit,/ Intgl_T_err,/Intgl_T_err_prev,/ Intgl_T_i,' /lambda,  /mu,  /m_0, /m_1, /m_2,/ Next_Power,&/Noise#/omega_n," /omega_p,+ %/Oven0}#/PARS-&/Power,/Power_i,( =/ +Power_values)'-#L#+0W#/Previous, /- // Proportional,/P_max, // RMS_Noise,/R_ho, /R_o,/S-/ SOLUTIONS-1 / stability,, / Start_Time, /Start_Ts, / Step_Density,/ Stop_Time, /Stop_Ts, /Time, / Time_Step,</ +Time_values)'.#N#+#/Type, /T_e,/T_err,/ T_err_prev,=/ +T_err_values)'/n#P#+.# /T_i,&  /T_o,/ T_o_Noise,/ T_o_prime,;/ +T_o_values)'08#M#+.#/ T_prime_err,/T_prime_err_prev,/ T_prime_i,%  /T_s,;/ +T_s_values)''#O#+/#'/Update*s#/V-. / AL1768D46E-?/ AL1769D46E-? / AL176AD46E-1 / AL176BD472-?/ AL176CD472-? / AL176DD472-1 / AL176ED477-?/ AL176FD477-?/ AL1770D477-? / AL1771D477-?/ AL1772D477-?/ AL177314FE,// AL1773D487-?/ AL17741501,// AL1774D487-? / AL17751505,// AL1775D487-1 / AL17761505,// AL1776D48B-?/ AL1777D48B-? / AL1778D48B-1 cC C  C  C  C  C C C C C C C C C C C  C  C  C  C  C  C  C  C  C #C !C RRC RC  RRC  RC #C !C  C  RC  C  RC !IIIIC  C  RC  C !IIIIC  C !IIIIC  RC #C !C  C  RRC #RC !RC  RC  RRC  C  C &RRC &RC &RRC  RC #C !C #C !C !C  RC  RC "C &Normal_Misc Macros Normal_ModelsU } M} * M} M} M}  M} ?M} M} M}  M} *M} CM}  M #xl@ @w<"I@@@@@@ " n@ Z@ c@ @X"@l@oU@rET@jBU@L@K@K"U@EP@@,p@bx2@x @ ""@@@SettingsController parameters !# Type=~ $?>%,C5  On-Off PIDBPID& ' '# Proportional=~ (@@<)j*C5  N/A W/CBW/C& ' '# Derivative=~ (9@<)|*C5'  N/A C/sBC/s& ' '# Integral=*Mb@?;))C5i  N/A CsBCs& ' '#Integral Limit=~ (?Z%HC5 (COn Off" N/ABOn& ' '#Hysteresis Band=~ ($@:)(C5 C N/ABN/A& ' ' +Ts ramp programme: , - . ! ' '   # , Start Time=~ $Y@ %s & ' ' # , Start Ts=~ (b@ %C & ' ' ' # , Stop Time=~ ([@J %8C0C-  #?# 4 4 4 4 4 4 4 # :h=A ;6;"D@/C$CC)C( 9 4!4!4!4!4!4!4!4!# ! :D='! ;@?C  ! &C! 4"4"4"4"4"4"4"4"# " <w=H" =dm?6CQC(\CC)+A " &/s" 4#4#4#4#4#4#4#4## # <d=j# >I O@XC(CC)C~+C)6CCC(O# &# 4$4$4$4$4$4$4$4($ Initial steady-state conditions:$ $ $ !$ 4$%4%4%4%4%4%4%4%# % ,T'_i=i% ;WC5M <C6C.*C7CFB % &C/s% 4&4&4&4&4&4&4&4&# & ,T_i=& ;b@#D%2BC  C59. B$9CEC.^PC6S\Cl#.~B & &C& 4'4'4'4'4'4'4'4'?'  Intgl_T_i=e'  @SC  8C."C6*C O7CkBB ' !sC' 4'(4(4(4(4(4(4(4(#( Power_i=X(  @FC5 +C:C6t*C)7B ( &W( 4)4)4)4)4)4)4)46).Interesting numbers for PID controlled system:) ) ) !) 4)*4*4*4*4*4*4*4*#* ,delta_p=0* ;d?CzCA* 9i.e. system is CGz?  under?C`:)\(?  over critically-"Bdampedi.e. system is underdamped* 4+4+4+4+4+4+4+4+#+ ,omega_p=0+ ;oHO7?CC'A + &/s+ 4,4,4,4,4,4,4,4,#, Integral stability=o, ;׆qp@]C  =CC'C7C E N/ABp, &^C ] AC, un Bstable integral level Bstable integral level, 4-4-4-4-4-4-4-4-/- 0- @- 3- 4-4-4-4-4-4-4-4-4.4.4. 4. 4. 4. 4. 4.4.4.4.4.4.4.4.4/4/4/ 4/ 4/ 4/ 4 / ATime /AT_s /AT_err/AT'_err/A Intgl(T_err) /APower /Ah /AT_o /AT_o'3/B+Intermediate results for Runge-Kutta method/B/B/B/A T_o_Noise04040 40 40 40 4 0 A(s) 0A(C) 0A(C)0A(C/s) 0A(sC) 0A(W)0A 0A(C)0A(C/s) 0Am_0 0Am_1 0Am_2 0Am_3 0A(C)14141 41 41 41 4~ 1 1b@C2C- #.i C2 /C0: ZC. $JC1UC. 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