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equations.cc
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// ****************************************************************************
// equations.cc DB48X project
// ****************************************************************************
//
// File Description:
//
// Implementation of the equations library
//
//
//
//
//
//
//
//
// ****************************************************************************
// (C) 2024 Christophe de Dinechin <[email protected]>
// This software is licensed under the terms outlined in LICENSE.txt
// ****************************************************************************
// This file is part of DB48X.
//
// DB48X is free software: you can redistribute it and/or modify
// it under the terms outlined in the LICENSE.txt file
//
// DB48X is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
// ****************************************************************************
#include "equations.h"
#include "expression.h"
#include "grob.h"
#include "parser.h"
#include "renderer.h"
#include "solve.h"
#include "tag.h"
#include "variables.h"
#include <algorithm>
RECORDER(equations, 16, "Equation objects");
RECORDER(equations_error, 16, "Error on equation objects");
// ============================================================================
//
// Equation definitions
//
// ============================================================================
static const cstring basic_equations[] =
// ----------------------------------------------------------------------------
// List of basic equations
// ----------------------------------------------------------------------------
// clang-format off
{
// ------------------------------------------------------------------------
"Columns and Beams", nullptr,
// ------------------------------------------------------------------------
//T#*: 21 vars 14 eqns 10 sims 10 secs
"Elastic Buckling", "{ "
" '(Pcr_kN)=(Ⓒπ²*(E_kPa)*(A_cm²))/sq((K*(L_m))/(r_cm))' "
" '(Pcr_kN)=(Ⓒπ²*(E_kPa)*(I_mm^4))/sq(K*(L_m))' "
" '(σcr_kPa)=(Pcr_kN)/(A_cm²)' "
" '(r_cm)²=(I_mm^4)/(A_cm²)' "
"}",
"Eccentric Columns", "{"
" '(σmax_kPa)=((P_kN)/(A_cm²))*(1+((ε_cm)*(c_cm))/sq(r_cm)*inv(cos(K/2*((L_m)/(r_cm))*sqrt((P_kN)/((E_kPa)*(A_cm²)))*1_r)))'"
" '(r_cm)²=(I_mm^4)/(A_cm²)'"
"}",
"Simple Deflection", "{"
" '(y_cm)=((P_kN)*((L_m)-(a_m)))*(x_m)/(6*(L_m)*(E_kPa)*(I_mm^4))*(x²+(L-a)²-L²)-((M_N*m)*x)/(E*I)*((c_m)-x²/(6*L)-L/3-c²/(2*L))-((w_N/m)*x)/(24*E*I)*(L³+x²*(x-2*L))'"
"}",
"Simple Slope", "{"
" '(θ_°)=(1_r)*(((P_N)*((L_m)-(a_m)))/(6*L*(E_kPa)*(I_mm^4))*(3*(x_m)²+(L-a)²-L²)-(M_N*m)/(E*I)*(c-(x²+c²)/(2*L)-L/3)-(w_N/m)/(24*E*I)*(L³+x²*(4*x-6*L)))'"
"}",
"Simple Moment", "{"
" '(Mx_N*m)=(P_N)*IFTE((x_m)≤(a_m);((L_m)-(a_m))*(x_m);(L-x)*a)/L+(M_N*m)*IFTE(x≤(c_m);x;x-L)/L+((w_N/m)*x*(L-x))/2'"
"}",
"Simple Shear", "{"
" '(V_N)=((P_N)*((L_m)-(a_m)))/L+(M_N*m)/L+((w_N/m)*(L-2*(x_m)))/2'"
"}",
"Cantilever Deflection", "{"
" '(y_m)=((P_N)*IFTE((x_m)≤(a_m);x;a)²)/(6*(E_kPa)*(I_mm^4))*IFTE(x≤a;x-3*a;a-3*x)+(M_N*m)*IFTE(x≤(c_m);x²;c*(2*x-c))/(2*E*I)-((w_N/m)*x²)/(24*E*I)*(6*L²-4*L*x+x²)'"
"}",
"Cantilever Slope", "{"
" '(θ_°)=(1_r)*((P_N)*IFTE((x_m)≤(a_m);x*(x-2*a);-a²)/(2*(E_kPa)*(I_mm^4))+(M_N*m)*IFTE(x≤(c_m);x;c)/(E*I)-((w_N/m)*x)/(6*E*I)*(3*L²-3*L*x+x²))'"
"}",
"Cantilever Moment", "{"
" '(Mx_N*m)=IFTE((x_m)≤(a_m);1;0)*(P_N)*((x_m)-(a_m))+IFTE(x≤(c_m);1;0)*(M_N*m)-(w_N/m)/2*(L²-2*L*x+x²)'"
"}",
"Cantilever Shear", "{"
" '(V_N)=IFTE((x_m)≤(a_m);1;0)*(P_N)+(w_N/m)*((L_m)-(x_m))'"
"}",
// ------------------------------------------------------------------------
"Electricity", nullptr,
// ------------------------------------------------------------------------
//T#*: 78 vars 67 eqns 32 sims 30 secs
"Coulomb’s Law & E Field", "{ "
" '(F_N)=1/(4*Ⓒπ*Ⓒε₀*εr)*((q1_C)*(q2_C)/(r_m)²)' "
" '(Er_(N/C))=(F_N)/(qtest_C)' "
"}",
"E Field Infinite Line", "{ "
" '(Er_(N/C))=1/(2*Ⓒπ*Ⓒε₀*εr)*((λ_(C/m))/(r_m))' "
" '(λ_(C/m))=(Q_C)/(L_m)' "
"}",
"E Field Finite Line", "{ "
" '(Er_(N/C))=1/(4*Ⓒπ*Ⓒε₀*εr)*((λ_(C/m))/(r_m)*(SIN(θ₁_r)-SIN(θ₂_r)))' "
" '(λ_(C/m))=(Q_C)/(L_m)' "
"}",
"E Field Infinite Plate", "{ "
" '(Ep_(N/C))=(σ_(μC/cm²))/(2*Ⓒε₀*εr)' "
" '(σ_(μC/cm²))=(Q_μC)/(A_(cm²))' "
"}",
"Ohm’s Law & Power", "{ "
" '(V_V)=(I_A)*(R_Ω)' "
" '(P_W)=(V_V)*(I_A)' "
" '(P_W)=(I_A)²*(R_Ω)' "
" '(P_W)=(V_V)²/(R_Ω)' "
"}",
"Volt Divider", "{ "
" '(V1_V)=(V_V)*((R1_Ω)/((R1_Ω)+(R2_Ω)))' "
"}",
"Current Divider", "{ "
" '(I1_A)=(I_A)*((R2_Ω)/((R1_Ω)+(R2_Ω)))' "
"}",
"Wire Resistance", "{ "
" '(R_Ω)=(ρ_(Ω*m))*(L_m)/(A_(m²))' "
"}",
"Resistivity & Conductivity", "{ "
" '(ρ_(Ω*m))=(ρ0_(Ω*m))*(1+UBASE(αT_K^-1)*(UBASE(T_K)-UBASE(T0_K)))' "
" '(σ_(S/m))=1/(ρ_(Ω*m))' "
"}",
"Series & Parallel R", "{ "
" '(Rs_Ω)=(R1_Ω)+(R2_Ω)'"
" '1/(Rp_Ω)=1/(R1_Ω)+1/(R2_Ω)' "
"}",
"Series & Parallel C", "{ "
" '1/(Cs_μF)=1/(C1_μF)+1/(C2_μF)' "
" '(Cp_μF)=(C1_μF)+(C2_μF)' "
"}",
"Series & Parallel L", "{ "
" '(Ls_mH)=(L1_mH)+(L2_mH)' "
" '1/(Lp_mH)=1/(L1_mH)+1/(L2_mH)' "
"}",
"Capacitive Energy", "{ "
" '(E_J)=(1/2)*(C_μF)*(V_V)²' "
" '(E_J)=(1/2)*(q_μC)*(V_V)' "
" '(E_J)=(q_μC)²/(2*(C_μF))' "
"}",
"Volumic Density Electric Energy", "{ "
" '(uE_(J/m^3))=(1/2)*Ⓒε₀*εr*(E_(V/m))²' "
"}",
"Inductive Energy", "{ "
" '(E_J)=(1/2)*(L_mH)*(I_A)²' "
"}",
"RLC Current Delay", "{ "
" 'TAN(φs_°)=((XL_Ω)-(XC_Ω))/(R_Ω)' "
" 'TAN(φp_°)=(1/(XC_Ω)-1/(XL_Ω))*(R_Ω)' "
" '(XC_Ω)=1/(((ω_(r/s))/(1_r))*(C_μF))' "
" '(XL_Ω)=(ω_(r/s))/(1_r)*(L_mH)' "
" '(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"}",
"DC Capacitor Current", "{ "
" '(I_A)=(C_μF)*((ΔV_V)/(Δt_s))' "
" '(ΔV_V)=(Vf_V)-(Vi_V)' "
" '(Δt_μs)=(tf_μs)-(ti_μs)' "
"}",
"Capacitor Charge", "{ "
" '(q_C)=(C_μF)*(V_V)' "
"}",
"DC Inductor Voltage", "{ "
" '(V_V)=-(L_mH)*((ΔIL_A)/(Δt_μs))' "
" '(ΔIL_A)=(ILf_A)-(ILi_A)' "
" '(Δt_μs)=(tf_μs)-(ti_μs)' "
"}",
"RC Transient", "{ "
" '(V_V)=(Vf_V)-((Vf_V)-(Vi_V))*EXP((-(t_ms))/((R_Ω)*(C_F)))' "
"}",
// Error in the placing of (R_Ω) that should be in the numerator
"RL Transient", "{ "
// " '(I_A)=1/(R_Ω)*((Vf_V)-((Vf_V)-(Vi_V))*EXP((-(t_μs))/((R_Ω)*(L_mH))))' "
" '(I_A)=1/(R_Ω)*((Vf_V)-((Vf_V)-(Vi_V))*EXP((-(t_μs)*(R_Ω))/(L_mH)))' "
"}",
// Modif of radian in eqn (4)
"Resonant Frequency", "{ "
" 'Qs=1/(R_Ω)*√((L_mH)/(C_μF))' "
" 'Qp=(R_Ω)*√((C_μF)/(L_mH))' "
" '(ω₀_(r/s))=2*(Ⓒπ_r)*(f0_Hz)' "
" '(ω₀_(r/s))=(1_r)/√((L_mH)*(C_μF))' "
"}",
"Plate Capacitor", "{ "
" '(C_μF)=Ⓒε₀*εr*(A_(cm²))/(d_cm)' "
" '(ΔV_V)=(Ein_(V/m))*(d_cm)' "
" '(Ein_(N/C))=(σ_(μC/cm²))/(Ⓒε₀*εr)' "
" '(σ_(μC/m²))=(Q_μC)/(A_(cm²))' "
"}",
"Cylindrical Capacitor", "{ "
" '(C_μF)=2*Ⓒπ*Ⓒε₀*εr*(L_cm)/(LN((ro_cm)/(ri_cm)))' "
" '(ΔV_V)=(Q_μC)*(LN((ro_cm)/(ri_cm)))/(2*Ⓒπ*Ⓒε₀*εr*(L_cm))' "
"}",
"Solenoid Inductance", "{ "
" '(L_mH)=Ⓒμ₀*μr*(n_cm^-1)²*(A_(cm²))*(h_cm)' "
"}",
"Toroid Inductance", "{ "
" '(L_mH)=Ⓒμ₀*μr*N²*(h_cm)/(2*Ⓒπ)*LN((ro_cm)/(ri_cm))' "
"}",
"Sinusoidal Voltage", "{ "
" '(V_V)=(Vmax_V)*SIN((ω_(r/s))*(t_μs)+(φ_°))' "
" '(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"}",
"Sinusoidal Current", "{ "
" '(I_A)=(Imax_A)*SIN((ω_(r/s))*(t_s)+(φ_°))' "
" '(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"}",
// Example of the following in
// https://en.wikipedia.org/wiki/Drift_velocity#Numerical_example
"Drift Speed & Current Density", "{ "
" '(vd_(m/s))=(I_A)/((n_(m^-3))*Ⓒqe*(A_(cm²)))' "
" '(J_(A/m²))=(vd_(m/s))*(ρ_(C/m^3))' "
" '(J_(A/m²))=(σ_(S/m))*(E_(V/m))' "
"}",
// Example of the following in
// https://en.wikipedia.org/wiki/Electron_mobility#Examples
// In Ge typical values can be μe=500_(cm²/(V*s)) μh=200_(cm²/(V*s)) meeff_kg='0.12*Ⓒme'
// mheff_kg='0.5*Ⓒme' ne=1.04e19_(cm^-3) nh=6.0e18_(m^-3) E=6.0e-9_V/m
"Electron & Hole Mobilities", "{ "
" '(J_(A/m²))=(Je_(A/m²))+(Jh_(A/m²))' "
" '(Je_(A/m²))=Ⓒqe*(ne_(m^-3))*(μe_(cm²/(V*s)))*(E_(V/m))' "
" '(Jh_(A/m²))=Ⓒqe*(nh_(m^-3))*(μh_(cm²/(V*s)))*(E_(V/m))' "
" '(μe_(cm²/(V*s)))=Ⓒqe*(τc_s)/(meeff_kg)' "
" '(μh_(cm²/(V*s)))=Ⓒqe*(τc_s)/(mheff_kg)' "
" '(σ_(S/m))=Ⓒqe*((μe_(cm²/(V*s)))*(ne_(m^-3))+(μh_(cm²/(V*s)))*(nh_(m^-3)))' "
"}",
// ------------------------------------------------------------------------
"Fluids", nullptr,
// ------------------------------------------------------------------------
//T#*: 30 vars 30 eqns 4 sims 4 secs in Fluids
"Pressure at Depth", "{ "
" '(P_kPa)=(P0_atm)+(ρ_(kg/m^3))*Ⓒg*(h_m)' "
"}",
// WARNING Errors in HP50G_AUR.pdf for eqns 7 & 8
// ("*" instead of"-") but not in HP50G
// WARNING doesn't work on the native HP50G
"Bernoulli Equation", "{ "
" '(ΔP_Pa)/(ρ_(kg/m^3))+((v2_(m/s))²-(v1_(m/s))²)/2+Ⓒg*(Δy_m)' "
" '(ΔP_Pa)/(ρ_(kg/m^3))+((v2_(m/s))²)*(1-((A2_(m²))/(A1_(m²)))²)/2+Ⓒg*(Δy_m)' "
" '(ΔP_Pa)/(ρ_(kg/m^3))+((v1_(m/s))²)*(((A1_(m²))/(A2_(m²)))²-1)/2+Ⓒg*(Δy_m)' "
" '(ΔP_Pa)=(P2_Pa)-(P1_Pa)' "
" '(Δy_m)=(y2_m)-(y1_m)' "
" '(M_(kg/s))=(ρ_(kg/m^3))*(Q_(m^3/s))' "
" '(Q_(m^3/s))=(A2_(m²))*(v2_(m/s))' "
" '(Q_(m^3/s))=(A1_(m²))*(v1_(m/s))' "
" '(A1_(m²))=(Ⓒπ*(D1_m)²)/4' "
" '(A2_(m²))=(Ⓒπ*(D2_m)²)/4' "
"}",
// WARNING doesn't work on the native HP50G
"Flow with Losses", "{ "
" '(M_(kg/s))*((ΔP_Pa)/(ρ_(kg/m^3))+((v2_(m/s))²-(v1_(m/s))²)/2+Ⓒg*(Δy_m)+(hL_((m/s)²)))=(W_W)' "
" '(M_(kg/s))*((ΔP_Pa)/(ρ_(kg/m^3))+(((v2_(m/s))²)*(1-((A2_(m²))/(A1_(m²)))²))/2+Ⓒg*(Δy_m)+(hL_((m/s)²)))=(W_W)' "
" '(M_(kg/s))*((ΔP_Pa)/(ρ_(kg/m^3))+(((v1_(m/s))²)*(((A1_(m²))/(A2_(m²)))²-1))/2+Ⓒg*(Δy_m)+(hL_((m/s)²)))=(W_W)' "
" '(ΔP_Pa)=(P2_Pa)-(P1_Pa)' "
" '(Δy_m)=(y2_m)-(y1_m)' "
" '(M_(kg/s))=(ρ_(kg/m^3))*(Q_(m^3/s))' "
" '(Q_(m^3/s))=(A2_(m²))*(v2_(m/s))' "
" '(Q_(m^3/s))=(A1_(m²))*(v1_(m/s))' "
" '(A1_(m²))=(Ⓒπ*(D1_m)²)/4' "
" '(A2_(m²))=(Ⓒπ*(D2_m)²)/4' "
"}",
// WARNING: Missing or undefined parameter "f" in eqn 1
// WARNING doesn't work on the native HP50G
// f represents probably the Fanning friction factor which is near 0 with
// the high RE number of the example the absolute roughness coefficient ϵ_in
// is defined as a variable in both the HP50G and the manual but here it is
// absent of all actual eqns
//24-11-12 I replace the call for FANNING by the corresponding calculation in eqn (2). Error in last eqn
"Flow In Full Pipes", "{ "
" '(ρ_(kg/m^3))*((Ⓒπ*(D_m)²)/4)*(vavg_(m/s))*((ΔP_Pa)/(ρ_(kg/m^3))+Ⓒg*(Δy_m)+(vavg_(m/s))²*(2*f*((L_m)/(D_m))+ΣK/2))=(W_W)' "
//" 'f=FANNING((ε_m)/(D_m);Reynolds)' "
" 'f=IFTE(Reynolds ≤ 2100;16/Reynolds;(4.781-((-2*log(((ϵ_m)/(D_m))/3.7+12/Reynolds)-4.781)²/((-2*log(((ϵ_m)/(D_m))/3.7+2.51*(-2*log(((ϵ_m)/(D_m))/3.7+12/Reynolds))/Reynolds))-2*(-2*log(((ϵ_m)/(D_m))/3.7+12/Reynolds))+4.781)))^-2)' "
" '(ΔP_Pa)=(P2_Pa)-(P1_Pa)' "
" '(Δy_m)=(y2_m)-(y1_m)' "
" '(M_(kg/s))=(ρ_(kg/m^3))*(Q_(m^3/s))' "
" '(Q_(m^3/s))=(A_(m²))*(vavg_(m/s))' "
" '(A_(m²))=Ⓒπ*(D_m)²/4' "
" 'Reynolds=(D_m)*(vavg_(m/s))*(ρ_(kg/m^3))/(μ_(kg/(m*s)))' "
" '(n_(m²/s))=(μ_(kg/(m*s)))/(ρ_(kg/m^3))' "
"}",
// ------------------------------------------------------------------------
"Forces And Energy", nullptr,
// ------------------------------------------------------------------------
//T#*: 37 vars 34 eqs 8 sims 8 secs
"Linear Mechanics", "{ "
"'(F_N)=(m_kg)*(a_(m/s²))' "
"'(Ki_J)=1/2*(m_kg)*(vi_(m/s))²' "
"'(Kf_J)=1/2*(m_kg)*(vf_(m/s))²' "
"'(W_J)=(F_N)*(x_m)' "
"'(W_J)=(Kf_J)-(Ki_J)' "
"'(P_W)=(F_N)*(v_(m/s))' "
"'(Pavg_W)=(W_J)/(t_s)' "
"'(vf_(m/s))=(vi_(m/s))+(a_(m/s²))*(t_s)' "
"}",
// Problems of radians units in eqns (2), (3) & (4) and remove radians from eqns (10), (11) & (12)
// And in eqns (10), (11) & (12 change units of N from HZ to rpm)
"Angular Mechanics", "{ "
"'(τ_(N*m))=(I_(kg*m²))*(α_(r/s²))/(1_r)' "
"'(Ki_J)=1/2*(I_(kg*m²))*((ωi_(r/s))²)/(1_r)²' "
"'(Kf_J)=1/2*(I_(kg*m²))*((ωf_(r/s))²)/(1_r)²' "
"'(W_J)=(τ_(N*m))*(θ_r)/(1_r)' "
"'(W_J)=(Kf_J)-(KI_J)' "
"'(P_W)=(τ_(N*m))*(ω_(r/s))/(1_r)' "
"'(Pavg_W)=(W_J)/(t_s)' "
"'(ωf_(r/s))=(ωi_(r/s))+(α_(r/s²))*(t_s)' "
"'(at_(m/s²))=(α_(r/s²))/(1_r)*(r_m)' "
"'(ω_(r/s))=2*(Ⓒπ)*(N_rpm)' "
"'(ωi_(r/s))=2*(Ⓒπ)*(Ni_rpm)' "
"'(ωf_(r/s))=2*(Ⓒπ)*(Nf_rpm)' "
"}",
"Centripetal Force", "{ "
"'(F_N)=(m_kg)*((ω_(r/s))/(1_r))²*(r_m)' "
"'(ω_(r/s))=(v_(m/s))*(1_r)/(r_m)' "
"'(ar_(m/s²))=(v_(m/s))²/(r_m)' "
"'(ω_(r/s))=2*(Ⓒπ_r)*(N_Hz)' "
"}",
"Hooke’s Law", "{ "
"'(F_N)=-(k_(N/m))*(x_m)' "
"'(W_J)=-1/2*(k_(N/m))*(x_m)²' "
"}",
"1D Elastic Collisions", "{ "
"'(v1f_(m/s))=(((m1_kg)-(m2_kg))/((m1_kg)+(m2_kg)))*(v1i_(m/s))' "
"'(v2f_(m/s))=((2*(m1_kg))/((m1_kg)+(m2_kg)))*(v1i_(m/s))' "
"}",
// The following sub-section was missing from v8.2
"Drag Force", "{ "
"'(F_N)=Cd*((ρ_kg/m^3)*(v_m/s)²/2)*(A_cm²)' "
"}",
// Formulas 3 & 4 corrected: (m1_kg)*(m2_kg) in the numerator instead of (m_kg) alone in the gravitational potential
"Gravitation Law", "{ "
"'(F_N)=ⒸG*((m1_kg)*(m2_kg))/(r_m)²' "
"'(W_J)=(UGf_J)-(UGi_J)' "
"'(UGf_J)=-ⒸG*((m1_kg)*(m2_kg))/(rf_m)' "
"'(UGi_J)=-ⒸG*((m1_kg)*(m2_kg))/(ri_m)' "
"}",
"Relativity Mass Energy", "{ "
"'(E_J)=(m_kg)*Ⓒc²' "
"}",
// ------------------------------------------------------------------------
"Gases", nullptr,
// ------------------------------------------------------------------------
//t#*: 38 vars 21 eqns 8 sims 8 secs
// Change all occurrences of gmol by mol
// In eq (1) units of T should be K instead of °C
"Ideal Gas", "{ "
"'(P_atm)*(V_l)=(n_mol)*ⒸR*UBASE(T_K)' "
"'(m_kg)=(n_mol)*(MW_(g/mol))' "
"}",
"Ideal Gas Law Change", "{ "
"'((Pf_Pa)*(Vf_l))/UBASE(Tf_K)=((Pi_Pa)*(Vi_l))/UBASE(Ti_K)' "
"}",
// Change all occurrences of gmol by mol
"Isothermal Expansion", "{ "
"'(W_J)=(n_mol)*ⒸR*(T_°C)*LN((Vf_l)/(Vi_l))' "
"'(m_kg)=(n_mol)*(MW_(g/mol))' "
"}",
"Polytropic Processes", "{ "
"'(Pf_atm)/(Pi_atm)=((Vf_(ft^3))/(Vi_(ft^3)))^(-n)' "
"'(Tf_°C)/(Ti_°C)=((Pf_atm)/(Pi_atm))^((n-1)/n)' "
"}",
"Isentropic Flow", "{ "
"'UBASE(T_K)/UBASE(T0_K)=2/(2+(k-1)*M²)' "
"'(P_kPa)/(P0_kPa)=((T_°C)/(T0_°C))^(k/(k-1))' "
"'(ρ_(kg/m^3))/(ρ0_(kg/m^3))=(UBASE(T_K)/UBASE(T0_K))^(1/(k-1))' "
"'(A_(cm²))/(At_(cm²))=(1/M)*(2/(k+1)*(1+(k-1)/2*M²))^((k+1)/(2*(k-1)))' "
"}",
// Change all occurrences of gmol by mol : Change °C for K in eqn 1
"Real Gas Law", "{ "
"'(P_atm)*(V_l)=(n_mol)*Z*ⒸR*UBASE(T_K)' "
"'(m_kg)=(n_mol)*(MW_(g/mol))' "
//"'Tr=UBASE(T_K)/UBASE(Tc_K)' " These 3 eqns will be substitute in a closed form for Z
//"'Pr=(P_Pa)/(Pc_Pa)' "
//"'ρr=0.27*(Pr/(Z*Tr))' "
//"'Z=1+(0.31506237-1.04670990/Tr-0.57832729/Tr^3)*ρr+(0.53530771-0.61232032/Tr)*ρr²+0.61232032*0.10488813*ρr^5/Tr+0.68157001*ρr²/Tr^3*(1+0.68446549*ρr²)*exp(-0.68446549*ρr²)' "
// change Ti for T and Pi for P in the following
"'Z=1+(0.31506237-1.04670990/(UBASE(T_K)/UBASE(Tc_K))-0.57832729/(UBASE(T_K)/UBASE(Tc_K))^3)*(0.27*(((P_Pa)/(Pc_Pa))/(Z*(UBASE(T_K)/UBASE(Tc_K)))))+(0.53530771-0.61232032/(UBASE(T_K)/UBASE(Tc_K)))*(0.27*(((P_Pa)/(Pc_Pa))/(Z*(UBASE(T_K)/UBASE(Tc_K)))))²+0.61232032*0.10488813*(0.27*(((P_Pa)/(Pc_Pa))/(Z*(UBASE(T_K)/UBASE(Tc_K)))))^5/(UBASE(T_K)/UBASE(Tc_K))+0.68157001*(0.27*(((P_Pa)/(Pc_Pa))/(Z*(UBASE(T_K)/UBASE(Tc_K)))))²/(UBASE(T_K)/UBASE(Tc_K))^3*(1+0.68446549*(0.27*(((P_Pa)/(Pc_Pa))/(Z*(UBASE(T_K)/UBASE(Tc_K)))))²)*exp(-0.68446549*(0.27*(((P_Pa)/(Pc_Pa))/(Z*(UBASE(T_K)/UBASE(Tc_K)))))²)' "
"}",
// Change all °C for K in eqn 1
"Real Gas State Change", "{ "
"'((Pf_atm)*(Vf_l))/(Zf*UBASE(Tf_K))=((Pi_atm)*(Vi_l))/(Zi*UBASE(Ti_K))' "
//"'Tri=UBASE(Ti_K)/UBASE(Tc_K)' " These 3 eqns will be substitute in a closed form for Zi
//"'Pri=(Pi_Pa)/(Pc_Pa)' "
//"'ρri=0.27*(Pri/(Zi*Tri))' "
//"'Zi=1+(0.31506237-1.04670990/Tri-0.57832729/Tri^3)*ρri+(0.53530771-0.61232032/Tri)*ρri²+0.61232032*0.10488813*ρri^5/Tri+0.68157001*ρri²/Tri^3*(1+0.68446549*ρri²)*exp(-0.68446549*ρri²)' "
"'Zi=1+(0.31506237-1.04670990/(UBASE(Ti_K)/UBASE(Tc_K))-0.57832729/(UBASE(Ti_K)/UBASE(Tc_K))^3)*(0.27*(((Pi_Pa)/(Pc_Pa))/(Zi*(UBASE(Ti_K)/UBASE(Tc_K)))))+(0.53530771-0.61232032/(UBASE(Ti_K)/UBASE(Tc_K)))*(0.27*(((Pi_Pa)/(Pc_Pa))/(Zi*(UBASE(Ti_K)/UBASE(Tc_K)))))²+0.61232032*0.10488813*(0.27*(((Pi_Pa)/(Pc_Pa))/(Zi*(UBASE(Ti_K)/UBASE(Tc_K)))))^5/(UBASE(Ti_K)/UBASE(Tc_K))+0.68157001*(0.27*(((Pi_Pa)/(Pc_Pa))/(Zi*(UBASE(Ti_K)/UBASE(Tc_K)))))²/(UBASE(Ti_K)/UBASE(Tc_K))^3*(1+0.68446549*(0.27*(((Pi_Pa)/(Pc_Pa))/(Zi*(UBASE(Ti_K)/UBASE(Tc_K)))))²)*exp(-0.68446549*(0.27*(((Pi_Pa)/(Pc_Pa))/(Zi*(UBASE(Ti_K)/UBASE(Tc_K)))))²)' "
//"'Trf=UBASE(Tf_K)/UBASE(Tc_K)' " These 3 eqns will be substitute in a closed form for Zf
//"'Prf=(Pf_Pa)/(Pc_Pa)' "
//"'ρrf= 0.27*(Prf/(Zf*Trf))' "
//"'Zf=1+(0.31506237-1.04670990/Trf-0.57832729/Trf^3)*ρrf+(0.53530771-0.61232032/Trf)*ρrf²+0.61232032*0.10488813*ρrf^5/Trf+0.68157001*ρrf²/Trf^3*(1+0.68446549*ρrf²)*exp(-0.68446549*ρrf²)' "
"'Zf=1+(0.31506237-1.04670990/(UBASE(Tf_K)/UBASE(Tc_K))-0.57832729/(UBASE(Tf_K)/UBASE(Tc_K))^3)*(0.27*(((Pf_Pa)/(Pc_Pa))/(Zf*(UBASE(Tf_K)/UBASE(Tc_K)))))+(0.53530771-0.61232032/(UBASE(Tf_K)/UBASE(Tc_K)))*(0.27*(((Pf_Pa)/(Pc_Pa))/(Zf*(UBASE(Tf_K)/UBASE(Tc_K)))))²+0.61232032*0.10488813*(0.27*(((Pf_Pa)/(Pc_Pa))/(Zf*(UBASE(Tf_K)/UBASE(Tc_K)))))^5/(UBASE(Tf_K)/UBASE(Tc_K))+0.68157001*(0.27*(((Pf_Pa)/(Pc_Pa))/(Zf*(UBASE(Tf_K)/UBASE(Tc_K)))))²/(UBASE(Tf_K)/UBASE(Tc_K))^3*(1+0.68446549*(0.27*(((Pf_Pa)/(Pc_Pa))/(Zf*(UBASE(Tf_K)/UBASE(Tc_K)))))²)*exp(-0.68446549*(0.27*(((Pf_Pa)/(Pc_Pa))/(Zf*(UBASE(Tf_K)/UBASE(Tc_K)))))²)' "
"}",
// Change all occurrences of gmol by mol
"Kinetic Theory", "{ "
"'(P_kPa)=((n_mol)*(MW_(g/mol))*(vrms_(m/s))²)/(3*(V_l))' "
"'(vrms_(m/s))=√((3*ⒸR*(T_°C))/(MW_(g/mol)))' "
"'(λ_nm)=1/(√(2)*Ⓒπ*((n_mol)*ⒸNA)/(V_l)*(d_nm)²)' "
"'(m_kg)=(n_mol)*(MW_(g/mol))' "
"}",
// ------------------------------------------------------------------------
"Heat Transfer", nullptr,
// ------------------------------------------------------------------------
//T#*: 31 vars 17 eqns 6 sims 6 secs
"Heat Capacity", "{ "
"'(Q_kJ)=(m_kg)*(c_(kJ/(kg*K)))*(UBASE(Tf_°C)-UBASE(Ti_°C))' "
"'(ΔT_°C)=(Tf_°C)-(Ti_°C)' "
"}",
"Thermal Expansion", "{ "
"'(δ_cm)=(α_(1/K))*(L_m)*(UBASE(Tf_°C)-UBASE(Ti_°C))' "
"'(ΔT_°C)=(Tf_°C)-(Ti_°C)' "
"}",
"Conduction", "{ "
"'(qr_W)=((k_(W/(m*K)))*(A_(m²))/(L_m))*(UBASE(Th_°C)-UBASE(Tc_°C))' "
"'(ΔT_°C)=(Th_°C)-(Tc_°C)' "
"}",
"Convection", "{ "
"'(qr_W)=(h_(W/(m²*K)))*(A_(m²))*(UBASE(Th_°C)-UBASE(Tc_°C))' "
"'(ΔT_°C)=(Th_°C)-(Tc_°C)' "
"}",
"Conduction & Convection", "{ "
"'(qr_W)=(A_(m²))*(UBASE(Th_°C)-UBASE(Tc_°C))/(1/(h1_(W/(m²*K)))+(L1_cm)/(k1_(W/(m*K)))+(L2_cm)/(k2_(W/(m*K)))+(L3_cm)/(k3_(W/(m*K)))+1/(h3_(W/(m²*K))))' "
"'(qr_W)=(A_(m²))*((Th_°C)-(Tc_°C))/(1/(h1_(W/(m²*K)))+(L1_cm)/(k1_(W/(m*K)))+(L2_cm)/(k2_(W/(m*K)))+(L3_cm)/(k3_(W/(m*K)))+1/(h3_(W/(m²*K))))' "
"'(U_(W/(m²*K)))=(qr_W)/((A_(m²))*(UBASE(Th_°C)-UBASE(Tc_°C)))' "
"'(ΔT_°C)=(Th_°C)-(Tc_°C)' "
"}",
//
// WARNING The db48x needs the Black-Body Integral function F0λ(T_K,λ_nm)
// defined in the HP50G (SP50G_AUR 3-82, 5-31)
// In eqn 1, change °C for K
// In eqn 2, change the calls for F0λ to explicit integral where the integration limits are from x1 to x2 with xi,j=Ⓒh*Ⓒc/((λi,j_nm)*Ⓒk*UBASE(T_K))
"Black Body Radiation", "{ "
"'(eb_(W/m²))=Ⓒσ*UBASE(T_K)^4' "
//"'f=F0λ((λ2_nm);(T_°C))-F0λ((λ1_nm);(T_°C))' "
"'f=15/Ⓒπ^4*∫(UBASE(Ⓒh*Ⓒc/((λ1_nm)*Ⓒk*UBASE(T_K)));UBASE(Ⓒh*Ⓒc/((λ2_nm)*Ⓒk*(T_K)));X^3/expm1(X);X)' "
"'(eb12_(W/m²))=f*(eb_(W/m²))' "
"'(λmax_nm)*UBASE(Tmax_°C)=Ⓒc3' "
"'(q_W)=(eb_(W/m²))*(A_(cm²))' "
"}",
// ------------------------------------------------------------------------
"Magnetism", nullptr,
// ------------------------------------------------------------------------
//T#*: 28 vars 14 eqns 13 sims 10 secs
// WARNING both HP50G & HP50G_AUR.pdf used a variables rw absent from all
// equations
"Straight Wire Infinite", "{ "
"'(B_T)=Ⓒμ₀*IFTE((r_m)≤(rw_m);μr*(r_m)*(I_A)/(2*Ⓒπ*(rw_cm)²);(I_A)/(2*Ⓒπ*(r_cm)))' "
"}",
// Correction factor 1/4 replace factor 1/2
"Straight Wire Finite", "{ "
"'(B_T)=Ⓒμ₀*IFTE((r_m)≤(rw_m);μr*(r_m)*(I_A)/(4*Ⓒπ*(rw_cm)²);(I_A)/(4*Ⓒπ*(r_cm)))*(COS(θ₁_r)-COS(θ₂_r))' "
"}",
"Force Between Wires", "{ "
"'(Fba_N)=(Ⓒμ₀*μr*(Ib_A)*(Ia_A)*(L_cm))/(2*Ⓒπ*(d_m))' "
"}",
"B Field In Infinite Solenoid", "{ "
"'(B_T)=Ⓒμ₀*μr*(I_A)*nl' "
"}",
"B Field In Finite Solenoid", "{ "
"'(B_T)=(1/2)*Ⓒμ₀*μr*(I_A)*nl*(COS(α2_°)-COS(α1_°))' "
"}",
"B Field In Toroid", "{ "
"'(B_T)=(Ⓒμ₀*μr*(I_A)*N)/(2*Ⓒπ)*(2/((ro_cm)+(ri_cm)))' "
"}",
"Hall Effect", "{ "
// Correction (24-10-05): (L_m) needs to be in the denominator and not in the numerator
"'(VH_V)=((I_A)*(B_T))/((n_(1/m^3))*ABS(q_C)*(L_m))' "
"}",
"Cyclotron Motion", "{ "
"'(Rc_m)=((m_kg)*(v_(m/s)))/(ABS(q_C)*(B_T))' "
"'(fc_Hz)=(ABS(q_C)*(B_T))/(2*Ⓒπ*(m_kg))' "
"'(T_s)=1/(fc_Hz)' "
"}",
"Helicoidal Motion", "{ "
"'(Rc_m)=((m_kg)*(v_(m/s))*SIN(θ_°))/(ABS(q_C)*(B_T))' "
"'(T_s)=(2*Ⓒπ*(Rc_m))/((v_(m/s))*ABS(SIN(θ_°)))' "
"'(Dpitch_m)=(v_(m/s))*COS(θ_°)*(T_s)' "
"}",
"Volumic Density Magnetic Energy", "{ "
"'(uB_(J/m^3))=(1/(2*Ⓒμ₀*μr))*(B_T)²' "
"}",
// ------------------------------------------------------------------------
"Motion", nullptr,
// ------------------------------------------------------------------------
//T#: 38 vars 34 eqns 10 sims 7 secs
"Linear Motion", "{ "
"'(x_m)=(x0_m)+(v0_(m/s))*(t_s)+1/2*(a_(m/s²))*(t_s)²' "
"'(x_m)=(x0_m)+(v_(m/s))*(t_s)-1/2*(a_(m/s²))*(t_s)²' "
"'(x_m)=(x0_m)+1/2*((v0_(m/s))+v_(m/s))*(t_s)' "
"'(v_(m/s))=(v0_(m/s))+(a_(m/s²))*(t_s)' "
"}",
//Ref.: https://fr.wikipedia.org/wiki/Pesanteur#cite_ref-10:~:text=sur%20les%20plantes.-,Variation%20en%20fonction%20du%20lieu,-%5Bmodifier%20%7C
// Reference: Commissions romandes de mathématique, de physique et de chimie, Formulaires et tables : Mathématiques, Physique, Chimie, Tricorne, 2000, 278
// Replacing φ by φ_° in eqn (6) AND rearranging units in eqn (6)
"Object In Free Fall", "{ "
"'(y_m)=(y0_m)+(v0_(m/s))*(t_s)-1/2*(gloc_(m/s²))*(t_s)²' "
"'(y_m)=(y0_m)+(v_(m/s))*(t_s)+1/2*(gloc_(m/s²))*(t_s)²' "
"'(v_(m/s))=(v0_(m/s))-(gloc_(m/s²))*(t_s)' "
"'(v_(m/s))²=(v0_(m/s))²-2*(gloc_(m/s²))*((y_m)-(y0_m))' "
"'gloc_(m/s²)=ⒸG*(Mp_kg)/(r_m)²' "
//"'gearth_(m/s²)=9.780327*(1+5.3024E-3*(SIN(φ_°))²-5.8E-6*(SIN(φ_°))²-3.086E-7*(h_m))' "
"'gearth_(m/s²)=(9.780327_(m/s²))*(1+5.3024E-3*(SIN(φ_°))²-5.8E-6*(SIN(φ_°))²-3.086E-7*(h_m)/(1_m))' "
"}",
"Projectile Motion", "{ "
"'(x_m)=(x0_m)+(v0_(m/s))*COS(θ0_°)*(t_s)' "
"'(y_m)=(y0_m)+(v0_(m/s))*SIN(θ0_°)*(t_s)-1/2*Ⓒg*(t_s)²' "
"'(vcx_(m/s))=(v0_(m/s))*COS(θ0_°)' "
"'(vcy_(m/s))=(v0_(m/s))*SIN(θ0_°)-Ⓒg*(t_s)' "
"'(R_m)=((v0_(m/s))²)/Ⓒg*SIN(2*(θ0_°))' "
"'(hmax_m)=((v0_(m/s))²)/(2*Ⓒg)*(SIN(θ0_r))²' "
"'(tf_s)=(2*(v0_(m/s)))/Ⓒg*SIN(θ0_r)' "
"}",
"Angular Motion", "{ "
// Due to "Inconsistent units" I had to rewrite all angular units of ω & α
// "'(θ_°)=(θ0_°)+(ω₀_rpm)*(t_s)+1/2*(α_(rpm²))*(t_s)²' "
// "'(θ_°)=(θ0_°)+(ω_rpm)*(t_s)-1/2*(α_(rpm²))*(t_s)²' "
// "'(θ_°)=(θ0_°)+1/2*((ω₀_rpm)+(ω_rpm))*(t_s)' "
// "'(ω_rpm)=(ω₀_rpm)+(α_(rpm²))/(1_turn)*(t_s)' "
"'(θ_°)=(θ0_°)+(ω₀_r/min)*(t_s)+1/2*(α_r/min²)*(t_s)²' "
"'(θ_°)=(θ0_°)+(ω_r/min)*(t_s)-1/2*(α_r/min²)*(t_s)²' "
"'(θ_°)=(θ0_°)+1/2*((ω₀_r/min)+(ω_r/min))*(t_s)' "
"'(ω_r/min)=(ω₀_r/min)+(α_r/min²)*(t_s)' "
"}",
// I rewrote the angular units in eqns (1) & (3)
"Uniform Circular Motion", "{ "
//"'(ω_rpm)=(v_(m/s))*(1_r)/(rc_cm)' "
"'(ω_r/min)=(v_(m/s))*(1_r)/(rc_cm)' "
"'(ar_(m/s²))=(v_(m/s))²/(rc_cm)' "
//"'(ω_rpm)=2*(Ⓒπ_r)*(N_rpm)' "
"'(ω_r/min)=2*(Ⓒπ_r/turn)*(N_rpm)' "
"}",
// 2 new eqns added (3) & (4), but the integration doesn't work
"Terminal Velocity", "{ "
"'(vt_(m/s))=√((2*(m_kg)*Ⓒg)/(Cd*(ρ_(kg/m^3))*(Ah_cm²)))' "
"'v_(m/s)=(vt_(m/s))*TANH((t_s)*Ⓒg/(vt_(m/s)))' "
"'tfr_s=ATANH(fr)/(Ⓒg/(vt_(m/s)))' "
"'xfr_ft=(vt_(m/s))*∫(0_s;tfr_s;TANH(t*Ⓒg/(vt_(m/s)));t)' "
// This last integration shall work since it is stripped of units but it doesn't
// "'xfr_ft=(vt_(m/s))*∫(0_s;tfr_s;TANH((t_s)*Ⓒg/(vt_(m/s)));t)' " that's the failing integral with units
// "'xfr_ft=(175.74722 3631_ft/s)*∫(0;10.00590 25332;TANH(t*Ⓒg/(175.74722 3631_ft/s)*(1_s));t)*(1_s)' " works in example 1
// "'xfr_m=(95.13182 74789_m/s)*∫(0;17.76964 17471;TANH(t*Ⓒg/(95.13182 74789_m/s)*(1_s));t)*(1_s)' " works in example 2
"}",
// New section added with new eqns (1), (3) & (4), but the integration doesn't work
//W=Fb+D <=> Vol*(ρ-ρf)*g=1/2*Cd*Ah*ρf*vt² => vt=IFTE('ρ<ρf';-1;1)*√(2*Vol/Ah*ABS(ρ/ρf-1)*g/Cd)
"Buoyancy & Terminal Velocity", "{ "
"'(vt_(m/s))=IFTE((ρ_(kg/m^3))<(ρf_(kg/m^3));-1;1)*√(2*(Vol_m^3)/(Ah_m²)*ABS((ρ_(kg/m^3))/(ρf_(kg/m^3))-1)*Ⓒg/Cd)' "
"'v_(m/s)=(vt_(m/s))*TANH((t_s)*Ⓒg/ABS(vt_(m/s)))' "
"'tfr_s=ATANH(fr)/(Ⓒg/ABS(vt_(m/s)))' "
"'xfr_m=(vt_(m/s))*∫(0_s;tfr_s;TANH(t*Ⓒg/(vt_(m/s)));t)' "
// This last integration shall work since it is stripped of units (but I tried and it also failed)
// "'xfr_m=(vt_(m/s))*∫(0_s;tfr_s;TANH((t_s)*Ⓒg/(vt_(m/s)));t)' " that's the failing integral wuth units
// "'xfr_ft=? (175.74722 3631_ft/s)*∫(0;10.00590 25332;t*TANH(t*Ⓒg/(175.74722 3631_ft/s)*(1_s));t)*(1_s)' " works in example 1
// "'xfr_ft=? (175.74722 3631_ft/s)*∫(0;10.00590 25332;t*TANH(t*Ⓒg/(175.74722 3631_ft/s)*(1_s));t)*(1_s)' " works in example 2
"}",
"Escape & Orbital Velocity", "{ "
"'(ve_(m/s))=√((2*ⒸG*(Mp_kg))/(R_m))' "
"'(vo_(m/s))=√((ⒸG*(Mp_kg))/(R_m))' "
"}",
// ------------------------------------------------------------------------
"Optics", nullptr,
// ------------------------------------------------------------------------
//T#*: 40 vars 38 eqns 11 sims 11 secs
"Refraction Law", "{ "
"'n1*SIN(θ₁_°)=n2*SIN(θ₂_°)' "
"'n1=Ⓒc/(v1_(m/s))' "
"'n2=Ⓒc/(v2_(m/s))' "
"}",
"Critical Angle", "{ "
"'SIN(θc_°)=n1/n2' "
"'n1=Ⓒc/(v1_(m/s))' "
"'n2=Ⓒc/(v2_(m/s))' "
"}",
//Ref.: https://mppolytechnic.ac.in/mp-staff/notes_upload_photo/AS273fiberoptics.pdf
"Fiber Optic", "{ "
// "'SIN(θc_°)=n2/n1' "
"'nf0=Ⓒc/(vf0_(m/s))' "
"'nf1=Ⓒc/(vf1_(m/s))' "
"'nf2=Ⓒc/(vf2_(m/s))' "
"'SIN(θ0_°)=(√(nf1²-nf2²))/nf0' "
"'NA=(√(nf1²-nf2²))/nf0'"
"}",
"Brewster’s Law", "{ "
"'TAN(θB_°)=n2/n1' "
"'(θB_°)+(θ₂_°)=90_°' "
"'n1=Ⓒc/(v1_(m/s))' "
"'n2=Ⓒc/(v2_(m/s))' "
"}",
// Third equn missing for magnification
"Spherical Reflection", "{ "
"'1/(u_cm)+1/(v_cm)=1/(f_cm)' "
"'(f_cm)=(r_cm)/2' "
"'m=-(v_cm)/(u_cm)' "
"}",
"Spherical Refraction", "{ "
"'n1/(u_cm)+n2/(v_cm)=(n2-n1)/(r_cm)' "
"'n1=Ⓒc/(v1_(m/s))' "
"'n2=Ⓒc/(v2_(m/s))' "
"}",
"Thin Lens", "{ "
"'1/(u_cm)+1/(v_cm)=1/(f_cm)' "
"'1/(f_cm)=(n-1)*(1/(r1_cm)-1/(r2_cm))' "
"'m=-(v_cm)/(u_cm)' "
"'n=Ⓒc/(vn_(m/s))' "
"}",
"Rayleigh’s Criterion", "{ "
"'SIN(θr_°)=UBASE(1.21966989*(λ_nm)/(d_m))' "
"'TAN(θr_°)=UBASE((y_m)/(L_m))' "
"}",
// Error in second eqn last "+" to be replaced by "*"
"Malus Law", "{ "
"'(I_(W/m²))/(I₀_(W/m²))=(COS(θ_°))²' "
//"'(Ix_(W/m²))/I₀x_(W/m²)=((fx_Hz)/(fx₀_Hz))*(1+Ⓒλc/Ⓒc*((fx₀_Hz)-(fx_Hz)))+(COS(θ_°))²' "
"'(Ix_(W/m²))/I₀x_(W/m²)=((fx_Hz)/(fx₀_Hz))*(1+Ⓒλc/Ⓒc*((fx₀_Hz)-(fx_Hz)))*(COS(θ_°))²' "
"'(I₀_(W/m²))=(1/(2*Ⓒμ₀*Ⓒc))*(E₀_(V/m))²' "
"}",
"2 Slits Young Interference", "{ "
"'(I_(W/m²))/(Imax_(W/m²))=4*(COS(Δφ_°)/2)²' "
"'(Δφ_r)=(2*(Ⓒπ_r)*(d_μm)*SIN(θ_°))/(λ_nm)' "
"'TAN(θ_°)=(y_m)/(L_m)' "
"'(Δyint_m)=(λ_nm)*(L_m)/(d_μm)' "
"}",
// Impose explicit radians units to Δα_° => Δα_r & Remove radians with *(1_r²)
"One Slit Diffraction", "{ "
//"'(I_(W/m²))/(Imax_(W/m²))=(IFTE(Δα_°;SIN(Δα/2)/(Δα/2);1))²' "
"'(I_(W/m²))/(Imax_(W/m²))=(1_r²)*(IFTE(Δα_r;SIN((Δα_r)/2)/((Δα_r)/2);1))²' "
"'(Δα_r)=(2*(Ⓒπ_r)*(a_μm)*SIN(θ_°))/(λ_nm)' "
"'TAN(θ_°)=(y_m)/(L_m)' "
"'(Δydiff_m)=2*(λ_nm)*(L_m)/(a_μm)' "
"}",
// ------------------------------------------------------------------------
"Oscillations", nullptr,
// ------------------------------------------------------------------------
//T#*: 25 vars 37 equs 7 sims 7 secs
"Mass‐Spring System", "{ "
"'(ω_(r/s))=(1_r)*√((k_(N/m))/(m_kg))' "
"'(T_s)=2*(Ⓒπ_r)/(ω_(r/s))' "
"'(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"}",
// Add radians units explicitely to θmax => (θmax_°)
"Simple Pendulum", "{ "
"'(ω_(r/s))=(1_r)*√(Ⓒg/(L_cm))' "
"'(Treal_s)=2*Ⓒπ*√((L_cm)/Ⓒg)*(Σ(x;0;5;((2·x)!÷((2↑x)·x!)²)²·sin((θmax_°)÷2)↑(2·x)))' "
"'(T_s)=2*(Ⓒπ_r)/(ω_(r/s))' "
"'(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"}",
"Conical Pendulum", "{ "
"'(ω_(r/s))=(1_r)*√(Ⓒg/(h_cm))' "
"'(h_m)=(L_m)*COS(θ_°)' "
"'(T_s)=2*(Ⓒπ_r)/(ω_(r/s))' "
"'(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"}",
"Torsional Pendulum", "{ "
"'(ω_(r/s))=(1_r)*√(((G_kPa)*(J_(cm^4)))/((L_cm)*(I_(kg*m²))))' "
"'(T_s)=2*(Ⓒπ_r)/(ω_(r/s))' "
"'(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"}",
// Modif of radian in eqns 2, 3, 4 & 5
"Simple Harmonic", "{ "
"'(x_cm)=(xm_cm)*COS((ω₀_(r/s))*(t_s)+(φ_°))' "
"'(v_(m/s))=-(ω₀_(r/s))/(1_r)*(xm_m)*SIN((ω₀_(r/s))*(t_s)+(φ_°))' "
"'(a_(m/s²))=-(ω₀_(r/s))²/(1_r)²*(xm_m)*COS((ω₀_(r/s))*(t_s)+(φ_°))' "
"'(ω₀_(r/s))²=(1_r)²*((k_(N/m))/(m_kg))' "
"'(E_J)=(1/2)*(m_kg)*((ω₀_(r/s))/(1_r)*(xm_m))²' "
"}",
// Modif of radian in eqns (3) & (6)
// Add radians to eqn (2); Eliminate radians in eqn (1) & 2 times in eqn (5) & in eqn (6)
// Ref.: https://scholar.harvard.edu/files/david-morin/files/waves_oscillations.pdf
"Underdamped Oscillations", "{ "
"'(x_m)=(xm_m)*EXP(-(γ_(r/s))*(t_s)/2/(1_r))*COS((ωu_(r/s))*(t_s)+(φ_°))' "
"'(γ_(r/s))=(1_r)*(b_(kg/s))/(m_kg)' "
"'(ω₀_(r/s))²=(1_r)²*((k_(N/m))/(m_kg))' "
"'(ωu_(r/s))²=(ω₀_(r/s))²*(1-((γ_(r/s))/(2*(ω₀_(r/s))))²)' "
"'(v_(m/s))=(xm_m)/(1_r)*EXP(-(γ_(r/s))*(t_s)/2/(1_r))*(-((γ_(r/s))/2)*COS((ωu_(r/s))*(t_s)+(φ_°))-(ωu_(r/s))*SIN((ωu_(r/s))*(t_s)+(φ_°)))' "
"'(a_(m²/s))=-((ω₀_(r/s))²/(1_r)²*(x_m)+(γ_(r/s))*(v_m/s)/(1_r))' "
"'(E_J)=(1/2)*(k_(N/m))*(x_m)²+(1/2)*(m_kg)*(v_(m/s))²' "
"'Q=(ω₀_(r/s))/(γ_(r/s))' "
"}",
// Ref.: https://scholar.harvard.edu/files/david-morin/files/waves_oscillations.pdf
"Driven Damped Oscillations", "{ "
"'(x_m)=(xp_m)*COS((ω_(r/s))*(t_s)+(φ_°))+(xh_m)*EXP(-(γ_(r/s))/(1_r)*(t_s)/2)*COS((ωu_(r/s))*(t_s)+(θ_°))' "
"'(γ_(r/s))=(1_r)*(b_(kg/s))/(m_kg)' "
"'(ω₀_(r/s))²=(1_r)²*((k_(N/m))/(m_kg))' "
"'(ωu_(r/s))²=(ω₀_(r/s))²*(1-((γ_(r/s))/(2*(ω₀_(r/s))))²)' "
"'(xp_m)²=((1_r)²*((Fd_N)/(m_kg)))²/(((ω₀_(r/s))²-(ω_(r/s))²)²+((γ_(r/s))*(ω_(r/s)))²)' "
"'TAN(φ_°)=-((γ_(r/s))*(ω_(r/s)))/((ω₀_(r/s))²-(ω_(r/s))²)' "
// The next 7th eqn doesn't fit on the screen, but it goes on the stack with Eq
"'(v_(m/s))=-(xp_m)*(ω_(r/s))/(1_r)*SIN((ω_(r/s))*(t_s)+(φ_°))+(xh_m)*EXP(-(γ_(r/s))/(1_r)*(t_s)/2)*(-((γ_(r/s))/(1_r)/2)*COS((ωu_(r/s))*(t_s)+(θ_°))-(ωu_(r/s))/(1_r)*SIN((ωu_(r/s))*(t_s)+(θ_°)))' "
"'(a_(m²/s))=-(((ω₀_(r/s))/(1_r))²*(x_m)+(γ_(r/s))/(1_r)*(v_m/s))+((Fd_N)/(m_kg))*COS((ω_(r/s))*(t_s))' "
"'(E_J)=(1/2)*(k_(N/m))*(x_m)²+(1/2)*(m_kg)*(v_(m/s))²' "
"'Q=(ω₀_(r/s))/(γ_(r/s))' "
"}",
// ------------------------------------------------------------------------
"Plane Geometry", nullptr,
// ------------------------------------------------------------------------
//T#*: 21 vars 31 eqns 6 sims 6 secs
"Circle", "{ "
"'(A_(cm²))=Ⓒπ*(r_cm)²' "
"'(C_cm)=2*Ⓒπ*(r_cm)' "
"'(I_(cm^4))=1/4*Ⓒπ*(r_cm)^4' "
"'(J_(cm^4))=1/2*Ⓒπ*(r_cm)^4' "
"'(Id_(cm^4))=(I_(cm^4))+(A_(cm²))*(d_cm)²' "
"}",
"Ellipse", "{ "
"'(A_(cm²))=Ⓒπ*(b_cm)*(h_cm)' "
"'(C_cm)²=(2*Ⓒπ)²*(((b_cm)²+(h_cm)²)/2)' "
"'(I_(cm^4))=1/4*Ⓒπ*(b_cm)*(h_cm)^3' "
"'(J_(cm^4))=1/4*Ⓒπ*(b_cm)*(h_cm)*((b_cm)²+(h_cm)²)' "
"'(Id_(cm^4))=(I_(cm^4))+(A_(cm²))*(d_cm)²' "
"}",
"Rectangle", "{ "
"'(A_(cm²))=(b_cm)*(h_cm)' "
"'(P_cm)=2*((b_cm)+(h_cm))' "
"'(I_(m^4))=1/12*(b_cm)*(h_cm)^3' "
"'(J_(cm^4))=1/12*(b_cm)*(h_cm)*((b_cm)²+(h_cm)²)' "
"'(Id_(cm^4))=(I_(cm^4))+(A_(cm²))*(d_cm)²' "
"}",
"Regular Polygon", "{ "
"'(A_(cm²))=(1/4*n*(L_yd)²)/TAN((180_°)/n)' "
"'(P_cm)=n*(L_cm)' "
"'(rs_cm)=((L_cm)/2)/TAN((180_°)/n)' "
"'(rv_cm)=((L_cm)/2)/SIN((180_°)/n)' "
"'(θ_°)=(n-2)/n*(180_°)' "
"'(β_°)=(360_°)/n' "
"}",
"Circular Ring", "{ "
"'(A_(cm²))=Ⓒπ*((ro_cm)²-(ri_cm)²)' "
"'(I_(cm^4))=Ⓒπ/4*((ro_cm)^4-(ri_cm)^4)' "
"'(J_(cm^4))=Ⓒπ/2*((ro_cm)^4-(ri_cm)^4)' "
"'(Id_(cm^4))=(I_(m^4))+(A_(m²))*(d_cm)²' "
"}",
// Error correction: change I_(cm^4) for Ix_(cm^4) in eqn (6)
"Triangle", "{ "
"'(A_(in²))=(b_cm)*(h_cm)/2' "
"'(P_cm)=(b_cm)+√((v_cm)²+(h_cm)²)+√(((b_cm)-(v_cm))²+(h_cm)²)' "
"'(Ix_(cm^4))=1/36*(b_cm)*(h_cm)^3' "
"'(Iy_(cm^4))=1/36*(b_cm)*(h_cm)*((b_cm)²+(v_cm)²-(b_cm)*(v_cm))' "
"'(J_(cm^4))=1/36*(b_cm)*(h_cm)*((b_cm)²+(v_cm)²+(h_cm)²-(b_cm)*(v_cm))' "
"'(Id_(cm^4))=(Ix_(cm^4))+(A_(in²))*(d_cm)²' "
"}",
// ------------------------------------------------------------------------
"Solid Geometry", nullptr,
// ------------------------------------------------------------------------
//T#*: 12 vars 18 eqns 4 sims 4 secs
"Cone", "{ "
"'(V_(cm^3))=(Ⓒπ/3)*(r_cm)²*(h_cm)' "
"'(A_(cm²))=Ⓒπ*(r_cm)²+Ⓒπ*(r_cm)*√((r_cm)²+(h_cm)²)' "
"'(Ixx_(kg*m²))=(3/20)*(m_kg)*(r_cm)²+(3/80)*(m_kg)*(h_cm)²' "
"'(Izz_(kg*m²))=(3/10)*(m_kg)*(r_cm)²' "
"'(Id_(kg*m²))=(Ixx_(kg*m²))+(m_kg)*(d_cm)²' "
"}",
//WARNING mass units in the example of HP50G_AUR (12.2) is lbs...
// is it lbf or lb (lb probably)
"Cylinder", "{ "
"'(V_(cm^3))=Ⓒπ*(r_cm)²*(h_cm)' "
"'(A_(cm²))=2*Ⓒπ*(r_cm)²+2*Ⓒπ*(r_cm)*(h_cm)' "
"'(Ixx_(kg*cm²))=(1/4)*(m_kg)*(r_cm)²+(1/12)*(m_kg)*(h_cm)²' "
"'(Izz_(kg*cm²))=(1/2)*(m_kg)*(r_cm)²' "
"'(Id_(kg*m²))=(Ixx_(kg*m²))+(m_kg)*(d_m)²' "
"}",
"Parallelepiped", "{ "
"'(V_(cm^3))=(b_cm)*(h_cm)*(t_cm)' "
"'(A_(cm²))=2*((b_cm)*(h_cm)+(b_m)*(t_cm)+(h_cm)*(t_cm))' "
"'(I_(kg*cm²))=(1/12)*(m_kg)*((h_cm)²+(t_cm)²)' "
"'(Id_(kg*cm²))=(I_(kg*cm²))+(m_kg)*(d_cm)²' "
"}",
"Sphere", "{ "
"'(V_(cm^3))=(4/3)*Ⓒπ*(r_cm)^3' "
"'(A_(cm²))=4*Ⓒπ*(r_cm)²' "
"'(I_(kg*m²))=(2/5)*(m_kg)*(r_cm)²' "
"'(Id_(kg*cm²))=(I_(kg*m²))+(m_kg)*(d_cm)²' "
"}",
// ------------------------------------------------------------------------
"Solid State Device", nullptr,
// ------------------------------------------------------------------------
//T#*: 54 vars 36 eqns 4 sims 4 secs
// **WARNING: Required function for the parameter ni_(cm^-3)=SIDENS(T_K)
// And Here q = Ⓒqe.
// ERROR in gm formula in HP50G_AUR.pdf: μm=μn (OK)
// ERROR in 4th formula of HP50G_AUR.pdf αR=>αF
//24-11-12 Replace the call for SIDENS by its explicit calculation in eqn (2)
"PN Step Junctions", "{ "
"'(Vbi_V)=(Ⓒk*(T_°C))/Ⓒqe*LN((NA_(cm^-3))*(ND_(cm^-3))/((ni_(cm^-3))²))' "
"'(ni_(cm^-3))=ⓁSiDensity(T_K)' "
//"'(ni_(cm^-3))=(8.35123e20_cm^-3)*exp(-(7555.17_K)/(T_K))' "
"'(xd_μ)²=((2*Ⓒεsi*Ⓒε₀)/Ⓒqe*((Vbi_V)-(Va_V))*(1/(NA_(cm^-3))+1/(ND_(cm^-3))))' "
"'(Cj_(pF/cm²))=(Ⓒεsi*Ⓒε₀)/(xd_μ)' "
"'(Emax_(V/m))=2*((Vbi_V)-(Va_V))/(xd_μ)' "
"'(BV_V)=(Ⓒεsi*Ⓒε₀*(E1_(V/m))²)/(2*Ⓒqe)*(1/(NA_(cm^-3))+1/(ND_(cm^-3)))' "
"'(J_(A/cm²))=(Js_(μA/cm²))*(EXP((Ⓒqe*(Va_V))/(Ⓒk*UBASE(T_K)))-1)' "
"'(Aj_(cm²))=((W_μ)+2*(ΔW_μ))*((L_μ)+2*(ΔL_μ))+Ⓒπ*((W_μ)+2*(ΔW_μ)+2*(ΔL_μ))*(xj_μ)+2*Ⓒπ*(xj_m)²' "
"'(I_mA)=(J_(A/cm²))*(Aj_(cm²))' "
"}",
// WARNING: In the 9th formula of HP50G it is μn and not μm as described in
// HP50G_AUR
//24-11-12 Replace the call for SIDENS by its explicit calculation in eqn (8) not anymore
//Error correction negative sign missing in the calculation of (φp_V)=-...
"NMOS Transistor", "{ "
"'(We_μ)=(W_μ)-2*(ΔW_μ)' "
"'(Le_μ)=(L_m)-2*(ΔL_μ)' "
"'(Cox_(pF/cm²))=(Ⓒεox*Ⓒε₀)/(tox_nm)' "
"'(IDS_mA)=(Cox_(pF/cm²))*(μn_((cm²)/(V*s)))*((We_μ)/(Le_μ))*(((VGS_V)-(Vt_V))*(VDS_V)-(VDS_V)²/2)*(1+(λ_(1/V))*(VDS_V))' "
"'(γ_(V^(1/2)))²=((2*Ⓒεsi*Ⓒε₀)*Ⓒqe*(NA_(cm^-3)))/(Cox_(pF/cm²))²' "
"'(Vt_V)=(Vt0_V)+(γ_(V^(1/2)))*(√(2*ABS(φp_V)-ABS(VBS_V))-√(2*ABS(φp_V)))' "
//"'(φp_V)=Ⓒk*UBASE(T_K)/Ⓒqe*LN((NA_(cm^-3))/(ni_(cm^-3)))' "
"'(φp_V)=-Ⓒk*UBASE(T_K)/Ⓒqe*LN((NA_(cm^-3))/(ni_(cm^-3)))' "
"'(ni_(cm^-3))=ⓁSiDensity(T_K)' "
//"'(ni_(cm^-3))=(8.35123e20_cm^-3)*exp(-(7555.17_K)/UBASE(T_K))' "
"'(gds_S)=(IDS_mA)*(λ_(V^-1))' "
"'(gm_(mA/V))²=((Cox_(pF/cm²))*(μn_((cm²)/(V*s)))*((We_m)/(Le_m))*(1+(λ_(V^-1))*(VDS_V))*2*(IDS_mA))' "
"'(VDsat_V)=(VGS_V)-(Vt_V)' "
"}",
// WARNING HP50G has precedence here for the presence of VBC in 2nd exp of
// 1st eqn (contrary to HP50G_AUR)
// WARNING HP50G has precedence here for the presence of αF instead of αR in
// 3rd eqn (contrary to HP50G_AUR)
"Bipolar Transistors", "{ "
"'(IE_mA)=-(IES_nA)*(exp((Ⓒqe*(VBE_V))/(Ⓒk*UBASE(T_K)))-1)+αR*(IES_nA)*(EXP((Ⓒqe*(VBC_V))/(Ⓒk*UBASE(T_K)))-1)' "
"'(IC_mA)=-(ICS_nA)*(exp((Ⓒqe*(VBC_V))/(Ⓒk*UBASE(T_K)))-1)+αF*(ICS_nA)*(EXP((Ⓒqe*(VBE_V))/(Ⓒk*UBASE(T_K)))-1)' "
"'(IS_nA)=αF*(IES_nA)' "
"'(IS_nA)=αR*(ICS_nA)' "
"'(IB_nA)+(IE_nA)+(IC_nA)=0_nA' "
"'(ICO_nA)=(ICS_nA)*(1-αF*αR)' "
"'(ICEO_nA)=(ICO_nA)/(1-αF)' "
"'(VCEsat_V)=((Ⓒk*UBASE(T_K)))/Ⓒqe*LN(MAX(((1+(IC_mA)/(IB_mA)*(1-αR)))/(αR*(1-(IC_mA)/(IB_mA)*((1-αF)/αF)));1E-100))' "
"}",
//24-11-12 Replace the call for SIDENS by its explicit calculation in eqn (2)
"JFETs", "{ "
"'(Vbi_V)=(Ⓒk*UBASE(T_K))/Ⓒqe*LN((ND_(cm^-3))/(ni_(cm^-3)))' "
"'(ni_(cm^-3))=ⓁSiDensity(T_K)' "
//"'(ni_(cm^-3))=(8.35123e20_cm^-3)*exp(-(7555.17_K)/UBASE(T_K))' "
"'(xdmax_μ)²=((2*Ⓒεsi*Ⓒε₀)/(Ⓒqe*(ND_(cm^-3)))*((Vbi_V)-(VGS_V)+(VDS_V)))' "
"'(G0_S)=Ⓒqe*(ND_(cm^-3))*(μn_((cm²)/(V*s)))*(((a_μ)*(W_μ))/(L_μ))' "
"'(ID_mA)=(G0_S)*((VDS_V)-((2/3)*√((2*Ⓒεsi*Ⓒε₀)/(Ⓒqe*(ND_(cm^-3))*(a_μ)²)))*(((Vbi_V)-(VGS_V)+(VDS_V))^(3/2)-((Vbi_V)-(VGS_V))^(3/2)))' "
"'(VDsat_V)=(Ⓒqe*(ND_(cm^-3))*(a_μ)²)/(2*Ⓒεsi*Ⓒε₀)-((Vbi_V)-(VGS_V))' "
"'(Vt_V)=(Vbi_V)-(Ⓒqe*(ND_(cm^-3))*(a_μ)²)/(2*Ⓒεsi*Ⓒε₀)' "
"'(gm_(mA/V))=(G0_S)*(1-√(((2*Ⓒεsi*Ⓒε₀)/(Ⓒqe*(ND_(cm^-3))*(a_μ)²))*((Vbi_V)-(VGS_V))))' "
"}",
// ------------------------------------------------------------------------
"Stress Analysis", nullptr,
// ------------------------------------------------------------------------
//T#*: 28 vars 16 eqns 4 sims 4 secs
//"Inconsistent units" in eqn (2), New trial with UBASE
"Normal Stress", "{ "
"'(σ_atm)=(E_atm)*ε' "
//"'ε=(δ_cm)/(L_m)' "
"'ε=UBASE((δ_cm)/(L_m))' "
"'(σ_Pa)=(P_N)/(A_cm²)' "
"}",
// ALL angles should be in radians in eqns 1 & 2, with a correction in eqn 1
"Shear Stress", "{ "
"'(τ_atm)=(G_atm)*(γ_r)/(1_r)' "
"'(γ_r)=((r_cm)*(φ_r))/(L_m)' "
"'(τ_atm)=((T_(cm*N))*(r_cm))/(J_(cm^4))' "
"}",
"Stress On An Element", "{ "
//Error in eqn (3) of HP50g_AUR
"'(σx1_kPa)=((σx_kPa)+(σy_kPa))/2+((σx_kPa)-(σy_kPa))/2*COS(2*(θ_°))+(τxy_kPa)*SIN(2*(θ_°))' "
"'(σx1_kPa)+(σy1_kPa)=(σx_kPa)+(σy_kPa)' "
//"'(τx1y1_kPa)=-(((σx_kPa)-(σy_kPa))/2)*SIN(2*(θ_°))+(τxy_kPa)*(σy_kPa)' "
"'(τx1y1_kPa)=-(((σx_kPa)-(σy_kPa))/2)*SIN(2*(θ_°))+(τxy_kPa)*COS(2*(θ_°))' "
"}",
// WARNING Error in formula 1 of HP50G_AUR square missing HP50G takes
// precedence
"Mohr’s Circle", "{ "
"'(σ1_atm)=((σx_atm)+(σy_atm))/2+√((((σx_atm)-(σy_atm))²)/2+(τxy_atm)²)' "
"'(σ1_atm)+(σ2_atm)=(σx_atm)+(σy_atm)' "
"'(SIN(2*(θp1_°)))²=(τxy_atm)²/((((σx_atm)-(σy_atm))/2)²+(τxy_atm)²)' "
"'(θp2_°)=(θp1_°)+(90_°)' "
"'(τmax_atm)=((σ1_atm)-(σ2_atm))/2' "
"'(θs_°)=(θp1_°)-(45_°)' "
"'(σavg_atm)=((σx_atm)+(σy_atm))/2' "
"}",
// ------------------------------------------------------------------------
"Waves", nullptr,
// ------------------------------------------------------------------------
//T#*: 39 vars 49 eqns 11 sims 9 secs
"Transverse Waves", "{ "
"'(y_cm)=(ym_cm)*SIN((k_(r/cm))*(x_cm)-(ω_(r/s))*(t_s)+(φ_r))' "
"'(v_(cm/s))=(λ_cm)*(f_Hz)' "
"'(k_(r/cm))=2*(Ⓒπ_r)/(λ_cm)' "
"'(ω_(r/s))=2*(Ⓒπ_r)*(f_Hz)' "
"'(vy_(cm/s))=-(ym_cm)*(ω_(r/s))/(1_r)*COS((k_(r/cm))*(x_cm)-(ω_(r/s))*(t_s)+(φ_r))' "
"'(ay_(cm/(s²)))=-((ω_(r/s))/(1_r))²*(ym_cm)' "
"}",