From 721a124f9671085c120581aaa66fbf6726b05ccc Mon Sep 17 00:00:00 2001 From: gloria-vytas <35671386+gloria-vytas@users.noreply.github.com> Date: Mon, 5 Aug 2019 20:36:31 -0400 Subject: [PATCH] Updated Merged.m Put all variables and definitions at the beginning of the code, added some constants and deleted others that were taken from the calculations in Appendix 3, and made sure that variables were consistent throughout equations. --- Merged.m | 400 ++++++++++++++++++------------------------------------- 1 file changed, 131 insertions(+), 269 deletions(-) diff --git a/Merged.m b/Merged.m index e818f9f..9217d84 100644 --- a/Merged.m +++ b/Merged.m @@ -1,55 +1,115 @@ clear ; close all; clc + +%VARIABLES: +%alpha_ec = Electric phase angle +%A_lad = area of ladder in metres^2 +%A_ps = Active primary slot area +%A_s2 = area of second slot in metres^2 +%beta = Damping coefficient +%B_g1 = Air gap flux density +%Bglk = Airgap flux density when s*Ge= 1 (which is the relative slip times the equivalent goodness factor) +%b_sp = Primary slot opening +%b_ss = Secondary slot opening +%b_s1 = Primary slot width +%b_s2 = Secondary slot width +%f_1 = Frequency of the primary +%f_2 = Secondary frequency or secondary slip frequency +%F_xn = Rated thrust +%f_xn = Shear secondary stress +%f_2r = Secondary required frequency +%G_e = Equivalent goodness factor +%G_ei= Effective goodness factor of iron - from equation 17 +%g_m = magnetic airgap +%h_sp = Primary tooth height +%h_ss = Secondary tooth height +%h_s1 = Active Primary slot height +%h_s2 = Secondary slot useful height +%I_1 = RMS value of primary phase current / RPS current per phase +%I_1r = Current in Primary winding +%I_1n = RMS phase current for rated thrust (from eq 40) +%j = unknown - should this be j_cor? +%j_cor = Current density in copper winding +%K_w1 = Primary winding factor + K_w1 = 0.933; %from table at end of thesis +%K_c = Carter coefficient + K_c = 1+(t_s/(2*pi*g_m))*((1+(b_sp/t_s))*log(1+(b_sp/t_s))+(1-(b_sp/t_s))*log(1-(b_sp/t_s))); %from LIM handbook, should be between 1.1-1.7 + K_c = 1.25; %from table at end of thesis +%K_ss=0.4 Total (Primary and Secondary) core magnetic saturation coefficient + K_ss = 0.4; %from table at end of thesis +%K_ladder = Ladder coefficient + K_ladder = 0.1; %from table at end of thesis +%K_l2 = Secondary leakage inductance coefficient +%K_r = Carter coefficient of rotor +%K_fill = Slot filling factor + K_fill = 0.6; %according to textbook +%L_m = Magnetization induction +%l_stack = stack width +%l_lad = ladder length, width of ladder, in metres +%l_ec = end-coil length per side +%w_1 = Number of turns per phase in Primary +%L_1l = Primary leakage inductance +%L_2l = Secondary leakage inductance +%lambda_diff1 = Airgap leakage specific permeance in Primary +%lambda_diff2 = Airgap leakage specific permeance in Secondary +%lambda_ecl = Primary end-coil leakage +%lambda_s1 = Slot specific (Nondimensional) permeance in Primary +%lambda_s2 = Slot specific (Nondimensional) permeance in Secondary +%Mu_0 = Permeability of Vacuum or Air + Mu_0 = 4*pi*10^(-7); %H/m, permeability constant of free space, tube will be at min 0.125 psi though +%N_s2 = Number of slots in secondary per primary length +%omega_1 = primany frequency in radians +%omega_2 = secondary frequency in radians +%p = number of poles +%PF = power factor calculated from EQ42 +%Phi_1n = calculated using equation 43 +%P_elm = Electromagnetic power from EQ45 +%q = q_1 = Number of slots per pole per phase +%R_1 = Primary resistance per phase +%R_2 = secondary resistance +%Rho_copper = resistivity of copper + Rho_copper = 1.72*10^(-8); %in ohm meters, at 20°C, annealed +%rho_Al = resistivity of aluminium + rho_Al = 3.125*10.^(-8); %from table at end of thesis +%s = Relative slip +%sigma_Al = density of Aluminium + sigma_Al = 3.5*10.^7; %from table at end of thesis +%t_s = Slot pitch + t_s = t/6; +%t_s2 = Secondary slot pitch +%t = Pole pitch +%theta_1m = Rated primary mmf per pole +%u_r = Required speed +%u_s = Synchronous speed from EQ 46 +%V_1line = line or supply voltage +%V_1r = Primary voltage +%V_1n = phase voltage +%w_1 = Number of turns per phase in Primary +%w_1r = unknown - should it be w_1? +%y = Coil span + %EQN 1 %Primary Resistance per Phase, -function R_1 = PrimResist(Rho_copper,l_stack,l_ec,j_cor,w_1,I_1r) +function R_1 = PrimResist(Rho_copper,l_stack,l_ec,j_cor,w_1,I_1r) R_1 = (2*Rho_copper*(l_stack+l_ec)*j_cor*(w_1)^2)/(w_1*I_1r) end -%R1 = Primary resistance per phase. -%Rhocopper = resistivity of copper -%lstack = stack width -%lec = end-coil length per side -%jcor = Current density in copper winding. -%w1 = Number if turns per phase in Primary. -%I1r = Current in Primary winding. %EQN 2 %Primary Leakage Inductance, function L_1l = PrimLeakInduc(Mu_0,p,q_1,lambda_s1,lambda_diff1,l_stack,lambda_ecl,l_ec,w_1) L_1l = ((2*Mu_0)/(p*q_1))*((lambda_s1+lambda_diff1)*l_stack+lambda_ecl*l_ec)*w_1 end -%L1l = Primary leakage inductance. -%mu0 = Permeability of Vacuum or Air. -%p = number of poles -%lambdas1 = Slot specific (Nondimensional) permeance in Primary. -%lambdadiff1 = Airgap leakage specific permeance in Primary. -%lstack = stack width -%lambdaecl = Primary end-coil leakage. -%lec = Primary end-coil leakage. -%w1 = Number if turns per phase in Primary. %EQN 3 %Primary Slot Specific Permeance (NonDimensional), function lambda_s1 = PrimSlotSpecPerm(h_s1,beta,h_sp,b_sp,b_s1) lambda_s1 = h_s1*(1+3*beta)/(12*b_s1)+(h_sp/b_sp) end -%lambdas1 = Slot specific (Nondimensional) permeance in Primary. -%hs1 = Active Primary slot height. -%beta = Damping coefficient. -%hsp = Primary tooth height. -%bsp = Primary slot opening. -%bs1 = Primary slot width. %EQN 4 %Secondary Slot Specific Permeance (NonDimensional), function lambda_s2 = SecSlotSpecPerm(h_s2,beta,h_ss,b_ss,b_s2) lambda_s2 = h_s2*(1+3*beta)/(12*b_s2)+(h_ss/b_ss) end -%lambdas2 = Slot specific (Nondimensional) permeance in Secondary -%hs2 = Secondary slot usefull height. -%beta = Damping coefficient. -%hss = Secondary tooth height. -%bss = Secondary slot width. -%bs2 = Secondary slot opening. %EQN 5 %Primary Airgap Leakage Specific Permeance, @@ -57,10 +117,6 @@ temp1 = K_c*(g/b_sp) lambda_diff1 = (5*temp1)/(5+4*temp1) end -%lambdadiff1 = Airgap leakage specific permeance in Primary -%Kc = Carter coefficient for dual slotting. -%g = mechanical airgap -%bsp = Primary slot opening. %EQN 6 %Secondary Airgap Leakage Specific Permeance, @@ -68,383 +124,189 @@ temp2 = K_c*(g/b_ss) lambda_diff2 = (5*temp2)/(5+4*temp2) end -%lambdadiff2 = Airgap leakage specific permeance in Secondary -%Kc = Carter coefficient for dual slotting. -%g = mechanical airgap -%bss = Secondary slot opening. %EQN 7 function lamda_ecl = EndCoilLeakagePermeance() - %beta is the damping coeffictient wich we would set as well - q1= 6; % varies depending on our decisions - beta=0.833;%from equations at end of thesis lamda_ecl= 0.3*((3*beta)-1)*q1; end %EQN 8 function L_m = MagnetizationInductance() - h_s2=0.018; %secondary slot height in metres - mu_0=1.257*10.^(-8) ; %permeability of vacuum or air - K_w1=0.933 ; %Primary winding factor - omega_1=617.216 ; %Number if turns per phase in Primary. - t =0.028 ; %Pole pitch - l_stack=6.881/1000; %stack width in metres - g=0.2/1000;% airgap in metres - p=3;%p= number of poles - K_c=1.25 ; %Carter coefficient - K_ss=0.4 ; %Total (Primary and Secondary) core magnetic saturation coefficient - - L_m= h_s2*((6*mu_0*(K_w1*omega_1)*t*l_stack)/((pi.^2)*K_c*g*(p*(1+K_ss)))); + L_m= h_s2*((6*mu_0*(K_w1*w_1)*t*l_stack)/((pi.^2)*K_c*g*(p*(1+K_ss)))); end %EQN 9 function r2 = SecondaryResistance() - Kwl= 0.933;%Primary winding factor - Wl=617.216;%Number if turns per phase in Primary. - Ns2=40; % Number of slots in secondary per primary length - Istack = 6.881/1000; %Stack width in metres - As2= 4.098*10.^(-5); %area of second slot in metres^2 - Ilad=0.004; %ladder length. width of ladder in metres - Alad= 8.77*10.^(-5);%area of ladder in metres^2 - pAluminium=3.125*10.^(-8); %resistivity of aluminium - - R2=12*pAluminium*((Kwl*Wl)/Ns2)*(Istack/As2 + 2*Ilad/Alad); + R2=12*rho_Al*((K_w1*w_1)/N_s2)*(l_stack/A_s2 + 2*l_lad/A_lad); +end %EQN 10 function L_2l = SecondaryLeakageInductance() -u0= 1.257*10.^(-6); %permeability of vacuum or air -Istack= 6.881/1000; % starck width in metres -Lambda_s2= 3.149; %Secondary Slot Specific permeance -Lambda_diff2= 0.15; % Secondary Airgap Leakage specific permeance -K_wl= 0.933; % Primary winding factor -w_l = 617.216; %numbr of turn per phase in primary -N_s2= 40; % Number of slots in secondary per primary length -K_ladder= 0.1; % Ladder coefficient -L_2l=24*u0(Istack*(Lambda_s2+Lambda_diff2))*(((K_wl*w_l).^2)/N_s2)*(1+K_ladder); +L_2l=24*Mu_0(l_stack*(lambda_s2+lambda_diff2))*(((K_w1*w_1).^2)/N_s2)*(1+K_ladder); +end %EQN 12 %electric phase angle function alpha_ec = ElecPhaseAngle(p,N_s2) -%alphaec = Electric Phase Angle. -%p = Number of poles. -%Ns2 = Number of slots in Secondary per Primary length. alpha_ec = 2*pi*p/N_s2 end - %EQN 14 %Secondary slot area function A_s2 = SecSlotArea(h_s2,b_s2) -%As2 = Area of Secondary slot. -%hs2 = Secondary slot usefull height. -%bs2 =Secondary slot width. A_s2 = h_s2*b_s2 end - %EQN 15 %Air gap flux density -function B_g1 = FluxDens(m_0,theta_1m, g, K_c, K_s, s, G_e) -B_g1 = (m_0*theta_1m)/(g*K_c*(1+K_s)*(1+(s^2*G_e^2))^0.5); +function B_g1 = FluxDens(Mu_0,theta_1m, g, K_c, K_s, s, G_e) +B_g1 = (Mu_0*theta_1m)/(g*K_c*(1+K_s)*(1+(s^2*G_e^2))^0.5); end -%B_g1 = Air gap flux density -%m_0 = 1.257*10.^(-6) = permeability of vacuum or air -%theta_1m = rated primary mmf per pole -%g = mechanical air gap -%K_c = 1.25 = Carter coefficient for sual slotting -%K_s = 0.4 = magnetic saturation factor -%s = relative slip -%G_e = equivalent goodness factor - %EQN 16 -function theta_1m = primarymmfpole(Kwl,omega1,I1,p) - theta_1m = (3*sqrt(2)*(Kwl*omega1)^2*I1)/(pi*p) +function theta_1m = primarymmfpole(K_w1,omega_1,I_1,p) + theta_1m = (3*sqrt(2)*(K_w1*omega_1)^2*I1)/(pi*p) end -%theta1m = rated primary mmf per pole -%Kwl = primary winding factor -%omegal1 = primany frequency in radians -%I1 = RMS value of primary phase current / RPS current per phase -%p = number of poles - - %EQN 19 -function K_l2 = CarterCoefSec2(omega2,L2l,R2) - K_l2 = sqrt(1+(omega2*L2l/R2)^2); +function K_l2 = CarterCoefSec2(omega_2,L2l,R2) + K_l2 = sqrt(1+(omega_2*L2l/R2)^2); end -%Kl2 = secondary leakage inductance coefficient -%omgea2 = secondary frequency in radians -%L2l = secondary leakage inductance -%R2 = secondary resistance - - %EQN 20 function B_glk = MagFluxDensity() - u_0= 1.257*10.^(-6); %permeability of vacuum or air - theta_lm =275.722;% Rated primary mmf per pole - g=0.2/1000;% airgap in metres - K_c=1.25 ; %Carter coefficient for sual slotting - K_s=0.4; % magnetic saturation factor - - B_glk= (u0*thetalm)/(g*Kc*(1+Ks)*sqrt(1+1.^2)); + B_glk= (Mu_0*theta_1m)/(g*K_c*(1+K_s)*sqrt(1+1.^2)); end - %EQN 21 -function K_wl = PrimaryWindingFactor(y) - t=0.028; %Pole pitch - %y= coil span - q= 2; %number of slot per pole per phase - K_wl = sin((pi/2)*(y/t))*((sin (pi/6))/(q*sin(pi/(6*q)))); +function K_w1 = PrimaryWindingFactor(y) + K_w1 = sin((pi/2)*(y/t))*((sin (pi/6))/(q*sin(pi/(6*q)))); end - %EQN 22 function w_1I_1 = CurrentPerTurnPerPhase() - theta_lm =275.722;% Rated primary mmf per pole - p=3;%p= number of poles - K_wl=0.933 ; %Primary winding factor - - w_1I_1= (theta_lm*pi*p)/(3*sqrt(2*K_wl)); + w_1I_1= (theta_1m*pi*p)/(3*sqrt(2*K_w1)); end - %EQN 23 function A_p = PrimaryActiveArea() - p=3;%p= number of poles - t =0.028 ; %Pole pitch - l_stack=6.881/1000; %stack width in metres - F_xn=10; %Rated thrust - f_xn=8.8*10.^3; %Shear secondary stress %Ap= Fxn/fxn %Ap=2*p*t.^2*(lstack/t) - A_p= 2*p*t*l_stack ; + A_p = 2*p*t*l_stack ; end - %EQN 24 function t_s = PrimarySlotPitch() - t=0.028; %Pole pitch t_s=t/6; end - %EQN 25 function omega_1 = AngFreq() - f_2r= 4;% Secondary required frequency - %w1 primary frequency in radians - omega_2r=2*pi*f_2r; - omega_1=omega_2r; + omega_2r = 2*pi*f_2r; + omega_1 = omega_2r; end - %EQN 26 function h_s2 = SecondarySlotUsefulHeight(G_ei) - %Gei= goodness factor of iron - g=0.2/1000; % air gap - K_c= 1.25;% Carter coefficient for dual slotting - K_s=0.4; % magnetic saturation factor - K_r=1.5; %Carter coefficient of rotor - K_l2=1.2; % Secondary leakage inductance coefficient - mu_0= 1257*10.^(-6); - omega_1= % primary frequency in radians - t=0.028; %Pole pitch - sigma_Al=3.5*10.^7; %density of Aluminium - b_s2 = 0.002;% Secondary slot width - tau_s2= 0.004;%Secondary slot pitch - - h_s2 = (G_ei*(pi.^2)*g*K_c*(1+K_s)*K_r*K_l2)/(mu_0*w_1*(t.^2)*sigma_Al*(1-b_s2/tau_s2)); + h_s2 = (G_ei*(pi.^2)*g*K_c*(1+K_s)*K_r*K_l2)/(Mu_0*w_1*(t.^2)*sigma_Al*(1-b_s2/t_s2)); end - %EQN 27 -function L_p = PrimLength(tau, p) -%p is the number of poles -%tau is the pole pitch - L_p = ((2*p)+1)*tau; +function L_p = PrimLength(t, p) + L_p = ((2*p)+1)*t; end - %EQN 28 -function A_ps = ActivePrimSlotArea(w1,I1,p,q,Jcor,Kfill) -%w1 is the Number of turns per phase in the primary -%I1 is the RMS value of the primary phase current/RPS current per phase -%p is the number of poles -%q is the number of slots per pole per phase -%Jcor is the current density in the copper winding -%Kfill is the slot filling factor (according to the textbook, should be 0.6) - A_ps = (w_1*I_1)/(p*q*J_cor*K_fill); +function A_ps = ActivePrimSlotArea(w_1,I_1,p,q,j_cor,K_fill) + A_ps = (w_1*I_1)/(p*q*j_cor*K_fill); end - %EQN 29 -function H_s1 = ActivePrimSlotHeight(A_ps,B_s1) -%Aps (from EQ28) -%Bs1 is the primary slot width - H_s1 = A_ps/B_s1; +function H_s1 = ActivePrimSlotHeight(A_ps,b_s1) + h_s1 = A_ps/b_s1; end - %EQN 30 -function F_nk = PeakNormalForce(B_glk, Mu_0, p, tau, l_stack) -%Bglk is the airgap flux density when s*Ge= 1 (which is the relative slip times the equivalent goodness factor) -%Mu0 is the permeability of the vacuum/air -%p is the number of poles -%tau is the pole pitch -%l is stack width - F_nk = ((B_glk^2)/2*Mu_0)*2*p*tau*l_stack; +function F_nk = PeakNormalForce(B_glk, Mu_0, p, t, l_stack) + F_nk = ((B_glk^2)/2*Mu_0)*2*p*t*l_stack; end - %EQN 31 -function F_x = ForwardForce(I_1,R_2,s,Ge,tau,f_1) -%I1 is the RMS value of the primary phase current/RPS current per phase -%R2 is secondary resistance -%s is relative slip -%Ge is the equivalent goodness factor -%tau is the pole pitch -%f1 is the frequency of the primary - F_x = (3*(I_1^2)*R_2*s*Ge)/(2*tau*f_1*(1+(s*Ge)^-2)); +function F_x = ForwardForce(I_1,R_2,s,G_e,t,f_1) + F_x = (3*(I_1^2)*R_2*s*G_e)/(2*tau*f_1*(1+(s*G_e)^-2)); end - %EQN 32 -function F_xn = ThrustForce(Tau,W_1,I_1,L_m) - % Tau is the pole pitch - % W1 is the number of turns per phase in primary - % I1 is the RMS value of primary phase current - % Lm is the magnetization induction - % Kl2 is the secondary leakage inductance coefficient - K_l2 = 1.2 ; - F_xn = ((3*pi)/(2*Tau)) * ((W_1*I_1).^2) * (L_m/K_l2) ; +function F_xn = ThrustForce(t,w_1,I_1,L_m) + F_xn = ((3*pi)/(2*t)) * ((w_1*I_1).^2) * (L_m/K_l2) ; end - %EQN 33 -function f_1r = PrimaryRequiredFrequency(f_2,U_r,Tau) - % f2 is the secondary frequency or secondary slip frequency - % Ur is the required speed - % Tao is the pole pitch - f_1r = f_2 + (U_r/(2*Tau)) ; +function f_1r = PrimaryRequiredFrequency(f_2,u_r,t) + f_1r = f_2 + (u_r/(2*t)) ; end - %EQN 34 function V_10 = AvailableVoltagePerPhase(V1line) - -% V1line is the line or supply voltage % This relation describes the RMS value of the voltage per each phase ... % based on the supply voltage - V_10 = 0.95*(V1line/(sqrt(3))) ; - end - %EQN35 function b_sp = PrimarySlotOpening(b_s1,g) - % bs1 is the primary slot width - % g is the mechanical airgap - b_sp = (b_s1/g) ; end - %EQN 38 %Relative slip function s = RelSlip(f_2, f_1) s = f_2/f_1; end -%f_2 = The secondary frequency or secondary slip frequency -%f_1 = The frequency of the primary - %EQN 39 %Primary voltage function V_1r = PrimVolt(I_1r, R_1, j, w_1r, L_1l, L_m, w_1, s, R_2, L_2l) V_1r = I_1r*(R_1 + j*w_1r*L_1l + (j*w_1*L_m* ((R_2/s+j*w_1*L_2l))/(R_2/s+(j*w_1* (L_m+L_2l))))); end -%V_1r = Primary voltage -%I_1r = Current in primary winding -%R_1 = Primary resistance per phase -%j = Current density -%w_1r = -%L_1l = Primary leakage inductance -%L_m = Magnetization inductance -%w_1 = Numebr of phases per turn in primary -%s = Relative slip -%R_2 = Secondary resistance -%L_2l = Secondary leakage inductance - %EQN 40 function I_1n = CurrentRatedThrust(w_1, I_1) -%w1 is the Number of turns per phase in the primary -%I1 is the RMS value of the primary phase current/RPS current per phase I_1n = (w_1*I_1)/w_1; end - %EQN 41 function P_1n = InputPower(V_1n,I_1n,PF) -%V1n is the phase voltage -%I1n is the RMS phase current for rated thrust (from eq 40) -%PF is the power factor calculated from EQ42, where the Phi1n is calculated -%from equation 43 P1n = 3*V_1n*I_1n*PF; end - %EQN 42 function PF = PowerFactor(Phi_1n) -%Phi1n is calculated using equation 43 PF = cos(Phi_1n); end - %EQN 43 -function Phi = Phi(Z_e) +function Phi_1n = Phi(Z_e) %Ze is the end effect impedence (calculated through eq 44) - Phi = arg(Z_e) + Phi_1n = arg(Z_e) end - %EQN 44 function EEI = EndEffectImpedence(R_1,omega_1,L_1l,L_m,R_2,s,L_2l) -%R1 is the primary resistance per phase -%omega1 is the primary frequency in radians -%L1l is the primary leakage inductance -%Lm is the magnetization inductance -%R2 is the secondary resistance -%s is the relative slip -%L2l is the secondary leakage inductance - EEI = R_1 + (omega_1*L_1l)j + ((omega_1*(L_m*((R_2/s)+(omega_1*L_2l)j)))j)/((R_2/s)+(omega_1+(L_2l+L_m))j); + EEI = R_1 + (omega_1*L_1l)*j + ((omega_1*(L_m*((R_2/s)+(omega_1*L_2l)*j)))*j)/((R_2/s)+(omega_1+(L_2l+L_m))*j); end - %EQN 45 -function P_elm = ElectromagPower(I_1n,R_2,S) -%I1n is the RMS phase current for rated thrust (from eq 40) -%R2 is the secondary resistance -%S is slip - P_elm = 3*(I_1n^2)*R_2*(1-S/S); +function P_elm = ElectromagPower(I_1n,R_2,s) + P_elm = 3*(I_1n^2)*R_2*(1-s/s); end - %EQN 46 -function SS = SynchronousSpeed(tau,f_1) -%tau is the pole pitch -%f1 is the frequency of the primary - SS = 2*tau*f_1; +function u_s = SynchronousSpeed(t,f_1) + u_s = 2*t*f_1; end - %EQN 47 -function F_x = MotorThrust(P_elm,Mu) -%Pelm is the electromagnetic power for EQ45 -%Mu(s) is the synchronous speed from EQ 46 - F_x = P_elm/Mu; +function F_x = MotorThrust(P_elm,u_s) + F_x = P_elm/u_s; end -