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 -