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Active_NRCRLH_Calcs.m
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259
Active_NRCRLH_Calcs.m
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%% General Setup
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clear;clc;close all
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format long
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f = linspace(0, 6, 1001)*1e9; % Frequency range
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w = 2*pi*f;
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c0 = 3e8;
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%% Input parameters
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Z0 = 50;
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% Desired frequency points for phase shifts and number of sub-units
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f1 = 2e9;
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f2 = 3.5e9;
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n = 3;
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w1 = 2*pi*f1;
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w2 = 2*pi*f2;
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% Desired phase shifts at w1 and w2
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theta1 = pi/2;% the positive number
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theta2 = -pi/2;% the negative number
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%% Calculate initial CRLH parameters
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LR = Z0*(theta1*(w1/w2) - theta2)/(n*w2*(1 - (w1/w2)^2));
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CR = (theta1*(w1/w2) - theta2)/(n*w2*Z0*(1 - (w1/w2)^2));
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LL = n*Z0*(1 - (w1/w2)^2)/(w1*(theta1 - (w1/w2)*theta2));
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CL = n*(1 - (w1/w2)^2)/(w1*Z0*(theta1 - (w1/w2)*theta2));
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disp(['LR = ' num2str(LR/1e-9, 4), ' nH'])
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disp(['CR = ' num2str(CR/1e-12, 4), ' pF'])
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disp(['LL = ' num2str(LL/1e-9, 4), ' nH'])
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disp(['CL = ' num2str(CL/1e-12, 4), ' pF'])
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Rse = 0*0.625; % Rse ~0.625 from observation of S21 in simulation
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Gsh = Rse/Z0^2; % Assume loss-balanced condition (not usually the case)
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%% Calculate desired maximum shunt conductance and Rn trend
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G_max = 1/(2*w2*LL);
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G_desired = -0.8*G_max;
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%% Solve for required NR trend with frequency
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% Solve eq. (5) at the two frequencies of interest
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Rn_req_f1 = (1 + [-1 1]*sqrt(1 - 4*G_desired^2*(w1*LL)^2))/(2*G_desired);
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Rn_req_f2 = (1 + [-1 1]*sqrt(1 - 4*G_desired^2*(w2*LL)^2))/(2*G_desired);
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Zn_soln1 = (1 + [-1]*sqrt(1 - 4*G_desired.^2*(w.*LL).^2))./(2*G_desired);
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Zn_soln2 = (1 + [1]*sqrt(1 - 4*G_desired.^2*(w.*LL).^2))./(2*G_desired);
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Rn_req_f1 = max(Rn_req_f1); % Get the value of required Rn with minimum absolute value
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Rn_req_f2 = max(Rn_req_f2);
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%% Model Rn as -A*exp(-a*w)
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a_rn = log(Rn_req_f1/Rn_req_f2)/(w1 - w2);
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af_rn = 2*pi*a_rn;
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A_rn = -Rn_req_f1/exp(a_rn*w1);
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Rn = -A_rn*exp(af_rn*f);
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disp(['A = ' num2str(A_rn)])
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disp(['alpha = ' num2str(a_rn*1e9) ' (*1e9)'])
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if abs(G_desired) == 0
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Rn = zeros(size(f));
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end
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%% Calculate series and shunt impedances
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Zse = 1i.*w.*LR + 1./(1i.*w.*CL) + Rse; % Zse always passive in this case
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Ysh = 1./(1i*w*LL + Rn) + 1i*w*CR + Gsh; % Ysh includes effects from Rn
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% Ysh = 1./(1i*w*LL) + 1i*w*CR + G_desired; % Perfectly flat G_sh for test
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% Ysh = 1./(1i*w*LL + real(Zn_soln1)) + 1i*w*CR + Gsh;
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Ysh_passive = 1./(1i*w*LL) + 1i*w*CR + Gsh; % Passive Y_sh for other cells
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%% Calculate ABCD parameters (symmetric, active and passive)
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A_sym = 1 + Zse.*Ysh/2;
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B_sym = Zse.*(1 + Zse.*Ysh/4);
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C_sym = Ysh;
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D_sym = 1 + Zse.*Ysh/2;
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A_sym_passive = 1 + Zse.*Ysh_passive/2;
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B_sym_passive = Zse.*(1 + Zse.*Ysh_passive/4);
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C_sym_passive = Ysh_passive;
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D_sym_passive = 1 + Zse.*Ysh_passive/2;
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%% Cascade ABCD parameters for a given value of 'n'
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if mod(n, 2) == 0 % For passive-active-passive symmetry
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error('n must be an odd number');
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end
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At = zeros(size(f));Bt = zeros(size(f));Ct = zeros(size(f));Dt = zeros(size(f)); %Initialize ABCD vectors
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for i = 1:length(f)
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if n == 1 %If n = 1, only the active part matters, no passive
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nPassive = 0; %Multiplies the active part by the identity matrix
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else
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nPassive = (n - 1)/2;
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end
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% For passive-active-passive case
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ABCDtemp = [A_sym_passive(i) B_sym_passive(i);C_sym_passive(i) D_sym_passive(i)]^nPassive*[A_sym(i) B_sym(i);C_sym(i) D_sym(i)]*[A_sym_passive(i) B_sym_passive(i);C_sym_passive(i) D_sym_passive(i)]^nPassive;
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% For active-passive case (one active then n-1 passive)
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% ABCDtemp = [A_sym(i) B_sym(i);C_sym(i) D_sym(i)]*[A_sym_passive(i) B_sym_passive(i);C_sym_passive(i) D_sym_passive(i)]^(n-1);
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% For the all-active case (does not have open stopband issue solved, will blow up near transition frequency)
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% ABCDtemp = [A_sym(i) B_sym(i);C_sym(i) D_sym(i)]^n;
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At(i) = ABCDtemp(1, 1);
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Bt(i) = ABCDtemp(1, 2);
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Ct(i) = ABCDtemp(2, 1);
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Dt(i) = ABCDtemp(2, 2);
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end
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%% Overall cascaded S-parameters
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[S11, S12, S21, S22] = SPARAMS.abcd2s(At, Bt, Ct, Dt, Z0);
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%% Plot S-parameters
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fig1 = figure(1);
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% yyaxis left
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plot(f/1e9, 20*log10(abs(S11)));
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ylim([-20 5])
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hold on
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% yyaxis right
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plot(f/1e9, 20*log10(abs(S21)));
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xline([w1 w2]/(2*pi*1e9))
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% ylim([-2 4])
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xlim([min(f) max(f)]/1e9)
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ylim([-40 5])
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grid on
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xlabel('Freq (GHz)')
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ylabel('S-Params (dB)')
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fig1.Position = [734 564 560 420];
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title('Cascaded S-Params')
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legend({'S_{11}', 'S_{21}'}, 'Location','best')
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disp(['S21 at w1 = ' num2str(20*log10(abs(S21(closestIdx(f, f1))))) ' dB'])
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disp(['S21 at w2 = ' num2str(20*log10(abs(S21(closestIdx(f, f2))))) ' dB'])
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%% Calculate and plot dispersion
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Betap = (acosd((At + Dt)/2));
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fig2 = figure(2);
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plot(f/1e9, real(Betap));
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ylim([0 180])
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xlim([min(f) max(f)]/1e9)
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xline([w1 w2]/(2*pi*1e9))
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grid on
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xlabel('Freq (GHz)')
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ylabel('\beta p (deg)')
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title('Dispersion')
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fig2.Position = [1295 564 560 420];
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%% Plot shunt conductance
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fig3 = figure(3);
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plot(f/1e9, real(Ysh));
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xline([w1 w2]/(2*pi*1e9))
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yline(G_desired)
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ylim([-0.01 0])
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xlim([min(f) max(f)]/1e9)
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xlabel('Freq (GHz)')
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ylabel('Shunt G (S)')
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title('Shunt Conductance vs. Frequency')
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fig3.Position = [1296 58 560 420];
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%% Plot required Rn
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fig4 = figure(4);
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% Calculate explicit solution across frequency eq. (5)
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Rn_exact_1 = (1 + sqrt(1 - 4.*G_desired.^2.*(w.*LL).^2))./(2.*G_desired);
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Rn_exact_2 = (1 - sqrt(1 - 4.*G_desired.^2.*(w.*LL).^2))./(2.*G_desired);
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plot(f/1e9, Rn)
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hold on
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plot(f/1e9, real(Rn_exact_1))
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plot(f/1e9, real(Rn_exact_2))
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ylim([-100 0])
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xlim([min(f) max(f)]/1e9)
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xline([w1 w2]/(2*pi*1e9))
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grid on
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xlabel('Freq (GHz)')
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ylabel('R_n(f) (\Omega)')
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legend({'Exponential Rn', 'Exact Rn soln. 1', 'Exact Rn soln. 2'}, 'Location','best')
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title('R_n(\omega) vs. Frequency')
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fig4.Position = [734 58 560 420];
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%% S-parameters of a single active unit cell
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[S11_single, S12_single, S21_single, S22_single] = SPARAMS.abcd2s(A_sym, B_sym, C_sym, D_sym, Z0);
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fig5 = figure(5);
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plot(f/1e9, 20*log10(abs(S11_single)))
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hold on
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plot(f/1e9, 20*log10(abs(S21_single)))
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Leq = (Rn.^2 + (w.*LL).^2)./(w.^2*LL); % Eq. (2)
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A_loss = Rse.*real(Ysh).*(w.*sqrt(Leq*CL)).^2;
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B_loss = w.*(CL*Rse + Leq.*Gsh);
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wL = 1./sqrt(LL*CL); Zc = sqrt(LL/CL);
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alpha_L = 0.5*(Rse./Zc + real(Ysh).*Zc);
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% alpha_L = (wL./2./w).*(CL*Rse + LL*real(Ysh));
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% IL_approx = 20*log10(abs(exp(-alpha_L)));
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IL_approx = -8.686*((Rse./Zc + real(Ysh).*Zc))/2;%-8.686*alpha_L;
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disp(['Approx S21 at w1 = ' num2str(IL_approx(closestIdx(f, f1))) ' dB'])
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disp(['Approx S21 at w2 = ' num2str(IL_approx(closestIdx(f, f2))) ' dB'])
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plot(f/1e9, IL_approx)
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% plot(f/1e9, IL_approx1)
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ylim([-30 2])
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grid on
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xlabel('Freq (GHz)')
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ylabel('S-Params (dB)')
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title('Unit Cell S-Params')
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legend({'S_{11}', 'S_{21}', 'S_{21approx}'}, 'Location','best')
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fig5.Position = [171 564 560 420];
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%% Bloch Impedance
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fig6 = figure(6);
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Z_B = Bt./sqrt(At.^2 - 1); % From full cascaded
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% Z_B = B_sym./sqrt(A_sym.^2 - 1); % From only active
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% Z_B = B_sym_passive./sqrt(A_sym_passive.^2 - 1); % From only passive
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plot(f/1e9, abs(real(Z_B)))
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hold on
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plot(f/1e9, abs(imag(Z_B)))
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ylim([0 100])
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grid on
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xlabel('Freq (GHz)')
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title('Bloch Impedance')
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legend({'re(Z_B)', 'im(Z_B)'}, 'Location','best')
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fig6.Position = [173 61 560 420];
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%% Insertion Phase
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% fig7 = figure(7);
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% plot(f/1e9, rad2deg(angle(S21)));
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% grid on
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%% Rollett (k-delta) stability test
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% fig111 = figure(111);
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% delt = S11.*S22 - S12.*S21;
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% K = (1 - abs(S11).^2 - abs(S22).^2 + abs(delt).^2)./(2*abs(S12.*S21));
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% plot(f/1e9, K)
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% hold on
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% plot(f/1e9, abs(delt))
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% grid on
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% ylim([0.5 1.5])
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% legend({'K', '|\Delta|'})
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maxGaindB = -4.343*(Rse./Zc - G_max*Zc);
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disp(['Max possible gain = ' num2str(maxGaindB) ' dB'])
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%% Plot group velocity
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% % vg = (diff(Betap)./diff(2*pi*f)).^(-1);
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% vg = -diff(unwrap(angle(S21)))./diff(2*pi*f);
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% figure(121)
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% plot(f(1:end-1)/1e9, vg)
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%% Plot Bsh to check change in transition frequency
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% figure(122)
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% fsh = 1/(2*pi*sqrt(CR*LL));
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% % plot(f/1e9, real(Ysh))
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% hold on
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% plot(f/1e9, imag(Ysh))
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% hold on
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% xline([f1 f2 fsh]/1e9)
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% grid on
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% xlim([f1/1e9 - 1, f2/1e9 + 1])
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function [idx, delta] = closestIdx(vec, val)
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[delta, idx] = min(abs(vec - val));
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end
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