% Section 5.1, L. Vandenberghe, S. Boyd, and A. El Gamal % "Optimizing dominant time constant in RC circuits" % Original by Lieven Vandenberghe % Adapted for CVX by Joelle Skaf - 11/25/05 % Modified by Michael Grant - 3/8/06 % % we consider the problem of sizing an interconnect wire that connects % a voltage source and conductance G to a capacitive load C. We divide the % wire into n segments of length li, and width xi, i = 1,...,n, which is % constrained as 0 <= xi <= Wmax. The total area of the interconnect wire % is therefore sum(li*xi). We use a pi-model of each wire segment, with % capacitors beta_i*xi and conductance alpha_i*xi. % To minimize the total area subject to the width bound and a bound Tmax on % dominant time constant, we solve the SDP % minimize sum_{i=1}^20 xi*li % s.t. Tmax*G(x) - C(x) >= 0 % 0 <= xi <= Wmax % % Circuit parameters % n=21; % number of nodes; n-1 is number of segments in the wire m=n-1; % number of segments beta = 0.5; % segment has two capacitances beta*xi alpha = 1; % conductance is alpha*xi per segment Go = 1; % driver conductance Co = 10; % load capacitance wmax = 1.0; % upper bound on x % % Construct the capacitance and conductance matrices % C(x) = C0 + x1 * C1 + x2 * C2 + ... + xn * Cn % G(x) = G0 + x1 * G1 + x2 * G2 + ... + xn * Gn % We assemble the coefficient matrices together as follows: % CC = [ C0(:) C1(:) C2(:) ... Cn(:) ] % GG = [ G0(:) G1(:) G2(:) ... Gn(:) ] % CC = zeros(n,n,m+1); GG = zeros(n,n,m+1); % constant terms CC(n,n,1) = Co; GG(1,1,1) = Go; % segment values for i = 1 : n - 1, CC(i, i, i+1) = beta; CC(i+1,i+1,i+1) = beta; GG(i, i, i+1) = +alpha; GG(i+1,i, i+1) = -alpha; GG(i, i+1,i+1) = -alpha; GG(i+1,i+1,i+1) = +alpha; end % Reshape for easy Matlab use CC = reshape(CC,n*n,m+1); GG = reshape(GG,n*n,m+1); % % Compute points the tradeoff curve, and the four sample points % npts = 50; delays = linspace(400,2000,npts); xdelays = [ 370, 400, 600, 1800 ]; xnpts = length(xdelays); areas = zeros(1,npts); xareas = zeros(1,xnpts); sizes = zeros(m,xnpts); for i = 1 : npts + xnpts, if i > npts, xi = i - npts; delay = xdelays(xi); disp( sprintf( 'Particular solution %d of %d (Tmax = %g)', xi, xnpts, delay ) ); else, delay = delays(i); disp( sprintf( 'Point %d of %d on the tradeoff curve (Tmax = %g)', i, npts, delay ) ); end % % Construct and solve the convex model % cvx_begin sdp quiet variable x(m) variable G(n,n) symmetric variable C(n,n) symmetric minimize( sum(x) ) G == reshape( GG * [ 1 ; x ], n, n ); C == reshape( CC * [ 1 ; x ], n, n ); delay * G - C >= 0; 0 <= x <= wmax; cvx_end if i <= npts, areas(i) = cvx_optval; else, xareas(xi) = cvx_optval; sizes(:,xi) = x; % % Plot the step response % figure(xi+2); A = -inv(C)*G; B = -A*ones(n,1); T = linspace(0,2000,1000); Y = simple_step(A,B,T(2),length(T)); hold off; plot(T,Y,'-'); hold on; xlabel('time'); ylabel('v'); % compute threshold delay, elmore delay, dominant time constant tthres=T(min(find(Y(n,:)>0.5))); GinvC=full(G\C); tdom=max(eig(GinvC)); telm=max(sum(GinvC')); plot(tdom*[1;1], [0;1], '--', telm*[1;1], [0;1],'--', ... tthres*[1;1], [0;1], '--'); text(tdom,0,'d'); text(telm,0,'e'); text(tthres,0,'t'); title(sprintf('Step responses at the 21 nodes for solution (%d), Tmax=%g', xi, delay )); end end % % Plot the tradeoff curve % figure(1) ind = isfinite(areas); plot(areas(ind), delays(ind)); xlabel('Area'); ylabel('Tdom'); title('Area-delay tradeoff curve'); hold on for k = 1 : xnpts, text( xareas(k), xdelays(k), sprintf( '(%d)', k ) ); end % % Draw wires for the four solutions % figure(2) m2 = 2 * m; x1 = reshape( [ 1 : m ; 1 : m ], 1, m2 ); x2 = x1( 1, end : -1 : 1 ); y = [ - 0.5 * sizes(x1,:) ; + 0.5 * sizes(x2,:) ; - 0.5 * sizes(1,:) ]; x1 = reshape( [ 0 : m - 1 ; 1 : m ], m2, 1 ); x2 = x1( end : -1 : 1, 1 ); x = [ x1 ; x2 ; 0 ]; h = fill( x, y, ones(4*m+1,1)*[0.9,0.8,0.7,0.6] ); hold on h2 = plot( x, y, '-' ); axis([ -0.1, m + 0.1, min(y(:))-0.25, max(y(:))+0.1 ]); colormap(gray); caxis([-1,1]); title('Solutions at points on the tradeoff curve'); legends = {}; for k = 1 : xnpts, set( h(k), 'EdgeColor', get( h2(k), 'Color' ) ); legends{k} = sprintf( 'Tmax=%g', xdelays(k) ); end legend(legends{:},4);

Point 1 of 50 on the tradeoff curve (Tmax = 400) Point 2 of 50 on the tradeoff curve (Tmax = 432.653) Point 3 of 50 on the tradeoff curve (Tmax = 465.306) Point 4 of 50 on the tradeoff curve (Tmax = 497.959) Point 5 of 50 on the tradeoff curve (Tmax = 530.612) Point 6 of 50 on the tradeoff curve (Tmax = 563.265) Point 7 of 50 on the tradeoff curve (Tmax = 595.918) Point 8 of 50 on the tradeoff curve (Tmax = 628.571) Point 9 of 50 on the tradeoff curve (Tmax = 661.224) Point 10 of 50 on the tradeoff curve (Tmax = 693.878) Point 11 of 50 on the tradeoff curve (Tmax = 726.531) Point 12 of 50 on the tradeoff curve (Tmax = 759.184) Point 13 of 50 on the tradeoff curve (Tmax = 791.837) Point 14 of 50 on the tradeoff curve (Tmax = 824.49) Point 15 of 50 on the tradeoff curve (Tmax = 857.143) Point 16 of 50 on the tradeoff curve (Tmax = 889.796) Point 17 of 50 on the tradeoff curve (Tmax = 922.449) Point 18 of 50 on the tradeoff curve (Tmax = 955.102) Point 19 of 50 on the tradeoff curve (Tmax = 987.755) Point 20 of 50 on the tradeoff curve (Tmax = 1020.41) Point 21 of 50 on the tradeoff curve (Tmax = 1053.06) Point 22 of 50 on the tradeoff curve (Tmax = 1085.71) Point 23 of 50 on the tradeoff curve (Tmax = 1118.37) Point 24 of 50 on the tradeoff curve (Tmax = 1151.02) Point 25 of 50 on the tradeoff curve (Tmax = 1183.67) Point 26 of 50 on the tradeoff curve (Tmax = 1216.33) Point 27 of 50 on the tradeoff curve (Tmax = 1248.98) Point 28 of 50 on the tradeoff curve (Tmax = 1281.63) Point 29 of 50 on the tradeoff curve (Tmax = 1314.29) Point 30 of 50 on the tradeoff curve (Tmax = 1346.94) Point 31 of 50 on the tradeoff curve (Tmax = 1379.59) Point 32 of 50 on the tradeoff curve (Tmax = 1412.24) Point 33 of 50 on the tradeoff curve (Tmax = 1444.9) Point 34 of 50 on the tradeoff curve (Tmax = 1477.55) Point 35 of 50 on the tradeoff curve (Tmax = 1510.2) Point 36 of 50 on the tradeoff curve (Tmax = 1542.86) Point 37 of 50 on the tradeoff curve (Tmax = 1575.51) Point 38 of 50 on the tradeoff curve (Tmax = 1608.16) Point 39 of 50 on the tradeoff curve (Tmax = 1640.82) Point 40 of 50 on the tradeoff curve (Tmax = 1673.47) Point 41 of 50 on the tradeoff curve (Tmax = 1706.12) Point 42 of 50 on the tradeoff curve (Tmax = 1738.78) Point 43 of 50 on the tradeoff curve (Tmax = 1771.43) Point 44 of 50 on the tradeoff curve (Tmax = 1804.08) Point 45 of 50 on the tradeoff curve (Tmax = 1836.73) Point 46 of 50 on the tradeoff curve (Tmax = 1869.39) Point 47 of 50 on the tradeoff curve (Tmax = 1902.04) Point 48 of 50 on the tradeoff curve (Tmax = 1934.69) Point 49 of 50 on the tradeoff curve (Tmax = 1967.35) Point 50 of 50 on the tradeoff curve (Tmax = 2000) Particular solution 1 of 4 (Tmax = 370) Particular solution 2 of 4 (Tmax = 400) Particular solution 3 of 4 (Tmax = 600) Particular solution 4 of 4 (Tmax = 1800)