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<h1>Elmore delay sizing for a straight wire (GP)</h1>
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<pre class="codeinput">
<span class="comment">% Boyd, "Problems in VLSI design" (Lecture)</span>
<span class="comment">% Written for CVX by Almir Mutapcic 02/08/06</span>
<span class="comment">%</span>
<span class="comment">% We consider the problem of finding optimal width profile</span>
<span class="comment">% for a straight wire segmented into N parts. We want to</span>
<span class="comment">% minimize the Elmore delay, subject to limits on wire width</span>
<span class="comment">% and the total area. We use a pi-model for each wire segment.</span>
<span class="comment">% Problem can be formulated as GP:</span>
<span class="comment">%</span>
<span class="comment">% minimize D</span>
<span class="comment">% s.t. w_min &lt;= w &lt;= w_max</span>
<span class="comment">% area &lt;= Amax</span>
<span class="comment">%</span>
<span class="comment">% where variables are widths w (and arrival times T that are used</span>
<span class="comment">% to formulate the overall delay D expression).</span>
<span class="comment">%</span>
<span class="comment">% Important: We label root node as 1, and all the other nodes as</span>
<span class="comment">% node_label_in_the_paper + 1 (due to Matlab's convention).</span>
<span class="comment">% Also label nodes with increasing numbers downstream.</span>
<span class="comment">%********************************************************************</span>
<span class="comment">% user supplied data (problem constants and tree topology)</span>
<span class="comment">%********************************************************************</span>
N = 10+1; <span class="comment">% number of segments (including the root node which is labeled as 1)</span>
<span class="comment">% parent node array for the straight wire</span>
<span class="comment">% specifies which node is a unique parent for node i (always have a tree)</span>
parent = [0:N-1];
<span class="comment">% problem constants</span>
Rsource = 0.1;
l = 1*ones(N-1,1);
alpha = 1*ones(N-1,1);
beta = 1*ones(N-1,1);
gamma = 1*ones(N-1,1);
<span class="comment">% load capacitance at each node</span>
Cload = [0; ones(N-1,1)];
<span class="comment">% minimum and maximum width and area specification</span>
Wmin = 1;
Wmax = 10;
Amax = 50;
<span class="comment">%********************************************************************</span>
<span class="comment">% derived data (computed from user's data)</span>
<span class="comment">%********************************************************************</span>
<span class="comment">% compute children cell array (evaluate who are children for each node)</span>
children = cell(N,1);
leafs = [];
<span class="keyword">for</span> node = [1:N]
children{node} = find(parent == node);
<span class="keyword">if</span> isempty(children{node})
leafs(end+1) = node; <span class="comment">% leafs have no children</span>
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="comment">%********************************************************************</span>
<span class="comment">% optimization</span>
<span class="comment">%********************************************************************</span>
disp(<span class="string">'Generating the tradeoff curve...'</span>)
Darray = []; widths = [];
<span class="keyword">for</span> Amax = [10.05 10.5 11 12:2:20 22.5 25:5:60]
fprintf( <span class="string">'Amax = %5.2f: '</span>, Amax );
cvx_begin <span class="string">gp</span> <span class="string">quiet</span>
<span class="comment">% optimization variables</span>
variable <span class="string">w(N-1)</span> <span class="comment">% wire width</span>
variable <span class="string">T(N)</span> <span class="comment">% arrival time (Elmore delay to node i)</span>
<span class="comment">% objective is the critical Elmore delay</span>
minimize( max( T(leafs) ) )
subject <span class="string">to</span>
<span class="comment">% wire segment resistance is inversely proportional to widths</span>
R = alpha.*l./w;
R = [Rsource; R];
<span class="comment">% wire segment capacitance is an affine function of widths</span>
C_bar = beta.*l.*w + gamma.*l;
C_bar = [0; C_bar];
<span class="comment">% compute common capacitances for each node (C_tilde in GP tutorial)</span>
C_tilde = cvx( zeros(N,1) );
<span class="keyword">for</span> node = [1:N]
C_tilde(node,1) = Cload(node);
<span class="keyword">for</span> k = parent(node)
<span class="keyword">if</span> k &gt; 0; C_tilde(node,1) = C_tilde(node,1) + C_bar(k); <span class="keyword">end</span>;
<span class="keyword">end</span>
<span class="keyword">for</span> k = children{node}
C_tilde(node,1) = C_tilde(node,1) + C_bar(k);
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="comment">% now compute total downstream capacitances</span>
C_total = C_tilde;
<span class="keyword">for</span> node = N:-1:1
<span class="keyword">for</span> k = children{node}
C_total(node,1) = C_total(node,1) + C_total(k,1);
<span class="keyword">end</span>
<span class="keyword">end</span>
<span class="comment">% generate Elmore delay constraints</span>
R(1)*C_total(1) &lt;= T(1,1);
<span class="keyword">for</span> node = 2:N
R(node)*C_total(node) + T(parent(node),1) &lt;= T(node,1);
<span class="keyword">end</span>
<span class="comment">% collect all the constraints</span>
sum(w.*l) &lt;= Amax;
Wmin &lt;= w &lt;= Wmax;
cvx_end
<span class="comment">% display and store computed values</span>
fprintf(<span class="string">'delay = %3.2f\n'</span>,cvx_optval);
Darray = [Darray cvx_optval];
widths = [widths w];
<span class="keyword">end</span>
<span class="comment">% indices of four taper designs on the tradeoff curve</span>
Amax = [10.05 10.5 11 12:2:20 22.5 25:5:60];
A11ind = find(Amax == 11);
A20ind = find(Amax == 20);
A35ind = find(Amax == 35);
A50ind = find(Amax == 50);
<span class="comment">% plot the tradeoff curve</span>
figure, clf
plot(Darray,Amax, <span class="keyword">...</span>
Darray(A11ind),Amax(A11ind),<span class="string">'ro'</span>,<span class="keyword">...</span>
Darray(A20ind),Amax(A20ind),<span class="string">'ro'</span>,<span class="keyword">...</span>
Darray(A35ind),Amax(A35ind),<span class="string">'ro'</span>,<span class="keyword">...</span>
Darray(A50ind),Amax(A50ind),<span class="string">'ro'</span>);
xlabel(<span class="string">'Elmore delay D'</span>); ylabel(<span class="string">'Amax'</span>);
disp(<span class="string">'Optimal tradeoff curve plotted.'</span>)
<span class="comment">% plot four taper designs</span>
figure, clf
w1 = widths(:,A50ind);
w2 = widths(:,A35ind);
w3 = widths(:,A20ind);
w4 = widths(:,A11ind);
plot_four_tapers(w1,w2,w3,w4);
</pre>
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<pre class="codeoutput">
Generating the tradeoff curve...
Amax = 10.05: delay = 255.72
Amax = 10.50: delay = 241.04
Amax = 11.00: delay = 228.67
Amax = 12.00: delay = 209.98
Amax = 14.00: delay = 184.90
Amax = 16.00: delay = 168.19
Amax = 18.00: delay = 156.01
Amax = 20.00: delay = 146.74
Amax = 22.50: delay = 137.78
Amax = 25.00: delay = 130.82
Amax = 30.00: delay = 120.77
Amax = 35.00: delay = 113.95
Amax = 40.00: delay = 109.06
Amax = 45.00: delay = 105.43
Amax = 50.00: delay = 102.96
Amax = 55.00: delay = 101.76
Amax = 60.00: delay = 101.60
Optimal tradeoff curve plotted.
</pre>
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