89 the dc voltage drop v v dc k at node k is equal to

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Intermediate Algebra
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Chapter 8 / Exercise 65
Intermediate Algebra
Mckeague
Expert Verified
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Intermediate Algebra
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Chapter 8 / Exercise 65
Intermediate Algebra
Mckeague
Expert Verified
The DC voltage drop V v dc k at node k is equal to the sum of the voltage drops across wires on the (unique) path from node k to the root. It can be expressed as V v dc k = m summationdisplay j =1 i dc j summationdisplay i ∈N ( j,k ) R i , (31) where N ( j,k ) consists of the indices of the branches upstream from nodes j and k , i.e. , i ∈ N ( j,k ) if and only if R i is in the path from node j to the root and in the path from node k to the root. The power supply noise at a node can be found as follows. The AC voltage at node k is equal to v ac k ( t ) = m summationdisplay j =1 i ac j ( t ) summationdisplay i ∈N ( j,k ) R i . We assume the AC current draws are independent, so the RMS value of v ac k ( t ) is given by the squareroot of the sum of the squares of the RMS value of the ripple due to each other node, i.e. , RMS( v ac k ) = m summationdisplay j =1 RMS( i ac j ) summationdisplay i ∈N ( j,k ) R i 2 1 / 2 . (32) The problem is to choose wire widths w i that minimize the total wire area n i = k w k l k subject to the following specifications: maximum allowable DC voltage drop at each node: V v dc k V dc max , k = 1 ,...,m, (33) where V v dc k is given by (31), and V dc max is a given constant. maximum allowable power supply noise at each node: RMS( v ac k ) V ac max , k = 1 ,...,m, (34) where RMS( v ac k ) is given by (32), and V ac max is a given constant. upper and lower bounds on wire widths: w min w i w max , i = 1 ,...,n, (35) where w min and w max are given constants. maximum allowable DC current density in a wire: summationdisplay j ∈M ( k ) i dc j slashBigg w k ρ max , k = 1 ,...,n, (36) where M ( k ) is the set of all indices of nodes downstream from resistor k , i.e. , j ∈ M ( k ) if and only if R k is in the path from node j to the root, and ρ max is a given constant. 90
maximum allowable total DC power dissipation in supply network: n summationdisplay k =1 R k summationdisplay j ∈M ( k ) i dc j 2 P max , (37) where P max is a given constant. These specifications must be satisfied for all possible i k ( t ) that satisfy (30). Formulate this as a convex optimization problem in the standard form minimize f 0 ( x ) subject to f i ( x ) 0 , i = 1 ,...,p Ax = b. You may introduce new variables, or use a change of variables, but you must say very clearly what the optimization variable x is, and how it corresponds to the problem variables w ( i.e. , is x equal to w , does it include auxiliary variables, . . . ?) what the objective f 0 and the constraint functions f i are, and how they relate to the objectives and specifications of the problem description why the objective and constraint functions are convex what A and b are (if applicable). 11.3 Optimal amplifier gains. We consider a system of n amplifiers connected (for simplicity) in a chain, as shown below. The variables that we will optimize over are the gains a 1 ,...,a n > 0 of the amplifiers. The first specification is that the overall gain of the system, i.e. , the product a 1 · · · a n , is equal to A tot , which is given.

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