89

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•
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.