3.3. TRANSFER MATRICES OF COMMON SWITCHING CIRCUITS 19

Figure 3.3: Circuit diagram structures for fanout, fanin, and crossover.

switching operators. Certain electronic technologies allow for fanin structures due to other circuit

element outputs being at a high-impedance or other state. As an example, certain current-mode

circuitry allow for a fanin to operate as an AND-type operation. In those cases, the fanin transfer

matrix is identical to that of the matrix characterizing the appropriate switching circuit gate. Many

voltage-mode circuits can only include fanin structures when the inputs are driven by disjoint

outputs thereby avoiding a short-circuit situation. In general, fanins are not allowed and thus we

model them in this case with null row vectors for the disallowed case. Null vectors occur in the

transfer matrix when those speciﬁc disallowed input cases are simply excluded in the calculation

of the matrix. For these reasons, the fanin transfer matrix is technology dependent.

In the case where fanins with diﬀering input stimulus values are disallowed, the transfer

matrix may be computed using the relationship in Equation 3.5 as:

F

I

D j00ih0j C j11ih1j D

2

6

6

4

1

0

0

0

3

7

7

5

1 0

C

2

6

6

4

0

0

0

1

3

7

7

5

0 1

D

2

6

6

4

1 0

0 0

0 0

0 1

3

7

7

5

3.3.3 CROSSOVER STRUCTURE TRANSFER MATRIX

In a later section, we will describe how the transfer matrix can be constructed for an interconnec-

tion of basic operators, or, a switching network. Because the network transfer matrix is dependent

upon the topology of the network, the case where conductors cross one another in the plane must

be accounted for. Multiple crossovers can be dealt with as a series of single crossovers, thus a

fundamental structure to be considered is the single crossing of two conductors that are electri-

cally isolated. Such a structure is depicted as the rightmost diagram in Figure 3.5. e crossover

matrix expresses the four input-output relationships h00j ! h00j, h01j ! h10j, h10j ! h01j, and

h11j ! h11j. Equation 3.5 indicates that the transfer matrix can be computed as:

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