60 IRE 4 S1ire standards on circuits definitions of terms for linear signal flow graphs 1960

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PROCEEDINGS OF THE IRE
IRE Standards on Circuits: Definitions of Terms
for Linear Signal Flow Graphs, 1960*
COMMITTEE PERSONNEL
60 IRE 4. S1
Committee on Circuits
J. T. BANGERT, Chairman 1958 1960
W. A. LYNCH, Chairman 1956-1958
S. J. MWASON, Vice-Chairman 1958-1960 J. T. BANGERT, Vice-Chairman 1956-1958
WV. R. Bennett R. Kahal C. H. Page
J. G. Brainerd B. N. Kinariwala E. H. Perkins
A. R. D'Heedene H. L. Krauss E. J. Robb
T. R. Finch J. G. Linvill J. J. Suran
R. M. Foster J. C. Logue A. P. Stern
W. H-. Huggins S. J. M\lasonX W. N. Tuttle
Standards Committee
1959-1960
R. F. SHEA, Chairman
J. G. KREER, JR., Vice-Chairman C. H. PAGE, Vice-Chairman L. G. CUMMING, Vice-Chairman
J. Avins R. J. Farber A. E. Kerwien R. C. MJoyer
AN. F. Bailey D. G. Fink E. R. Kretzmer J. H. Mulligan, Jr.
M. W. Baldwin, Jr. G. L. Fredenidall G. S. Ley A. A. Oliner
J. T. Bangert E. A. Gerber Wayne M\lason 1\1. L. Phillips
XW. R. Bennett A. B. Glenn D. E. Maxwell R. L. Pritchard
J. G. Brainerd V. M. Graham P. Mertz P. A. Redlhead
P. S. Carter R. A. Hackbusch H. I. Metz R. Serrell
A. G. Clavier R. T. Haviland H. R. Mimno W. A. Shipmau
S. Doba, Jr. A. G. Jensen E. Mittelman H. R. Terhune
R. D. Elbourn R. WV. Johnston L. H. Montgomery, Jr. E. Weber
G. A. Espersen 1. Kerney G. A. Morton R. B. Wilcox
W. T. Wintringham
Definitions Coordinator
C. H. Page
Branch. A line segment joining two nodes, or joining one
node to itself.
V\ntv* n]Q Mv,,n tad Rvwrh.
Branch Transmittance. The ratio of branch output signal
to branch input signal.
Cascade Node (Branch). A node (branch) not contained
Branch Input Signal. The signal, xX, at the input end of in a loop.
branch jk. Cofactor (or Path Cofactor). See Path (Loop) Factor.
Branch Output Signal (of branch jk). The component of Dependent Node. A node having one or more incoming
signal Xk contributed to node k via branch jk. branches.
* Approved bv the IRE Standards Committee, January 14, 1960. Reprints of this Standard 60 IRE 4.S1, may be purchased while available
from the Institute of Radio Engineers, 1 East 79th Street, New York, N. Y., at $0.25 per copy. A 20 per cent discount will be allowed for
100 or more copies mailed to one address.
1960 1611
PROCEEDINGS OF T1f11EIRE
Directed Branch. A branch having an assigned direction.
Note: In identifying the branch direction, the
branch, jk, may be thought of as outgoing from node
j and incoming at node k. Alternatively, branch jk
may be thought of as originating or having its input
at node j, and terminating or having its output at node
k. The assigned direction is conveniently indicated
by an arrow pointing from node j toward node ke
Feedback Node (Branch). A node (branch) contained in
a loop.
Graph Determinant. One plus the sum of the loop-set
transmittances of all non-touching loop sets contained in
the graph.
Note 1: The graph determinant is conveniently ex
pressed in the form:
A (t ,Li>+ , LiLj LiLjLk +* e
where Li is the loop transmittance of the ith loop of
the graph, and the first summation is over all of the
different loops of the graph, the second is over all of
the different pairs of non-touching loops, and the
third is over all the different triplets of non-touching
loops, etc.
Note 2: The graph determinant may be written al-
ternatively as
A = [( Li)(I - L2) . . . (1 Ln)t
where Li, L2,, L,,° are the loop transmittances of
the n different loops in the graph, and where the
dagger indicates that, after carrying out the mutlti
plications within the brackets, a term will be dropped
if it contains the transmittance product of two
touching loops.
Note 3 The graph determinant reduces to the re
turn difference for a graph having only one loop.
Note 4: The graph determinant is equal to the de
terminant of the coefficient equations.
Graph Transmittance. The ratio of signal at some speci-
fied dependent node, to the signal applied at some speci
fied source node.
Note: The graph transmittance is the weighted suim
of the path transmittances of the different open paths
from the designated source node to the designated de
pendent node, where the weight for each path is the
path factor divided by the graph determinant.
Loop (Feedback Loop). A simple closed path.
Loop Factor. See Path Factor.
Loop Graph. A signal flow graph each of whose branches
is contained in at least one loop.
Note: Any loop graph embedded in a general graph
can be found by removing the cascade branches,
Loop-Set Transmittance. The product of the negatives
of the loop transmittances of the loops in a set.
Loop Transmittance. The product of the branci trans
mittances ir a loop.
Loop Transmittance of a Branch The loop transmit-
tance of an interior node inserted ir that branch
Note: A branch may always be replaced by an
equivalent sequence of branches, thereby creating irn
terior nodes.
Loop Transmittance of a Node. T5he graph transmittance
from the source node to the sink node created by splitting
the designated node
Node. One of the set of discrete points in a flow graph,
Node Absorption. A flow-graph transformation whereby
one or more dependent nodes disappear and the resulting
graph is equivalent with respect to the remaining node
signals.
Note: For example, a circuit analog of node absorp
tion is the Star-Delta trarisformation
Node Signal. A variable, Xk, associated with node kJ
Non-Touching Loop Set. A set of loops no two of which
have a common node.
Open Path. A path along which no node appears more
than once.
Path. Any continuous succession of branches, traversed
in the indicated branch directions.
Path (Loop) Factor. The graph determinant of that part
of the graph not touching the specified path (loop).
Note I A paith (loop) factor is obtainable from the
graph determinant by strikinig out all terms containing
transmittance products of loops which touch that
path (loop).
Note 2: For loop Lk the loop factor is
dA/dLk-
Path Transmittance. The product of the branch trans-
mittances in that path.
Return Difference. One minus the loop transmittance.
Signal Flow Graph. A network of directed branches in
which each dependent node signal is the algebraic sum of
the incoming branch signals at that node.
Note: Thus, Xltk+X2tk I xXXn1.k=Xk at each
dependent node k, where tjk is the branch transmit-
tance of braneh jk.
Sink Node. A node having only incoming branches.
Source Node. A node havng only outgoing branches.
Split Node. A node that has been separat1ed ito a source
node and a sink node.
Note : Splitting a node nterrupts all signal trans-
mission through that node.
Note 2: In splitting a node, all incoming branches
are associated with the resulting sink node and all
outgoing branches with the resulting source node.
162 September