Mason IRE 1956

PROCEEDINGS OF THE IRE

CONCLUSION

The elliptic function transformation (1). is used herefor the purpose of locating zeros and poles of a low-passfilter network function. Charts of the type shown inFigs. 7 to 12 may be prepared for any range of applica-tion whenever desired. The compactness of the expres-sions that give the tolerance and other characteristicquantities makes the preparation of these charts whichrepresent a whole group of network functions withmany singularities a matter of evaluating only a few

terms together with a few rational operations. Thesecharts, after they are prepared, will be very helpful fordesign purposes. For instance, if a required attenuationbeyond twice the cut-off frequency must be greater than13 db, Fig. 10 indicates that a filter function with thecharge arrangement of Fig. 3(b) and values of a and cof 0.810 and 0.673 respectively will satisfy the require-ment. The locations of all poles and zeros of this filterare determined in the z-plane. The locations of zeros andpoles in the s-plane may be found by applying the in-verse transformation.

Feedback Theory -Further Properties of SignalFlow Graphs*SAMUEL J. MASONt

Summary-A way to enhanceWriting gain at a glance.Dr. Tustin extendedProof appended.Examples illustrativePray not frustrative.

BACKGROUNDT HERE ARE many different paths to the solution

of a set of linear equations. The formal methodinvolves inversion of a matrix. We know, how-

ever, that there are many different ways of inverting amatrix: determinantal expansion in minors, systematicreduction of a matrix to diagonal form, partitioninginto submatrices, and so forth, each of which has itsparticular interpretation as a sequence of algebraicmanipulations within the original equations. A deter-minantal expansion of special interest is

D = >, alia2ja3k ... an (1)where amp is the element in the mth row and pthcolumn of a) determinant having n rows, and the sum-mation is taken over all possible permutations of thecolumn subscripts. (The sign of each term is positiveor negative in accord with an even or odd number ofsuccessive adjacent column-subscript interchanges re-quired to produce a given permutation.) Since the solu-tion of a set of linear equations involves ratios of deter-minantal quantities, (1) suggests the general idea that

* Original manuscript received by the IRE, August 16, 1955;revised manuscript received, February 27, 1956. This work was sup-ported in part by the Army (Signal Corps), the Air Force (Office ofScientific Research, Air Research and Development Command),and the Navy (Office of Naval Research).

t Dept. of Elect. Engrg. and Res. Lab. of Electronics, M.I.T.,Cambridge, Mass.

a linear system analysis problem should be interpretableas a search for all possible combinations of somethingor other, and that the solution should take the form ofa sum of products of the somethings, whatever they are,divided by another such sum of products. Hence, in-stead of undertaking a sequence of operations, we canfind the solution by looking for certain combinationsof things. The method will be especially useful if thesecombinations have a simple interpretation in the con-text of the problem.

aAb3tc --2

Fig. 1-An electrical network graph.

As a concrete illustration of the idea, consider theelectrical network graph shown in Fig. 1. For simplicity,let the branch admittances be denoted by letters a, b,c, d, and e. This particular graph has three independentnode-pairs. First locate all possible sets of three brancheswhich do not contain closed loops and write the sum oftheir branch admittance products as the denominatorof (2).

ab + ac + bc + bdabd + abe + acd + ace + ade + bcd + bce + bde

Now locate all sets of two branches which do not form

920 July

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Mason: Further Properties of Signal Flow Graphs

closed loops and which also do not contain any paths fromnode 1 to ground or from node 2 to ground. Write the sumof their branch admittance products as the numerator ofexpression (2). The result is the transfer impedancebetween nodes 1 and 2, that is, the voltage at node 2when a unit current is injected at node 1. Any imped-ance of a branch network can be found by this process.'So much for electrical network graphs. Our main

concern in this paper is with signal flow graphs,2 whosebranches are directed. Tustin3 has suggested that thefeedback factor for a flow graph of the form shown inFig. 2 can be formulated by combining the feedback loop

The purposes of this paper are: to extend the methodto a general form applicable to any flow graph; topresent a proof of the general result; and to illustratethe usefulness of such flow graph techniques by applica-tion to practical linear analysis problems. The proofwill be given last. It is tempting to add, at this point,that a better understanding of linear analysis is a greataid in problems of nonlinear analysis and linear or non-linear design.

A BRIEF STATEMENT OF SOME ELEMENTARYPROPERTIES OF LINEAR SIGNAL

FLOW GRAPHSa b c d e f g

* 9- t a, t 0 t PI t PI t t PI eK)jh

Fig. 2-The flow graph of an automatic control system.

gains in a certain way. The three loop gains areT,= bchT2= cdiT3 =

and the forward path gain isGo= abcdefg.

A signal flow graph is a network of directed brancheswhich connect at nodes. Branch jk originates at node jand terminates upon node k, the direction from j to kbeing indicated by an arrowhead on the branch. Eachbranch jk has associated with it a quantity called thebranch gain gjk and each node j has an associated quan-tity called the node signal xi. The various node signals(3a) are related by the associated equations

(3b)(3c)

A X jg1k = Xk,i

k = 1, 2, 3,

The graph shown in Fig. 3, for example, has equations

(4)The gain of the complete system is found to be

[1- (Ti + T2)](1 - T3) (5)and expansion of the denominator yields

Go .~~~~(6)1-(Ti + T2+ T3)+ (T1T3+ T2T3)Tustin recognized the denominator as unity plus thesum of all possible products of loop gains taken one ata time (T1+T2+Ts), two at a time, (T1T3+T2T3),three at a time, and so forth, excluding products ofloops that touch or partially coincide. The productsT1T2 and T1T2T3 are properly and accordingly missingin this particular example. The algebraic sign alter-nates, as shown, with each succeeding group of products.

Tustin did not take up the general case but gave ahint that a graph having several different forward pathscould be handled by considering each path separatelyand superposing the effects. Detailed examination of thegeneral problem shows, in fact, that the form of (6)must be modified to include possible feedback factors inthe numerator. Otherwise (6) applies only to thosegraphs in which each loop touches all forward paths.

1 Y. H. Ku, "Resume of Maxewll's and Kirchoff's rules for net-work analysis," J. Frank. Inst., vol. 253, pp. 211-224; March,1952.

2 S. J. Mason, "Feedback theory-some properties of signal flowgraphs," PROC. IRE, vol. 41, pp. 1144-1156; September, 1953.

3 A. Tustin, "Direct Current Machines for Control Systems,"The Macmillan Company, New York, pp. 45-46, 1952.

ax, + dx3 = X2bx2 + fx4 = X3eX2 + Cx3 = X4

gX3 + hx4 = X5.We shall need certain definitions. A source is a node

having only outgoing branches (node 1 in Fig. 3). Asink is a node having only incoming branches. A pathis any continuous succession of branches traversed inthe indicated branch directions. A forward path is apath from source to sink along which no node is encoun-tered more than once (abch, aeh, aefg, abg, in Fig. 3).

d f

e 9

Fig. 3-A simple signal flow graph.

A feedback loop is a path that forms a closed cycle alongwhich each node is encountered once per cycle (bd, cf,def, but not bcfd, in Fig. 3). A path gain is the product ofthe branch gains along that path. Theloop gain of afeedback loop is the product of the gains of the branchesforming that loop. The gain of a flow graph is the signalappearing at the sink per unit signal applied at thesource. Only one source and one sink need be considered,since sources are superposable and sinks are independ-ent of each other.

Additional terminology will be introduced as needed.

(7)

(8a)(8b)(8c)(8d)

1956 921


PROCEEDINGS OF TIHE IRE

GENERAL FORMULATION OF FLOW GRAPH GAINTo begin with an example, consider the graph shown

in Fig. 4. This graph exhibits three feedback loops,whose gains are

T1-=h (9a)(9b)(9c)

T2= fgT3 de

and two forward paths, whose gains areG1 = abG2= ceb.

'h

(a) .~I f -Ga-abc+d(I - bf)I - ce -hf-cg-dgfe +aecg

e( b) -- - - --

-*----a T = ae

(c) 4-- -- --A---T2=bfb,b

(lOa) g(lOb) (d) -k--C- -

e f g(e) -

de 9

(f ) *---10- --, a

* T,= c g

_

T4=dgfe

- rT1 T3 = aecgc

a(g k o G,=abc,A =I

b c',

bFig. 4-A flow graph with three feedback loops.

To find the graph gain, first locate all possible sets ofnontouching loops and write the algebraic sum of theirgain products as the denominator of (11).

G1(1- T1 - T2) +G2( 1- T1)G= -

1 - T -T2-T3+ T,T3(11)

Each term of the denominator is the gain product ofa set of nontouching loops. The algebraic sign of theterm is plus (or minus) for an even (or odd) number ofloops in the set. The graph of Fig. 4 has no sets of threeor more nontouching loops. Taking the loops two at atime we find only one permissible set, T1T3. When theloops are taken one at a time the question of touchingdoes not arise, so that each loop in the graph is itselfan admissible "set." For completeness of form we mayalso consider the set of loops taken "none at a time" and,by analogy with the zeroth power of a number, interpretits gain product as the unity term in the denominatorof (11). The numerator contains the sum of all forwardpath gains, each multiplied by a factor. The factor fora given forward path is made up of all possible sets ofloops which do not touch each other and which also donot touch that forward path. The first forward path(G1 = ab) touches the third loop, and T3 is thereforeabsent from the first numerator factor. Since the secondpath (G2=ceb) touches both T2 and T3, only T1 entersthe second factor.The general expression for graph gain may be writ-

ten as

-f( h ) 0 .1 . .- -oG =d ,A =I bf

I Ns b

d

Fig. 5-Identification of paths and loop sets.

GkAk

k (12a)

whereinGk = gain of the kth forward pathA = 1 - Pml + L Pm2 - LPm3 +

m m m

PmrT = gain product of the mth possible combina-tion of r nontouching loops

Ak= the value of A for that part of the graph

(12b)(12c)

(12d)

not touching the kth forward path. (12e)The form of (12a) suggests that we call A the determi-nant of the graph, and call Ak the cofactor of forwardpath k.A subsidiary result of some interest has to do with

graphs whose feedback loops form nontouching sub-graphs. To find the loop subgraph of any flow graph,simply remove all of those branches not lying in feed-back loops, leaving all of the feedback loops, and noth-ing but the feedback loops. In general, the loop sub-graph may have a number of nontouching parts.The useful fact is that the determinant of a completeflow graph is equal to the product of the determinants ofeach of the nontouching parts in its loop subgraph.

A =I-T1-T T2 -T 3 -T4 +rTI 3

922 Jlty


Mason: Further Properties of Signal Flow Graphse

(a) y D

d

(b) *d

f

(c)c

(d).

d(l-be) +abcI-b- be

a + beI-ad-be-cf- bcd-afe

c +abI-od- be -oabf -cd -cf

i(I-g-h-cd+gh)+ae(l-h)+ bf (i-g)+adf +bceI -g-h-cd+gh

flow graph are in cause-and-effect form, each variableexpressed explicitly in terms of others, and since physi-cal problems are often very conveniently formulated injust this form, the study of flow graphs assumes practi-cal slgnificance.

Z5ilI

(a)

eI i 2 e2 i3 e3I 2 :4 I * e3

-zI -Y2 -Z 3 -Y4

( b)I a b

(e ) f

I e d

g ( -hi -jc-hbcd +hijc) +a ie (1-jc)+ a bcde1- fg -hi-jc -faie - hbcd -fabcde+ fghi +fg jc+hijc +foiejc+fghbcd- f ghijc

Fig. 6-Sample flow graphs and their gain expressions.

ILLUSTRATIVE EXAMPLES OF GAIN EVALUATIONBY INSPECTION OF PATHS AND LooP SETS

Eq. (12) is formidable at first sight but the idea issimple. More examples will help illustrate its simplicity.Fig. 5 (on the previous page) shows the first of these dis-played in minute detail: (a) the graph to be solved; (b)-(f) the loop sets contributing to A; (g) and (h) the for-

i e 2 e2 3 e3

Fig. 7-The transfer impedance of a ladder.

Consider the ladder network shown in Fig. 7(a). Theproblem is to find the transfer impedance e3/il. One pos-sible formulation of the problem is indicated by the flowgraph Fig. 7(b). The associated equations state thatei =zl(il-i2), i2 =y2(el-e2), and so forth. By inspectionof the graph,

e3 Zly2Z3y4Zsii 1 + Z1y2 + Y2Z3 + Z3Y4 + Y4Z5 + Zly2Z3y4 + Zly2y4Z6 + y2Z3y4Z6 (13a)

or, with numerator and denominator multiplied byYlYsY5 = 1 /z1z3z6,

e3 Y2Y4ii Yly3y5 + Y2Y3YS + YlY2YS + yly4y5 + Y1Y3Y4 + y2y4y5 + Y2Y3y4 + yly2y4 (13b)

ward paths and their cofactors. Fig. 6 gives several ad-ditional examples on which you may wish to practiceevaluating gains by inspection.

ILLUSTRATIVE APPLICATIONS OF FLOW GRAPHTECHNIQUES TO PRACTICAL

ANALYSIS PROBLEMSThe study of flow graphs is a fascinating topological

game and therefore, from one viewpoint, worthwhile inits own right. Since the associated equations of a linear

This result can be checked by the branch-combinationmethod mentioned at the beginning of this paper.A different formulation of the problem is indicated by

the graph of Fig. 7(c), whose equations state thati3-= y5e3, e2= e3+z4i3, i2=i3+y3e2, and so forth. In thephysical problem i1 is the primary cause and e3 thefinal effect. We may, however, choose a value of e3 andthen calculate the value of i1 required to produce thate3. The resulting equations will, from the analysis view-point, treat e3 as a primary cause (source) and il as

9231956



the final effect (sink) produced by the chain of calcu-lations. This does not in any way alter the physical roleof ii. The new graph (c) may appear simpler to solvethan that of (b). Since graph (c) contains no feedbackloops, the determinant and path cofactors are all equalto unity. There are many forward paths, however, andcareful inspection is required to identify the sum oftheir gains as

- = ylz2y3z4y5 + y1z2y3 + y1z2y6 + y1z4y6e3

+ y3Z4y5 + yl + y3 + y5

(a) (b)

(13c)which proves to be, as it should, the reciprocal of (13b).Incidentally, graph (c) is obtainable directly from graph(b), as are all other possible cause-and-effect formula-tions involving the same variables, by the process ofpath inversion discussed in a previous paper.2 Thisexample points out the two very important facts: 1)the primary physical source does not necessarily appearas a source node in the graph, and 2) of two possibleflow graph formulations of a problem, the one havingfewer feedback loops is not necessarily simpler to solveby inspection, since it may also have a much more com-plicated set of forward paths.

Fig. 8(a) offers another sample analysis problem,determination of the voltage gain of a feedback ampli-fier. One possible chain of cause-and-effect reasoning,which leads from the circuit model, Fig. 8(b), to theflow graph formulation, Fig. 8(c), is the following. Firstnotice that eg, is the difference of e1 and ek. Next expressii as an effect due to causes eg1 and ek, using superposi-tion to write the gains of the two branches enteringnode i1. The dependency of eg2 upon il follows directly.Now, e2 would be easy to evaluate in terms of eithereg2 or if if the other were zero, so superpose the twoeffects as indicated by the two branches entering nodee2. At this point in the formulation ek and if are as yetnot explicitly specified in terms of other variables. It isa simple matter, however, to visualize ek as the super-position of the voltages in Rk caused by i1 and if, and toidentify if as the superposition of two currents in Rfcaused by ek and e2. This completes the graph.The path from ek to eg0 to i1 may be lumped in

parallel with the branch entering i1 from ek. This simpli-fication, convenient but not necessary, yields the graphshown in Fig. 9. We could, of course, have expressed ilin terms of e1 and ek at the outset and arrived at Fig. 9directly. All simplifications of a graph are themselves

(C)Fig. 8-Voltage gain of a feedback amplifier. (a) A feedback ampli-

fier; (b) The midband linear incremental circuit model; (c) Apossible flow graph.

Rk

Fig. 9-Elimination of superfluous nodes e,j and e,2.

possible formulations. The better our perception ofthe workings of a circuit, the fewer variables will weneed to introduce at the outset and the simpler will bethe resulting flow graph structure.

In discussing the feedback amplifier of Fig. 8(a) itis common practice to neglect the loading effect of thefeedback resistor Rf in parallel with Rk, the loadingeffect of Rf in parallel with R2, and the leakage trans-mission from ek to e2 through Rf. Such an approximationis equivalent to the removal of the branches from ekto if and if to e2 in Fig. 9. It is sometimes dangerous tomake early approximations, however, and in this case nioappreciable labor is saved, since we can write the exactanswer by inspection of Fig. 9:

lAlA2RlR2 RAR-1ARkr2R2e2 (r1 + R1)(r2 + R2) L Rf I (r1 + R1)(Rf)(r2 + R2)el 1 + (/l1 + 1)Rk RA, r2R2 +(l + 1)/2R,RkR2 (g1 + 1)Rkr2R2r+ + +R++ +++ 1 Rf Rf1(r2 R2) Rf1(ri + RI)(r2 + R2) Rf,(r, + Ri)(r2 R2)

(14)

924 July


Mason: Further Properties of Signal Flow Graphs

The two forward paths are elile2 and elilekife2, the firsthaving a cofactor due to loop ekif. The principal feed-back loop is ile2ifek and its gain is the fifth term of thedenominator. Physical interpretations of the variouspaths and loops could be discussed but our main pur-pose, to illustrate the formulation of a graph and theevaluation of its gain by inspection, has been covered.As a final example, consider the calculation of micro-

wave reflection from a triple-layered dielectric sand-wich. Fig. 10(a) shows the incident wave A, the re-flection B, and the four interfaces between adjacentregions of different material. The first and fourth inter-faces, of course, are those between air and solid. Letri be the reflection coefficient of the first interface,relating the incident and reflected components of tan-gential electric field. It follows from the continuity oftangential E that the interface transmission coefficientis 1 +r1, and from symmetry that the reflection coeffi-cient from the opposite side of the interface is the nega-tive of ri. A suitable flow graph is sketched in Fig.10(b). Node signals along the upper row are right-going waves just to the left or right of each interface,those on the lower row are left-going waves, and quanti-ties d are exponential phase shift factors accounting forthe delay in traversing each layer.Apart from the first branch r1, the graph has the

same structure as that of Fig. 6(e). Hence the reflectiv-ity of the triple layer will be

B--= ri + (1 + ri) (1 - rl)G (15)A

where G is in the same form as the gain of Fig. 6(e). Weshall not expand it in detail. The point is that theanswer can be written by inspection of the paths andloops in the graph.

PROOF OF THE GENERAL GAIN EXPRESSIONIn an earlier paper2 a quantity A was definied as

A = (1 - Ti')(I - T2') ... (1 - Tn') (16)for a graph having n nodes, where

Tk' =loop gain of the kth node as computed with allhigher-numbered nodes split.

Splitting a node divides that node into a new source anda new sink, all branches entering that node going withthe new sink and all branches leaving that node goingwith the new source. The loop gain of a node was de-fined as the gain from the new source to the new sink,when that node is split. It was also shown that A, as com-puted according to (16), is independent of the order inwhich the nodes are numbered, and that consequently A isa linear function of each branch gain in the graph. It fol-lows that A is equal to unity plus the algebraic sum oJvarious branch-gain products.We shall first show that each term of A, other than

the unity term, is a product of the gains of nontouching

A

B

In

)

r I( I) (2) (3) (4)

(a)

I-r, d12 +r2 d23 1+ r3 d34

AP (-r -2 P.r 3} X3 4

I-r1 d12 I-r2 d23 1-r3 d34(b)

Fig. 10-A wave reflection problem. (a) Reflection of wavesfrom a triple-layer; (b) A possible flow graph.

k-ak

(a)

(c)

(b)

,* o~~~~~~~~~~~~~~~~I-

(d)

Fig. 1I-Two touching paths.

feedback loops. This can be done by contradiction. Con-sider two branches which either enter the same nodeor leave the same node, as shown in Fig. 11(a) and (c).Imagine these branches imbedded in a larger graph, theremainder of which is not shown. Call the branch gainska and kb. Now consider the equivalent replacements(b) and (d). The new node may be numbered zero,whence To'=O, the other T' quantities in (16) are un-changed, and A is therefore unaltered. If both brancheska and kb appear in a term of the A of graph (a) thenthe square of k must appear in a term of the A of graph(b). This is impossible since A must be a linear functionof branch gain k. Hence no term of A can contain thegains of two touching paths.Now suppose that of the several nontouching paths

appearing in a given term of A, some are feedback loopsand some are open paths. Destruction of all otherbranches eliminates some terms from A but leaves thegiven term unchanged. It follows from (16) and thedefinitions of Tk', however, that the A for the subgraphcontaining only these nontouching paths is just

A = (1 - T1)(1 - T2) . . . (1 - Tm) (17)where Tk is the gain of the kth feedback loop in the sub-graph. Hence the open path gains cannot appear in thegiven term and it follows that each term of A is theproduct of gains of nontouching feedback loops. More-over, it is clear from the structure of A that a term inany subgraph A must also appear as a term in the A ofthe complete graph, and conversely, every term of A is a

1956 925



term of some subgraph A. Hence, to identify all possibleterms in A we must look for all possible subgraphs com-prising sets of nontouching loops. Eq. (17) also showsthat the algebraic sign of a term is plus or minus inaccord with an even or odd number of loops in thatterm. This verifies the form of A as given in (12c) and(12d).We shall next establish the general expression for

graph gain (12a). The following notation will prove con-venient. Consider the graph shown schematically inFig. 12, with node n+1 given special attention. LetA'= the A for the complete graph of n+1 nodes.A = the value of A with node n+1 split or removed.T=the loop gain of node n+1.

n+I FIRST n NODES

Fig. 12 A flow graph with one node placed strongly in evidence.

There will in general be several different feedback loopscontaining node n+ 1. LetTk= gain of the kth feedback loop containing noden+1,

Ak = the value of A for that part of the graph nottouching loop Tk.

With the above notation, we have from (16) that

(18)A

Remembering that any A is the algebraic sum of gainproducts of nontouching loops, we find it possible to write

A' = A - E TkAk.k

(19)

Eq. (19) represents the count of all possible nontouch-ing loop sets in A'. The addition of node n+1 createsnew loops Tk but the only new loop sets of A' not al-ready in A are the nontouching sets TkAIc. The negativesign in (19) suffices to preserve the sign rule, since theproduct of Tk and a positive term of Ak will contain anodd number of loops.

Substitution of (19) into (18) yields the generalresult:

E TkAkk

T = (20)

With node n+1 permanently split, T is just the source-to-sink gain of the graph and Tk is the kth forward path.This verifies (12a).

ACKNOWLEDGMENTThe writer is indebted to Prof. K. Wildes and also

to J. Cruz, L. Boffi, G. Amster, and other students inhis 1955 summer-term class in subject 6.633, ElectronicCircuit Theory, M.I.T.; for helpful suggestions.

CORRECTIONJames R. Wait, author of the Correspondence item

"The Radiation Pattern of an Antenna Mounted on aSurface of Large Radius of Curvature," which appearedon page 694 of the MAy, 1956 issue of PROCEEDINGS, hasrequested that the following text, modified in editing,be reinstated in its original form.The last sentence in the first paragraph should read:

"It is the purpose of the present note to extendand apply the Van der Pol-Bremmer theory to

calculate the radiation pattern of a dipole or a sloton a conducting sphere of large radius."The last sentence of the article should read:

"It is interesting to compare this value with the6 db field strength reduction in the tangent planefrom a slot on a flat ground plane which is abruptlytruncated."Mr. Wait has also informed the editors that in

(2), the second bracketed term should be zhJ(')(z).

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Mason IRE 1956