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9 SIGNAL-FLOW GRAPHS AND APPLICATIONS A signal-¯ow graph is a graphical means of portraying the relationship among the variables of a set of linear algebraic equations. S. J. Mason originally introduced it to represent the cause-and-effect of linear systems. Associated terms are de®ned in this chapter along with the procedure to draw the signal-¯ow graph for a given set of algebraic equations. Further,signal-¯ow graphs of microwave networks are obtained in terms of their S-parameters and associated re¯ection coef®cients. The manipula- tion of signal-¯ow graphs is summarized to ®nd the desired transfer functions. Finally,the relations for transducer power gain,available power gain,and operating power gain are formulated in this chapter. Consider a linear network that has N input and output ports. It is described by a set of linear algebraic equations as follows: V i P N j1 Z ij I j i 1; 2; FFF; N 9:1 This says that the effect V i at the ith port is a sum of gain times causes at its N ports. Hence, V i represents the dependent variable (effect) while I j are the independent variables (cause). Nodes or junction points of the signal-¯ow graph represent these variables. The nodes are connected together by line segments called branches with an arrow on each directed toward the dependent node. Coef®cient Z ij is the gain of a branch that connects the ith dependent node with jth independent node. Signal can be transmitted through a branch only in the direction of the arrow. 354 Radio-Frequency and Microwave Communication Circuits: Analysis and Design Devendra K. Misra Copyright # 2001 John Wiley & Sons,Inc. ISBNs: 0-471-41253-8 (Hardback); 0-471-22435-9 (Electronic) Basic properties of the signal-¯ow graph can be summarized as follows: A signal-¯ow graph can be used only when the given system is linear. A set of algebraic equations must be in the form of effects as functions of causes before its signal-¯ow graph can be drawn. A node is used to represent each variable. Normally,these are arranged from left to right,following a succession of inputs (causes) and outputs (effects) of the network. Nodes are connected together by branches with an arrow directed toward the dependent node. Signals travel along the branches only in the direction of the arrows. Signal I k traveling along a branch that connects nodes V i and I k is multiplied by the branch-gain Z ik . Dependent node (effect) V i is equal to the sum of the branch gain times corresponding independent nodes (causes). Example 9.1: Output b 1 of a system is caused by two inputs a 1 and a 2 as represented by the following algebraic equation. Find its signal-¯ow graph. b 1 S 11 a 1 S 12 a 2 There are two independent variables and one dependent variable in this equation. Locate these three nodes and then connect node a 1 and a 2 with b 1 ,as shown in Figure 9.1. Arrows on two branches are directed toward effect b 1 . Coef®cients of input (cause) are shown as the gain of that branch. Example 9.2: Output b 2 of a system is caused by two inputs a 1 and a 2 as represented by the following algebraic equation. Find its signal-¯ow graph. b 2 S 21 a 1 S 22 a 2 There are again two independent variables and one dependent variable in this equation. Locate these three nodes and then connect node a 1 with b 2 and a 2 with b 2 , as shown in Figure 9.2. Arrows on two branches are directed toward effect b 2 . Coef®cients of input (cause) are shown as the gain of that branch. Figure 9.1 Signal-¯ow graph representation of Example 9.1. SIGNAL-FLOW GRAPHS AND APPLICATIONS 355 Example 9.3: Input±output characteristics of a two-port network are given by the following set of linear algebraic equations. Find its signal-¯ow graph. b 1 S 11 a 1 S 12 a 2 b 2 S 21 a 1 S 22 a 2 There are two independent variables a 1 and a 2 and two dependent variables b 1 and b 2 in this set of equations. Locate these four nodes and then connect node a 1 and a 2 with b 1 . Similarly,connect a 1 and a 2 with b 2 ,as shown in Figure 9.3. Arrows on the branches are directed toward effects b 1 and b 2 . Coef®cients of each input (cause) are shown as the gain of that branch. Example 9.4: The following set of linear algebraic equations represents the input± output relations of a multiport network. Find the corresponding signal-¯ow graph. X 1 R 1 1 2 s X 2 X 2 À4X 1 R 2 À 7Y 1 À 1 s 4 X 2 Y 1 s s 2 3 X 2 Y 2 10X 1 À sY 1 Figure 9.2 Signal-¯ow graph representation of Example 9.2. Figure 9.3 Signal-¯ow graph representation of Example 9.3. 356 SIGNAL-FLOW GRAPHS AND APPLICATIONS In the ®rst equation, R 1 and X 2 represent the causes while X 1 is the effect. Hence, the signal-¯ow graph representing this equation can be drawn as illustrated in Figure 9.4. Now,consider the second equation. X 1 is the independent variable in it. Further, X 2 appears as cause as well as effect. This means that a branch must start and ®nish at the X 2 node. Hence,when this equation is combined with the ®rst one,the signal- ¯ow graph will look as illustrated in Figure 9.5. Next,we add to it the signal-¯ow graph of the third equation. It has Y 1 as the effect and X 2 as the cause. It is depicted in Figure 9.6. Figure 9.4 Signal-¯ow graph representation of the ®rst equation of Example 9.4. Figure 9.5 Signal-¯ow graph representation of the ®rst two equations of Example 9.4. Figure 9.6 Signal-¯ow graph representation of the ®rst three equations of Example 9.4. SIGNAL-FLOW GRAPHS AND APPLICATIONS 357 Finally,the last equation has Y 2 as a dependent variable,and X 1 and Y 1 are two independent variables. A complete signal-¯ow graph representation is obtained after superimposing it as shown in Figure 9.7. 9.1 DEFINITIONS AND MANIPULATION OF SIGNAL-FLOW GRAPHS Before we proceed with manipulation of signal-¯ow graphs,it will be useful to de®ne a few remaining terms. Input and Output Nodes: A node that has only outgoing branches is de®ned as an input node or source. Similarly,an output node or sink has only incoming branches. For example, R 1 , R 2 ,and Y 1 are the input nodes in the signal-¯ow graph shown in Figure 9.8. This corresponds to the ®rst two equations of Example 9.4. There is no output node (exclude the dotted branches) in it because X 1 and X 2 have both Figure 9.7 Complete signal-¯ow graph representation of Example 9.4. Figure 9.8 Signal-¯ow graph with R 1 , R 2 ,and Y 1 as input nodes. 358 SIGNAL-FLOW GRAPHS AND APPLICATIONS outgoing as well as incoming branches. Nodes X 1 and X 2 in Figure 9.8 can be made the output nodes by adding an outgoing branch of unity gain to each one. This is illustrated in Figure 9.8 with dotted branches. It is equivalent to adding X 1 X 1 and X 2 X 2 in the original set of equations. Thus,any non-output node can be made an output node in this way. However,this procedure cannot be used to convert these nodes to input nodes because that changes the equations. If an incoming branch of unity gain is added to node X 1 then the corresponding equation is modi®ed as follows: X 1 X 1 R 1 1 2 s X 2 However, X 1 can be made an input node by rearranging it as follows. The corresponding signal-¯ow graph is illustrated in Figure 9.9. It may be noted that now R 1 is an output node: R 1 X 1 À 1 2 s X 2 Path: A continuous succession of branches traversed in the same direction is called the path. It is known as a forward path if it starts at an input node and ends at an output node without hitting a node more than once. The product of branch gains along a path is de®ned as the path gain. For example,there are two forward paths between nodes X 1 and R 1 in Figure 9.9. One of these forward paths is just one branch connecting the two nodes with path gain of 1. The other forward path is X 1 to X 2 to R 1 . Its path gain is 4=2 s. Loop: A loop is a path that originates and ends at the same node without encountering other nodes more than once along its traverse. When a branch originates and terminates at the same node,it is called a self-loop. The path gain of a loop is de®ned as the loop gain. Figure 9.9 Signal-¯ow graph with R 1 as an output,and X 1 , R 2 ,and Y 1 as the input nodes. DEFINITIONS AND MANIPULATION OF SIGNAL-FLOW GRAPHS 359 Once the signal-¯ow graph is drawn,the ratio of an output to input node (while other inputs,if there are more than one,are assumed to be zero) can be obtained by using rules of reduction. Alternatively, Mason's rule may be used. However,the latter rule is prone to errors if the signal-¯ow graph is too complex. The reduction rules are generally recommended for such cases,and are given as follows. Rule 1: When there is only one incoming and one outgoing branch at a node (i.e., two branches are connected in series),it can be replaced by a direct branch with branch gain equal to the product of the two. This is illustrated in Figure 9.10. Rule 2: Two or more parallel paths connecting two nodes can be merged into a single path with a gain that is equal to the sum of the original path gains,as depicted in Figure 9.11. Rule 3: A self-loop of gain G at a node can be eliminated by multiplying its input branches by 1=1 À G. This is shown graphically in Figure 9.12. Rule 4: A node that has one output and two or more input branches can be split in such a way that each node has just one input and one output branch,as shown in Figure 9.13. Figure 9.10 Graphical illustration of Rule 1. Figure 9.12 Graphical illustration of Rule 3. Figure 9.11 Graphical illustration of Rule 2. 360 SIGNAL-FLOW GRAPHS AND APPLICATIONS Rule 5: It is similar to Rule 4. A node that has one input and two or more output branches can be split in such a way that each node has just one input and one output branch. This is shown in Figure 9.14. Mason's Gain Rule: Ratio T of the effect (output) to that of the cause (input) can be found using Mason's rule as follows: Ts P 1 Á 1 P 2 Á 2 P 3 Á 3 ÁÁÁ Á 9:1:1 where P i is the gain of the ith forward path, Á 1 À P L1 P L2À P L3ÁÁÁ 9:1:2 Á 1 1 À P L1 1 P L2 1 À P L3 1 ÁÁÁ 9:1:3 Á 2 1 À P L1 2 P L2 2 À P L3 2 ÁÁÁ 9:1:4 Á 3 1 À P L1 3 ÁÁÁ 9:1:5 F F F P L1 stands for the sum of all ®rst-order loop gains; P L2 is the sum of all second-order loop gains,and so on. P L1 1 denotes the sum of those ®rst-order loop gains that do not touch path P 1 at any node; P L2 1 denotes the sum of those Figure 9.13 Graphical illustration of Rule 4. Figure 9.14 Graphical illustration of Rule 5. DEFINITIONS AND MANIPULATION OF SIGNAL-FLOW GRAPHS 361 second-order loop gains that do not touch the path of P 1 at any point; P L1 2 consequently denotes the sum of those ®rst-order loops that do not touch path P 2 at any point,and so on. First-order loop gain was de®ned earlier. Second-order loop gain is the product of two ®rst-order loops that do not touch at any point. Similarly,third-order loop gain is the product of three ®rst-order loops that do not touch at any point. Example 9.5: A signal-¯ow graph of a two-port network is given in Figure 9.15. Using Mason's rule,®nd its transfer function Y =R. There are three forward paths from node R to node Y . Corresponding path gains are found as follows: P 1 1 ? 1 ? 1 s 1 ? 1 ? s s 2 ? 3 ? 1 3s s 1s 2 P 2 1 ? 6 ? 1 6 and, P 3 1 ? 1 ? 1 s 1 ? À4 ? 1 À 4 s 1 Next,it has two loops. The loop gains are L 1 À 3 s 1 and, L 2 À 5s s 2 Using Mason's rule,we ®nd that Y R P 1 P 2 1 À L 1 À L 2 L 1 L 2 P 3 1 À L 2 1 À L 1 À L 2 L 1 L 2 Figure 9.15 Signal-¯ow graph of Example 9.5. 362 SIGNAL-FLOW GRAPHS AND APPLICATIONS 9.2 SIGNAL-FLOW GRAPH REPRESENTATION OF A VOLTAGE SOURCE Consider an ideal voltage source E S 0 in series with source impedance Z S ,as shown in Figure 9.16. It is a single-port network with terminal voltage and current V S and I S ,respectively. It is to be noted that the direction of current-¯ow is assumed as entering the port,consistent with that of the two-port networks considered earlier. Further,the incident and re¯ected waves at this port are assumed to be a S and b S , respectively. Characteristic impedance at the port is assumed to be Z o . Using the usual circuit analysis procedure,total terminal voltage V S can be found as follows: V S V in S V ref S E S Z S I S E S Z S I in S I ref S E S Z S V in S À V ref S Z o 9:2:1 where superscripts ``in'' and ``ref'' on V S and I S are used to indicate the corresponding incident and re¯ected quantities. This equation can be rearranged as follows: 1 Z S Z o V ref S E S À 1À Z S Z o V in S or, V ref S Z o Z o Z S E S À Z o À Z S Z o Z S V in S Dividing it by 2Z o p ,we ®nd that V ref S 2Z o p Z o Z o Z S E S 2Z o p À Z o À Z S Z o Z S V in S 2Z o p Figure 9.16 Incident and re¯ected waves at the output port of a voltage source. SIGNAL-FLOW GRAPH REPRESENTATION OF A VOLTAGE SOURCE 363 [...]...
9:2:5 From (9.2.2), the signal ¯ow graph for a voltage source can be drawn as shown in Figure 9.17 9.3 SIGNAL-FLOW GRAPH REPRESENTATION OF A PASSIVE SINGLE-PORT DEVICE Consider load impedance ZL as shown in Figure 9.18 It is a single-port device with port voltage and current VL and IL, respectively The incident and re¯ected waves at the port are assumed to be aL and bL, respectively Further, characteristic... SIGNAL-FLOW GRAPH REPRESENTATION OF A PASSIVE Figure 9.18 365 A passive one-port circuit or, ZL ref ZL in À 1 VL VL Zo Zo
9:3:1 ZL À Zo in in V ÀL VL ZL Zo L
9:3:2 1 or, ref VL After dividing it by p 2Zo, we then use (7.6.15) and (7.6.16) to ®nd that bL À L aL
9:3:3 ref VL bL p 2Zo
9:3:4 in VL aL p 2Zo
9:3:5 where, and, ÀL Z L À Zo Z L Zo
9:3:6... 9.20 Draw the signal-¯ow graph and determine the re¯ection coef®cient at its input port using Mason's rule 366 SIGNAL-FLOW GRAPHS AND APPLICATIONS Figure 9.19 Signal-¯ow graph of a one-port passive device Figure 9.20 Two-port network with termination As shown in Figure 9.21, combining the signal-¯ow graphs of a passive load and that of the two-port network obtained in Example 9.3, we can get the representation... terminates its output, as shown in Figure 9.22 Draw its signal-¯ow graph and ®nd the output re¯ection coef®cient Àout A signal-¯ow graph of this circuit can be drawn by combining the results of the previous example with those of a voltage source representation obtained in section 9.2 This is illustrated below in Figure 9.23 The output re¯ection coef®cient is de®ned as the ratio of b2 to a2 with load . REPRESENTATION OF A PASSIVE SINGLE-PORT DEVICE Consider load impedance Z L as shown in Figure 9.18. It is a single-port device with port voltage and current V. o V in S or, V ref S Z o Z o Z S E S À Z o À Z S Z o Z S V in S Dividing it by 2Z o p ,we ®nd that V ref S 2Z o p Z o Z o
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