Lecture 12 Date:

Transcription

1 Information Technolog Delhi Lecture 2 Date: The Signal Flow Graph

2 Information Technolog Delhi Signal Flow Graph Q: Using individual device scattering parameters to analze a comple microwave network results in a lot of mess math! Isn t there an easier wa? A: Yes! We can represent a microwave network with its signal flow graph and then decompose this graph using a standard set of rules results into simpler analsis. It provides a sort of a graphical wa to do algera! Signal Flow Graph (SFG) can also help us understand the fundamental phsical ehavior of a network or device. It can even help us approimate the network in a wa that makes it simpler to analze and/or design!

3 Information Technolog Delhi To understand the significance of SFG, let us consider a comple 3-port microwave network constructed of 5 simpler microwave devices S n is the scattering matri of each device, and S is the overall scattering matri of the entire 3-port network The S-parameter (S) of the whole network can e otained from the knowledge of S-parameter of individual devices Tedious Algera! Alternative is SFG ased solution!

4 Information Technolog Delhi Signal flow graphs are helpful in three was! W It provide us with a graphical means of solving large sstems of simultaneous equations. W We ll see that it can provide us with a road map of the wave propagation paths throughout a HF device or network. If we re paing attention, we can glean great phsical insight as to the inner working of the device represented the graph. Wa 3 It provide us with a quick and accurate method for approimating a network or device. We will find that we can often replace a rather comple graph with a much simpler one that is almost equivalent. We find this to e ver helpful when designing microwave components. From the analsis of these approimate graphs, we can often determine design rules or equations that are tractale, and allow us to design components with (near) optimal performance.

5 Information Technolog Delhi Some definitions! Ever SFG consists of a set of nodes. These nodes are connected ranches, which are simpl contours with a specified direction. Similarl, each ranch has an associated comple value. j -j Q: What could this possil have to do with RF/microwave engineering?

6 Information Technolog Delhi In high frequenc applications, each port of a device is represented two nodes the a node and the node. The a node simpl represents the value of the normalized amplitude of the wave incident on that port, evaluated at the plane of that port: a n V z z Z n n np 0n Similarl, the node simpl represents the normalized amplitude of the wave eiting that port, evaluated at the plane of that port: n V z z Z n n np Note then that the total voltage at a port is simpl: 0n 0 V z z a Z n n np n n n

7 Information Technolog Delhi The value of the ranch connecting two nodes is simpl the value of the scattering parameter relating these two voltage values. a n V z z Z n n np 0n m V z z Z m m mp The signal flow graph aove is simpl a graphical representation of the equation: a S m n mn Moreover, if multiple ranches enter a node, then the voltage represented that node is the sum of the values from each ranch. For eample, following SFG represents: S mn 0m S S 3 = S a + S a +S a S 2 a 3

8 Information Technolog Delhi Now, consider a two-port device with a scattering matri S: So that: S S S a a S 2 S 2 22 = S a + S a = S + S We can then graphicall represent a two-port device as: S 2 2 S S 22 S 2

9 Information Technolog Delhi Now, consider a two-port device where the second port is terminated some load Γ L : Additional Equation Γ V z z = z z L V 2 2 2P L 2 2 2P a = 2 L 2 Therefore, the signal flow graph of this terminated network is: S 2 2 S S 22 S 2 Γ L

10 Information Technolog Delhi Now consider cascading of two different two-port networks Here: ΓL a = 2 SFG = a S 2 S 2 S S 22 S S 22 S 2 S 2 Γ L

11 Information Technolog Delhi Now consider networks connected with a transmission line segment: scattering matri Z 0 l SFG S e 0 jl e jl S 2 -jβl e S 2 S S 22 S S 22 S 2 -jβl e S 2 Γ L

12 Information Technolog Delhi 2 2 S 2 -jβl e S 2 S S 22 S S 22 S 2 -jβl e S 2 Γ L Note that there is one (and onl one!) independent variale in this graphical representation (i.e., SFG) This is the onl node of the SFG that does not have an incoming ranches. As a result, its value depends on no other node values in the SFG Independent nodes in the SFG are called sources!

13 Information Technolog Delhi Independent nodes in the SFG are called sources! This makes sense phsicall (do ou see wh?) The node value represents the comple amplitude of the wave incident on the one-port network. If this value is zero, then no power is incident on the network the rest of the nodes (i.e., wave amplitudes) will e zero! Now, sa we wish to determine, for eample:. The reflection coefficient Γ in of the one-port device 2. The total current at port of second network (i.e., network ) 3. The power asored the load at port 2 of the second () network.

14 Information Technolog Delhi In the first case, we need to determine the value of dependent node : in a For the second case, we must determine the value of wave amplitudes and : a I Z 0 For the third and final case, the values of nodes and 2 are required: P as a2 2 How do we determine the values of these wave amplitude nodes? solve the simultaneous equations that descrie this network. Decompose (reduce) the SFG!

15 Information Technolog Delhi SFG reduction is a method for simplifing the comple paths of that SFG into a more direct (ut equivalent!) form. Reduction is reall just a graphical method of decoupling the simultaneous equations that are descried the SFG. SFGs can e reduced appling one of four simple rules. Q: Can these rules e applied in an order? A: YES! The rules can onl e applied when/where the structure of the SFG allows. You must search the SFG for structures that allow a rule to e applied, and the SFG will then e (a little it) reduced. You then search for the net valid structure where a rule can e applied. Eventuall, the SFG will e completel reduced! It s a it like solving a puzzle. Ever SFG is different, and so each requires a different reduction procedure. It requires a little thought, ut with a little practice, the reduction procedure can e easil mastered You ma find its kind of a fun! (TRUST ME)

16 Information Technolog Delhi Series Rule Consider these two comple equations: a a2 These two equations can comined to form an equivalent set of equations: a a a a Graphicall the can e represented as: 2 Equivalent SFGs

17 Information Technolog Delhi This last discussion leads us to our first SFG reduction rule: Rule - Series Rule If a node has one (and onl one!) incoming ranch, and one (and onl one!) outgoing ranch, the node can e eliminated and the two ranches can e comined, with the new ranch having a value equal to the product of the original two. Eample: a 0.3 j Can e reduced to 0.3 j0.3

18 Information Technolog Delhi Parallel Rule Consider these two comple equations: a a a The equation can also e epressed as: These equations can e epressed in terms of SFG as: a Equivalent SFGs

19 Information Technolog Delhi This last discussion leads us to our second SFG reduction rule: Rule 2 - Parallel Rule If two nodes are connected parallel ranches and the ranches have the same direction the ranches can e comined into a single ranch, with a value equal to the sum of each two original ranches. Eample: Can e reduced to a 0.5

20 How aout: Indraprastha Institute of Information Technolog Delhi What aout this signal flow graph? Can it e transformed into a So that a 0.3 Asolutel not! NEVER DO THIS!! Asolutel not! NEVER DO THIS EITHER!!

21 Information Technolog Delhi Actuall from this SFG we can onl conclude that a a SFG can e of the form 0.3 a a a Branches that egin and end at the same node are called self-loops In practical situations, self-loop node will alwas have at least one other incoming ranch

22 Information Technolog Delhi Practical eample of node with self-loop: j 0.3

23 Information Technolog Delhi Self-Loop Rule Consider the comple equation: SFG a SFG A little it of algera allows us to determine the value of node : a a 2 Equivalent

24 Information Technolog Delhi This last discussion leads us to our third SFG reduction rule: Rule 3 Self-Loop Rule A self-loop can e eliminate multipling all of the ranches feeding the self-loop node ( S sl ), where S sl is the value of the self loop ranch. Eample j0.4 can e simplified eliminating the selfloop multipl oth of the two ranches feeding the self-loop node :.25 S 0.2 sl 0.6* Simplified and Reduced j0.4*.25 j0.5

25 Information Technolog Delhi Eample 0.06 Here S sl j Simplified j / 0.94 Onl the incoming ranches are modified the self-loop rule! Here, the 0.3 ranch is eiting the self-loop node and therefore doesn t get modified. Onl the j ranch(incoming at node ) to the self-loop node are modified the self-loop rule!