Tài liệu RF và mạch lạc lò vi sóng P9 docx

The tion of signal-¯ow graphs is summarized to ®nd the desired transfer functions.Finally,the relations for transducer power gain,available power gain,and operatingpower gain are formula

in terms of their S-parameters and associated re¯ection coef®cients The tion of signal-¯ow graphs is summarized to ®nd the desired transfer functions.Finally,the relations for transducer power gain,available power gain,and operatingpower gain are formulated in this chapter.

manipula-Consider a linear network that has N input and output ports It is described by aset of linear algebraic equations as follows:

ViˆPNjˆ1ZijIj i ˆ 1; 2; ; N …9:1†

This says that the effect Viat the ith port is a sum of gain times causes at its N ports.Hence, Vi represents the dependent variable (effect) while Ij are the independentvariables (cause) Nodes or junction points of the signal-¯ow graph represent thesevariables The nodes are connected together by line segments called branches with

an arrow on each directed toward the dependent node Coef®cient Zijis the gain of abranch 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

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 ofcauses before its signal-¯ow graph can be drawn

 A node is used to represent each variable Normally,these are arranged fromleft to right,following a succession of inputs (causes) and outputs (effects) ofthe network

 Nodes are connected together by branches with an arrow directed toward thedependent node

 Signals travel along the branches only in the direction of the arrows

 Signal Iktraveling along a branch that connects nodes Viand Ikis multiplied bythe branch-gain Zik Dependent node (effect) Vi is equal to the sum of thebranch gain times corresponding independent nodes (causes)

Example 9.1: Output b1 of a system is caused by two inputs a1 and a2 asrepresented by the following algebraic equation Find its signal-¯ow graph

b1ˆ S11a1‡ S12a2There are two independent variables and one dependent variable in this equation.Locate these three nodes and then connect node a1 and a2 with b1,as shown inFigure 9.1 Arrows on two branches are directed toward effect b1 Coef®cients ofinput (cause) are shown as the gain of that branch

Example 9.2: Output b2 of a system is caused by two inputs a1 and a2 asrepresented by the following algebraic equation Find its signal-¯ow graph

b2ˆ S21a1‡ S22a2There are again two independent variables and one dependent variable in thisequation Locate these three nodes and then connect node a1with b2and a2with b2,

as shown in Figure 9.2 Arrows on two branches are directed toward effect b2.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 thefollowing set of linear algebraic equations Find its signal-¯ow graph.

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

Figure 9.2 Signal-¯ow graph representation of Example 9.2

Figure 9.3 Signal-¯ow graph representation of Example 9.3

In the ®rst equation, R1and X2represent the causes while X1is the effect Hence,the signal-¯ow graph representing this equation can be drawn as illustrated in Figure9.4.

Now,consider the second equation X1 is the independent variable in it Further,

X2appears as cause as well as effect This means that a branch must start and ®nish

at the X2node 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 Y1 as theeffect and X2 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 Y2 as a dependent variable,and X1 and Y1 are twoindependent variables A complete signal-¯ow graph representation is obtained aftersuperimposing it as shown in Figure 9.7.

9.1 DEFINITIONS AND MANIPULATION OF SIGNAL-FLOW GRAPHSBefore we proceed with manipulation of signal-¯ow graphs,it will be useful tode®ne a few remaining terms

Input and Output Nodes: A node that has only outgoing branches is de®ned as aninput node or source Similarly,an output node or sink has only incoming branches.For example, R1, R2,and Y1 are the input nodes in the signal-¯ow graph shown inFigure 9.8 This corresponds to the ®rst two equations of Example 9.4 There is nooutput node (exclude the dotted branches) in it because X1 and X2 have both

Figure 9.7 Complete signal-¯ow graph representation of Example 9.4

Figure 9.8 Signal-¯ow graph with R1, R2,and Y1as input nodes

outgoing as well as incoming branches.

Nodes X1 and X2 in Figure 9.8 can be made the output nodes by adding anoutgoing branch of unity gain to each one This is illustrated in Figure 9.8 withdotted branches It is equivalent to adding X1ˆ X1and X2 ˆ X2in the original set ofequations 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 nodesbecause that changes the equations If an incoming branch of unity gain is added tonode X1 then the corresponding equation is modi®ed as follows:

X1ˆ X1‡ R1‡2 ‡ s1 X2However, X1 can be made an input node by rearranging it as follows Thecorresponding signal-¯ow graph is illustrated in Figure 9.9 It may be noted that now

R1 is an output node:

R1ˆ X1 2 ‡ s1 X2Path: A continuous succession of branches traversed in the same direction is calledthe path It is known as a forward path if it starts at an input node and ends at anoutput node without hitting a node more than once The product of branch gainsalong a path is de®ned as the path gain For example,there are two forward pathsbetween nodes X1 and R1 in Figure 9.9 One of these forward paths is just onebranch connecting the two nodes with path gain of 1 The other forward path is X1to

X2 to R1 Its path gain is 4=…2 ‡ s†

Loop: A loop is a path that originates and ends at the same node withoutencountering other nodes more than once along its traverse When a branchoriginates 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 R1as an output,and X1, R2,and Y1as 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 (whileother inputs,if there are more than one,are assumed to be zero) can be obtained byusing rules of reduction Alternatively, Mason's rule may be used However,thelatter rule is prone to errors if the signal-¯ow graph is too complex The reductionrules 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 withbranch 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 asingle path with a gain that is equal to the sum of the original path gains,as depicted

Figure 9.10 Graphical illustration of Rule 1

Figure 9.12 Graphical illustration of Rule 3

Figure 9.11 Graphical illustration of Rule 2

Rule 5: It is similar to Rule 4 A node that has one input and two or more outputbranches can be split in such a way that each node has just one input and one outputbranch This is shown in Figure 9.14.

Mason's Gain Rule: Ratio T of the effect (output) to that of the cause (input) can befound using Mason's rule as follows:

T…s† ˆP1
1‡ P2
2
‡ P3
3‡


…9:1:1†where Piis the gain of the ith forward path,

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 P1 at any point; PL…1†…2†consequently denotes the sum of those ®rst-order loops that do not touch path P2atany point,and so on.

First-order loop gain was de®ned earlier Second-order loop gain is the product oftwo ®rst-order loops that do not touch at any point Similarly,third-order loop gain isthe 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 gainsare found as follows:

P3ˆ 1 ? 1 ? s ‡ 11 ? … 4† ? 1 ˆ s ‡ 14Next,it has two loops The loop gains are

s ‡ 1and,

L2ˆ s ‡ 25sUsing Mason's rule,we ®nd that

9.2 SIGNAL-FLOW GRAPH REPRESENTATION OF A VOLTAGESOURCE

Consider an ideal voltage source ES€ 0 in series with source impedance ZS,asshown in Figure 9.16 It is a single-port network with terminal voltage and current

VSand IS,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 aS and bS,respectively Characteristic impedance at the port is assumed to be Zo

Using the usual circuit analysis procedure,total terminal voltage VScan be found

Vref

S ˆ Zo

Zo‡ ZSES Zo ZS

Zo‡ ZSVSinDividing it byp2Zo,we ®nd that

Vref S

Zo‡ ZS

Vin S



2Zop

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

Using (7.6.15) and (7.6.16),this can be written as follows:

where,

bSˆ VSref2Zo

1 ‡ZZLo

aLˆ VLin2Zo

Figure 9.18 A passive one-port circuit

SIGNAL-FLOW GRAPH REPRESENTATION OF A PASSIVE 365

As shown in Figure 9.21,combining the signal-¯ow graphs of a passive load andthat of the two-port network obtained in Example 9.3,we can get the representationfor this network.

Its input re¯ection coef®cient is given by the ratio of b1 to a1 For Mason's rule,

we ®nd that there are two forward paths, a1! b1 and a1! b2! aL! bL!

a2! b1 There is one loop, b2! aL! bL ! a2! b2,which does not touch theformer path The corresponding path and loop gains are

P1ˆ S11

P2ˆ S21? 1 ? L ? 1 ? S12ˆ LS21S12and,

L1ˆ 1 ? L ? 1 ? S22ˆ LS22

Figure 9.20 Two-port network with termination

Figure 9.19 Signal-¯ow graph of a one-port passive device

Figure 9.21 Signal-¯ow graph representation of the network shown in Figure 9.20

A signal-¯ow graph of this circuit can be drawn by combining the results of theprevious example with those of a voltage source representation obtained in section9.2 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 loaddisconnected and the source impedance ZS terminates port-1 (the input) Hence,there are two forward paths from a2 to b2, a2! b2 and a2! b1! aS! bS!

a1! b2 There is only one loop (because the load is disconnected), b1!

aS! bS! a1 ! b1 The path and loop gains are found as follows:

P1ˆ S22

P2ˆ S12? 1 ? S? 1 ? S21ˆ SS12S21

Figure 9.22 Two-port network with the source and the termination

Figure 9.23 Signal-¯ow graph representation of the network shown in Figure 9.22

SIGNAL-FLOW GRAPH REPRESENTATION OF A PASSIVE 367

L1ˆ 1 ? S? 1 ? S11ˆ SS11Therefore,from Mason's rule,we have

Power output of the source ˆ jbSj2Power reflected back into the source ˆ jaSj2Hence,power delivered by the source,Pd,is

Pdˆ jbSj2 jaSj2Similarly,we can write the following relations for power at the load:

Power incident on the load ˆ jaLj2Power reflected from the load ˆ jbLj2ˆ j Lj2jaLj2

where L is the load re¯ection coef®cient

Hence,power absorbed by the load,PL,is given by

PLˆ jaLj2 jbLj2ˆ jaLj2…1 j Lj2† ˆ jbSj2…1 j Lj2†

Figure 9.24 Signal-¯ow graph for Example 9.8

2…1 j Lj2†For PL to be a maximum,its denominator 1 S Lis minimized Let us analyzethis term more carefully using the graphical method It says that there is a phasor

S L at a distance of unity from the origin that rotates with a change in L ( Sisassumed constant),as illustrated in Figure 9.25 Hence,the denominator has an

Figure 9.25 Graphical representation of (1 S L)

SIGNAL-FLOW GRAPH REPRESENTATION OF A PASSIVE 369

extreme value whenever S L is a pure real number If this number is positive realthen the denominator is minimum On the other hand,the denominator is maximizedwhen S L is a negative real number.

In order for S L to be a positive real number,the load re¯ection coef®cient Lmust be complex conjugate of the source re¯ection coef®cient S In other words,

PLˆ jbGj2

…1 j Sj2†ˆ Pavswhere Pavs is maximum power available from the source It is a well-knownmaximum power transfer theorem of electrical circuit theory

Example 9.9: Draw the signal-¯ow graph of the three-port network shown in Figure9.26 Find the transfer characteristics b3=bSusing the graph

For a three-port network,the scattering equations are given as follows:

b1ˆ S11a1‡ S12a2‡ S13a3

b2ˆ S21a1‡ S22a2‡ S23a3

b3ˆ S31a1‡ S32a2‡ S33a3For the voltage source connected at port-1,we have

bSˆ bG‡ SaS

Figure 9.26 Three-port network of Example 9.9

Since the wave coming out from the source is incident on port-1, bSis equal to a1.Therefore,we can write

a1ˆ bG‡ Sb1For the load ZL connected at port-2,

bL ˆ LaLAgain,the wave emerging from port-2 of the network is incident on the load.Similarly,the wave re¯ected back from the load is incident on port-2 of the network.Hence,

b2ˆ aLand,

bL ˆ a2Therefore,the above relation at the load ZL can be modi®ed as follows:

a2ˆ Lb2Following a similar procedure for the load ZDconnected at port-3,we ®nd that

bDˆ DaD) a3ˆ Db3Using these equations,a signal-¯ow graph can be drawn for this circuit as shown

L8ˆ S13S31 D STherefore,various terms of Mason's rule can be evaluated as follows:

1 ˆ 1 S22 L

2ˆ 1P

L…1† ˆ L1‡ L2‡ L3‡ L4‡ L5‡ L6‡ L7‡ L8P

L…2† ˆ L1L2‡ L1L3‡ L2L3‡ L3L4‡ L1L7‡ L2L8P

L…3† ˆ L1L2L3Using Mason's rule we can ®nd

b3

bSˆ

P1
1‡ P2
2

1 PL…1† ‡PL…2† PL…3†

Note that this signal-¯ow graph is too complex,and therefore,one can easily miss

a few loops In cases like this,it is prudent to simplify the signal-¯ow graph usingthe ®ve rules mentioned earlier

Figure 9.27 Signal-¯ow graph of the network shown in Figure 9.26

9.4 POWER GAIN EQUATIONS

We have seen in the preceding section that maximum power is transferred when loadre¯ection coef®cient Lis conjugate of the source re¯ection coef®cient S It can begeneralized for any port of the network In the case of an ampli®er,maximum powerwill be applied to its input port if its input re¯ection coef®cient in is complexconjugate of the source re¯ection coef®cient S If this is not the case then input will

be less than the maximum power available from source Similarly,maximumampli®ed power will be transferred to the load only if load re¯ection coef®cient

L is complex conjugate of output re¯ection coef®cient out Part of the poweravailable at the output of the ampli®er will be re¯ected back if there is a mismatch.Therefore,the power gain of an ampli®er can be de®ned in at least three differentways as follows:

Transducer power gain; GTˆ PL

Pavsˆ

Power delivered to the loadPower available from the sourceOperating power gain; GPˆPPL

inˆPower delivered to the loadPower input to the networkAvailable power gain; GAˆPAVN

Pavs ˆ

Power available from the networkPower available from the sourcewhere

Pavsˆ Power available from the source

Pinˆ Power input to the network

PAVNˆ Power available from the network

PLˆ Power delivered to the load

Therefore,the transducer power gain will be equal to that of operating power gain ifthe input re¯ection coef®cient in is complex conjugate of the source re¯ectioncoef®cient S That is,

POWER GAIN EQUATIONS 373