Lecture 9 Microwve Network nlysis. Nssiri - NL June 9, 003 Microwve Physics nd Techniques UC June 003
-Prmeter Mesurement Technique VVM: The vector voltmeter mesures the mgnitude of reference nd test voltge nd the difference in phse between the voltges. ecuse it cn mesure phse, it llows us to directly mesure the -prmeters of circuit Unfortuntely, the use of the directionl couplers nd test cbles connecting the mesuring system to the vector voltmeter introduces unknown ttenution nd phse shift into the mesurements. These cn be compensted for by mking dditionl clibrtion mesurements. Microwve Physics nd Techniques UC June 003
Reflection mesurements: or ignl Gen N-NC (HP 8657) (HP 8508) VVM N-NC N-NC NC cble NC cble N-NC N-NC 0 d 0 d (HP 778D Dul Directionl Coupler) N-NC DUT Mtched lod Microwve Physics nd Techniques UC June 003 3
Reflection mesurements: or From the setup, it is seen tht the voltge t chnnel of the VVM ( D ) is proportionl to the mplitude of the voltge wve entering the device under test (DUT) ( D ). imilrly, the voltge t chnnel ( D ) is proportionl to the mplitude of the voltge wve reflected from DUT (b D ). Thus we cn write D D K K D b Where K nd K re constnts tht depend on the connecting cbles. ince D is zero becuse of the mtched lod t port, is given by D D D b D D K K Microwve Physics nd Techniques UC June 003 4
Reflection mesurements: or To find K nd K it is necessry to mke second mesurement with known DUT. This is clled clibrtion mesurement. If the DUT is removed nd replced by short circuit, the voltge t chnnel ( s ) nd chnnel ( s ) re given by K K b Where s is the mplitude of the voltge wve entering the short nd b s is the mplitude of the voltge wve reflected from the short. However, for short circuit the rtio of these mplitudes is (reflection coefficient of short). Thus b K K Microwve Physics nd Techniques UC June 003 5
Microwve Physics nd Techniques UC June 003 6 Reflection mesurements: or K K D D Note: since VVM displys quntities in terms of mgnitude nd phse we cn rewrite s ( ) π φ φ Γ Γ D D D D D D φ Γ φ Γ
Trnsmission mesurements: or Genertor Chnnel Chnnel Mtched lod Mtched lod N-NC DUT NC cble Mtched lod The DUT is connected directly between two directionl couplers. Voltge t of the VVM is proportionl to the voltge wve incident on the DUT while the voltge t of the VVM is proportionl to voltge wve trnsmitted through the DUT. Microwve Physics nd Techniques UC June 003 7
Microwve Physics nd Techniques UC June 003 8 Trnsmission mesurements: or D D D D b K K D D D D K K b To find out the constnts clibrtion mesurement must be mde. Remove the DUT nd connect both directionl couplers directly together. The Known DUT in this cse is just zero-length guide with trnsmission coefficient of unity. The mesured voltges re: E E E E b K K E E E E K K b where E E K K
Microwve Physics nd Techniques UC June 003 9 Trnsmission mesurements: or E E D D ( ) E T T D E D θ θ D D D D T θ E E E E T θ where
cttering Prmeters cttering Prmeters (-Prmeters) plys mjor role is network nlysis This importnce is derived from the fct tht prcticl system chrcteriztions cn no longer be ccomplished through simple open- or short-circuit mesurements, s is customrily in low-frequency pplictions. In the cse of short circuit with wire; the wire itself possesses n inductnce tht cn be of substntil mgnitude t high frequency. lso open circuit leds to cpcitive loding t the terminl. Microwve Physics nd Techniques UC June 003 0
cttering Prmeters In either cse, the open/short-circuit conditions needed to determine Z-, Y-, h-, nd CD-prmeters cn no longer be gurnteed. Moreover, when deling with wve propgtion phenomen, it is not desirble to introduce reflection coefficient whose mgnitude is unity. For instnce, the terminl discontinuity will cuse undesirble voltge nd/or current wve reflections, leding to oscilltion tht cn result in the destruction of the device. With -prmeters, one hs proper tool to chrcterize the two-port network description of prcticlly ll RF devices without hrm to DUT. Microwve Physics nd Techniques UC June 003
Definition of cttering Prmeters -prmeters re power wve descriptors tht permit us to define the input-output reltions of network in terms of incident nd reflected power wves. [] b b n normlized incident power wves b n normlized reflected power wves Microwve Physics nd Techniques UC June 003
Definition of cttering Prmeters n Z o ( V + Z I ) ( ) n o n b n Z o ( V Z I ) ( ) n o n Index n refers either to port number or. The impednce Z 0 is the chrcteristic impednce of the connecting lines on the input nd output side of the network. Microwve Physics nd Techniques UC June 003 3
Definition of cttering Prmeters Inverting () leds to the following voltge nd current expressions: V + n Z o ( b ) ( 3) n n I n Z o ( b ) ( 4) n n Microwve Physics nd Techniques UC June 003 4
Microwve Physics nd Techniques UC June 003 5 Recll the equtions for power: Definition of cttering Prmeters { } ( ) () 5 n n n n n b I V P * Re Isolting forwrd nd bckwrd trveling wve components in (3) nd (4), we see ( ) ( ) 7 6 + + n n n n n n I Z Z V b I Z Z V o o o o
Definition of cttering Prmeters We cn now define -prmeters: b b () 8 Microwve Physics nd Techniques UC June 003 6
Definition of cttering Prmeters b b b b 0 0 0 0 Refkected powe wve t Incident power wve t Trnsmitted powe wve t port Incident power wve t port Refkected powe wve t Incident power wve t Trnsmitted powe wve t port Incident power wve t port Microwve Physics nd Techniques UC June 003 7 port port port port 9 ( ) 0 ( ) ( ) ( )
Observtions: 0, nd 0 no power wves re returned to the network t either port or port. However, these conditions cn only be ensured when the connecting trnsmission line re terminted into their chrcteristic impednces. ince the -prmeters re closely relted to power reltions, we cn express the normlized input nd output wves in terms of time verged power. The verge power t port is given by + + o o V V P in Z Z ( ) ( ) Γ ( 3) Microwve Physics nd Techniques UC June 003 8
cttering Prmeters The reflection coefficient t the input side is expressed in terms of under mtched output ccording: V + b Γin V 0 ( 4) This lso llow us to redefine the VWR t port in terms of s VWR + ( 5) Microwve Physics nd Techniques UC June 003 9
cttering Prmeters We cn identify the incident power in (3) nd express it in terms of : + V Pinc Z o Mximl vilble power from the genertor Microwve Physics nd Techniques UC June 003 0 ( 6) The totl power t port (under mtched output condition) expressed s combintion of incident nd reflected powers: P P inc + P refl in ( ) ( ) b Γ ( 7)
cttering Prmeters If the reflected coefficient, or, is zero, ll vilble power from the source is delivered to port of the network. n identicl nlysis t port gives P out ( ) ( ) b Γ ( 8) Microwve Physics nd Techniques UC June 003
Mening of -Prmeters -prmeters cn only be determined under conditions of perfect mtching on the input or the output side. 0 Z o [] V G Z o Z o Z L b b Mesurement of nd by mtching the line impednce Z o t port through corresponding lod impednce Z L Z o Microwve Physics nd Techniques UC June 003
Mening of -Prmeters This configurtion llows us to compute by finding the input reflection coefficient: Γ in Z Z in in Z o + Z o ( 9) Tking the logrithm of the mgnitude of gives us the return loss in d RL 0log ( 0) Microwve Physics nd Techniques UC June 003 3
Mening of -Prmeters With port properly terminted, we find b 0 ( V + Z I ) ( Z ) V o Z + + I V o o 0 ( ) ince 0, we cn set to zero the positive trveling voltge nd current wves t port. Replcing V by the genertor voltge V G minus the voltge drop over the source impednce Z o, V G -Z o I gives V V V G V G Microwve Physics nd Techniques UC June 003 4 ( )
The forwrd power gin is G o Mening of -Prmeters V V G ( 3) If we reverse the mesurement procedure nd ttch genertor voltge V G to port nd properly terminte port, we cn determine the remining two - prmeters, nd. Z o Z o 0 [] Z o Z o V G b b Microwve Physics nd Techniques UC June 003 5
Mening of -Prmeters To compute we need to find the output reflection coefficient Γ out in similr wy for : b Γ out 0 Z Z out out + Z Z Microwve Physics nd Techniques UC June 003 6 o o ( 4) ( V + Z I ) ( Z ) V o Z + + I V V V V or V V V G G o o 0 ( 5) ( ) 6 G ( 7) Reverse power gin G
Determintion of T-network elements Find the -prmeters nd resistive elements for the 3-d ttenutor network. ssume tht the network is plced into trnsmission line section with chrcteristic line impednce of Z o 50 Ω R R Port R 3 Port Microwve Physics nd Techniques UC June 003 7
Determintion of T-network elements n ttenutor should be mtched to the line impednce nd must meet the requirement 0. R R R 3 50Ω Port Port Z in R3( R + 50Ω) ( R + R + 50Ω) R + 50Ω 3 ecuse of symmetry, it is cler tht R R. Circuit for nd Microwve Physics nd Techniques UC June 003 8
Determintion of T-network elements We now investigte the voltge V V - t port in terms of V V +. R R 50Ω R 3 V R3( R + 50Ω) ( R3 + R + 50Ω) R3( R + 50Ω) + ( R + R + 50Ω) 3 50Ω V 50 R R Ω + Port Port Microwve Physics nd Techniques UC June 003 9
Determintion of T-network elements For 3 d ttenution, we require V V 0. 707 V V G etting the rtio of V /V to 0.707 nd using the input impednce expression, we cn determine R nd R 3 R R R 3 Z o 8. 58Ω + Z 4. 4Ω o Microwve Physics nd Techniques UC June 003 30
Determintion of T-network elements Note: the choice of the resistor network ensures tht t the input nd output ports n impednce of 50 Ω is mintined. This implies tht this network cn be inserted into 50 Ω trnsmission line section without cusing undesired reflections, resulting in n insertion loss. Microwve Physics nd Techniques UC June 003 3
Chin cttering Mtrix To extend the concept of the -prmeter presenttion to cscded network, it is more efficient to rewrite the power wve expressions rrnged in terms of input nd output ports. This results in the chin scttering mtrix nottion. Tht is, b T T T T b ( 8) It is immeditely seen tht cscding of two dulport networks becomes simple multipliction. Microwve Physics nd Techniques UC June 003 3
Chin cttering Mtrix b Port [ T ] [ ] T Port b b b Cscding of two networks nd Microwve Physics nd Techniques UC June 003 33
Microwve Physics nd Techniques UC June 003 34 Chin cttering Mtrix If network is described by ( ) 9 b T T T T b nd network by ( ) 30 b T T T T b
Microwve Physics nd Techniques UC June 003 35 Chin cttering Mtrix ( ) 3 b b Thus, for the combined system, we conclude ( ) 3 b T T T T T T T T b
Chin cttering Mtrix The conversion from -mtrix to the chin mtrix nottion is similr s described before. T T T T b 0 ( 33) ( 34) ( ) Microwve Physics nd Techniques UC June 003 36 ( 3) - ( 35)
Chin cttering Mtrix Conversely, when the chin scttering prmeters re given nd we need to convert to -prmeters, we find the following reltions: b ( T T T T ) T 0 T T T T T ( 38) ( 39) b b T T T T Microwve Physics nd Techniques UC June 003 37 ( 36) ( 37)
Conversion between Z- nd -Prmeters To find the conversion between the -prmeters nd the Z-prmeters, let us begin with defining -prmeters reltion in mtrix nottion { b } [ ]{ } ( 40) Multiplying by Z gives o Z o {} { } [ ]{ } [ ]{ } b V Z V ( 4) o + { } { } + dding V Z o to both sides results in + + V + V + E V 4 { } [ ]{ } { } ([ ] [ ]){ } V ( ) + Microwve Physics nd Techniques UC June 003 38
Conversion between Z- nd -Prmeters To compre this form with the impednce expression { V } [ Z ]{ I } We hve to express {V + } in term of {I}. ubtrct [}{V + } from both sides of { + } V Z {} o { + } [ ]{ + } V V Z { } { b} { + } V Z [ E ] [ ] o 43 ( ) Z { I } ( ) o I ( ) { } ( 44) o Microwve Physics nd Techniques UC June 003 39
Conversion between Z- nd -Prmeters ubstituting (44) into (4) yields { V } [ ] + [ E ] or ( ){ + V } Z [ ] + [ E ] o ( )([ ] [ ] E ) { I } ( 45) [ Z ] Z ([ ] + [ E ])[ ( E ] [ ]) ( 46 ) o Explicitly Z Z Z Z + Z o + Z o ( )( ) + + Microwve Physics nd Techniques UC June 003 40 ( 47)
Prcticl Network nlysis Microwve Physics nd Techniques UC June 003 4
Criteri for Distortionless Trnsmission Liner Networks Constnt mplitude over bndwidth of interest Liner phse over bndwidth of interest Mgnitude Phse Frequency Frequency Microwve Physics nd Techniques UC June 003 4
Liner Versus Nonliner ehvior in 360 * f * t Time t o * in 360 * f ( t - t ) Time phse shift to * 360 * f Liner behvior: input nd output frequencies re the sme (no dditionl frequencies creted) output frequency only undergoes mgnitude nd phse chnge f Frequency Input DUT Output f Frequency Time Nonliner behvior: output frequency my undergo frequency shift (e.g. with mixers) dditionl frequencies creted (hrmonics, inter-modultion) f Frequency Microwve Physics nd Techniques UC June 003 43
Mgnitude Vrition with Frequency f ( t ) sinωt + sin3ωt + sin5ωt 3 5 Time Time Liner Network Mgnitude Frequency Frequency Frequency Microwve Physics nd Techniques UC June 003 44
Phse Vrition with Frequency f ( t ) sinωt + sin3ωt + sin5ωt 3 5 Time Liner Network Time Mgnitude Frequency 0-80 Frequency Frequency -360 Microwve Physics nd Techniques UC June 003 45
Criteri for Distortionless Trnsmission Nonliner Networks turtion, crossover, inter-modultion, nd other nonliner effects cn cuse signl distortion Time Time Frequency Frequency Microwve Physics nd Techniques UC June 003 46
The Need for oth Mgnitude nd Phse. Complete chrcteriztion of liner networks 4. Time Domin Chrcteriztion. Complex impednce needed to design mtching circuits Mg Time High Frequency Trnsistor Model 5. Vector ccurcy Enhncement 3. se Complex vlues needed for device modeling Collector Emitter Mesured Error ctul Microwve Physics nd Techniques UC June 003 47
High-Frequency Device Chrcteriztion Lightwve nlogy Incident Trnsmitted Reflected Microwve Physics nd Techniques UC June 003 48
Trnsmission Line Review Low frequencies Wvelength >> wire length Current (I) trvels down wires esily for efficient power trnsmission Voltge nd current not dependent on position I High frequencies Wvelength or << wire (trnsmission line) length Need trnsmission-line structures for efficient power trnsmission Mtching to chrcteristic impednce (Z0) is very importnt for low reflection Voltge dependent on position long line Microwve Physics nd Techniques UC June 003 49
Trnsmission Line Terminted with Z o Z s Z o Z o chrcteristic impednce of trnsmission line Z o Vinc Vrefl 0! (ll the incident power is bsorbed in the lod) For reflection, trnsmission line terminted in Zo behves like n infinitely long trnsmission line Microwve Physics nd Techniques UC June 003 50
Trnsmission Line Terminted with hort, Open Z s Z o Vinc Vrefl o In phse (0 ) for open o Out of phse (80 ) for short For reflection, trnsmission line terminted in short or open reflects ll power bck to source Microwve Physics nd Techniques UC June 003 5
Trnsmission Line Terminted with 5Ω Z s Z o ZL 5 Ω Vinc Vrefl tnding wve pttern does not go to zero s with short or open Microwve Physics nd Techniques UC June 003 5
High-Frequency Device Chrcteriztion Incident R Reflected REFLECTION Trnsmitted TRNMIION Reflected Incident R Trnsmitted Incident R WR -Prmeters, Reflection Coefficient Γ, ρ Return Loss Impednce, dmittnce R+jX, G+j Gin / Loss -Prmeters, Trnsmission Coefficient Τ,τ Insertion Phse Group Dely Microwve Physics nd Techniques UC June 003 53
Reflection Prmeters Reflection Coefficient Γ Return loss -0 log(ρ), V reflected V incident ρ Φ ρ Γ Emx Emin Z L Z O Z L + ZO Voltge tnding Wve Rtio VWR Emx Emin + ρ - ρ No reflection (ZL Zo) 0 d ρ RL VWR Microwve Physics nd Techniques UC June 003 54 Full reflection (ZL open, short) 0 d
Trnsmission Prmeters V Incident DUT V Trnsmitted Trnsmission Coefficient Τ V Trnsmitted VIncident τ φ Insertion Loss (d) - 0 Log V Trns V Inc - 0 log τ Gin (d) 0 Log V Trns V Inc 0 log τ Microwve Physics nd Techniques UC June 003 55
Devition from Liner Phse Use electricl dely to remove liner portion of phse response o Phse 45 /Div RF filter response Frequency Liner electricl length dded (Electricl dely function) + yields Frequency Devition from liner phse Frequency o Phse /Div Low resolution High resolution Microwve Physics nd Techniques UC June 003 56
Low-Frequency Network Chrcteriztion H-prmeters V hi + hv V hi + hv Y-prmeters I yv + yv I yv + yv Z-prmeters V zi + zi V zi + zi h V I V0 h V V I0 (requires short circuit) (requires open circuit) ll of these prmeters require mesuring voltge nd current (s function of frequency) Microwve Physics nd Techniques UC June 003 57
Limittions of H, Y, Z Prmeters (Why use -prmeters?) H,Y, Z prmeters Hrd to mesure totl voltge nd current t device ports t high frequencies ctive devices my oscillte or self-destruct with shorts opens -prmeters Relte to fmilir mesurements (gin, loss, reflection coefficient...) Reltively esy to mesure Cn cscde -prmeters of multiple devices to predict system performnce nlyticlly convenient CD progrms Flow-grph nlysis Cn compute H, Y,or Z prmeters from - prmeters if desired Incident Reflected b Trnsmitted b DUT Port Port Reflected Trnsmitted Incident b + b + Microwve Physics nd Techniques UC June 003 58
Mesuring -Prmeters Incident Trnsmitted Forwrd b Z 0 Reflected DUT Lod b 0 Reflected Incident Trnsmitted Incident b 0 b 0 Reflected Incident Trnsmitted Incident b 0 b 0 Z 0 Lod 0 b DUT Trnsmitted Microwve Physics nd Techniques UC June 003 59 Reflected Incident b Reverse
. Wht is the difference between network nd spectrum nlyzers? Hrd: getting (ccurte) trce Esy: interpreting results Esy: getting trce Hrd: interpreting results 8563 PECTRUM NLYZER 9 khz - 6.5 GHz mplitude Rtio Mesures known signl Power Mesures unknown signls Frequency Network nlyzers: mesure components, devices, circuits, sub-ssemblies contin source nd receiver disply rtioed mplitude nd phse (frequency or power sweeps) Frequency pectrum nlyzers: mesure signl mplitude chrcteristics (crrier level, sidebnds, hrmonics...) re receivers only (single chnnel) cn be used for sclr component test (no phse) with trcking gen. or ext. source(s) Microwve Physics nd Techniques UC June 003 60
ignl eprtion Mesuring incident signls for rtioing 50 Ω 50 Ω 6 d 6 d plitter usully resistive non-directionl brodbnd Coupled signl Min signl Coupler directionl low loss good isoltion, directivity hrd to get low freq performnce Microwve Physics nd Techniques UC June 003 6
Forwrd Coupling Fctor Coupling, forwrd ource -0 dm.0 mw Z 0 0 dm mw -.046 dm.99 mw Exmple of 0 d Coupler Coupling Fctor (d) -0 log P coupling forwrd P incident Microwve Physics nd Techniques UC June 003 6
Directionl Coupler Isoltion (Reverse Coupling Fctor) Coupling, reverse -50 dm.0000 mw this is n error signl during mesurements ource Z 0 0 dm mw.046 dm.99 mw Exmple of 0 d Coupler "turned round" P coupled reverse Isoltion Fctor (d) -0 log P incident Microwve Physics nd Techniques UC June 003 63
Directionl Coupler Directivity Directivity (d) 0 log P coupled forwrd Pcoupled reverse Directivity Coupling Fctor Isoltion Directivity (d) Isoltion (d) - Coupling Fctor (d) Exmple of 0 d Coupler with 50 d isoltion: Directivity 50 d - 0 d 30 d Microwve Physics nd Techniques UC June 003 64
Mesuring Coupler Directivity the Esy Wy.0 (0 d) (reference) Coupler Directivity 35 d Good pproximtion for coupling fctors 0 d short ource.08 (35 d) (normlized) Directivity 35 d - 0 d 35 d ource lod ssume perfect lod Microwve Physics nd Techniques UC June 003 65
Nrrowbnd Detection - Tuned Receiver DC / DP est sensitivity / dynmic rnge Provides hrmonic / spurious signl rejection Improve dynmic rnge by incresing power, decresing IF bndwidth, or verging Trde off noise floor nd mesurement speed 0 MHz 6.5 GHz Microwve Physics nd Techniques UC June 003 66
Comprison of Receiver Techniques 0 d rodbnd (diode) detection 0 d Nrrowbnd (tunedreceiver) detection -50 d -50 d -00 d -60 dm ensitivity higher noise floor flse responses -00 d < -00 dm ensitivity high dynmic rnge hrmonic immunity Dynmic rnge mximum receiver power - receiver noise floor Microwve Physics nd Techniques UC June 003 67
Dynmic Rnge nd ccurcy Dynmic rnge is very importnt for mesurement ccurcy! Error (d, deg) 00 0 Error Due to Interfering ignl phse error + mgn (d) - mgn (d) phse (± deg) 0. mgn error 0.0 0.00 0-5 -0-5 -0-5 -30-35 -40-45 -50-55 -60-65 -70 Interfering signl (d) Microwve Physics nd Techniques UC June 003 68
Mesurement Error Modeling ystemtic errors due to imperfections in the nlyzer nd test setup re ssumed to be time invrint (predictble) cn be chrcterized (during clibrtion process) nd mthemticlly removed during mesurements Rndom errors vry with time in rndom fshion (unpredictble) cnnot be removed by clibrtion min contributors: instrument noise (source phse noise, IF noise floor, etc.) switch repetbility connector repetbility Drift errors Mesured Dt re due to instrument or test-system performnce chnging fter clibrtion hs been done re primrily cused by temperture vrition cn be removed by further clibrtion(s) Errors: YTEMTIC RNDOM DRIFT Unknown Device Microwve Physics nd Techniques UC June 003 69
ystemtic Mesurement Errors R Directivity Crosstlk DUT Frequency response reflection trcking (/R) trnsmission trcking (/R) ource Mismtch Lod Mismtch ix forwrd nd six reverse error terms yields error terms for two-port devices Microwve Physics nd Techniques UC June 003 70
Types of Error Correction Two min types of error correction: response (normliztion) simple to perform only corrects for trcking errors stores reference trce in memory, then does dt divided by memory vector requires more stndrds requires n nlyzer tht cn mesure phse ccounts for ll mjor sources of systemtic error thru HORT OPEN thru LOD M Microwve Physics nd Techniques UC June 003 7
ignl Flow Computtions Complicted networks cn be efficiently nlyzed in mnner identicl to signls nd systems nd control. Z 0 b Z L b Γ L in generl i Γ ij b j Microwve Physics nd Techniques UC June 003 7
ignl Flow Grphs sic Rules: We ll follow certin rules when we build up network flow grph.. Ech vrible,,, b, nd b will be designted s node.. Ech of the -prmeters will be brnch. 3. rnches enter dependent vrible nodes, nd emnte from the independent vrible nodes. 4. In our -prmeter equtions, the reflected wves b nd b re the dependent vribles nd the incident wves nd re the independent vribles. 5. Ech node is equl to the sum of the brnches entering it. Microwve Physics nd Techniques UC June 003 73
ignl Flow Grphs Let s pply these rules to the two -prmeters equtions b b + + First eqution hs three nodes: b,, nd. b is dependent node nd is connected to through the brnch nd to node through the brnch. The second eqution is similr. b b Microwve Physics nd Techniques UC June 003 74
Complete Flow Grph for -Port ignl Flow Grphs b b The reltionship between the trveling wves is now esily seen. We hve incident on the network. Prt of it trnsmits through the network to become prt of b. Prt of it is reflected to become prt of b. Menwhile, the wve entering port two is trnsmitted through the network to become prt of b s well s being reflected from port two s prt of b. y merely following the rrows, we cn tell wht s going on in the network. This technique will be ll the more useful s we cscde networks or dd feedbck pths. Microwve Physics nd Techniques UC June 003 75
rrngement for ignl Flow nlysis Z G b V G I G Z 0 Z L b s b Γ b s b b s Z G Z o + Z b o V Γ L Γ G Microwve Physics nd Techniques UC June 003 76 b Γ L
nlysis of Most Common Circuit Z b s Γ Γ L V Z 0 [] Z 0 Z L b b b s b Γ Γ L b b s b s + Γ Microwve Physics nd Techniques UC June 003 77 L Γ L Γ
b s Γ b Γ L Γ L Γ in b Γ L + Γ b s L Γ L Γ Microwve Physics nd Techniques UC June 003 78 Note: Only 0 ensures tht cn be mesured. b + Γ L Γ L
cttering Mtrix The scttered-wve mplitudes re linerly relted to the incident wve mplitudes. Consider the N port junction If the only incident wve is V + then V V + is the reflection coefficient The totl voltge is port is V V + 3 V - 3 + V + V V - 4 V + 4 Port 4 V - 5 Port 5 Port N Port 3 Port Port V + 5 V - N V - V + N Wves will lso be scttered out of other ports. We will hve V + V - V + + V n n V n n, 3, 4,... N Microwve Physics nd Techniques UC June 003 79
If ll ports hve incident wve then cttering Mtrix V V... V N... N... N 3 3... N 3............ N N... NN V V... V + + + N or [ ] [ ][ + V V ] [ ] is clled the scttering mtrix i + for V 0 ( k j ) ij v v + j k Microwve Physics nd Techniques UC June 003 80
cttering Mtrix If we choose the equivlent Z 0 equl to then the incident power is given by + V n nd the scttering will be symmetricl. With this choice + V V + V, I I + I nd V V + ( V + I ) ( V I ) + Microwve Physics nd Techniques UC June 003 8
cttering Mtrix V + nd V - re the vribles in the scttering mtrix formultion; but they re liner combintion of V nd I. Other normliztion re ν V Z o i I Z o Just s in the impednce mtrix there re severl properties of the scttering mtrix we wnt to consider.. shift of the reference plnes. mtrix for reciprocl devices 3. mtrix for the lossless devices Microwve Physics nd Techniques UC June 003 8
cttering Mtrix Exmple: two-port network Equivlent Circuit Z Z t l l t + V I Z 3 I V + Port Port ssume TE 0 modes t t nd t pply KVL: V Z V Z I I + Z 3 + Z I 3 I + Z 3 + Z I 3 I Microwve Physics nd Techniques UC June 003 83
If Z Z Z 3 Z Z Z V I Z Z I 0 cttering Mtrix Then we hve V V nd Z Z I I + Z [ V ] [ Z ][ I ] + Z I I Z Z Z Z Z Microwve Physics nd Techniques UC June 003 84
This cn be trnsformed into n dmittnce mtrix cttering Mtrix Y I I Y Y Y Y V V Y Y Y Y Microwve Physics nd Techniques UC June 003 85
cttering Mtrix Trveling Wve: V V + ( ) ( ) ( ) x e, V V δx + x + V e x δx I imilrly for current: + ( x ) I ( x ) I ( x ) Reflection Coefficient: V + Z ( x ) V ( x ) o Z o Γ ( ) V V x + ( ) x ( x ) Microwve Physics nd Techniques UC June 003 86
ν Introduce normlized vribles: ( x ) V ( x ) Z, ( x ) Z I ( x ) ν i o tht o cttering Mtrix o ( ) ( ) ( ) ( ) ( ) ( ) nd ( ) ( ) ( ) x x + b x i x x b x b x Γ x x This defines single port network. Wht bout -port? -port b b + + Microwve Physics nd Techniques UC June 003 87
cttering Mtrix Ech reflected wve (b,b ) hs two contributions: one from the incident wve t the sme port nd nother from the incident wve t the other port. How to clculte -prmeters? b 0 Input reflected coefficient with output mtched. b 0 Reverse trnsmission coefficient with input mtched. b 0 Trnsmission coefficient with output mtched. b 0 Output reflected coefficient with input mtched. Microwve Physics nd Techniques UC June 003 88