Lecture 9 Microwave Network Analysis A. Nassiri - ANL June 19, Microwave Physics and Techniques UCSB June

Size: px
Start display at page:

Download "Lecture 9 Microwave Network Analysis A. Nassiri - ANL June 19, 2003. Microwave Physics and Techniques UCSB June 2003 1"

Transcription

1 Lecture 9 Microwve Network nlysis. Nssiri - NL June 9, 003 Microwve Physics nd Techniques UC June 003

2 -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

3 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

4 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

5 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

6 Microwve Physics nd Techniques UC June 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

7 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

8 Microwve Physics nd Techniques UC June 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

9 Microwve Physics nd Techniques UC June 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

10 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

11 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

12 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

13 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

14 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

15 Microwve Physics nd Techniques UC June 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 ( ) ( ) n n n n n n I Z Z V b I Z Z V o o o o

16 Definition of cttering Prmeters We cn now define -prmeters: b b () 8 Microwve Physics nd Techniques UC June 003 6

17 Definition of cttering Prmeters b b b b 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 port port port port 9 ( ) 0 ( ) ( ) ( )

18 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

19 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

20 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 ( 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)

21 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

22 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

23 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

24 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 ( )

25 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

26 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 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

27 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

28 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

29 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

30 Determintion of T-network elements For 3 d ttenution, we require V V V V G etting the rtio of V /V to 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

31 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

32 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

33 Chin cttering Mtrix b Port [ T ] [ ] T Port b b b Cscding of two networks nd Microwve Physics nd Techniques UC June

34 Microwve Physics nd Techniques UC June 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

35 Microwve Physics nd Techniques UC June Chin cttering Mtrix ( ) 3 b b Thus, for the combined system, we conclude ( ) 3 b T T T T T T T T b

36 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 ( 3) - ( 35)

37 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 ( 36) ( 37)

38 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

39 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

40 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 ( 47)

41 Prcticl Network nlysis Microwve Physics nd Techniques UC June 003 4

42 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

43 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

44 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

45 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

46 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

47 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

48 High-Frequency Device Chrcteriztion Lightwve nlogy Incident Trnsmitted Reflected Microwve Physics nd Techniques UC June

49 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

50 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

51 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

52 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

53 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

54 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 Full reflection (ZL open, short) 0 d

55 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

56 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

57 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

58 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

59 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 Reflected Incident b Reverse

60 . 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 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

61 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

62 Forwrd Coupling Fctor Coupling, forwrd ource -0 dm.0 mw Z 0 0 dm mw 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

63 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

64 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

65 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

66 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

67 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

68 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 Interfering signl (d) Microwve Physics nd Techniques UC June

69 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

70 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

71 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

72 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

73 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

74 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

75 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

76 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 b Γ L

77 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 L Γ L Γ

78 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 Note: Only 0 ensures tht cn be mesured. b + Γ L Γ L

79 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 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

80 If ll ports hve incident wve then cttering Mtrix V V... V N... N... N N 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

81 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

82 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

83 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

84 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

85 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

86 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

87 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

88 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

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

4.5 Signal Flow Graphs

4.5 Signal Flow Graphs 3/9/009 4_5 ignl Flow Grphs.doc / 4.5 ignl Flow Grphs Reding Assignment: pp. 89-97 Q: Using individul device scttering prmeters to nlze comple microwve network results in lot of mess mth! Isn t there n

More information

Understanding Basic Analog Ideal Op Amps

Understanding Basic Analog Ideal Op Amps Appliction Report SLAA068A - April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

EE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form- example shown

EE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form- example shown EE247 Lecture 4 Ldder type filters For simplicity, will strt with ll pole ldder type filters Convert to integrtor bsed form exmple shown Then will ttend to high order ldder type filters incorporting zeros

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Pulsed-IV Pulsed-RF Measurements Using a Large Signal Network Analyzer

Pulsed-IV Pulsed-RF Measurements Using a Large Signal Network Analyzer Pulsed-IV Pulsed-RF Mesurements Using Lrge Signl Network Anlyzer Seok Joo Doo*, Ptrick Roblin* #, Sunyoung Lee*, Dominique Chillot* + nd Mrc Vnden Bossche + *The Ohio Stte University, * + on leve from

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.

4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors. 4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Increasing Q of Waveguide Pulse-Compression Cavities

Increasing Q of Waveguide Pulse-Compression Cavities Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide Pulse-Compression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Project 6 Aircraft static stability and control

Project 6 Aircraft static stability and control Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The

More information

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

How To Set Up A Network For Your Business

How To Set Up A Network For Your Business Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING Sung Joon Kim*, Dong-Chul Che Kore Aerospce Reserch Institute, 45 Eoeun-Dong, Youseong-Gu, Dejeon, 35-333, Kore Phone : 82-42-86-231 FAX

More information

Answer, Key Homework 10 David McIntyre 1

Answer, Key Homework 10 David McIntyre 1 Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your

More information

Hillsborough Township Public Schools Mathematics Department Computer Programming 1

Hillsborough Township Public Schools Mathematics Department Computer Programming 1 Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

A Network Management System for Power-Line Communications and its Verification by Simulation

A Network Management System for Power-Line Communications and its Verification by Simulation A Network Mngement System for Power-Line Communictions nd its Verifiction y Simultion Mrkus Seeck, Gerd Bumiller GmH Unterschluerscher-Huptstr. 10, D-90613 Großhersdorf, Germny Phone: +49 9105 9960-51,

More information

Vector differentiation. Chapters 6, 7

Vector differentiation. Chapters 6, 7 Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

6 Energy Methods And The Energy of Waves MATH 22C

6 Energy Methods And The Energy of Waves MATH 22C 6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this

More information

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management Journl of Mchine Lerning Reserch 9 (2008) 2079-2 Submitted 8/08; Published 0/08 Vlue Function Approximtion using Multiple Aggregtion for Multittribute Resource Mngement Abrhm George Wrren B. Powell Deprtment

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

More information

How To Network A Smll Business

How To Network A Smll Business Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix. APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The

More information

Thin crustal layering in Northern France: observations and modelling of the PMP spectral content

Thin crustal layering in Northern France: observations and modelling of the PMP spectral content Ceophys. 1. Int. (1989) 99, 229-246 Thin crustl lyering in Northern Frnce: observtions nd modelling of the PMP spectrl content Anne Pul nd Florence Nicollin2 I Lbortoire de Gtophysique Interne et Tectonophysique,

More information

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl

More information

Version 001 Summer Review #03 tubman (IBII20142015) 1

Version 001 Summer Review #03 tubman (IBII20142015) 1 Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03

More information

Second-Degree Equations as Object of Learning

Second-Degree Equations as Object of Learning Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct Second-Degree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,

More information

Engineer-to-Engineer Note

Engineer-to-Engineer Note Engineer-to-Engineer Note EE-265 Technicl notes on using Anlog Devices DSPs, processors nd development tools Contct our technicl support t dsp.support@nlog.com nd t dsptools.support@nlog.com Or visit our

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207.

Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. MIT OpenCourseWre http://ocw.mit.edu Hus, Hermnn A., nd Jmes R. Melcher. Electromgnetic Fields nd Energy. Englewood Cliffs, NJ: Prentice-Hll, 1989. ISBN: 9780132490207. Plese use the following cittion

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

COMPONENTS: COMBINED LOADING

COMPONENTS: COMBINED LOADING LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of

More information

Applications to Physics and Engineering

Applications to Physics and Engineering Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

ClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment

ClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment ClerPeks Customer Cre Guide Business s Usul (BU) Services Pece of mind for your BI Investment ClerPeks Customer Cre Business s Usul Services Tble of Contents 1. Overview...3 Benefits of Choosing ClerPeks

More information

EasyMP Network Projection Operation Guide

EasyMP Network Projection Operation Guide EsyMP Network Projection Opertion Guide Contents 2 About EsyMP Network Projection Functions of EsyMP Network Projection... 5 Vrious Screen Trnsfer Functions... 5 Instlling the Softwre... 6 Softwre Requirements...6

More information

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd

More information

Section 5.2, Commands for Configuring ISDN Protocols. Section 5.3, Configuring ISDN Signaling. Section 5.4, Configuring ISDN LAPD and Call Control

Section 5.2, Commands for Configuring ISDN Protocols. Section 5.3, Configuring ISDN Signaling. Section 5.4, Configuring ISDN LAPD and Call Control Chpter 5 Configurtion of ISDN Protocols This chpter provides instructions for configuring the ISDN protocols in the SP201 for signling conversion. Use the sections tht reflect the softwre you re configuring.

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

More information

Performance Monitoring Fundamentals: Demystifying Performance Assessment Techniques

Performance Monitoring Fundamentals: Demystifying Performance Assessment Techniques Performnce Monitoring Fundmentls: Demystifying Performnce Assessment Techniques Roert C. Rice, PhD Rchelle R. Jyringi Dougls J. Cooper, PhD Control Sttion, Inc. Deprtment of Chemicl Engineering Control

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

QoS Mechanisms C HAPTER 3. 3.1 Introduction. 3.2 Classification

QoS Mechanisms C HAPTER 3. 3.1 Introduction. 3.2 Classification C HAPTER 3 QoS Mechnisms 3.1 Introduction In the previous chpter, we introduced the fundmentl QoS concepts. In this chpter we introduce number of key QoS mechnisms tht enble QoS services. At the end of

More information

Simulation of operation modes of isochronous cyclotron by a new interative method

Simulation of operation modes of isochronous cyclotron by a new interative method NUKLEONIKA 27;52(1):29 34 ORIGINAL PAPER Simultion of opertion modes of isochronous cyclotron y new intertive method Ryszrd Trszkiewicz, Mrek Tlch, Jcek Sulikowski, Henryk Doruch, Tdeusz Norys, Artur Srok,

More information

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd

More information

trademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007

trademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007 trdemrk nd symbol guidelines trdemrk guidelines The trdemrk Cn be plced in either of the two usul configurtions but horizontl usge is preferble. Wherever possible the trdemrk should be plced on blck bckground.

More information

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00 COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

Data replication in mobile computing

Data replication in mobile computing Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

An efficient integral equation technique for the analysis of arbitrarily shaped capacitive waveguide circuits

An efficient integral equation technique for the analysis of arbitrarily shaped capacitive waveguide circuits RADIO SCIENCE, VOL. 46,, doi:10.1029/2010rs004458, 2011 An efficient integrl eqution technique for the nlysis of rbitrrily shped cpcitive wveguide circuits F. D. Quesd Pereir, 1 P. Ver Cstejón, 1 A. Álvrez

More information