Fields and Waves I ecture 3 Input Impedance n Transmissin ines K. A. Cnnr Electrical, Cmputer, and Systems Engineering Department Rensselaer Plytechnic Institute, Try, NY
These Slides Were Prepared by Prf. Kenneth A. Cnnr Using Original Materials Written Mstly by the Fllwing: Kenneth A. Cnnr ECSE Department, Rensselaer Plytechnic Institute, Try, NY J. Darryl Michael GE Glbal Research Center, Niskayuna, NY Thmas P. Crwley Natinal Institute f Standards and Technlgy, Bulder, CO Sheppard J. Saln ECSE Department, Rensselaer Plytechnic Institute, Try, NY ale Ergene ITU Infrmatics Institute, Istanbul, Turkey Jeffrey Braunstein Chung-Ang University, Seul, Krea Materials frm ther surces are referenced where they are used. Thse listed as Ulaby are figures frm Ulaby s textbk. 10 September 2006 Fields and Waves I 2
http://memry.lc.gv/ammem/index.html 10 September 2006 Fields and Waves I 3
Overview Review Henry Farny Sng f the Talking Wire ltages and Currents n Transmissin ines Standing Waves Input Impedance ssy Transmissin ines w ss Transmissin ines 10 September 2006 Fields and Waves I 4
What d we knw s far? Slutins lk like ω 2π β ω lc ω με u λ 2π ω 2πf T 2π u λ β f ( m β ) Acs ωt z u 1 1 lc με ε ε r ε μ μ μ r Figure frm http://www.emc.maricpa.edu/ 10 September 2006 Fields and Waves I 5
Phase elcity Phase cnst A simple way t find the phase velcity Identify sme feature f the sine wave. Fr this we chse cnstant phase ωt m βz cnst Determine it s velcity. Since the phase is a cnstant, we knw that ( ) ω t m t β z 0 ω β z m z ω u β t t 0 10 September 2006 Fields and Waves I 6
Phasr Ntatin Fr ease f analysis (changes secnd rder partial differential equatin int a secnd rder rdinary differential equatin), we use phasr ntatin. ( m ) { m jβz} The term in the brackets is the phasr. ( jωt) f (,) z t Acsωt βz Re Ae e f () z Ae m j β z 10 September 2006 Fields and Waves I 7
Phasr Ntatin T cnvert t space-time frm frm the phasr frm, multiply by and take the real part. e j t ω m j β z jωt ( ω m β ) f (,) z t Re( Ae e ) Acs t z If A is cmplex A Ae j θ A ( ω m β θ ) jθa m jβz jωt f (,) z t Re( Ae e e ) Acs t z A 10 September 2006 Fields and Waves I 8
Wrkspace What is the phasr f the time derivative? ([ jβz] jωt) vzt (,) Re e e ([ β ] ω ) t vzt j z j t (,) Re e e? t 10 September 2006 Fields and Waves I 9
Transmissin ines Incident Wave Mismatched lad Reflected Wave Standing wave due t interference 10 September 2006 Fields and Waves I 10
10 September 2006 Fields and Waves I 11 Transmissin ines Phasr ltage Slutin Phasr Frm f the Wave Equatin: c l z t c l z 2 2 2 2 2 2 2 ω where: z j e ± β m General Slutin: z j z j e e β β
Transmissin ines - Standing Wave Derivatin e jβ z e jβ z Frward Wave cs( ω t β z ) TIME DOMAIN Backward Wave cs( ω t β z ) max ccurs when Frward and Backward Waves are in Phase CONSTRUCTIE INTERFERENCE min ccurs when Frward and Backward Waves are ut f Phase DESTRUCTIE INTERFERENCE 10 September 2006 Fields and Waves I 12
Transmissin ine ltages and Currents General Slutin vz e j β () z e jβz iz () e e jβz jβz e jβz e jβz The latter expressin is derived frm vz () jωli() z z Can we easily explain the minus sign? 10 September 2006 Fields and Waves I 13
Wrkspace 10 September 2006 Fields and Waves I 14
Reflectin Cefficient Derivatin Define the Reflectin Cefficient: m Γ m Maximum Amplitude when in Phase: max m m max m (1 Γ ) Similarly: (1 Γ ) min m Standing Wave Rati (SWR) max min 1 1 Γ Γ 10 September 2006 Fields and Waves I 15
Anther General Frm fr the Slutin Using the reflectin cefficient Nte that we will be rewriting the slutin in different frms v z e j β () z Γ e jβz iz e jβz () Γ e jβz 10 September 2006 Fields and Waves I 16
Transmissin ines Bth frm Ulaby 10 September 2006 Fields and Waves I 17
10 September 2006 Fields and Waves I 18 Reflectin Cefficient et z0 at the OAD ) (1 0 0 j j lad e e Γ β β ( ) I lad Γ Γ 1
Reflectin Cefficient This is a key relatinship At the lad I lad lad ( 1 Γ ) ( 1 Γ ) Γ 10 September 2006 Fields and Waves I 19
Wrkspace in? ut? 10 September 2006 Fields and Waves I 20
Wrkspace 10 September 2006 Fields and Waves I 21
Wrkspace Shrt Circuit ad 10 September 2006 Fields and Waves I 22
Wrkspace Open Circuit ad 10 September 2006 Fields and Waves I 23
Shrt Circuit ad Fr 0, we have Γ 0 0 1 ( ) j z v() z e Γ e e e β jβz jβz jβz jβz e csβz jsin βz jβz e csβz jsin βz ( ) vz () j2sinβz 10 September 2006 Fields and Waves I 24
Shrt Circuit ad Cnvert t space-time frm ( j ω t ) ( ( ) jωt 2 β ) vzt (,) Re vze () Re j sin ze (( ) jωt j2 βz e ) 2 βz( j βz βz) This is a standing wave ( ) Re sin Re sin cs sin vzt (,) sin zsin t 2 β ω 10 September 2006 Fields and Waves I 25
Shrt Circuit ad What are the vltage maxima and minima? vzt (,) sin zsin t 2 β ω Where are they? The standing wave pattern is the envelpe f this functin. 10 September 2006 Fields and Waves I 26
Reflectin 50 Ω 80 m 2 pp 1.5 MHz ~ RG-58 93 Ω Nte that we are free t chse either the lad end r the surce end as z0 z0 z80 z'80 z'0 10 September 2006 Fields and Waves I 27
Reflectin Cefficient Γ Determine the reflectin cefficient at the lad, and the standing wave rati, SWR. Start with a shrt circuit lad and then cnsider a 25 Ohm lad. Then d an pen circuit and 93 Ohm lad. Assume that the frward traveling wave has an amplitude f 10 lts. Sketch the standing wave pattern fr vltage and current fr the shrt circuit lad. Include numbers fr amplitudes and distances. Under what cnditins d yu get a vltage maximum at the lad? a minimum? Can yu answer this in general? If the lad is a 3.3 nf capacitr, what is the reflectin cefficient at the lad? Where is the lcatin f the first minimum? T answer this, we need a bit mre develpment. 10 September 2006 Fields and Waves I 28
First Sketch the Standing Wave Patterns by Hand The reflectin cefficient 0 Γ 50 0 50 1 Γ 25 50 25 50 1 0333. 3 Γ 50 50 1 Γ 93 50 93 50 43 03. 143 10 September 2006 Fields and Waves I 29
Using Matlab fr the ltage Standing Wave Patterns 10 September 2006 Fields and Waves I 30
10 September 2006 Fields and Waves I 31
Current Standing Wave Patterns Can we use what we just displayed t find the current standing wave patterns? Yes, because the reflectin cefficient fr current is always just the negative f the vltage reflectin cefficient 10 September 2006 Fields and Waves I 32
Standing Wave Pattern We have just seen that: Minimum ccurs at OAD fr 0 Is it als true that: Or, in general, that: Maximum ccurs at OAD fr Γ 0 > 0 < 0 > < Γ 0 Max at OAD Min at OAD IF is REA 10 September 2006 Fields and Waves I 33
10 September 2006 Fields and Waves I 34 Standing Wave Pattern If z at OAD and z0 at SOURCE, ) z ( 2 j e ) z ( β Γ Γ ) ( 2 z j j e e Γ Γ β θ When Phase π, the FIRST MINIMUM ccurs ( ) π β θ Γ z 2 ( ) λ π θ λ Γ 4 4 z Other MINs are displaced by λ/2 Phase f the reflectin cefficient
A Repeat f HW2 Experiment (600kHz) What did yu see at the 20 ndes? Time Delay Amplitude Did any f yu try an pen r shrt circuit? 10 September 2006 Fields and Waves I 35
Change the Frequency t 1.5MHz 10 September 2006 Fields and Waves I 36
Cntinuing 10 September 2006 Fields and Waves I 37
Capacitive ad? 10 September 2006 Fields and Waves I 38
Capacitive ad Nw we can answer the questin abut where the first minimum is lcated fr a capacitive lad. Γ 1 jωc 1 jωc 1 1 jωc jωc 1exp tan 1 ωc 1 ( ωc ) 2 a a jb jb a a jb jb a a jb jb a j2abb 2 2 a b 2 2 ( a b ) exp tan 2 2 1 a b 2 2 2ab a b 2 2 exp tan 1 2ab a b 2 2 10 September 2006 Fields and Waves I 39
Capacitive ad Fr the specific case here Γ 1 jωc 1 jωc j 32. 2 50 j32. 2 50 04. j091. 1e j06362. π λ z Γ θπ λ λ. 6362π λ λ. 1591λ 0. 09λ 4 2 4 4π 4 10 September 2006 Fields and Waves I 40
Standing Wave Pattern fr Capacitive ad Reflectin Cefficient Γ 1e j 0. 6362π ad End 10 September 2006 Fields and Waves I 41
PSpice Input Impedance Fr the same surce and line, but different lad: R1 50 OFF 0 AMP 10 FREQ 1meg 1 T1 R1 300 0 0 R2 50 OFF 0 AMP 10 FREQ 1meg 2 T2 R2 50 0 0 10 September 2006 Fields and Waves I 42
Changing the ad The vltages and currents at the input change the input impedance changes. 5.0 0 SE>> -5.0 10 (T2:A) (T2:B) 0-10 10.0us 10.5us 11.0us 11.5us 12.0us 12.5us 13.0us (T1:A) (T1:B) Time 10 September 2006 Fields and Waves I 43
Changing the ength and ine Prperties Frm the standing wave patterns r the expressins fr the vltages and the currents n the line, we can see that the rati f the vltage t the current will depend n the length f the line and the line prperties. 10 September 2006 Fields and Waves I 44
Wrkspace 10 September 2006 Fields and Waves I 45
Input Impedance What des in in, lk like? I in Define: When is cmplex, s is. T address the input impedance, we need t generalize the reflectin cefficient. Γ( z) e e jβ z jβ z Γ e j2β z Γ j 2 z e β if z 0 at OAD 10 September 2006 Fields and Waves I 46
Anther Frm fr the General Slutin Using the Generalized Reflectin Cefficient ( 1 Γ ) vz () e jβz () z iz e jβz () () ( 1 Γ z ) 10 September 2006 Fields and Waves I 47
10 September 2006 Fields and Waves I 48 Input Impedance Previusly, we have seen: )) ( (1 ) ( ) ( ) ( ˆ z e z z z z j Γ β What abut I? )) ( (1 ) ( ) ( ) ( ˆ z e z z z I z j Γ β Als, ) z ( j 2 e ) z ( β Γ Γ
Input Impedance Frm the Rati (the generalized impedance): ˆ( z) Iˆ( z) 1 Γ( z) 1 Γ( z) ( z) We are primarily interested in z0 value treat cnnectin t rest f circuit as 2 prt with, in ( z 0) 1 Γ( z 0) 1 Γ( z 0) 10 September 2006 Fields and Waves I 49
10 September 2006 Fields and Waves I 50 Input Impedance After lts f algebra, ne can shw: ) tan( ) tan( 0) ( j j z in β β Special Case example: 0 (shrt circuit) ) tan( ) tan( 0 ) tan( 0 0) ( j j j z in β β β
Input Impedance - SHORT CIRCUIT in ( z 0) j tan( β ) Can change in by changing these tw parameters Fix β, vary - different effects ary β, fix - get same effects Nte that is the length f the Transmissin ine 10 September 2006 Fields and Waves I 51
Input Impedance Shrt Circuit Fr varying frequency, the input impedance is imaginary and can achieve any value. 10 September 2006 Fields and Waves I 52
Input Impedance - T Cnsider sme ther cases 80 Ω l 80 m 1 Peak ~ in RG 58 10 September 2006 Fields and Waves I 53
Input Impedance - T Open Circuit Case small j j tan( β ) tan( β ) small 93Ω - lts f cmplex algebra, but straight frward 10 September 2006 Fields and Waves I 54
Using the Input Impedance We knw in (z0) - treat as 2-PORT 80 Ω 1 in in ltage Divider Pwer 1 2 Re { I } in in 1 2 in Re in in 1 2 Re 2 in in 10 September 2006 Fields and Waves I 55
10 September 2006 Fields and Waves I 56
Using the Input Impedance In a ssless Transmissin ine, P in flws int the Transmissin ine and it is dissipated at the 2 OAD What is the vltage at the lad? ˆ ( z) e jβ z (1 Γ( z)) ˆ jβ z ( z 0) e (1 Γ( z in P in 1 2 1 Γ(0) in 0)) Can then plug back and get the full phasr expressin 10 September 2006 Fields and Waves I 57
Using the Input Impedance The full frm f the vltage z e j β () z e jβz All infrmatin is nw available t determine the vltage and current everywhere n the line. Yu will be ding this n the prject. 10 September 2006 Fields and Waves I 58
Special Cases Recall that the standing wave pattern repeated every half wavelength. Thus, we expect that this will als happen fr in. First, cnsider the trivial case f 0. j tan β in j tan β Nw let the line be a half wavelength lng 2π λ tan β tan tan( π) λ 2 0 in 0 0 10 September 2006 Fields and Waves I 59
Special Cases Thus, fr a line that is exactly an integer number f half wavelengths lng in Thus, if yu have a transmissin line with the wrng characteristic impedance, yu can match the lad t the surce by selecting a length equal t a half wavelength. 10 September 2006 Fields and Waves I 60
Special Cases If the line is an dd multiple f a quarter wavelength, we als get an interesting result. j tan β in j tan β j tan β j tan β 2π λ π tan β tan tan λ 4 2 Thus, such a transmissin line wrks like an impedance transfrmer and has a real input impedance. 2 10 September 2006 Fields and Waves I 61
10 September 2006 Fields and Waves I 62 Tday s Majr Result Input Impedance ) tan( ) tan( 0) ( j j z in β β