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Notes 2 5317-6351 Transmission Lines Part 1 (TL Theory).pptx

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This document provides notes on transmission line theory. Some key points: - Transmission line theory is needed when the length of a line is significant compared to a wavelength. - Transmission lines have per-unit-length parameters of capacitance, inductance, resistance, and conductance. - The telegrapher's equations describe voltage and current on a transmission line as a function of position and time. - Waves on a transmission line travel at the phase velocity, which depends on the transmission line's characteristics. - The characteristic impedance Z0 of a transmission line relates the amplitudes of voltage and current waves traveling on the line.

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Prof. David R. Jackson Dept. of ECE Notes 2 ECE 5317-6351 Microwave Engineering Fall 2019 Transmission Lines Part 1: TL Theory 1 Adapted from notes by Prof. Jeffery T. Williams
Transmission-Line Theory We need transmission-line theory whenever the length of a line is significant compared to a wavelength. 2 L
Transmission Line 2 conductors 4 per-unit-length parameters: C = capacitance/length [F/m] L = inductance/length [H/m] R = resistance/length [/m] G = conductance/length [ /m or S/m]  Dz 3
Transmission Line (cont.) z D   , i z t + + + + + + + - - - - - - - - - -   , v z t x x x B 4 Note: There are equal and opposite currents on the two conductors. (We only need to work with the current on the top conductor, since we have chosen to put all of the series elements there.) + - + -   , i z z t  D   , i z t   , v z z t  D   , v z t R z D L z D G z D C z D z
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) i z t v z t v z z t i z t R z L z t v z z t i z t i z z t v z z t G z C z t    D  D  D    D   D   D D  D  Transmission Line (cont.) 5 + - + -   , i z z t  D   , i z t   , v z z t  D   , v z t R z D L z D G z D C z D z
Hence ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) v z z t v z t i z t Ri z t L z t i z z t i z t v z z t Gv z z t C z t  D      D   D    D    D  D  Now let Dz  0: v i Ri L z t i v Gv C z t               “Telegrapher’s Equations” TEM Transmission Line (cont.) 6
To combine these, take the derivative of the first one with respect to z: 2 2 2 2 v i i R L z z z t i i R L z t z v v v R Gv C L G C t t t                                                          Switch the order of the derivatives. TEM Transmission Line (cont.) 7 i v Gv C z t       
  2 2 2 2 ( ) 0 v v v RG v RC LG LC z t t                  The same differential equation also holds for i. Hence, we have: 2 2 2 2 v v v v R Gv C L G C z t t t                            TEM Transmission Line (cont.) 8 Note: There is no exact solution in the time domain, in the lossy case.
  2 2 2 ( ) ( ) 0 d V RG V RC LG j V LC V dz           2 2 2 2 ( ) 0 v v v RG v RC LG LC z t t                  TEM Transmission Line (cont.) Time-Harmonic Waves: 9 j t    
Note that     2 2 2 ( ) d V RG V j RC LG V LC V dz       2 ( ) ( )( ) RG j RC LG LC R j L G j C           = = Z R j L Y G j C       series impedance / unit length parallel admittance / unit length Then we can write: 2 2 ( ) d V ZY V dz  TEM Transmission Line (cont.) 10
Define We have: Solution: 2 ZY   ( ) z z V z Ae Be        1/2 ( )( ) R j L G j C           2 2 2 d V z V z dz   Then TEM Transmission Line (cont.)  is called the “propagation constant”. 11 Question: Which sign of the square root is correct?
Principal square root: TEM Transmission Line (cont.) /2 j z r e         12  1/2 ( )( ) R j L G j C       We choose the principal square root.   Re 0 z  ( )( ) R j L G j C       Re 0   Hence (Note the square-root (“radical”) symbol here.) j z re   1/2 1/2 4 2, 4 2 1 1 , 2 2 j j j j               Examples :
Denote: TEM Transmission Line (cont.) Re 0     13 ( )( ) R j L G j C       [np/m]   attenuationcontant j        rad/m   phaseconstant   1/m   propagationconstant
TEM Transmission Line (cont.) 14 There are two possible locations for the complex square root: 0, 0     Hence: Re Im       R j L   Re Im G j C   Re Im    Re Im ( )( ) R j L G j C       The principal square root must be in the first quadrant.
TEM Transmission Line (cont.) 15   z z j z V z Ae Ae e         Wave traveling in +z direction:   j      Wave is attenuating as it propagates.   z z j z V z Ae Ae e         Wave traveling in -z direction:   j      Wave is attenuating as it propagates.
TEM Transmission Line (cont.) 16    10 10 dB 20log 20log 20 0.4343 8.686 out in z V V A e A z z                             Attenuation in     10 log 0.4343 ln x x  Note: Attenuation in dB/m:   z j z V z Ae e      dB/m 8.686  Attenuationin
Wavenumber Notation 17 ( )( ) R j L G j C       "propagationconstant"   z z j z V z Ae Ae e           z jk z z j z V z Ae Ae e        ( )( ) z k j R j L G j C       "propagationwavenumber" z jk   j      z k j    
TEM Transmission Line (cont.) 0 0 ( ) z z j z V z V e V e e            Forward travelling wave (a wave traveling in the positive z direction):             0 0 0 ( , ) Re Re cos z j z j t j z j z j t z v z t V e e e V e e e e V e t z                          2 g     2 g    The wave “repeats” when: Hence: 18 g  0 t  z 0 z V e    “snapshot” of wave
Phase Velocity Let’s track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest of the wave. 0 ( , ) cos( ) z v z t V e t z           19 z (phase velocity) p v
Phase Velocity (cont.) 0 constant            t z dz dt dz dt Set Hence p v      Im )( p v R j L G j C       In expanded form: 20
Characteristic Impedance Z0 0 ( ) ( ) V z Z I z    0 0 ( ) ( ) z z V z V e I z I e           so 0 0 0 V Z I    Assumption: A wave is traveling in the positive z direction. (Note: Z0 is a number, not a function of z.) 21   I z    V z    z
Use first Telegrapher’s Equation: v i Ri L z t        so dV RI j LI ZI dz       Hence 0 0 z z V e ZI e           Characteristic Impedance Z0 (cont.) 22 0 0 ( ) ( ) z z V z V e I z I e           Recall:
From this we have: Use: 0 0 0 V Z Z Z I ZY       Characteristic Impedance Z0 (cont.) Z R j L Y G j C       23 Both are in the first quadrant 0 0 / Re 0 ZY Z Z ZY Z          first quadrant right - half plane 0 Z Z Y  (principal square root)
Hence, we have Characteristic Impedance Z0 (cont.) 24 0 Z R j L Z Y G j C       0 Re 0 Z 
  0 0 0 0 j z j j z z z z j z V e e V z V e V V e e e e e                                   0 0 cos c , R os e j t z z V e t v z t V z z V z e e t                       In the time domain: Wave in +z direction Wave in -z direction General Case (Waves in Both Directions) 25
Backward-Traveling Wave 0 ( ) ( ) V z Z I z     0 ( ) ( ) V z Z I z     so A wave is traveling in the negative z direction. Note: The reference directions for voltage and current are chosen the same as for the forward wave. 26 z   I z    V z   
General Case 0 0 0 0 0 ( ) 1 ( ) z z z z V z V e V e I z V e V e Z                     A general superposition of forward and backward traveling waves: Most general case: 27 z   I z   V z  
       0 0 0 0 0 0 0 z z z z V z V e V e V V I z e e Z j R j L G j C R j L Z G j Z C                                  2 m g     [m/s] p v    Guided wavelength: Phase velocity: Summary of Basic TL formulas 28 dB/m 8.686  Attenuationin   I z   V z  
Lossless Case 0, 0 R G   ( )( ) j R j L G j C j LC             so 0 LC      0 R j L Z G j C      0 L Z C  1 p v LC  p v    (independent of freq.) (real and independent of freq.) 29
Lossless Case (cont.) 1 p v LC  If the medium between the two conductors is lossless and homogeneous (uniform) and is characterized by (, ), then (proof given later): LC   The speed of light in a dielectric medium is 1 d c   Hence, we have that: p d v c  In the lossless case the phase velocity does not depend on the frequency, and it is always equal to the speed of light (in the material). (proof given later) 30 0 r r LC k k           and
  0 0 z z V z V e V e         Where do we assign z = 0 ? The usual choice is at the load. Amplitude of voltage wave propagating in negative z direction at z = 0. Amplitude of voltage wave propagating in positive z direction at z = 0. Terminated Transmission Line 31     0 0 0 0 V V V V         V z    V z  Terminating impedance (load)   V z   I z 0 z  z L Z  
  0 0 0 z V V z e                 0 0 0 0 z z z z V z V z e V z e            0 0 0 z V z V e        0 0 0 z V V z e       Terminated Transmission Line (cont.)   0 0 z z V z V e V e         Hence Can we use z = z0 as a reference plane? 32   0 0 0 z V z V e      Terminating impedance (load)   V z   I z 0 z  z L Z   0 z z 
Terminated Transmission Line (cont.)       0 0 z z V z V e V e         Compare: This is simply a change of reference plane, from z = 0 to z = z0. 33           0 0 0 0 z z z z V z V z e V z e          Terminating impedance (load)   V z   I z 0 z  z L Z   0 z z 
  0 0 z z V z V e V e         What is V(-d) ?   0 0 d d V d V e V e           0 0 0 0 d d V V I d e e Z Z         Propagating forwards Propagating backwards Terminated Transmission Line (cont.) d  distance away from load The current at z = -d is then: 34 (This does not necessarily have to be the length of the entire line.) Terminating impedance (load)   I d  z d     0 z  z L Z   V d 
    2 0 0 1 d d L V I d e e Z           2 0 0 0 0 0 1 d d d d V V d V e V e V e e V                       Similarly, L  load reflection coefficient Terminated Transmission Line (cont.) 35     2 0 1 d d L V d V e e         or 0 0 L V V       I d  z d     0 z  z L Z   V d  (-d) = reflection coefficient at z = -d   2 d L d e      
        2 0 2 0 0 1 1 d d L d d L V d V e e V I d e e Z                Z(-d) = impedance seen “looking” towards load at z = -d. Terminated Transmission Line (cont.) 36           2 0 0 2 1 1 1 1 d L d L V d d e Z d Z Z I d e d                                 Note: If we are at the beginning of the line, we will call this the “input impedance”.   I d  z d     0 z  z L Z   V d    Z d 
At the load (d = 0):   0 1 0 1 L L L Z Z Z             Terminated Transmission Line (cont.) 0 0 L L L Z Z Z Z     37    2 d L d e       At any point on the line(d > 0):
Thus,   2 0 0 0 2 0 0 1 1 d L L d L L Z Z e Z Z Z d Z Z Z e Z Z                                     Terminated Transmission Line (cont.)       2 0 0 2 1 1 1 1 d L d L d e Z d Z Z d e                              Recall 38
Simplifying, we have:       0 0 0 tanh tanh L L Z Z d Z d Z Z Z d             Terminated Transmission Line (cont.)                           2 0 2 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1 cosh sinh cosh sinh d L d L L L d d L L L L d d L L d d L L L L Z Z e Z Z Z Z Z Z e Z d Z Z Z Z Z Z e Z Z e Z Z Z Z e Z Z e Z Z Z e Z Z e Z d Z d Z Z d Z d                                                                                        Hence, we have 39
        2 0 2 0 0 1 1 j d j d L j d j d L V d V e e V I d e e Z                Impedance is periodic with period g/2:     2 1 2 1 2 1 2 / 2 g g d d d d d d              Terminated Lossless Transmission Line j j        Note:       tanh tanh tan d j d j d      The tan function repeats when       0 0 0 tan tan L L Z jZ d Z d Z Z jZ d             40 Lossless:   2 0 2 1 1 j d L j d L e Z d Z e              
      2 0 2 0 0 0 1 1 tanh tanh d L d L L L e Z d Z e Z Z d Z Z Z d                             0 0 2 L L L g p Z Z Z Z v            Summary for Lossy Transmission Line 41   I d  z d     0 z  z L Z   V d    Z d  ( )( ) j R j L G j C          
      2 0 2 0 0 0 1 1 tan tan j d L j d L L L e Z d Z e Z jZ d Z Z jZ d                            0 0 2 2 L L L g d p d Z Z Z Z k v c                 Summary for Lossless Transmission Line 42   I d  z d     0 z  z L Z   V d    Z d  LC k       
0 0 0 L L L Z Z Z Z      No reflection from the load Matched Load (ZL=Z0)   0 Z d Z z   for any 43   2 0 2 1 1 d L d L e Z d Z e                 I d  z d     0 z  z L Z   V d    Z d  
1 L    Always imaginary! Short-Circuit Load (ZL=0) 44       0 0 0 tan tan L L Z jZ d Z d Z Z jZ d                 0 tan Z d jZ d    0 0 L L L Z Z Z Z       Z d  z d   0 z  z   I d    V d    Lossless Case
Note: 2 g d d       sc Z d jX   S.C. can become an O.C. with a g/4 transmission line. Short-Circuit Load (ZL=0)   0 tan sc X Z d   45     0 tan Z d jZ d    sc X Inductive Capacitive / g d  0 1/ 4 1/ 2 3/ 4 Lossless Case   g d    lossless
0 0 1 / 1 1 / L L L Z Z Z Z      Always imaginary! Open-Circuit Load (ZL=) 46 1 L    0 0 L L L Z Z Z Z           0 0 0 tan tan L L Z jZ d Z d Z Z jZ d                 0 cot Z d jZ d               0 0 0 1 / tan / tan L L j Z Z d Z d Z Z Z j d             or   Z d  z d   0 z  z   I d    V d    Lossless Case
2 g d d     O.C. can become a S.C. with a g/4 transmission line. Open-Circuit Load (ZL=) 47   oc Z d jX     0 cot oc X Z d        0 cot Z d jZ d     Note: Inductive Capacitive oc X / g d  0 1/ 4 1/ 2 Lossless Case   g d    lossless
Using Transmission Lines to Synthesize Loads A microwave filter constructed from microstrip line. This is very useful in microwave engineering. 48 We can obtain any reactance that we want from a short or open transmission line.
49 Find the voltage at any point on the line.       0 0 0 tanh tanh L in L Z Z l Z Z l Z Z Z l                  in Th in Th Z V l V Z Z          At the input: Voltage on a Transmission Line 0 , Z    V z   I z   0 z  L Z l Th Z Th V   in Z Th V Th Z in Z   V l     
    0 2 1 z z L V z V e e         0 0 L L L Z Z Z Z         2 0 1 l l in L Th in Th Z V l V e e V Z Z                     2 2 1 1 z l z in L Th l in Th L Z e V z V e Z Z e                         At z = -l: Hence 0 2 1 1 l in Th l in Th L Z V V e Z Z e                     50 Voltage on a Transmission Line (cont.)  Incident (forward) wave (not the same as the initial wave from the source!)
Let’s derive an alternative form of the previous result.   2 0 2 1 1 l L in l L e Z Z l Z e                               2 2 0 2 0 2 2 2 0 0 2 2 0 2 0 0 2 0 2 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 l L l l L L l l l L Th L L Th l L l L l Th L Th l L l Th Th L in in T Th h h T e Z Z e e Z e Z e e Z Z e Z e Z Z e Z Z e Z Z Z Z Z e Z Z Z Z Z Z Z Z                                                                                          2 2 0 0 1 1 l L l Th L Th e Z Z e Z Z                   51 Start with: Voltage on a Transmission Line (cont.)
    2 0 2 0 1 1 z l z L Th l Th s L Z e V z V e Z Z e                         2 0 2 0 1 1 l in L l in Th L s Th Z Z e Z Z Z Z e                       where 0 0 Th s Th Z Z Z Z     Therefore, we have the following alternative form for the result: Hence, we have 52 (source reflection coefficient) Voltage on a Transmission Line (cont.)     2 2 1 1 z l z in L Th l in Th L Z e V z V e Z Z e                         Recall: Substitute
      2 0 2 0 1 1 z z l L Th l Th s L Z e V z V e Z Z e                          The “initial” voltage wave that would exist if there were no reflections from the load (we have a semi-infinite transmission line or a matched load). 53 Voltage on a Transmission Line (cont.) This term accounts for the multiple (infinite) bounces. 0 , Z    V z   I z   0 z  L Z l Th Z Th V   in Z
            2 2 2 2 2 2 0 0 1 l l L L s l l l l Th L s L L s L s Th e e Z V l V e e e e Z Z                                                        Wave-bounce method (illustrated for z = -l): 54 Voltage on a Transmission Line (cont.) We add up all of the bouncing waves. 0 , Z    0 z  L Z l Th Z Th V   in Z   V l 
          2 2 2 2 2 2 2 0 0 1 1 l l L S L S l l l Th L L S L S Th e e Z V l V e e e Z Z                                                      Geometric series: 2 0 1 1 , 1 1 n n z z z z z           2 l L s z e      55 Group together alternating terms: Voltage on a Transmission Line (cont.)             2 2 2 2 2 2 0 0 1 l l L L s l l l l Th L s L L s L s Th e e Z V l V e e e e Z Z                                                       
or   2 0 2 0 2 1 1 1 1 l L s Th d Th L l L s e Z V l V Z Z e e                                            2 0 2 0 1 1 l L Th l Th L s Z e V l V Z Z e                    This agrees with the previous result (setting z = -l). The wave-bounce method is a very tedious method – not recommended. Hence 56 Voltage on a Transmission Line (cont.)     2 0 2 0 1 1 z l z L Th l Th s L Z e V z V e Z Z e                          Previous (alternative)result :
At a distance d from the load:                * * * 2 0 2 2 * 2 * 0 2 2 2 0 0 2 4 2 2 * 2 * * 0 0 1 Re 2 1 Re 1 1 2 1 1 Re 1 Re 2 2 z z z L L z z z z z L L L P z V z I z V e e e Z V V e e e e e Z Z                                                           2 2 0 2 4 0 1 1 2 z z L V P z e e Z         If Z0  real (low-loss transmission line): Time-Average Power Flow         2 0 2 0 0 1 1 z z L z z L V z V e e V I z e e Z j                       * 2 * 2 * 2 2 z z L L z z L L e e e e                pure imaginary Note: 57   I z z   0 z  z L Z   V z P(z) = power flowing in + z direction (please see the note)
Low-loss line     2 2 0 2 4 0 2 2 2 0 0 2 2 0 0 1 1 2 1 1 2 2 z z L z z L V P z e e Z V V e e Z Z                  Power in forward-traveling wave Power in backward-traveling wave     2 2 0 0 1 1 2 L V P z Z     Lossless line ( = 0) Time-Average Power Flow (cont.) 58   I z z   0 z  z L Z   V z Note: For a very lossy line, the total power is not the difference of the two individual powers.
0 0 0 tan tan L T in T T L Z jZ d Z Z Z jZ d            2 4 4 2 g g g d           0 0 T in T L jZ Z Z jZ         0 2 0 0 0 in in T L Z Z Z Z Z       Quarter-Wave Transformer 2 0T in L Z Z Z  so 0 0 T L Z Z Z  Hence (This requires ZL to be real.) 59 Matching condition 0T Z L Z 0 Z in Z / 4 g d   Lossless line
Match a 100  load to a 50  transmission line at a given frequency. 0 100 50 70.7 T Z    0 0 2 2 2 g d r r k k               0 70.7 T Z   60 50 / 4 g    100 L Z   0T Z 0 c f   Lossless line Quarter-Wave Transformer (cont.) LC k        Note: Example
  2 0 1 L j j z L V z V e e              2 0 2 0 1 1 L j z j z L j j z j z L V z V e e V e e e                      max 0 min 0 1 1 L L V V V V         Voltage Standing Wave 61 z 1+ L  1 1- L  0 ( ) V z V  / 2 g z  D  0 z  2 2 / 2 g z z    D   D    I z z   0 z  z L Z   V z Lossless Case Note: The voltage repeats every g. The magnitude repeats every g /2. Note: The voltage changes by a minus sign after g /2.
  max min V V  Voltage Standing Wave Ratio VSWR Voltage Standing Wave Ratio 1 1 L L      VSWR z 1+ L  1 1- L  0 ( ) V z V  / 2 g z  D  0 z  62     max 0 min 0 1 1 L L V V V V           I z z   0 z  z L Z   V z
At high frequency, discontinuity effects can become important. Limitations of Transmission-Line Theory Bend Incident Reflected Transmitted The simple TL model does not account for the bend. 63 L Z 0 Z Th Z  
At high frequency, radiation effects can also become important. When will radiation occur? We want energy to travel from the generator to the load, without radiating. Limitations of Transmission-Line Theory (cont.) 64 This is explored next. L Z 0 Z Th Z  
The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, as long as the metal thickness is large compared with a skin depth. The fields are confined to the region between the two conductors. Limitations of Transmission-Line Theory (cont.) 65 Coaxial Cable r  a b z
The twin lead is an open type of transmission line – the fields extend out to infinity. The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair”.) Limitations of Transmission-Line Theory (cont.) Having fields that extend to infinity is not the same thing as having radiation, however! 66  
The infinite twin lead will not radiate by itself, regardless of how far apart the lines are (this is true for any transmission line). The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency.   * 1 ˆ Re 0 2 t S P E H dS             S + - Limitations of Transmission-Line Theory (cont.) 67 Incident Reflected No attenuation on an infinite lossless line h
A discontinuity on the twin lead will cause radiation to occur. Note: Radiation effects usually increase as the frequency increases. Limitations of Transmission-Line Theory (cont.) 68 Incident wave Pipe Obstacle Reflected wave h Bend Incident wave Bend Reflected wave h
To reduce radiation effects of the twin lead at discontinuities: 1) Reduce the separation distance h (keep h << ). 2) Twist the lines (twisted pair). Limitations of Transmission-Line Theory (cont.) CAT 5 cable (twisted pair) 69 h

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Notes 2 5317-6351 Transmission Lines Part 1 (TL Theory).pptx

  • 1. Prof. David R. Jackson Dept. of ECE Notes 2 ECE 5317-6351 Microwave Engineering Fall 2019 Transmission Lines Part 1: TL Theory 1 Adapted from notes by Prof. Jeffery T. Williams
  • 2. Transmission-Line Theory We need transmission-line theory whenever the length of a line is significant compared to a wavelength. 2 L
  • 3. Transmission Line 2 conductors 4 per-unit-length parameters: C = capacitance/length [F/m] L = inductance/length [H/m] R = resistance/length [/m] G = conductance/length [ /m or S/m]  Dz 3
  • 4. Transmission Line (cont.) z D   , i z t + + + + + + + - - - - - - - - - -   , v z t x x x B 4 Note: There are equal and opposite currents on the two conductors. (We only need to work with the current on the top conductor, since we have chosen to put all of the series elements there.) + - + -   , i z z t  D   , i z t   , v z z t  D   , v z t R z D L z D G z D C z D z
  • 5. ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) i z t v z t v z z t i z t R z L z t v z z t i z t i z z t v z z t G z C z t    D  D  D    D   D   D D  D  Transmission Line (cont.) 5 + - + -   , i z z t  D   , i z t   , v z z t  D   , v z t R z D L z D G z D C z D z
  • 6. Hence ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) v z z t v z t i z t Ri z t L z t i z z t i z t v z z t Gv z z t C z t  D      D   D    D    D  D  Now let Dz  0: v i Ri L z t i v Gv C z t               “Telegrapher’s Equations” TEM Transmission Line (cont.) 6
  • 7. To combine these, take the derivative of the first one with respect to z: 2 2 2 2 v i i R L z z z t i i R L z t z v v v R Gv C L G C t t t                                                          Switch the order of the derivatives. TEM Transmission Line (cont.) 7 i v Gv C z t       
  • 8.   2 2 2 2 ( ) 0 v v v RG v RC LG LC z t t                  The same differential equation also holds for i. Hence, we have: 2 2 2 2 v v v v R Gv C L G C z t t t                            TEM Transmission Line (cont.) 8 Note: There is no exact solution in the time domain, in the lossy case.
  • 9.   2 2 2 ( ) ( ) 0 d V RG V RC LG j V LC V dz           2 2 2 2 ( ) 0 v v v RG v RC LG LC z t t                  TEM Transmission Line (cont.) Time-Harmonic Waves: 9 j t    
  • 10. Note that     2 2 2 ( ) d V RG V j RC LG V LC V dz       2 ( ) ( )( ) RG j RC LG LC R j L G j C           = = Z R j L Y G j C       series impedance / unit length parallel admittance / unit length Then we can write: 2 2 ( ) d V ZY V dz  TEM Transmission Line (cont.) 10
  • 11. Define We have: Solution: 2 ZY   ( ) z z V z Ae Be        1/2 ( )( ) R j L G j C           2 2 2 d V z V z dz   Then TEM Transmission Line (cont.)  is called the “propagation constant”. 11 Question: Which sign of the square root is correct?
  • 12. Principal square root: TEM Transmission Line (cont.) /2 j z r e         12  1/2 ( )( ) R j L G j C       We choose the principal square root.   Re 0 z  ( )( ) R j L G j C       Re 0   Hence (Note the square-root (“radical”) symbol here.) j z re   1/2 1/2 4 2, 4 2 1 1 , 2 2 j j j j               Examples :
  • 13. Denote: TEM Transmission Line (cont.) Re 0     13 ( )( ) R j L G j C       [np/m]   attenuationcontant j        rad/m   phaseconstant   1/m   propagationconstant
  • 14. TEM Transmission Line (cont.) 14 There are two possible locations for the complex square root: 0, 0     Hence: Re Im       R j L   Re Im G j C   Re Im    Re Im ( )( ) R j L G j C       The principal square root must be in the first quadrant.
  • 15. TEM Transmission Line (cont.) 15   z z j z V z Ae Ae e         Wave traveling in +z direction:   j      Wave is attenuating as it propagates.   z z j z V z Ae Ae e         Wave traveling in -z direction:   j      Wave is attenuating as it propagates.
  • 16. TEM Transmission Line (cont.) 16    10 10 dB 20log 20log 20 0.4343 8.686 out in z V V A e A z z                             Attenuation in     10 log 0.4343 ln x x  Note: Attenuation in dB/m:   z j z V z Ae e      dB/m 8.686  Attenuationin
  • 17. Wavenumber Notation 17 ( )( ) R j L G j C       "propagationconstant"   z z j z V z Ae Ae e           z jk z z j z V z Ae Ae e        ( )( ) z k j R j L G j C       "propagationwavenumber" z jk   j      z k j    
  • 18. TEM Transmission Line (cont.) 0 0 ( ) z z j z V z V e V e e            Forward travelling wave (a wave traveling in the positive z direction):             0 0 0 ( , ) Re Re cos z j z j t j z j z j t z v z t V e e e V e e e e V e t z                          2 g     2 g    The wave “repeats” when: Hence: 18 g  0 t  z 0 z V e    “snapshot” of wave
  • 19. Phase Velocity Let’s track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest of the wave. 0 ( , ) cos( ) z v z t V e t z           19 z (phase velocity) p v
  • 20. Phase Velocity (cont.) 0 constant            t z dz dt dz dt Set Hence p v      Im )( p v R j L G j C       In expanded form: 20
  • 21. Characteristic Impedance Z0 0 ( ) ( ) V z Z I z    0 0 ( ) ( ) z z V z V e I z I e           so 0 0 0 V Z I    Assumption: A wave is traveling in the positive z direction. (Note: Z0 is a number, not a function of z.) 21   I z    V z    z
  • 22. Use first Telegrapher’s Equation: v i Ri L z t        so dV RI j LI ZI dz       Hence 0 0 z z V e ZI e           Characteristic Impedance Z0 (cont.) 22 0 0 ( ) ( ) z z V z V e I z I e           Recall:
  • 23. From this we have: Use: 0 0 0 V Z Z Z I ZY       Characteristic Impedance Z0 (cont.) Z R j L Y G j C       23 Both are in the first quadrant 0 0 / Re 0 ZY Z Z ZY Z          first quadrant right - half plane 0 Z Z Y  (principal square root)
  • 24. Hence, we have Characteristic Impedance Z0 (cont.) 24 0 Z R j L Z Y G j C       0 Re 0 Z 
  • 25.   0 0 0 0 j z j j z z z z j z V e e V z V e V V e e e e e                                   0 0 cos c , R os e j t z z V e t v z t V z z V z e e t                       In the time domain: Wave in +z direction Wave in -z direction General Case (Waves in Both Directions) 25
  • 26. Backward-Traveling Wave 0 ( ) ( ) V z Z I z     0 ( ) ( ) V z Z I z     so A wave is traveling in the negative z direction. Note: The reference directions for voltage and current are chosen the same as for the forward wave. 26 z   I z    V z   
  • 27. General Case 0 0 0 0 0 ( ) 1 ( ) z z z z V z V e V e I z V e V e Z                     A general superposition of forward and backward traveling waves: Most general case: 27 z   I z   V z  
  • 28.        0 0 0 0 0 0 0 z z z z V z V e V e V V I z e e Z j R j L G j C R j L Z G j Z C                                  2 m g     [m/s] p v    Guided wavelength: Phase velocity: Summary of Basic TL formulas 28 dB/m 8.686  Attenuationin   I z   V z  
  • 29. Lossless Case 0, 0 R G   ( )( ) j R j L G j C j LC             so 0 LC      0 R j L Z G j C      0 L Z C  1 p v LC  p v    (independent of freq.) (real and independent of freq.) 29
  • 30. Lossless Case (cont.) 1 p v LC  If the medium between the two conductors is lossless and homogeneous (uniform) and is characterized by (, ), then (proof given later): LC   The speed of light in a dielectric medium is 1 d c   Hence, we have that: p d v c  In the lossless case the phase velocity does not depend on the frequency, and it is always equal to the speed of light (in the material). (proof given later) 30 0 r r LC k k           and
  • 31.   0 0 z z V z V e V e         Where do we assign z = 0 ? The usual choice is at the load. Amplitude of voltage wave propagating in negative z direction at z = 0. Amplitude of voltage wave propagating in positive z direction at z = 0. Terminated Transmission Line 31     0 0 0 0 V V V V         V z    V z  Terminating impedance (load)   V z   I z 0 z  z L Z  
  • 32.   0 0 0 z V V z e                 0 0 0 0 z z z z V z V z e V z e            0 0 0 z V z V e        0 0 0 z V V z e       Terminated Transmission Line (cont.)   0 0 z z V z V e V e         Hence Can we use z = z0 as a reference plane? 32   0 0 0 z V z V e      Terminating impedance (load)   V z   I z 0 z  z L Z   0 z z 
  • 33. Terminated Transmission Line (cont.)       0 0 z z V z V e V e         Compare: This is simply a change of reference plane, from z = 0 to z = z0. 33           0 0 0 0 z z z z V z V z e V z e          Terminating impedance (load)   V z   I z 0 z  z L Z   0 z z 
  • 34.   0 0 z z V z V e V e         What is V(-d) ?   0 0 d d V d V e V e           0 0 0 0 d d V V I d e e Z Z         Propagating forwards Propagating backwards Terminated Transmission Line (cont.) d  distance away from load The current at z = -d is then: 34 (This does not necessarily have to be the length of the entire line.) Terminating impedance (load)   I d  z d     0 z  z L Z   V d 
  • 35.     2 0 0 1 d d L V I d e e Z           2 0 0 0 0 0 1 d d d d V V d V e V e V e e V                       Similarly, L  load reflection coefficient Terminated Transmission Line (cont.) 35     2 0 1 d d L V d V e e         or 0 0 L V V       I d  z d     0 z  z L Z   V d  (-d) = reflection coefficient at z = -d   2 d L d e      
  • 36.         2 0 2 0 0 1 1 d d L d d L V d V e e V I d e e Z                Z(-d) = impedance seen “looking” towards load at z = -d. Terminated Transmission Line (cont.) 36           2 0 0 2 1 1 1 1 d L d L V d d e Z d Z Z I d e d                                 Note: If we are at the beginning of the line, we will call this the “input impedance”.   I d  z d     0 z  z L Z   V d    Z d 
  • 37. At the load (d = 0):   0 1 0 1 L L L Z Z Z             Terminated Transmission Line (cont.) 0 0 L L L Z Z Z Z     37    2 d L d e       At any point on the line(d > 0):
  • 38. Thus,   2 0 0 0 2 0 0 1 1 d L L d L L Z Z e Z Z Z d Z Z Z e Z Z                                     Terminated Transmission Line (cont.)       2 0 0 2 1 1 1 1 d L d L d e Z d Z Z d e                              Recall 38
  • 39. Simplifying, we have:       0 0 0 tanh tanh L L Z Z d Z d Z Z Z d             Terminated Transmission Line (cont.)                           2 0 2 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1 cosh sinh cosh sinh d L d L L L d d L L L L d d L L d d L L L L Z Z e Z Z Z Z Z Z e Z d Z Z Z Z Z Z e Z Z e Z Z Z Z e Z Z e Z Z Z e Z Z e Z d Z d Z Z d Z d                                                                                        Hence, we have 39
  • 40.         2 0 2 0 0 1 1 j d j d L j d j d L V d V e e V I d e e Z                Impedance is periodic with period g/2:     2 1 2 1 2 1 2 / 2 g g d d d d d d              Terminated Lossless Transmission Line j j        Note:       tanh tanh tan d j d j d      The tan function repeats when       0 0 0 tan tan L L Z jZ d Z d Z Z jZ d             40 Lossless:   2 0 2 1 1 j d L j d L e Z d Z e              
  • 41.       2 0 2 0 0 0 1 1 tanh tanh d L d L L L e Z d Z e Z Z d Z Z Z d                             0 0 2 L L L g p Z Z Z Z v            Summary for Lossy Transmission Line 41   I d  z d     0 z  z L Z   V d    Z d  ( )( ) j R j L G j C          
  • 42.       2 0 2 0 0 0 1 1 tan tan j d L j d L L L e Z d Z e Z jZ d Z Z jZ d                            0 0 2 2 L L L g d p d Z Z Z Z k v c                 Summary for Lossless Transmission Line 42   I d  z d     0 z  z L Z   V d    Z d  LC k       
  • 43. 0 0 0 L L L Z Z Z Z      No reflection from the load Matched Load (ZL=Z0)   0 Z d Z z   for any 43   2 0 2 1 1 d L d L e Z d Z e                 I d  z d     0 z  z L Z   V d    Z d  
  • 44. 1 L    Always imaginary! Short-Circuit Load (ZL=0) 44       0 0 0 tan tan L L Z jZ d Z d Z Z jZ d                 0 tan Z d jZ d    0 0 L L L Z Z Z Z       Z d  z d   0 z  z   I d    V d    Lossless Case
  • 45. Note: 2 g d d       sc Z d jX   S.C. can become an O.C. with a g/4 transmission line. Short-Circuit Load (ZL=0)   0 tan sc X Z d   45     0 tan Z d jZ d    sc X Inductive Capacitive / g d  0 1/ 4 1/ 2 3/ 4 Lossless Case   g d    lossless
  • 46. 0 0 1 / 1 1 / L L L Z Z Z Z      Always imaginary! Open-Circuit Load (ZL=) 46 1 L    0 0 L L L Z Z Z Z           0 0 0 tan tan L L Z jZ d Z d Z Z jZ d                 0 cot Z d jZ d               0 0 0 1 / tan / tan L L j Z Z d Z d Z Z Z j d             or   Z d  z d   0 z  z   I d    V d    Lossless Case
  • 47. 2 g d d     O.C. can become a S.C. with a g/4 transmission line. Open-Circuit Load (ZL=) 47   oc Z d jX     0 cot oc X Z d        0 cot Z d jZ d     Note: Inductive Capacitive oc X / g d  0 1/ 4 1/ 2 Lossless Case   g d    lossless
  • 48. Using Transmission Lines to Synthesize Loads A microwave filter constructed from microstrip line. This is very useful in microwave engineering. 48 We can obtain any reactance that we want from a short or open transmission line.
  • 49. 49 Find the voltage at any point on the line.       0 0 0 tanh tanh L in L Z Z l Z Z l Z Z Z l                  in Th in Th Z V l V Z Z          At the input: Voltage on a Transmission Line 0 , Z    V z   I z   0 z  L Z l Th Z Th V   in Z Th V Th Z in Z   V l     
  • 50.     0 2 1 z z L V z V e e         0 0 L L L Z Z Z Z         2 0 1 l l in L Th in Th Z V l V e e V Z Z                     2 2 1 1 z l z in L Th l in Th L Z e V z V e Z Z e                         At z = -l: Hence 0 2 1 1 l in Th l in Th L Z V V e Z Z e                     50 Voltage on a Transmission Line (cont.)  Incident (forward) wave (not the same as the initial wave from the source!)
  • 51. Let’s derive an alternative form of the previous result.   2 0 2 1 1 l L in l L e Z Z l Z e                               2 2 0 2 0 2 2 2 0 0 2 2 0 2 0 0 2 0 2 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 l L l l L L l l l L Th L L Th l L l L l Th L Th l L l Th Th L in in T Th h h T e Z Z e e Z e Z e e Z Z e Z e Z Z e Z Z e Z Z Z Z Z e Z Z Z Z Z Z Z Z                                                                                          2 2 0 0 1 1 l L l Th L Th e Z Z e Z Z                   51 Start with: Voltage on a Transmission Line (cont.)
  • 52.     2 0 2 0 1 1 z l z L Th l Th s L Z e V z V e Z Z e                         2 0 2 0 1 1 l in L l in Th L s Th Z Z e Z Z Z Z e                       where 0 0 Th s Th Z Z Z Z     Therefore, we have the following alternative form for the result: Hence, we have 52 (source reflection coefficient) Voltage on a Transmission Line (cont.)     2 2 1 1 z l z in L Th l in Th L Z e V z V e Z Z e                         Recall: Substitute
  • 53.       2 0 2 0 1 1 z z l L Th l Th s L Z e V z V e Z Z e                          The “initial” voltage wave that would exist if there were no reflections from the load (we have a semi-infinite transmission line or a matched load). 53 Voltage on a Transmission Line (cont.) This term accounts for the multiple (infinite) bounces. 0 , Z    V z   I z   0 z  L Z l Th Z Th V   in Z
  • 54.             2 2 2 2 2 2 0 0 1 l l L L s l l l l Th L s L L s L s Th e e Z V l V e e e e Z Z                                                        Wave-bounce method (illustrated for z = -l): 54 Voltage on a Transmission Line (cont.) We add up all of the bouncing waves. 0 , Z    0 z  L Z l Th Z Th V   in Z   V l 
  • 55.           2 2 2 2 2 2 2 0 0 1 1 l l L S L S l l l Th L L S L S Th e e Z V l V e e e Z Z                                                      Geometric series: 2 0 1 1 , 1 1 n n z z z z z           2 l L s z e      55 Group together alternating terms: Voltage on a Transmission Line (cont.)             2 2 2 2 2 2 0 0 1 l l L L s l l l l Th L s L L s L s Th e e Z V l V e e e e Z Z                                                       
  • 56. or   2 0 2 0 2 1 1 1 1 l L s Th d Th L l L s e Z V l V Z Z e e                                            2 0 2 0 1 1 l L Th l Th L s Z e V l V Z Z e                    This agrees with the previous result (setting z = -l). The wave-bounce method is a very tedious method – not recommended. Hence 56 Voltage on a Transmission Line (cont.)     2 0 2 0 1 1 z l z L Th l Th s L Z e V z V e Z Z e                          Previous (alternative)result :
  • 57. At a distance d from the load:                * * * 2 0 2 2 * 2 * 0 2 2 2 0 0 2 4 2 2 * 2 * * 0 0 1 Re 2 1 Re 1 1 2 1 1 Re 1 Re 2 2 z z z L L z z z z z L L L P z V z I z V e e e Z V V e e e e e Z Z                                                           2 2 0 2 4 0 1 1 2 z z L V P z e e Z         If Z0  real (low-loss transmission line): Time-Average Power Flow         2 0 2 0 0 1 1 z z L z z L V z V e e V I z e e Z j                       * 2 * 2 * 2 2 z z L L z z L L e e e e                pure imaginary Note: 57   I z z   0 z  z L Z   V z P(z) = power flowing in + z direction (please see the note)
  • 58. Low-loss line     2 2 0 2 4 0 2 2 2 0 0 2 2 0 0 1 1 2 1 1 2 2 z z L z z L V P z e e Z V V e e Z Z                  Power in forward-traveling wave Power in backward-traveling wave     2 2 0 0 1 1 2 L V P z Z     Lossless line ( = 0) Time-Average Power Flow (cont.) 58   I z z   0 z  z L Z   V z Note: For a very lossy line, the total power is not the difference of the two individual powers.
  • 59. 0 0 0 tan tan L T in T T L Z jZ d Z Z Z jZ d            2 4 4 2 g g g d           0 0 T in T L jZ Z Z jZ         0 2 0 0 0 in in T L Z Z Z Z Z       Quarter-Wave Transformer 2 0T in L Z Z Z  so 0 0 T L Z Z Z  Hence (This requires ZL to be real.) 59 Matching condition 0T Z L Z 0 Z in Z / 4 g d   Lossless line
  • 60. Match a 100  load to a 50  transmission line at a given frequency. 0 100 50 70.7 T Z    0 0 2 2 2 g d r r k k               0 70.7 T Z   60 50 / 4 g    100 L Z   0T Z 0 c f   Lossless line Quarter-Wave Transformer (cont.) LC k        Note: Example
  • 61.   2 0 1 L j j z L V z V e e              2 0 2 0 1 1 L j z j z L j j z j z L V z V e e V e e e                      max 0 min 0 1 1 L L V V V V         Voltage Standing Wave 61 z 1+ L  1 1- L  0 ( ) V z V  / 2 g z  D  0 z  2 2 / 2 g z z    D   D    I z z   0 z  z L Z   V z Lossless Case Note: The voltage repeats every g. The magnitude repeats every g /2. Note: The voltage changes by a minus sign after g /2.
  • 62.   max min V V  Voltage Standing Wave Ratio VSWR Voltage Standing Wave Ratio 1 1 L L      VSWR z 1+ L  1 1- L  0 ( ) V z V  / 2 g z  D  0 z  62     max 0 min 0 1 1 L L V V V V           I z z   0 z  z L Z   V z
  • 63. At high frequency, discontinuity effects can become important. Limitations of Transmission-Line Theory Bend Incident Reflected Transmitted The simple TL model does not account for the bend. 63 L Z 0 Z Th Z  
  • 64. At high frequency, radiation effects can also become important. When will radiation occur? We want energy to travel from the generator to the load, without radiating. Limitations of Transmission-Line Theory (cont.) 64 This is explored next. L Z 0 Z Th Z  
  • 65. The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, as long as the metal thickness is large compared with a skin depth. The fields are confined to the region between the two conductors. Limitations of Transmission-Line Theory (cont.) 65 Coaxial Cable r  a b z
  • 66. The twin lead is an open type of transmission line – the fields extend out to infinity. The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair”.) Limitations of Transmission-Line Theory (cont.) Having fields that extend to infinity is not the same thing as having radiation, however! 66  
  • 67. The infinite twin lead will not radiate by itself, regardless of how far apart the lines are (this is true for any transmission line). The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency.   * 1 ˆ Re 0 2 t S P E H dS             S + - Limitations of Transmission-Line Theory (cont.) 67 Incident Reflected No attenuation on an infinite lossless line h
  • 68. A discontinuity on the twin lead will cause radiation to occur. Note: Radiation effects usually increase as the frequency increases. Limitations of Transmission-Line Theory (cont.) 68 Incident wave Pipe Obstacle Reflected wave h Bend Incident wave Bend Reflected wave h
  • 69. To reduce radiation effects of the twin lead at discontinuities: 1) Reduce the separation distance h (keep h << ). 2) Twist the lines (twisted pair). Limitations of Transmission-Line Theory (cont.) CAT 5 cable (twisted pair) 69 h