1. Dr. Sandra Cruz-Pol ECE Dept. UPRM
Transmission Lines
Dr. Sandra Cruz-Pol
ECE Dept. UPRM

2. Introduction to Transmission Lines (T.L.)
Hi frequency or hi power requires T.L.
TEM waves propagate thru T.L.
We will develop T.L. theory to see how waves propagate thru them

3. Seem to be in parallel, but these V are not equal!
Exercise 11.3
Seem to be in parallel, but these V are not equal!
A 40-m long TL has Vg=15 Vrms, Zo=30+j60 W, and VL=5e-j48o Vrms. If the line is matched to the load and the generator, find: the input impedance Zin, the sending-end current Iin and Voltage Vin, the propagation constant g.
Answers:
Don’t worry about the details, I’ll teach you about solving this type of problems pretty soon. ZL Zg Vg +
Vin -
Iin VL
Zo=30+j60
g=a +j b
40 m

4. Transmission Lines TL parameters (R’,L’,C’, G’)
TL Equations (for V and I)
3 Concepts:
Input Impedance,
Reflection Coefficient and
Characteristic Impedance
SWR, Power
Smith Chart
Applications - Quarter-wave transformer, Slotted line, Single stub
Microstrips

5. Transmission Lines (TL)
TL have 2 conductors in parallel with a dielectric separating them
They transmit TEM waves inside the lines
outer conductor
for insulation

6. Common Transmission Lines
Two-wire (ribbon)
Microstrip
twinlead
Stripline (Triplate)
Coaxial

7. Other TL (higher order) [Chapter 12]

8. Fields inside the TL
V proportional to E,
I proportional to H

9. Distributed parameters
The parameters that characterize the TL are given in terms of per length.
R = ohms/meter
L = Henries/ m
C = Farads/m
G = mhos/m
At high frequencies we’re dealing with wavelengths comparable to the size of the circuits.

10. Common Transmission Lines
R’, L’, G’, and C’ depend on the particular transmission line structure and the material properties. R, L, G, and C can be calculated using fundamental EMAG techniques.
Parameter
Two-Wire Line
Coaxial Line
Parallel-Plate Line
Unit R’ L’ G’ C’

11. TL representation

12. Distributed line parameters
Using KVL:

13. Distributed parameters
Taking the limit as Dz tends to 0 leads to
Similarly, applying KCL to the main node gives

14. Wave equation Using phasors The two expressions reduce to
Wave Equation for voltage

15. Transmission Lines TL parameters (R’,L’,C’, G’)
TL Equations (for V and I)
3 Concepts:
Input Impedance,
Reflection Coefficient and
Characteristic Impedance
SWR, Power
Smith Chart
Applications - Quarter-wave transformer, Slotted line, Single stub
Microstrips

16. TL Equations
Note that these are the wave eq. for voltage and current inside the lines.
The propagation constant is g and the wavelength and velocity are

17. Waves move through line
The general solution is
In time domain is z

18. Waves move through line
For Current
Similarly for time-domain, I z

19. Transmission Lines TL parameters (R’,L’,C’, G’)
TL Equations (for V and I)
3 Concepts:
Input Impedance,
Reflection Coefficient and
Characteristic Impedance
SWR, Power
Smith Chart
Applications - Quarter-wave transformer, Slotted line, Single stub
Microstrips

20. We will define 3 concepts:
Characteristic impedance, Zo
Reflection Coefficient, GL
Input Impedance, Zin

21. We will define 3 concepts:
Characteristic impedance, Zo
Reflection Coefficient, GL
Input Impedance, Zin

22. Characteristic Impedance of a Line, Zo
Is the ratio of positively traveling voltage wave to current wave at any point on the line z

23. Characteristic Impedance, Zo

24. Find the characteristic impedance and propagation constant for each:
Different cases of TL
Find the characteristic impedance and propagation constant for each:
Lossless
Distortionless
Lossy
Transmission line
Transmission line
Transmission line

25. Lossless Lines (R=0=G)
Have perfect conductors and a perfect dielectric medium between them.
Propagation:
Wavelength & Velocity:
Impedance

26. Distortionless line (R/L = G/C)
Is one in which the attenuation is independent on frequency.
Propagation:
Velocity:
Impedance

27. Characteristic Impedance
Summary
g = a + jb
Propagation Constant Zo
Characteristic Impedance
General
(Lossy)
Lossless
Distortionless
RC = GL

28. P.E. 11.2
A telephone line has R=30 W/km, L=100 mH/km, G=0, and C= 20mF/km. At 1kHz, FIND: the characteristic impedance of the line, the propagation constant, the phase velocity.
Is it distortionless?
Solution: No

29. We will define 3 concepts:
Characteristic impedance, Zo
Reflection Coefficient, GL
Input Impedance, Zin

30. We will define 3 concepts:
Characteristic impedance, Zo
Reflection Coefficient, GL
Input Impedance, Zin

31. Reflection coefficient at the load, GL
Load is usually taken at z=0
and generator at z= -l
z=0
z= -l

32. For a Lossless TL terminated with a load
Then,
Similarly,
The impedance anywhere along the line is given by
The impedance at the load end, ZL, is given by

33. Terminated, Lossless TL
Solving for GL
Conclusion: The reflection coefficient is a function of the load impedance and the characteristic impedance.
Recall for the lossless case,
Then

34. Definition: Matched line
Means that Zo=ZL
Therefore there are no reflection!
GL=0

35. What happens when you connect the wrong TL to a speaker?

36. Terminated, Lossless TL
Using convention the coordinate system, z = - l , at input. -z
z = - l
Rewriting the expressions for voltage and current, we have
Rearranging,

37. Voltage anywhere on the Line
Recall, -z
z = - l
Input voltage= sending end,
Load Voltage = receiving end
Voltage quaterwave from
matchedload

38. We will define 3 concepts:
Characteristic impedance, Zo
Reflection Coefficient, GL
Input Impedance, Zin

39. Impedance (Lossless line)
The impedance anywhere along the line is given by
The reflection coefficient at any point along the line:
Then, the impedance can be written as.
After some algebra, an alternative expression for the impedance is given by
Conclusion: The load impedance is “transformed” as we move away from the load.

40. Impedance (Lossy line)
The impedance anywhere along the line is given by
The reflection coefficient can be modified as follows
Then, the impedance can be written as
After some algebra, an alternative expression for the impedance is given by
Conclusion: in lossy lines, we end up with the hyperbolic tangent.

41. Example: Matched Case
A TL has Vg=10 Vrms, Zo=50W If the line is matched to the load and the generator, find: the input impedance Zin, the sending-end Voltage Vin
Answers: ZL Zg Vg +
Vin -
Iin VL Zo
g=j b l

42. Example: Matched Case
A l/8 long TL has VL=5 < 30o, Zo=50W If the line is matched to the load, find: the input impedance Zin, the sending-end Voltage Vin, the propagation constant g.
Answers: ZL Zg Vg +
Vin -
Iin VL Zo
g=j b l
Here i don’t know what’s Zg, so i can’t use Thevenin, need to use V eq twice, find V+

43. Exercise 1: Not Matched
A 2cm lossless TL has VL=10 ej30o, Zg=60 W, ZL=50 W and Zo=100W, l=10cm . Find: the input impedance Zin, the sending-end Voltage Vin,
Use this equation at load and at input, find V+
Find Vin
Find Zin (at input) ZL Zg Vg +
Vin -
Iin VL Zo
g=j b l

44. Exercise 2: using formulas
A 2cm lossless TL has Vg=10 Vrms, Zg=60 W, ZL=100+j80 W and Zo=40W, l=10cm . find: the input impedance Zin, the sending-end Voltage Vin, ZL Zg Vg +
Vin -
Iin VL Zo
g=j b l
Voltage Divider:

45. Example 3: Not matched to load
A generator with 10Vrms and Rg=50, is connected to a 75W load thru a 0.8l, 50W-lossless line.
Find VL ZL Zg Vg +
Vin -
Iin VL Zo
g=j b l

46. Exercise 11.3: Matched Case
A 40-m long TL has Vg=15 Vrms, Zo=30+j60 W, and VL=5e-j48o Vrms. If the line is matched to the load and the generator, find: the input impedance Zin, the sending-end current Iin and Voltage Vin, the propagation constant g.
Answers: ZL Zg Vg +
Vin -
Iin VL
Zo=30+j60
g=a +j b
40 m

47. Transmission Lines TL parameters (R’,L’,C’, G’)
TL Equations (for V and I)
3 Concepts:
Input Impedance,
Reflection Coefficient and
Characteristic Impedance
SWR, Power
Smith Chart
Applications - Quarter-wave transformer, Slotted line, Single stub
Microstrips

48. Power
The average input power at a distance l from the load is given by
which can be reduced to
The first term is the incident power and the second is the reflected power. Maximum power is delivered to load if G=0

49. SWR or VSWR or s
Whenever there is a reflected wave, a standing wave will form out of the combination of incident and reflected waves.
The (Voltage) Standing Wave Ratio - SWR (or VSWR) is defined as

50. Summary
Input Impedance
Reflection Coef
SWR

51. Three (3) common Cases of line-load combinations:
Shorted Line (ZL=0)
Open-circuited Line (ZL=∞)
Matched Line (ZL = Zo)

52. Standing Waves –Short (ZL=0)
Voltage maxima
So substituting in V(z) -z
-l/4
-l/2 -l
|V(z)|
*Voltage minima occurs at same place that impedance has a minimum on the line

53. Standing Waves –Open(ZL=∞)
So substituting in V(z)
Voltage minima
|V(z)| -z
-l/4
-l/2 -l

54. Standing Waves –Matched (ZL = Zo)
So substituting in V(z)
|V(z)| -z
-l/4
-l/2 -l

55. Java applets http://www.amanogawa.com/transmission.html

56. Transmission Lines TL parameters (R’,L’,C’, G’)
TL Equations (for V and I)
3 Concepts:
Input Impedance,
Reflection Coefficient and
Characteristic Impedance
SWR, Power
Smith Chart
Applications - Quarter-wave transformer, Slotted line, Single stub
Microstrips

57. The Smith Chart

58. Smith Chart Commonly used as graphical representation of a TL.
Used in hi-tech equipment for design and testing of microwave circuits
One turn (360o) around the SC = to l/2

59. What can be seen on the screen?
Network Analyzer
What can be seen on the screen?

60. Smith Chart Use the reflection coefficient real and imaginary parts .
and define the normalized ZL: Gi
|G| Gr

61. Now relating to z =r +jx After some algebra, we obtain two eqs.
Similar to general equation of a circle of radius a, center at (h,k)
Circles of r
Circles of x

62. Examples of circles of r and x
Circles of x

63. Examples
Find
z= r+jx
given G
Also Find G
Given Z

64. Examples of circles of r and x
Circles of x

65. Find SWR on the SC Numerically s = r on the +axis of Gr in the SC
Proof:

66. Moving on the TL on the SmithC
A lossless TL is represented as a circle of constant radius, |G|, or constant s
Moving along the line from the load toward the generator, the phase decrease, therefore, in the SC equals to moves clockwisely.
To generator

67. One-turn on the Smith Chart
One turn (360o) around the SC = to l/2 because in the formula below, if you substitute length for half-wavelength, the phase changes by 2p, which is one turn.
Find the point in the SC where G=+1,-1, j, -j, 0, 0.5
What is r and x for each case?

68. Fun facts : Admittance in the SC
The admittance, y=YL/Yo where Yo=1/Zo, can be found by moving ½ turn (l/4) on the TL circle

69. Vmax and Vmin on the SmithC
The Gr +axis, where r > 0 corresponds to Vmax
The Gr -axis, where r < 0 corresponds to Vmin
Vmin
Vmax (Maximum impedance)

70. Exercise: using S.C.
A 2cm lossless TL has Vg=10 Vrms, Zg=60 W, ZL=100+j80 W and Zo=40W, l=10cm . find: the input impedance Zin, the sending-end Voltage Vin,
Load is at S.C.
Move .2l and arrive to .4179l
Read ZL Zg Vg +
Vin -
Iin VL Zo
g=j b
2 cm
Voltage Divider:

71. zL
zin
0.2l

72. Exercise: cont….using S.C.
A 2cm lossless TL has Vg=10 Vrms, Zg=60 W, ZL=100+j80 W and Zo=40W, l=10cm . find: the input impedance Zin, the sending-end Voltage Vin,
Distance from the load (.2179l) to the nearest minimum & max
Move to horizontal axis toward the generator and arrive to .5l (Vmax) and to .25l for the Vmin.
Distance to min= =.282l
Distance to 2st voltage maximum is .282l +.25l=.482 See drawing ZL Zg Vg +
Vin -
Iin VL Zo
g=j b
2 cm

73. Exercise : using formulas
A 2cm lossless TL has Vg=10 Vrms, Zg=60 W, ZL=100+j80 W and Zo=40W, l=10cm . find: the input impedance Zin, the sending-end Voltage Vin, ZL Zg Vg +
Vin -
Iin VL Zo
g=j b
2 cm
Voltage Divider:

74. Distance to first Vmax:
Another example:
A 26cm lossless TL is connected to load ZL=36-j44 W and Zo=100W, l=10cm . find: the input impedance Zin
Load is at S.C.
Move .1l and arrive to .527l (=.027l)
Read ZL Zg Vg +
Vin -
Iin VL Zo
g=j b
26cm
Distance to first Vmax:

75. Exercise 11.4
A 70 W lossless line has s =1.6 and at the load qG =300o. If the line is 0.6l long, obtain G, ZL, Zin and the distance of the first minimum voltage from the load.
Answer
The load is located at:
Move to l and draw line from
center to this place, then read where it crosses you TL circle.
Distance to Vmin in this case, lmin =.5l-.3338l= ~

77. Java Applet : Smith Chart

78. The end of INEL 4151
Material extra for (INEL 4155)

79. Transmission Lines TL parameters (R’,L’,C’, G’)
TL Equations (for V and I)
3 Concepts:
Input Impedance,
Reflection Coefficient and
Characteristic Impedance
SWR, Power
Smith Chart
Applications - Quarter-wave transformer, Slotted line, Single stub
Microstrips

80. Applications Slotted line as a frequency meter Impedance Matching
If ZL is Real: Quarter-wave Transformer (l/4 Xmer)
If ZL is complex: Single-stub tuning (use admittance Y)
Microstrip lines

81. HP Network Analyzer in Standing Wave Display
Slotted Line
Used to measure frequency and load impedance
HP Network Analyzer in Standing Wave Display

83. Slotted line example
Given s, the distance between adjacent minima, and lmin for an “air” 100W transmission line, Find f and ZL
s=2.4, lmin=1.5 cm, lmin-min=1.75 cm
Solution: =8.6GHz
Draw a circle on r=2.4, that’s your T.L.
move from Vmin to zL

84. Quarter-wave transformer …for impedance matchingZL
Zin
Zo , g
l= l/4
Conclusion: **A piece of line of l/4 can be used to change the impedance to a desired value (e.g. for impedance matching)

85. Single Stub Tuning …for impedance matching
A stub is connected in parallel to sum the admittances
Use a reactance from a short-circuited stub or open-circuited stub to cancel reactive part
Zin=Zo therefore z =1 or y=1 (this is our goal!)

86. Single Stub Basics
We work with Y, because in parallel connections they add.
YL (=1/ZL) is to be matched to a TL having characteristic admittance Yo by means of a "stub" consisting of a shorted (or open) section of line having the same characteristic admittance Yo

87. Single Stub Steps
First, the length l is adjusted so that the real part of the admittance at the position where the stub is attached is equal to Yo or yline = 1+jb
Then the length of the shorted stub is adjusted so that it's susceptance cancels that of the line, or ystub= -jb

88. Example: Single Stub
A 75W lossless line is to be matched to a 100-j80 W load with a shorted stub. Calculate the distance from the load, the stub length, and the necessary stub admittance.
Answer:
Change to:
=0.094l (1+j.96) or next intersection:0.272l,
Short stub: =0.126l
With ystub= -j.96/75 =-j.0128 mhos

90. Example: Single Stub
A 50W lossless T.L. is 20 m long and terminated into a 120+j220W load.   To perfectly match , what should be the length and location of a short-circuited stub line. Assume an operating frequency of 10MHz.
Answer:  l=30m, zL= 2.4+j4.4 (circulito rojo). 
Trazo raya por el centro y leo yL al otro lado (circulito azul).
La yL está en la posición 0.472l.
Trazo círculo amarillo, esa es mi T.L. donde está la carga. Busco donde interseca el circulo de r=1(circulo turquesa).
Lo interseca en 1+j3.2 Ese es mi 1+jb.  Y está en la posicion 0.214l.
Por tanto la distancia desde la carga al segmento(stub) = (debido a cambio de escala) +.214= .242 l.
Ahora miro el círculo del segmento= circulo grande en el SC (donde Gamma =1).y busco donde jb= -j3.2 (abajo ver flecha roja).
Para buscar su posición, trazo línea desde el centro hasta afuera y leo posición está en .2958l. El segmento empieza en carga de corto circuito (ver palabra roja que dice short) está en .25l. Por lo tanto el largo del segmento es
= .0485l

91. Transmission Lines TL parameters (R’,L’,C’, G’)
TL Equations (for V and I)
3 Concepts:
Input Impedance,
Reflection Coefficient and
Characteristic Impedance
SWR, Power
Smith Chart
Applications - Quarter-wave transformer, Slotted line, Single stub
Microstrips

92. Microstrips

93. Microstrips analysis equations & Pattern of EM fields

94. Microstrip Design Equations
Falta un radical en eeff

95. Microstrip Design Curves

96. Example
A microstrip with fused quartz (er=3.8) as a substrate, and ratio of line width to substrate thickness is w/h=0.8, find:
Effective relative permittivity of substrate
Characteristic impedance of line
Wavelength of the line at 10GHz
Answer: eeff=2.75, Zo=86.03 W, l=18.09 mm

97. Diseño de microcinta:
Dado (er=4) para el substrato, y h=1mm halla w para Zo=50 W y cuánto es eeff?
Solución: Suponga que como Z es pequena w/h>2