1. Chapter 10. Transmission lines

2. Concept of transmission lines
Transmission lines are used to transmit electric energy and signals from one point to another.
Transmitter to an antenna
connections between computers in a network
hydroelectric generating plant and a substation several miles away
interconnect between components of a stereo system
CATV service provider and your TV set
Connections between devices on a circuit board

3. Example – Cable TV

4. Example – Computer network

5. Example – Electric power transmission line

6. Example – Printed circuit board

7. Example – Printed circuit board

8. Types of transmission lines
Microstrip line
Coaxial cable
Two-wire transmission line

9. Signal propagation in tx-line
E-field +
The speed of electromagnetic wave is the same as that of light. + V -

10. Example – Line trace  E  H      + - ZL
H-field due to current  E  + V - H ZL     
The voltage source generates time-varying electric field, thereby displacement current flows. The current, in turn, generates time-varying magnetic field. The H-field generates E-field by Faraday’s law. In this way, E and H fields propagate in the right direction.

11. Parallel plate waveguide

12. 11.1 Physical description of transmission line propagation

13. Transmission line circuits modeling
i (z, t) +
v (z, t) - z
i (z, t)
i (z+z, t) +
L z
v (z, t)
C z
v (z+ z,t) - z

14. Transmission line eq. solution

15. Transmission line parameter - examples
Coax a b
Parallel Plate W d

16. Parallel wire a + - D
Coplanar waveguide

17. Properties of transmission linesH + V -
E/H fields propagating in the same direction keep the ratio E+/H+ constant.
V+/I+ wave propagating in the same direction keeps that ratio constant. The ratio is called a characteristic impedance of the transmission line.→ Z0
If the ratio is broken arriving at the load, a reflected wave is generated.
propagation H E

18. Reflection coefficient+ V -

19. Load voltage on an unmatched transmission line
Impedance mismatched + V -
Vin
Vout R R2
R=1k Ohm
MLIN R1
R=20 Ohm
VtPulse
SRC1 t + V -
Z0= 50 
Zs = 20  + V -
Z0= 50 
ZL= 1k  + V -
0.5m + V -

20. Load voltage on a matched transmission line
Impedance matched + V -
Vin
Vout R R2
R=50 Ohm
MLIN R1
VtPulse
SRC1 t + V -
Z0= 50 
Zs = 1  + V -
Z0= 50 
ZL= 50  + V -
0.5m + V -

21. Typical narrow band signals

22. Frequency domain solution
β : propagation constant, vp : speed of light

23. Phasor representation+ V -

24. Transmission line terminated with short, open
Out of phase (180 ) for short V
inc
Vrefl
Zs = Zo
In phase (0 ) for open o
Vrefl
For reflection, a transmission line terminated in a short or open reflects all power back to source

25. Transmission Line Terminated with 25 Ω
Zs = Zo
ZL = 25 W V
inc
Vrefl
Standing wave pattern does not go to zero as with short or open

26. Equivalent input impedance

27. Input impedance of short

28. Input impedance of open

29. 11.12 Some transmission line examples
case 1) matched load

30. case 2) unmatched load

31. Input impedance of ¼ wavelength line
Quarter wavelength transformer

32. Smith chart
The chart contains circles of which points correspond to the load impedances that have the same resistances or reactances.
The characteristic impedance of the transmission line used for measurement is Z0
Normalized impedances are used in a Smith chart.

33. Network analyzer

34. Rectilinear impedance plane
Smith chart review
Z-plane
+jX 90 o
Γ-plane
Z-to-Γ transform
Polar plane
1.0 .8 .6 +R
¥ ® .4
180 o + - .2 o
-jX
Rectilinear impedance plane
-90 o
Constant X
Z = Zo
Smith Chart maps rectilinear impedance plane onto polar plane L
Constant R G =
Z = 0
(short)
Z =
(open) L L G G = 1
±180 O = 1 O
Smith Chart

35. Constant resistance, reactance circlesx
r=0
r=0.5
r=1
r=2 R
0.5 1 2 x 2
x=1
x=0.5 1
x=2
0.5 R
0.5 1
x=-2
x=-1
x=-0.5 2

36. Constant admittance circles
Y-plane
+jB
+jG
Z-plane
+jX
If the impedance chart undergoes point-symmetric displacement, admittance chart can be obtained.
-jX

37. Impedance matching
Maximum power delivered when applied to matched load
Minimum reflections occur when the transmission line and the load are matched.
Zs = 20  + V -
Z0= 50 
ZL= 1k 
Zs = 50 
Z0= 50 
ZL= 50 

38. Ringing ~
Signal source
Load
Mismatched load

39. Impedance matching - Digital
Source matching ~
Load matching ~

40. Matching with lumped elements
L-section matching networks. (a) Network for zL inside the 1 + jx circle. (b) Network for zL outside the 1 + jx circle.

41. Basic Smith chart operation
1. Translation
2. Add series element L C

42. 3. Add shunt element L C

43. Impedance-admittance chart
ZL= 200-j 100 
Z0= 100 
f = 500MHz
0.0 1
0.2
Add series L
Add shunt C
0.5
1.2

44. Real capacitor characteristics
Valid range

45. Real inductor characteristics
Valid range

46. Single stub tuning ZL= 60-j 80  Z0= 50  f = 2GHz Translate by ‘d’1 1
0.314
0.314
0.422
d should be adjusted to take the zL on the 1+jb circle.

47. Add shunt stub (shorted)
To take the zL from the 1+jb circle to the origin, a shunt stub is used.

49. Example p.11.20
(a) Determine SWR on the transmission line of Fig Note that the dielectric is air:
(b) Find the input impedance

50. (c) If ωL = 10Ω, find Is.
Znet = 20+jωL+Zin = 20 + j − j37.5 = 81.8 − j27.5.
Is = 100/(81.8 − j27.5) = j0.37A
(d) What value of L will produce a maximum value for |Is| at ω = 1 Grad/s?
L = 37.5/ω = 37.5nH.
(e) supplied by the source: Ps = (1/2)Re{VsIs∗} = (1/2)(100)(1.22) = 61.1W
(f) delivered to ZL = 40+j30 Ω: The power delivered to the load will be the same as the power delivered to the input impedance

51. 11.14 Transient analysis