1. July, 2003© 2003 by H.L. Bertoni1 I. Introduction to Wave Propagation Waves on transmission lines Plane waves in one dimension Reflection and transmission at junctions Spatial variations for harmonic time dependence Impedance transformations in space Effect of material conductivity

2. July, 2003© 2003 by H.L. Bertoni2 Waves on Transmission Lines Equivalent circuits using distributed C and L Characteristic wave solutions Power flow

3. July, 2003© 2003 by H.L. Bertoni3 Examples of Transmission Lines I(z,t) + V(z,t) - z I(z,t) + V(z,t) - Dielectric Conductors Strip Line Coaxial Line Two-Wire Line (Twisted Pair)

4. July, 2003© 2003 by H.L. Bertoni4 Properties of Transmission Lines (TL’s) Two wires having a uniform cross-section in one (z) dimension Electrical quantities consist of voltage V(z,t) and current I(z,t) that are functions of distance z along the line and time t Lines are characterized by distributed capacitance C and inductance L between the wires –C and L depend on the shape and size of the conductors and the material between them

5. July, 2003© 2003 by H.L. Bertoni5 Capacitance of a Small Length of Line I(t) + V(t) - l Open circuit E

6. July, 2003© 2003 by H.L. Bertoni6 Inductance of a Small Length of Line I(t) + V(t) - l Short circuitB

7. July, 2003© 2003 by H.L. Bertoni7 C and L for an Air Filled Coaxial Line a b

8. July, 2003© 2003 by H.L. Bertoni8 C and L for Parallel Plate Line w h z

9. July, 2003© 2003 by H.L. Bertoni9 Two-Port Equivalent Circuit of Length  z I(z,t) + V(z,t) - z z+  z z L  z C  z I(z,t) + V(z,t) - + I(z +  z,t) V(z+  z,t) -

10. July, 2003© 2003 by H.L. Bertoni10 Transmission Line Equations

11. July, 2003© 2003 by H.L. Bertoni11 Conditions for Existence of TL Solution

12. July, 2003© 2003 by H.L. Bertoni12 F(t-z/v) Is a Wave Traveling in +z Direction V(z,0)=F[(-1/v)(z)] V(z,t)=F[(-1/v)(z-vt)] a z -a a+vt z -a+vt vt t = 0 t > 0

13. July, 2003© 2003 by H.L. Bertoni13 G(t+z/v) Is a Wave Traveling in -z Direction V(z,0)=G[(1/v)(z)] a 2a z t = 0 V(z,t)=G[(1/v)(z+vt)] 2a-vt z -vt a-vt t > 0

14. July, 2003© 2003 by H.L. Bertoni14 Example of Source Excitation ∞ z V S (t) + 0 R S I(0,t) V(0,t) + V S (t) I(0,t) R S V(0,t) 0 z ∞

15. July, 2003© 2003 by H.L. Bertoni15 Receive Voltage Further Along Line + V S (t) ∞ z V S (t) + 0 l R S V(l,t) Scope R S V(-l,t) -l 0 z ∞ Scope

16. July, 2003© 2003 by H.L. Bertoni16 Power Carried by Waves P(z,t) I(z,t) V(z,t) z

17. July, 2003© 2003 by H.L. Bertoni17 Summary of Solutions for TL’s Solutions for V and I consists of the sum of the voltages and current of two waves propagating in ±z directions For either wave, the physical current flows in the direction of propagation in the positive wire Semi-infinite segment of TL appears at its terminals as a resistance of value Z (even though the wires are assumed to have no resistance) The waves carry power independently in the direction of wave propagation

18. July, 2003© 2003 by H.L. Bertoni18 Plane Waves in One Dimension Electric and magnetic fields in terms of voltage and current Maxwell’s equations for 1-D propagation Plane wave solutions Power and polarization

19. July, 2003© 2003 by H.L. Bertoni19 Electric Field and Voltage for Parallel Plates w h z y E x (z,t) + V(z,t) - x

20. July, 2003© 2003 by H.L. Bertoni20 Magnetic Field and Current for Parallel Plates w h z y H y (z,t) or B y (z,t) I(z,t) x

21. July, 2003© 2003 by H.L. Bertoni21 Maxwell’s Equations in 1-D

22. July, 2003© 2003 by H.L. Bertoni22 Plane Waves: Solutions to Maxwell Equations

23. July, 2003© 2003 by H.L. Bertoni23 Power Density Carried by Plane Waves E Direction of propagation H

24. July, 2003© 2003 by H.L. Bertoni24 Polarization

25. July, 2003© 2003 by H.L. Bertoni25 Examples of Polarization E x z H y x z E y H

26. July, 2003© 2003 by H.L. Bertoni26 Summary of Plane Waves Plane waves are polarized with fields E and H perpendicular to each other and to the direction of propagation Wave velocity is the speed of light in the medium E x H watts/m 2 is the power density carried by a plane wave

27. July, 2003© 2003 by H.L. Bertoni27 Reflection and Transmission at Junctions Junctions between different propagation media Reflection and transmission coefficients for 1-D propagation Conservation of power, reciprocity Multiple reflection/transmission

28. July, 2003© 2003 by H.L. Bertoni28 Junctions Between Two Regions 0 z I(0 -,t) I(0 +,t) TL 1 V(0 -,t) + V(0 +,t) TL 2 E x (0 -,t) E x (0 +,t) H y (0 -,t) H y (0 +,t) Medium 1 Medium 2 x z

29. July, 2003© 2003 by H.L. Bertoni29 Reflection and Transmission Incident wave E x In (z,t)=F 1 (t-z/v 1 ) H y In (z,t) Transmitted wave Reflected wave v 1 and  1 v 2 and  2 x z

30. July, 2003© 2003 by H.L. Bertoni30 Reflection and Transmission Coefficients

31. July, 2003© 2003 by H.L. Bertoni31 Reflection and Transmission, cont.

32. July, 2003© 2003 by H.L. Bertoni32 Reflected and Transmitted Power

33. July, 2003© 2003 by H.L. Bertoni33 Conservation of Power and Reciprocity

34. July, 2003© 2003 by H.L. Bertoni34 Termination of a Transmission Line I(0 -,t) TL V(0 -,t) + R L 0 z

35. July, 2003© 2003 by H.L. Bertoni35 Reflections at Multiple Interfaces Incident wave E x In (z,t)=F 1 (t-z/v 1 ) Transmitted H y In (z,t) waves Reflected waves Multiple internal reflections v 1 and  1 v 2 and  2 v 3 and  3 x 0 l z

36. July, 2003© 2003 by H.L. Bertoni36 Scattering Diagram for a Layer 1                                              l z 2l/v 2 4l/v 2 t                                        

37. July, 2003© 2003 by H.L. Bertoni37 Summary of Reflection and Transmission The planar interface between two media is analogous to the junction of two transmission lines At a single interface (junction) the equation T = 1 +  is a statement of the continuity of electric field (voltage) The ratio of reflected to incident power =   Power is conserved so that the ratio of transmitted to incident power = 1 -   The reciprocity condition implies that reflected and transmitted power are the same for incidence from either medium At multiple interfaces, delayed multiple interactions complicate the description of the reflected and transmitted fields for arbitrary time dependence

38. July, 2003© 2003 by H.L. Bertoni38 Spatial Variations for Harmonic Time Dependence Traveling and standing wave representations of the z dependence Period average power Impedance transformations to account for layered materials Frequency dependence of reflection from a layer

39. July, 2003© 2003 by H.L. Bertoni39 Harmonic Time Dependence at z = 0

40. July, 2003© 2003 by H.L. Bertoni40 Traveling Wave Representation

41. July, 2003© 2003 by H.L. Bertoni41 Standing Wave Representation

42. July, 2003© 2003 by H.L. Bertoni42 Variation of the Voltage Magnitude | V + | z 0 z

43. July, 2003© 2003 by H.L. Bertoni43 Standing Wave Before a Conductor I SC , v short 0 z Incident wave E x In (z) H y In (z) E x Re (z) Reflected wave x Perfect conductor 0 z

44. July, 2003© 2003 by H.L. Bertoni44 Standing Wave Before a Conductor, cont.  I SC -  z

45. July, 2003© 2003 by H.L. Bertoni45 Period Averaged Power

46. July, 2003© 2003 by H.L. Bertoni46 Reflection From a Load Impedance V + V - Z L 0 z I(0) V(0) + Z L 0 z

47. July, 2003© 2003 by H.L. Bertoni47 Summary of Spatial Variation for Harmonic Time Dependence Field variation can be represented by two traveling waves or two standing waves The magnitude of the field for a pure traveling wave is independent of z The magnitude of the field for a pure standing wave is periodic in z with period The period average power is the algebraic sum of the powers carried by the traveling waves The period average power is independent of z no matter if the wave is standing or traveling The fraction of the incident power carried by a reflected wave is  

48. July, 2003© 2003 by H.L. Bertoni48 Impedance Transformations in Space Impedance variation in space Using impedance for material layers Frequency dependence of reflection from a brick wall Quarter wave matching layer

49. July, 2003© 2003 by H.L. Bertoni49 Defining Impedance Along a TL I(0) Z IN V(0) + Z L -l 0 z

50. July, 2003© 2003 by H.L. Bertoni50 Properties of the Impedance Transform

51. July, 2003© 2003 by H.L. Bertoni51 Using Transform for Layered Media Incident wave E x In (z) E x TR (z) Transmitted H y In (z) wave E x Re (z) Reflected wave v 1,  1 v 2,  2 v 3,  3 x 0 l z Z IN (l) Z L =  3 Z=  2

52. July, 2003© 2003 by H.L. Bertoni52 Circuit Solution for Reflection Coefficient

53. July, 2003© 2003 by H.L. Bertoni53 Example 1: Reflection at a Brick Wall w

54. July, 2003© 2003 by H.L. Bertoni54 Example 1: Reflection at a Brick Wall, cont. 0 0.25 0.50 0.75 1.0 1.25 1.50 1.75 2.0 f GHz   

55. July, 2003© 2003 by H.L. Bertoni55 Example 2: Quarter Wave Layers Incident wave E x In (z) E x TR (z) Transmitted H y In (z) wave E x Re (z) Reflected wave v 1,  1 v 2,  2 v 3,  3 l=  k 2 )=   x 0 z

56. July, 2003© 2003 by H.L. Bertoni56 Example 2: Quarter Wave Layers, cont.

57. July, 2003© 2003 by H.L. Bertoni57 Summary of Impedance Transformation The impedance repeats every half wavelength in space, and is inverted every quarter wavelength Impedances can be cascaded to find the impedance seen by an incident wave Reflection from a layer has periodic frequency dependence with minima (or maxima) separated by  f = v 2 /(2w) Quarter wave layers can be used impedance matching to eliminate reflections

58. July, 2003© 2003 by H.L. Bertoni58 Effect of Material Conductivity Equivalent circuit for accounting for conductivity Conductivity of some common dielectrics Effect of conductivity on wave propagation

59. July, 2003© 2003 by H.L. Bertoni59 G, C, L for Parallel Plate Line w h z

60. July, 2003© 2003 by H.L. Bertoni60 Equivalent Circuit for Harmonic Waves + I(z) V(z) - z z+  z z I(z) + V(z) + I(z +  z) V(z+  z) j  L  z j  C  z G

61. July, 2003© 2003 by H.L. Bertoni61 Harmonic Fields and Maxwell’s Equations w h z y Hy(z)Hy(z) I(z) x + V(z) E x (z)

62. July, 2003© 2003 by H.L. Bertoni62 Maxwell’s Equations With Medium Loss

63. July, 2003© 2003 by H.L. Bertoni63 Constants for Some Common Materials When conductivity exists, use complex dielectric constant given by  =  o (  r - j  ") where  " =  o and  o  10 -9 /36  Material*  r  mho/m)  " at 1 GHz Lime stone wall7.50.030.54 Dry marble8.80.22 Brick wall40.020.36 Cement4 - 60.3 Concrete wall6.50.081.2 Clear glass4 - 60.005 - 0.1 Metalized glass5.02.545 Lake water810.0130.23 Sea Water813.359 Dry soil2.5---- Earth7 - 300.001 - 0.030.02 - 0.54 * Common materials are not well defined mixtures and often contain water.

64. July, 2003© 2003 by H.L. Bertoni64 Incorporating Material Loss Into Waves

65. July, 2003© 2003 by H.L. Bertoni65 Wave Number and Impedance

66. July, 2003© 2003 by H.L. Bertoni66 Effect of Loss on Traveling Waves  V +   V +  e  z

67. July, 2003© 2003 by H.L. Bertoni67 Attenuation in dB

68. July, 2003© 2003 by H.L. Bertoni68 Effect of Loss on Traveling Waves, cont.    V+V+eV+V+e z

69. July, 2003© 2003 by H.L. Bertoni69 Loss Damps Out Reflection in Media Traveling wave amplitude zz Reflecting boundary Incident wave Reflected wave

70. July, 2003© 2003 by H.L. Bertoni70 Effect of Damping on the |  | for a Wall 0 0.25 0.50 0.75 1.0 1.25 1.50 1.75 2.0 f GHz    

71. July, 2003© 2003 by H.L. Bertoni71 Summary of Material Loss Conductivity is represented in Maxwell’s equations by a complex equivalent dielectric constant The wavenumber k =  j  and wave impedance  then have imaginary parts The attenuation length = 1/  Loss in a medium damps out reflections within a medium