PART 1: Wave propagation 1

2  PROPAGATION & REFLECTION OF PLANE WAVES  ELECTRIC AND MAGNETIC FIELDS FOR PLANE WAVE  PLANE WAVE IN LOSSY DIELECTRICS – IMPERFECT DIELECTRICS  PLANE WAVE IN LOSSLESS (PERFECT) DIELECTRICS  PLANE WAVE IN FREE SPACE  PLANE WAVE IN CONDUCTORS  POWER AND THE POYNTING VECTOR Introduction

PROPAGATION & REFLECTION OF PLANE WAVES Will discuss the effect of propagation of EM wave in four medium : Free space ; Lossy dielectric ; Lossless dielectric (perfect dielectric) and Conducting media. Also will be discussed the phenomena of reflections at interface between differe nt media. Ex : EM wave is radio wave, TV signal, radar radiation and optical wave in opti cal fiber. Three basics characteristics of EM wave : These propagation phenomena for a type traveling wave called pl ane wave can be explained or derived by Maxwell’s equations. - travel at high velocity - travel following EM wave characteristics - travel outward from the source 3

ELECTRIC AND MAGNETIC FIELDS FOR PLAN E WAVE From Maxwell’s equations : 4

Assume the medium is free of charge : From vector identity and taking the curl of (1)and substituting (1) and (2) 5

In Cartesian coordinates : Assume that : (i)Electric field only has x component (ii)Propagate in the z direction Similarly in the same way, from vect or identity and taking the curl of (2)a nd substituting (1) and (2) 6

The solution for this equation : Incidence wave propagate in +z direction Reflected wave propagate i n -z direction To find H field : 7

On the right side equ ation : Equating components on both side = y component 8

Hence : These equations of EM wave are called PLANE WAVE. Main characteristics of EM wave : (i)Electric field and magnetic field always perpendicular. (ii) NO electric or magnetic fields component in the direction of propagation. (iii) will provides information on the direction of propagation. 9

PLANE WAVE IN LOSSY DIELECTRICS – IMPER FECT DIELECTRICS Assume a media is charged free, ρ v =0 (1) (2) Taking the curl of (2) : 10

From vector identity : (1) (2) Where : (4) Define : (5) Equating (4) and (5) for Re and Im parts : 11

Magnitude for (5) ; Magnitude for (4) ; Equate (8) and (9) : Hence :  is known as attenuation constant as a measure of the wave is attenuated while traveling in a m edium. Add (10) and (6) : 12

Substract (10) and (6) :  is phase constant If the electric field propagate in +z direction and has component x, the equation o f the wave is given by : And the magnetic field : (13) (14) 13

where ; (15) (14) (15) Conclusions that can be made for the wave propagating in lossy dielectrics mater ial : (i) E and H fields amplitude will be attenuated by (ii) E leading H by Intrinsic impedance : where ; (16) (17) 14

Wave velocity ; Loss tangent ; Loss tangent values will determine types of media : tan θ small (σ / ωε < 0.1) – good dielectric – low loss tan θ large (σ /ωε > 10 ) - good conductor – high loss (18) Another factor that determined the characteristic of the media is operating frequency. A m edium can be regarded as a good conductor at low frequency might be a good dielectric at higher frequency. From (17) and (18) 15

x z y Graphical representation of E field in lossy dielectric (14) (15) 16

PLANE WAVE IN LOSSLESS (PERFECT) DIELECTRICS Characteristics: Substitute in (11) and (12) : The zero angle means that E and H fields are in phase at each fixed location. (19) (20) (21) (22) 17

PLANE WAVE IN FREE SPACE Characteristics: Free space is nothing more than the perfect dielectric media : Substitute in (20) and (21) : (20) (21) (22) where (23) (24) (25) (26) 18

(14) (15) The field equations for E and H obtained : x y z (at t = 0) E x + kos(-  z) H y + kos(-  z) (27) (28) E and H fields and the direction of propagation : Generally : 19

PLANE WAVE IN CONDUCTORS In conductors : or With the characteristics : (29) Substitute in (11 and (12) : E leads H by 45 0 The field equations for E and H obtained : (30) (32) (33) (31) 20

It is seen that in conductors and waves are attenuated by From the diagram is referred to as the skin depth. It refers to the amplitude of th e wave propagate to a conducting media is reduced to e -1 or 37% from its initial va lue E 0 E0E0 x z In a distance : It can be seen that at higher freque ncies is decreasing. (34) 21

Ex : A lossy dielectric has an intrinsic impedance of at the particular frequency. If at that particular frequency a plane wave that propagate in a medium has a magnetic field given by : Find and. Solution : From intrinsic impedance, the magnitude of E field : It is seen that E field leads H field : 22

To find : ; and we know Hence: 23

POWER AND THE POYNTING VECTOR From vector identity: Dot product (36) with : (35) (36) (37) Change in (37) and use (38), equation (37) becomes : (38) (39) 24

(35) And from (35): (39) (40) Therefore (39) becomes: where: (41) Integration (41) throughout volume v : (42) 25

Using divergence theorem to (42): (42) Total energy flow leavi ng the volume The decrease of the energy densi ties of energy stored in the electr ic and magnetic fields Dissipated ohmi c power Equation (43) shows Poynting Theorem and can be written a s : 26

Poynting theorem states that the total power flow leaving the volume is equal to t he decrease of the energy densities of energy stored in the electric and magnetic fi elds and the dissipated ohmic power. Stored electric fie ld Stored magnetic fi eld Ohmic losses Output power Input power The theorem can be explained as shown in the diagram below : 27

Given for lossless dielectric, the electric and magnetic fields are : The Poynting vector becomes: 28

To find average power density : Integrate Poynting vector and divide with interval T = 1/f : Average power through area S : 29

Given for lossy dielectric, the electric and magnetic fields are : The Poynting vector becomes: Average power : 30

Ex. 2: A uniform plane wave propagate in a lossless dielectric in the +z direction. T he electric field is given by : The average power density measured was Find: (i)Dielectric constant of the material if (ii)Wave frequency (iii)Magnetic field equation Solution: (i) Average power : 31

For lossless dielectric : (ii) Wave frequency : 32

(iii) Magnetic field equation : 33

34 Chapter 2. Transmission Line Theory

Transmission Lines A transmission line is a distributed-parameter network, where voltages and currents can vary in magnitude and phase over the length of the line. Lumped Element Model for a Transmission Line Transmission lines usually consist of 2 parallel conductors. A short segment Δz of transmission line can be modeled as a lumped-element circuit. Figure 2.1 Voltage and current definitions and equivalent circuit for an incremental length of transmission line. (a) Voltage and current definitions. (b) Lumped-element equivalent circuit.

36 R = series resistance per unit length for both conductors L = series inductance per unit length for both conductors G = shunt conductance per unit length C = shunt capacitance per unit length Applying KVL and KCL,

37 Dividing (2.1) by Δz and Δz  0,  Time-domain form of the transmission line, or telegrapher, equation. For the sinusoidal steady-state condition with cosine-based phasors,

38 Wave Propagation on a Transmission Line By eliminating either I(z) or V(z): where the complex propagation constant. (α = attenuation constant, β = phase constant) Traveling wave solutions to (2.4): Wave propagation in +z directon Wave propagation in - z directon

39 Applying (2.3a) to the voltage of (2.6), If a characteristic impedance, Z 0, is defined as (2.6) can be rewritten

40 Converting the phasor voltage of (2.6) to the time domain: The wavelength of the traveling waves: The phase velocity of the wave is defined as the speed at which a constant phase point travels down the line,

41 Lossless Transmission Lines R = G = 0 givesor The general solutions for voltage and current on a lossless transmission line:

42 The wavelength on the line: The phase velocity on the line:

Field Analysis of Transmission Lines Transmission Line Parameters Figure 2.2 Field lines on an arbitrary TEM transmission line.

44 The time-average stored magnetic energy for 1 m section of line: The circuit theory gives  Similarly, 

45 Power loss per unit length due to the finite conductivity (from (1.130)) Circuit theory  (H || S) Time-average power dissipated per unit length in a lossy dielectric (from (1.92))

46 The Telegrapher Equations Derived form Field Analysis of a Coaxial Line Eq. (2.3) can also be obtained from ME. A TEM wave on the coaxial line: E z = H z = 0. Due to the azimuthal symmetry, no φ-variation  ə/əφ = 0 The fields inside the coaxial line will satisfy ME. where

47 Since the z-components must vanish, From the B.C., E φ = 0 at ρ = a, b  E φ = 0 everywhere

48 The voltage between 2 conductors The total current on the inner conductor at ρ = a

49 Propagation Constant, Impedance, and Power Flow for the Lossless Coaxial Line From Eq. (2.24) For lossless media, The wave impedance The characteristic impedance of the coaxial line Ex 2.1

50 Power flow ( in the z direction) on the coaxial line may be computed from the Poynting vector as The flow of power in a transmission line takes place entirely via the E & H fields between the 2 conductors; power is not transmitted through the conductors themselves.

The Terminated Lossless Transmission Lines The total voltage and current on the line

52 The total voltage and current at the load are related by the load impedance, so at z = 0 The voltage reflection coefficient: The total voltage and current on the line:

53 It is seen that the voltage and current on the line consist of a superposition of an incident and reflected wave.  standing waves When Γ= 0  matched. For the time-average power flow along the line at the point z:

54 When the load is mismatched, not all of the available power from the generator is delivered to the load. This “loss” is return loss (RL): RL = -20 log|Γ| dB If the load is matched to the line, Γ= 0 and |V(z)| = |V 0 + | (constant)  “ flat ”. When the load is mismatched,

55 A measure of the mismatch of a line, called the voltage standing wave ratio (VSWR) (1< VSWR<∞) From (2.39), the distance between 2 successive voltage maxima (or minima) is l = 2π/2β = λ/2 (2βl = 2π), while the distance between a maximum and a minimum is l = π/2β = λ/4. From (2.34) with z = -l,

56 At a distance l = -z,  Transmission line impedance equation

57 Special Cases of Terminated Transmission Lines Short-circuited line Z L = 0  Γ= -1

58 Figure 2.6 (a) Voltage, (b) current, and (c) impedance (R in = 0 or  ) variation along a short-circuited transmission line.

59 Open-circuited line Z L = ∞  Γ= 1

60 Figure 2.8 (a) Voltage, (b) current, and (c) impedance (R in = 0 or  ) variation along an open- circuited transmission line.

61 Terminated transmission lines with special lengths. If l = λ/2, Z in = Z L. If the line is a quarter-wavelength long, or, l = λ/4+ nλ/2 (n = 1,2,3…), Z in = Z 0 2 /Z L.  quarter-wave transformer Figure 2.9 Reflection and transmission at the junction of two transmission lines with different characteristic impedances.

The Smith Chart A graphical aid that is very useful for solving transmission line problems. Derivation of the Smith Chart Essentially a polar plot of the Γ(= |Γ|e jθ ). This can be used to convert from Γto normalized impedances (or admittances), and vice versa, using the impedance (or admittance) circles printed on the chart.

63 Figure 2.10 The Smith chart.

64 If a lossless line of Z 0 is terminated with Z L, z L = Z L /Z 0 (normalized load impedance), Let Γ= Γ r +jΓ i, and z L = r L + jx L.

65 The Smith chart can also be used to graphically solve the transmission line impedance equation of (2.44). If we have plotted |Γ|e jθ at the load, Z in seen looking into a length l of transmission line terminates with z L can be found by rotating the point clockwise an amount of 2 βl around the center of the chart.

66 Smith chart has scales around its periphery calibrated in electrical lengths, toward and away from the “generator”. The scales over a range of 0 to 0.5 λ.

67 Ex 2.2 Z L = 40+j70, l = 0.3λ, find Γ l, Γ in and Z in Figure 2.11 Smith chart for Example 2.2.

68 The Combined Impedance-Admittance Smith Chart Since a complete revolution around the Smith chart corresponds to a line length of λ/2, a λ/4 transformation is equivalent to rotating the chart by 180 °. Imaging a give impedance (or admittance) point across the center of the chart to obtain the corresponding admittance (or impedance) point.

69 Ex 2.3 Z L = 100+j50, Y L, Y in ? when l = 0.15λ Figure 2.12 ZY Smith chart with solution for Example 2.3.

70 The Slotted Line A transmission line allowing the sampling of E field amplitude of a standing wave on a terminated line. With this device the SWR and the distance of the first voltage minimum from the load can be measured, from this data Z L can be determined. Z L is complex  2 distinct quantities must be measured. Replaced by vector network analyzer.

71 Figure 2.13 An X-band waveguide slotted line.

72 Assume for a certain terminated line, we have measured the SWR on the line and l min, the distance from the load to the first voltage minimum on the line. Minimum occurs when The phase of Γ = Load impedance

73 Ex 2.4 With a short circuit load, voltage minima at z = 0.2, 2.2, 4.2 cm With unknown load, voltage minima at z = 0.72, 2.72, 4.72 cm λ = 4 cm, If the load is at 4.2 cm, l min = 4.2 – 2.72 = 1.48 cm = 0.37 λ

74 Figure 2.14 Voltage standing wave patterns for Example 2.4. (a) Standing wave for short-circuit load. (b) Standing wave for unknown load.

75 Figure 2.15 Smith chart for Example 2.4.

The Quarterwave Transformer Impedance Viewpoint For βl = (2π/λ)(λ/4) = π/2 In order for Γ = 0, Z in = Z 0

77 Figure 2.16 The quarter-wave matching transformer.

78 Ex 2.5 Frequency Response of a Quarter-Wave Transformer R L = 100, Z 0 = 50

79 Figure 2.17 Reflection coefficient versus normalized frequency for the quarter-wave transformer of Example 2.5.

80 The Multiple Reflection Viewpoint Figure 2.18 (p. 75) Multiple reflection analysis of the quarter- wave transformer.

81

82 Numerator

Generator and Load Mismatches Because both the generator and load are mismatched, multiple reflections can occur on the line. In the steady state, the net result is a single wave traveling toward the load, and a single reflected wave traveling toward the generator. In Fig. 2.19, where z = -l,

84 Figure 2.19 Transmission line circuit for mismatched load and generator.

85 The voltage on the line: Power delivered to the load: By (2.67) &

86 Case 1: the load is matched to the line, Z l = Z 0, Γ l = 0, SWR = 1, Z in = Z 0, Case 2: the generator is matched to the input impedance of a mismatched line, Z in = Z g If Z g is fixed, to maximize P l,

87 or Therefore, R in = R g and X in = -X g, or Z in = Z g * Under these conditions Finally, note that neither matching for zero reflection (Z l = Z 0 ), nor conjugate matching (Z in = Z g * ), necessary yields a system with the best efficiency.