chapter 2 transmission line theory l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Chapter 2. Transmission Line Theory PowerPoint Presentation
Download Presentation
Chapter 2. Transmission Line Theory

Loading in 2 Seconds...

play fullscreen
1 / 58

Chapter 2. Transmission Line Theory - PowerPoint PPT Presentation


  • 1390 Views
  • Uploaded on

Chapter 2. Transmission Line Theory. Sept. 29 th , 2008. 2.1 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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

Chapter 2. Transmission Line Theory


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
2 1 transmission lines
2.1 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.
slide3

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.

slide4
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,
slide5
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,
wave propagation on a transmission line
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

slide7
Applying (2.3a) to the voltage of (2.6),
  • If a characteristic impedance, Z0, is defined as
  • (2.6) can be rewritten
slide8
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,
lossless transmission lines
Lossless Transmission Lines
  • R = G = 0 gives or
  • The general solutions for voltage and current on a lossless transmission line:
slide10
The wavelength on the line:
  • The phase velocity on the line:
2 2 field analysis of transmission lines
2.2 Field Analysis of Transmission Lines
  • Transmission Line Parameters

Figure 2.2 (p. 53)Field lines on an arbitrary TEM transmission line.

slide12
The time-average stored magnetic energy for 1 m section of line:
  • The circuit theory gives

  • Similarly,

slide13
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))
slide14
Circuit theory 
  • Ex 2.1 Transmission line parameters of a coaxial line
  • Table 2.1
the telegrapher equations derived form field analysis of a coaxial line
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: Ez = Hz = 0.
  • Due to the azimuthal symmetry, no φ-variation

 ə/əφ = 0

  • The fields inside the coaxial line will satisfy ME.

where

slide16

Since the z-components must vanish,

From the B.C., Eφ = 0 at ρ = a, b Eφ = 0 everywhere

slide17

The voltage between 2 conductors

The total current on the inner conductor at ρ = a

propagation constant impedance and power flow for the lossless coaxial line
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

slide19
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.
2 3 the terminated lossless transmission lines
2.3 The Terminated Lossless Transmission Lines

The total voltage and current on the line

slide21
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:
slide22
It is seen that the voltage and current on the line consist of a superposition of an incident and reflected wave.  standing waves
  • WhenΓ= 0  matched.
  • For the time-average power flow along the line at the point z:
slide23
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)| = |V0+| (constant)  “flat”.
  • When the load is mismatched,
slide24
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,
slide25
At a distance l = -z,

 Transmission line impedance equation

special cases of terminated transmission lines
Special Cases of Terminated Transmission Lines
  • Short-circuited line

ZL = 0  Γ= -1

slide27

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

slide28
Open-circuited line

ZL = ∞  Γ= 1

slide29

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

slide30
Terminated transmission lines with special lengths.
  • If l = λ/2, Zin = ZL.
  • If the line is a quarter-wavelength long, or, l = λ/4+ nλ/2 (n = 1,2,3…), Zin = Z02/ZL.  quarter-wave transformer
slide31

Figure 2.9 (p. 63)Reflection and transmission at the junction of two transmission lines with different characteristic impedances.

2 4 the smith chart
2.4 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|ejθ).
  • 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.
slide34
If a lossless line of Z0 is terminated with ZL, zL = ZL/Z0 (normalized load impedance),
  • Let Γ= Γr +jΓi, and zL = rL + jxL.
slide35
The Smith chart can also be used to graphically solve the transmission line impedance equation of (2.44).
  • If we have plotted |Γ|ejθ at the load, Zin seen looking into a length l of transmission line terminates with zLcan be found by rotating the point clockwise an amount of 2βl around the center of the chart.
slide36
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.
ex 2 2 z l 40 j70 l 0 3 find l in and z in
Ex 2.2 ZL = 40+j70,l = 0.3λ, find Γl, Γin and Zin

Figure 2.11 (p. 67)Smith chart for Example 2.2.

the combined impedance admittance smith chart
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.
ex 2 3 z l 100 j50 y l y in when l 0 15
Ex 2.3ZL = 100+j50, YL, Yin ? when l = 0.15λ

Figure 2.12 (p. 69)ZY Smith chart with solution for Example 2.3.

the slotted line
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 ZL can be determined.
  • ZL is complex  2 distinct quantities must be measured.
  • Replaced by vector network analyzer.
slide42
Assume for a certain terminated line, we have measured the SWR on the line and lmin, the distance from the load to the first voltage minimum on the line.
  • Minimum occurs when
  • The phase of Γ =
  • Load impedance
slide43
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, lmin = 4.2 – 2.72 = 1.48 cm = 0.37 λ
slide44

Figure 2.14 (p. 71)Voltage standing wave patterns for Example 2.4. (a) Standing wave for short-circuit load. (b) Standing wave for unknown load.

2 5 the quarterwave transformer
2.5 The Quarterwave Transformer

Impedance Viewpoint

  • For βl = (2π/λ)(λ/4) = π/2
  • In order for Γ = 0, Zin = Z0
slide49

Figure 2.17 (p. 74)Reflection coefficient versus normalized frequency for the quarter-wave transformer of Example 2.5.

the multiple reflection viewpoint
The Multiple Reflection Viewpoint

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

2 6 generator and load mismatches
2.6 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,
slide55
The voltage on the line:
  • Power delivered to the load:

By (2.67) &

slide56
Case 1: the load is matched to the line, Zl = Z0, Γl = 0, SWR = 1, Zin = Z0,
  • Case 2: the generator is matched to the input impedance of a mismatched line, Zin = Zg
  • If Zg is fixed, to maximize Pl,
slide57
or

or

  • Therefore, Rin = Rg and Xin = -Xg, or Zin = Zg*
  • Under these conditions
  • Finally, note that neither matching for zero reflection (Zl = Z0), nor conjugate matching (Zin = Zg*), necessary yields a system with the best efficiency.