Chapter 1 Theory and Applications of Transmission Lines - Part 2 [PDF]

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Chapter 1 Theory and Applications of Transmission Lines Part 2: Smith Chart and Impedance Matching

Huynh Phu Minh Cuong, PhD [email protected]

Department of Telecommunications Faculty of Electrical and Electronics Engineering Ho Chi Minh city University of Technology 4/10/2017

Cuong Huynh, Ph.D.Telecommunications Engineering DepartmentHCMUT

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Smith Chart and Impedance Matching Outline  Introduction  Smith Chart   

Smith Chart Description Smith Chart Characteristics Z-Y Smith Chart

 Smith Chart Applications    

Determining Impedance and Reflection Coefficients Determining VSWR Input Impedance of a Complex Circuit Input Impedance of a Terminated Transmission Line

 Impedance Matching  



Quarter-wave Transformer Matching with Lumped Elements Single-Stub Matching Networks 2

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1. Introduction  Many of calculations required to solve T.L. problems involve the use of complicated equations.  Smith Chart, developed by Phillip H. Smith in 1939, is a graphical aid that can be very useful for solving T.L. problems.  The Smith chart, however, is more than just a graphical technique as it provides a useful way of visualizing transmission line phenomenon without the need for detailed numerical calculations.  A microwave engineer can develop a good intuition about transmission line and impedance-matching problems by learning to think in terms of the Smith chart.  From a mathematical point of view, the Smith chart is simply a representation of all possible complex impedances with respect to coordinates defined by the reflection coefficient.  The domain of definition of the reflection coefficient is a circle of radius 1 in the complex plane. This is also the domain of the Smith chart. 4/10/2017

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1. Introduction

Phillip Hagar Smith (1905–1987): graduated from Tufts College in 1928, invented the Smith Chart in 1939 while he was working for the Bell Telephone Laboratories. 4/10/2017

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2. Smith Chart The initial goal of the Smith chart is to represent a reflection coefficient and its corresponding normalized impedance by a point, from which the conversion between them can be easily achieved. To do so, we start from the general definition of reflection coefficient Z  R  jX

Y=1/Z=G+jB

z

Z R X   j  r  jx Z0 Z0 Z0

y

Y G B   j  g  jb Y0 Y0 Y0



Z  Z0  Re( )  j Im(  ) Z  Z0



z 1 z 1 4/10/2017

z

1  1 

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2. Smith Chart Now we can write z  1   as 1 

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2. Smith Chart

 Resistance circles

 r  Center :  ,0  1 r 

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1 Radius : 1 r

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2. Smith Chart

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2. Smith Chart  Reactance circles

 1 Center :  1,   x

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1 Radius : x

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2. Smith Chart Resistance circles r-circles

Unit circle

Matching point Shorted point Opened point

Reactance circles x-circles

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2. Smith Chart For the constant r circles: 1. The centers of all the constant r circles are on the horizontal axis – real part of the reflection coefficient. 2. The radius of circles decreases when r increases. 3. All constant r circles pass through the point r =1, i = 0. 4. The normalized resistance r =  is at the point r =1, i = 0.

z = r+jx



=r+i

For the constant x (partial) circles: 1. The centers of all the constant x circles are on the r =1 line. The circles with x > 0 (inductive reactance) are above the r axis; the circles with x < 0 (capacitive) are below the r axis. 2. The radius of circles decreases when absolute value of x increases. 3. The normalized reactances x =  are at the point r =1, i = 0 4/10/2017

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2. Smith Chart

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2. Smith Chart

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2. Smith Chart

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2. Smith Chart

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2. Smith Chart Constant circle

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7.4 Smith Chart: 2. Smith Basic Chart Procedures

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2. Smith Chart

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2. Smith Chart

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2. Smith Chart: Y Smith Chart z 1  z 1

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1 1 y 1 y    1 y 1 1 y

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z 1  : Z-Smith C. z 1 y 1   : Y  Smith C. y 1  The Smith chart can be used with normalized impedances or with normalized admittances. As an impedance chart, the Smith chart consists of rL and xL circles, the resistance and reactance of a normalized load impedance zL, respectively.  When used as an admittance chart, the rL circles become gL circles and the xL circles become bL circles, where gL and bL are the conductance and susceptance of the normalized load admittance yL, respectively. Engineering DepartmentHCMUT 20

3. Smith Chart Applications

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3. Smith Chart Applications

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3. Smith Chart Applications

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3. Smith Chart Applications Given R and ZR  Find the Voltage Standing Wave Ratio (VSWR) The Voltage standing Wave Ratio or VSWR is defined as

The normalized impedance at a maximum location of the standing wave pattern is given by

This quantity is always real and ≥ 1. The VSWR is simply obtained on the Smith chart, by reading the value of the (real) normalized impedance, at the location dmax where  is real and positive. 4/10/2017

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3. Smith Chart Applications

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3. Smith Chart Applications

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3. Smith Chart Applications

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3. Smith Chart Applications A. Given 𝑍 𝑑 , find Γ 𝑑 𝑍 𝑑 .

or

Given Γ 𝑑 , find

B. Given Γ𝐿 𝑎𝑛𝑑 𝑍𝐿 , find Γ 𝑑 and 𝑍 𝑑 . Given Γ 𝑑 and 𝑍 𝑑 , find Γ𝐿 𝑎𝑛𝑑 𝑍𝐿 . C. Find dmax and dmin (maximum and minimum locations for the VSW pattern). D. Find the VSWR. E. Given 𝑍 𝑑 , find 𝑌 𝑑 𝑍 𝑑 . 4/10/2017

or

Given 𝑌 𝑑 , find

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3. Smith Chart Applications A. Given 𝒁 𝒅 , find 𝜞 𝒅 1.

Normalize the impedance: 𝒛 𝒅 =

2. 3. 4.

𝒁 𝒅 𝑹 𝑿 = +𝒋 = 𝒓 + 𝒋𝒙 𝒁𝟎 𝒁𝟎 𝒁𝟎

Find the circle of constant normalized resistance r. Find the circle of constant normalized reactance x. Find the interaction of the two curves indicates the reflection coefficient in the complex plane. The chart provides directly magnitude and the phase angle of Γ 𝑑 .

Example 1: Find Γ 𝑑 given 𝑍 𝑑 = 25 + 𝑗100Ω and 𝑍0 = 50Ω 29 4/10/2017

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3. Smith Chart Applications

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3. Smith Chart Applications A. Given 𝜞 𝒅 , find 𝒁 𝒅 1. Determine the complex point representing the given reflection coefficient Γ 𝑑 on the chart. 2. Read the value of normalized resistance r and the normalized reactance x that correspond to the reflection coefficient point. 3. The normalized impedance is: 𝒛 𝒅 = 𝒓 + 𝒋𝒙 4. The actual impedance is: 𝒁 𝒅 = 𝒛 𝒅 𝒁𝟎

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3. Smith Chart Applications B. Given 𝜞𝑳 𝒂𝒏𝒅 𝒁𝑳 , find 𝜞 𝒅 and 𝒁 𝒅 The magnitude of the reflection coefficient is constant along a lossless T.L. terminated by a specific load, since: Γ 𝑑

1. 2. 3.

4.

= Γ𝐿 𝑒 −𝑗2𝛽𝑑 = Γ𝐿

Identify the load reflection coefficient Γ𝐿 and the normalized load impedance 𝑍𝐿 on the Smith Chart. Draw the circle of constant coefficient amplitude Γ 𝑑 = Γ𝐿 Starting from the point representing the load, travel on the circle in the clockwise direction by an angle 𝜃 = 2𝛽𝑑. The new location on the chart corresponds to location d on the T.L. Here the value of Γ 𝑑 and 𝑍 𝑑 can be read from the chart.

Example: Find Γ 𝑑 and 𝑍 𝑑 given 𝑍𝐿 = 25 + 𝑗100Ω, 𝑍0 = 50Ω and 𝑑 = 0.18𝜆 32 4/10/2017

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3. Smith Chart Applications

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3. Smith Chart Applications Example 3: Find Γ 𝑑 and 𝑍 𝑑 given 𝑍𝑅 = 100 − 𝑗50Ω , 𝑍0 = 50Ω and 𝑑 = 0.1𝜆

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3. Smith Chart Applications C. Given 𝜞𝑳 𝒂𝒏𝒅 𝒁𝑳 , find 𝒅𝒎𝒂𝒙 and 𝒅𝒎𝒊𝒏 1.

2. 3.

4.

Identify the load reflection coefficient Γ𝐿 and the normalized load impedance 𝑍𝐿 on the Smith Chart. Draw the circle of constant coefficient amplitude Γ 𝑑 = Γ𝐿 The circle intersects the real axis of the reflection coefficient at two points which identify dmax (when Γ 𝑑 = real positive) and dmin (when Γ 𝑑 = real negative). The Smith chart provides an outer graduation where the distances normalized to the wavelength can be read directly.

Example 4: Find dmax and dmin for 𝑍𝐿 = 100 + 𝑗50Ω, 𝑍0 = 50Ω and 𝑑 = 0.18𝜆 35 4/10/2017

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3. Smith Chart Applications

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3. Smith Chart Applications D. Given 𝜞𝑳 𝒂𝒏𝒅 𝒁𝑳 , find 𝑽𝑺𝑾𝑹 𝑉𝑚𝑎𝑥 1 + Γ𝐿 = The VSWR is defined as: 𝑉𝑆𝑊𝑅 = 𝑉𝑚𝑖𝑛 1 − Γ𝐿

The normalized impedance at the maximum location of the SW pattern is given by: 𝑧 𝑑𝑚𝑎𝑥

1 + Γ 𝑑𝑚𝑎𝑥 1 + Γ𝐿 = = = 𝑉𝑆𝑊𝑅 1 − Γ 𝑑𝑚𝑎𝑥 1 − Γ𝐿

This quantity is always real and greater than 1. The VSWR is simply obtained on the Smith Chart by reading the value of real normalized impedance at the location dmax where Γ is real and positive. Example 5: Find VSWR for 𝑍𝐿 = 25 ± 𝑗100Ω, 𝑍0 = 50Ω. 37 4/10/2017

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3. Smith Chart Applications E. Given 𝒁 𝒅 , find 𝒀 𝒅  The normalized impedance and admittance are defined as: 1+Γ 𝑑 𝑧 𝑑 = 1−Γ 𝑑  Since: Γ 𝑑 +

1−Γ 𝑑 𝑦 𝑑 = 1+Γ 𝑑

𝜆 𝜆 = −Γ 𝑑 → 𝑧 𝑑 + =𝑦 𝑑 4 4

 The actual values are given by: 𝜆 𝜆 𝑍 𝑑+ = 𝑍0 𝑧 𝑑 + 4 4

𝜆 𝜆 𝑌 𝑑+ = 𝑌0 𝑦 𝑑 + = 4 4

𝜆 𝑦 𝑑+4 𝑍0

Example 6: Find YL given 𝑍𝐿 = 25 ± 𝑗100Ω, 𝑍0 = 50Ω. 4/10/2017

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3. Smith Chart Applications

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2. Smith Chart: Y Smith Chart z 1  z 1

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1 1 y 1 y    1 y 1 1 y

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z 1  : Z-Smith C. z 1 y 1   : Y  Smith C. y 1  The Smith chart can be used with normalized impedances or with normalized admittances. As an impedance chart, the Smith chart consists of rL and xL circles, the resistance and reactance of a normalized load impedance zL, respectively.  When used as an admittance chart, the rL circles become gL circles and the xL circles become bL circles, where gL and bL are the conductance and susceptance of the normalized load admittance yL, respectively. Engineering DepartmentHCMUT 40

4. Impedance Matching Maximum power transfer: TL is terminated by Zo Zo

Impedance Matching

 Using lump elements  Using transmission lines  ADS Smith Chart tool 4/10/2017

Impedance Matching Network

ZL

 Matching with Lumped Elements  Single-Stub Matching Networks  Quarter-wave Transformer

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4. Impedance Matching  The purpose of the matching network is to eliminate reflections at terminal MM’ for wave incident from the source. Even though multiple reflections may occur between AA’ and MM’, only a forward travelling wave exists on the feedline.

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4. Impedance Matching A. Quarter wavelength Transformer Matching:

𝑍0 = 50

𝑍𝐿 = 40 Ω

 In case of complex impedance:

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4. Impedance Matching B. Lumped-Element Matching: choose d and YS to achieve a match at MM’.

 The input admittance at MM’ can be written as: 𝑌𝑖𝑛 = 𝑌𝑑 + 𝑌𝑠 = 𝐺𝑑 + 𝑗𝐵𝑑 + 𝑗𝐵𝑠  To achieve a matched condition at MM’, it is necessary that 𝑦𝑖𝑛 = 1, which translates into two specific conditions, namely: 𝑔𝑑 = 1 𝑏𝑠 = −𝑏𝑑 4/10/2017

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4. Impedance Matching B. Lumped-Element Matching: choose d and YS to achieve a match at MM’.

Example 9: A load impedance 𝑍𝐿 = 25 − 𝑗50Ω is connected to a 50Ω T.L. Insert a shunt element to eliminate reflections towards the sending end of the line. Specify the insert location d (in wavelengths), the type of element and its value, given that 𝑓 = 100𝑀𝐻𝑧.

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4. Impedance Matching C. Single Stub Matching: choose d and length of stub l to achieve a match at MM’.

Example 10: Repeat Example 9 but use a shorted stub to match the load impedance.

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More Examples Example 11: A 50Ω lossless line 0.6 long is terminated in a load with 𝑍𝐿 = (50 + 𝑗25)Ω . At 0.3 from load, a resistor with resistance 𝑅 = 30Ω is connected as shown in following figure. Use the Smith Chart to find 𝑍𝑖𝑛 .

𝟎. 𝟏𝟗𝟒𝝀

𝒛𝑳 = 𝟏 + 𝒋𝟎. 𝟓

𝒚𝑨 = 𝟏. 𝟑𝟕 + 𝒋𝟎. 𝟒𝟓

𝒚𝑩 = 𝟑. 𝟎𝟒 + 𝒋𝟎. 𝟒𝟓 𝒛𝒊𝒏 = 𝟏. 𝟗 − 𝒋𝟏. 𝟒

𝟎. 𝟑𝟗𝟒𝝀 𝒁𝒊𝒏 = (𝟗𝟓 − 𝒋𝟕𝟎)𝜴 4/10/2017

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More Examples Example 12: Use the Smith Chart to find 𝑍𝑖𝑛 of the 50Ω feedline shown in following figure. 𝟎. 𝟎𝟖𝟖𝝀

𝒛𝟏 = 𝟏 + 𝒋

𝒚𝒊𝒏𝟏 = 𝟏. 𝟗𝟕 + 𝒋𝟏. 𝟎𝟐 𝒚𝒋𝒖𝒏𝒄 = 𝟑. 𝟗𝟒

𝒚𝒊𝒏𝟐 = 𝟏. 𝟗𝟕 − 𝒋𝟏. 𝟎𝟐 𝒛𝒊𝒏 = 𝟏. 𝟔𝟓 − 𝒋𝟏. 𝟕𝟗 𝒛𝟐 = 𝟏 − 𝒋

𝒁𝒊𝒏 = (𝟖𝟐. 𝟓 − 𝒋𝟖𝟗. 𝟓)𝜴 4/10/2017

𝟎. 𝟒𝟏𝟐𝝀

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More Examples Example 13: A 50Ω lossless line is to be matched to an antenna with 𝑍𝐿 = (75 − 𝑗20)Ω using a shorted stub. Use the Smith Chart to determine the stub length and distance between the antenna and stub. 𝟎. 𝟎𝟕𝟕𝝀 𝒚 = 𝒋𝟎. 𝟓𝟐 𝟎. 𝟎𝟒𝟏𝝀

𝒛𝟏 = 𝟏. 𝟓 − 𝒋𝟎. 𝟒

𝒛𝟏 = 𝟏. 𝟓 − 𝒋𝟎. 𝟒

𝒚 = −𝒋𝟎. 𝟓𝟐 𝒅𝟏 = 𝟎. 𝟏𝟎𝟒𝝀, 𝒍𝟏 = 𝟎. 𝟏𝟕𝟑𝝀 4/10/2017

𝟎. 𝟑𝟐𝟕𝝀 𝒅𝟐 = 𝟎. 𝟑𝟏𝟒𝝀, 𝒍𝟐 = 𝟎. 𝟑𝟐𝟕𝝀 49

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Homework Homework 1: A 6-m section of 150Ω lossless line is driven by a source with 𝑣𝑔 𝑡 = 5 cos 8𝜋 × 107 𝑡 − 300 (𝑉) And 𝑍𝑔 = 150Ω. If the line, which has a relative permittivity 𝜀𝑟 = 2.25 is terminated in a load 𝑍𝐿 = (150 − 𝑗50)Ω, find: a. 𝜆 on the line. Note that: 𝜆 = 𝑣𝑃

𝑓

where 𝑣𝑃 = 𝑐

𝜀𝑟 .

b. The reflection coefficient at the load. c. The input impedance. d. The input voltage Vi and time-domain voltage vi(t).

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Homework Homework 2: Two half-wave dipole antennas, each with impedance of 75Ω are connected in parallel through a pair of T.L. and the combination is connected to a feed T.L. as shown in the following figure. All lines are 50Ω lossless. a. Calculate 𝑍𝑖𝑛1 b. Calculate 𝑍𝑖𝑛 of the feed line.

Homework 3: A 50Ω lossless line is to be matched to an antenna with 𝑍𝐿 = (100 + 𝑗50)Ω using a shorted stub. Use the Smith Chart to determine the stub length and distance between the antenna and stub. Homework 4: Generate a plot of 𝑍0 as a function of strip width w from 0.05mm to 5mm for a microstrip line fabricated on a 0.7mm thick substrate with 𝜀𝑟 = 9.8. 51 4/10/2017

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Homework Homework 5: A 50Ω lossless line of length 𝑙 = 0.15𝜆 connects a 300MHz generator with 𝑉𝑔 = 300𝑉 and 𝑍𝑔 = 50Ω to a load 𝑍𝐿 = 75Ω.

a. Compute 𝑍𝑖𝑛 b. Compute 𝑉𝑖 and 𝐼𝑖 . 1

c. Compute the time-average power delivered to the line, 𝑃𝑖𝑛 = 2 ℝ𝑒 𝑉𝑖 𝐼𝑖 . d. Compute 𝑉𝐿 , 𝐼𝐿 and the time-average power delivered to the load, 𝑃𝐿 = 1 ℝ𝑒 𝑉𝐿 𝐼𝑙 . 2

e. Compute the time-average power delivered by the generator and time-average power dissipated by in 𝑍𝑔

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Additional Homework Homework 6: In addition to not dissipating power, a lossless line has two important features: (1) It is dispersionless (vp is independent of frequency). (2) Its characteristic impedance Z0 is real. Sometimes it is not possible to design a T.L. such that 𝑅′ ≪ 𝜔𝐿′ and 𝐺′ ≪ 𝜔𝐶′ but it is possible to choose the dimensions of the line and its material properties so as to satisfy the condition 𝑅’𝐶’ = 𝐿’𝐺’ (distortionless line). Such a line is called a distortionless line because despite the fact that it is not lossless, it nonetheless possesses the previous mentioned features of the lossless line. Show that for a distortionless line:

𝛼 = 𝑅′

𝐶′ 𝐿′

𝛽 = 𝜔 𝐿′ 𝐶′ 𝑍0 =

𝐿′ 𝐶′

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Additional Homework Homework 7: A 300Ω lossless line is connected to a complex load composed of a resistor 𝑅 = 600Ω and an inductor with 𝐿 = 0.02𝑚𝐻. At 10MHz, determine:

a. Reflection coefficient at load Γ𝐿 ? b. Voltage Standing Wave Ratio (VSWR). c. Location of voltage maximum nearest the load. d. Location of current maximum nearest the load. Homework 8: On a 150Ω lossless line, the following observations were noted: distance of first voltage minimum from load is 3cm, distance of first voltage maximum from load is 9cm and VSWR=9. Find 𝑍𝐿 ? Homework 9: A load with impedance 𝑍𝐿 = 25 − 𝑗50Ω is to be connected to a lossless T.L. with characteristic impedance 𝑍0 with chosen 𝑍0 such that the VSWR is the smallest possible. What should 𝑍0 be? 54 4/10/2017

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Additional Homework Homework 10: A voltage generator with 𝑣𝑔 𝑡 = 5 cos 2𝜋 × 109 𝑡

(𝑉)

and internal impedance is 𝑍𝑔 = 50Ω is connected to a 50Ω lossless T.L. The line length is 5cm and the line is terminated in a load with impedance 𝑍𝐿 = 100 − 𝑗100Ω. Determine: a. Reflection coefficient at load Γ𝐿 ?

b. 𝑍𝑖𝑛 at the input of the T.L. c. The input voltage 𝑣𝑖 𝑡 and input current 𝑖𝑖 𝑡 ? Homework 11: A 75Ω load is preceded by a 𝜆 4 section of 50Ω T.L which itself preceded by another 𝜆 4 section of 100Ω T.L. What is the input impedance?

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Additional Homework Homework 12: A 100MHz FM broadcast station uses a 300Ω T.L. between the transmitter and a tower-mounted half-wave dipole antenna. The antenna impedance is 73Ω. You are asked to design a quarter-wavelength transformer to match the antenna to the line. a. Determine the length and characteristic impedance of the quarter-wavelength section? b. If the quarter-wavelength is a two-wire line with 𝐷 = 2.5𝑐𝑚 and the wires are embedded in polystyrene with 𝜀𝑟 = 2.6. Determine the physical length of the quarter-wave section and the radius of the two wire conductor. Note that the characteristic parameters of T.Ls are given in the following table:

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Additional Homework Homework 13: Consider the circuit below. A generator with 𝑅0 = 75Ω is connected to a complex of 𝑍𝐿 = 100 + 𝑗100Ω through a T.L. of arbitrary length with 𝑍0 = 75Ω and 𝑣𝑃 = 0.8𝑐. Using the Smith Chart, evaluate the line for stub matching. The generator is operating at 100MHz. Find a. The electrical length of 𝜆 of the T.L. b. The normalized load impedance. c. The closest stub location as measured from the load. d. The length of the stub at the closest location. e. The lumped load element value that could take the place of the stub at the nearest location.

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Additional Homework Homework 14: A Vector Network Analyzer (VNA) is attached to the end of a lossless, 15m long T.L. (50Ω, 𝜖𝑟 = 2.3) operating at 220MHz. The VNA shows an input impedance of 𝑍𝑖𝑛 = 75 − 𝑗35Ω. Using the Smith Chart: a. Find the VSWR on the line. b. Find the normalized, denormalized and equivalent circuit of the load impedance 𝑍𝐿 at the far end of the line. The equivalent circuit must show the correct schematic symbols (L and/or R and/or C) and the values of each symbol. c. Find the normalized load admittance YL at the far end of the line. The length of the stub at the closest location. d. Find the distance in meters from the load to the first matching point. e. What is the normalized admittance at the first match point? f. Find the shortest stub to match the susceptance found at the first match point. Give the length of the stub in meters. g. If fabrication of a coaxial stub was not feasible but a lumped matching element was necessary, draw the component schematic symbol and give its value.

h. After the matching network is connected, where do standing waves exist and where 58 do they not exist in this system? What is the SWR at the input to the line? 4/10/2017

Cuong Huynh, Ph.D.Telecommunications Engineering DepartmentHCMUT

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