Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 1
Trinh Xuan Dung, PhD
Department of Telecommunications
Faculty of Electrical and Electronics Engineering
Ho Chi Minh city University of Technology
Chapter 2
Smith Chart and Impedance Matching
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering
Contents
2
1. Introduction
2. Smith Chart
3. Smith Chart Applications
4. Y Smith Chart
5. Impedance Matching
Problems
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering
1. Introduction
3
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.
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering
1. Introduction
4CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering
2. Smith Chart
5
We start from the general definition of reflection coefficient:
𝒁(𝒙)
𝛤 =𝑍 − 𝑍0𝑍 + 𝑍0
= 𝑍 𝑍0
− 1
𝑍 𝑍0+ 1
=𝑧 − 1
𝑧 + 1
Now z can be written as:
𝑧 =1 + 𝛤
1 − 𝛤=1 + Re 𝛤 + 𝑗 Im 𝛤
1 − Re 𝛤 − 𝑗 Im 𝛤=1 − Re2 𝛤 − Im2 𝛤 + 2𝑗 Im 𝛤
1 − Re 𝛤 2 + Im2 𝛤
𝑧 = 𝑟 + 𝑗𝑥where: . Then: 𝑟 =1 − Re2 𝛤 − Im2 𝛤
1 − Re 𝛤 2 + Im2 𝛤𝑥 =
2 Im 𝛤
1 − Re 𝛤 2 + Im2 𝛤
These equations can be re-arranged into:
Re 𝛤 −𝑟
1 + 𝑟
2
+ Im2 𝛤 =1
1 + 𝑟
2
Re 𝛤 − 1 2 + Im 𝛤 −1
𝑥
2
=1
𝑥
2
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 6
2. Smith Chart
Re 𝛤 −𝑟
1 + 𝑟
2
+ Im2 𝛤 =1
1 + 𝑟
2
: Resistance circles
Center: 𝑟
1+𝑟, 0
Radius: 1
1+𝑟
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 7
2. Smith Chart
: Reactance circles
Center: 1,1
𝑥
Radius: 1
𝑥
Re 𝛤 − 1 2 + Im 𝛤 −1
𝑥
2
=1
𝑥
2
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 8
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 9
2. Smith ChartFor 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 rincreases.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.
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
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 10
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 11
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 12
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 13
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 14
2. Smith Chart
Constant circle
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 15
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 16
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 17
2. Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 18
3. Smith Chart Applications
A. Given 𝑍 𝑑 , find Γ 𝑑 or Given Γ 𝑑 , find 𝑍 𝑑 .
B. Given Γ𝐿 𝑎𝑛𝑑 𝑍𝐿, find Γ 𝑑 and 𝑍 𝑑 .
Given Γ 𝑑 and 𝑍 𝑑 , find Γ𝑅 𝑎𝑛𝑑 𝑍𝑅.
C. Find dmax and dmin (maximum and minimum locations for the VSW pattern).
D. Find the VSWR.
E. Given 𝑍 𝑑 , find 𝑌 𝑑 or Given 𝑌 𝑑 , find 𝑍 𝑑 .
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 19
3. Smith Chart ApplicationsA. Given 𝒁 𝒅 , find 𝜞 𝒅
1. Normalize the impedance:
2. Find the circle of constant normalized resistance r.
3. Find the circle of constant normalized reactance x.
4. 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
𝒛 𝒅 =𝒁 𝒅
𝒁𝟎=
𝑹
𝒁𝟎+ 𝒋
𝑿
𝒁𝟎= 𝒓 + 𝒋𝒙
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 20
3. Smith Chart Applications
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 21
3. Smith Chart Applications
A. Given 𝜞 𝒅 , find 𝒁 𝒅
1. Determine the complex point representing the given reflection coefficient Γ 𝑑on 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:
4. The actual impedance is:
𝒛 𝒅 = 𝒓 + 𝒋𝒙
𝒁 𝒅 = 𝒛 𝒅 𝒁𝟎
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 22
3. Smith Chart ApplicationsB. 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:
1. Identify the load reflection coefficient Γ𝐿 and the normalized load impedance 𝑍𝐿on the Smith Chart.
2. Draw the circle of constant coefficient amplitude Γ 𝑑 = Γ𝐿
3. Starting from the point representing the load, travel on the circle in the
clockwise direction by an angle 𝜃 = 2𝛽𝑑.
4. 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
Γ 𝑑 = Γ𝐿𝑒−𝑗2𝛽𝑑 = Γ𝐿
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 23
3. Smith Chart Applications
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 24
3. Smith Chart Applications
Example 3: Find Γ 𝑑 and 𝑍 𝑑given 𝑍𝑅 = 100 − 𝑗50Ω , 𝑍0 =50Ω and 𝑑 = 0.1𝜆
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 25
3. Smith Chart ApplicationsC. Given 𝜞𝑳 𝒂𝒏𝒅 𝒁𝑳, find 𝒅𝒎𝒂𝒙 and 𝒅𝒎𝒊𝒏
1. Identify the load reflection coefficient Γ𝐿 and the normalized load impedance 𝑍𝐿on the Smith Chart.
2. Draw the circle of constant coefficient amplitude Γ 𝑑 = Γ𝐿
3. 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).
4. 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
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 26
3. Smith Chart Applications
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 27
3. Smith Chart ApplicationsD. Given 𝜞𝑳 𝒂𝒏𝒅 𝒁𝑳, find 𝑽𝑺𝑾𝑹
The VSWR is defined as:
The normalized impedance at the maximum location of the SW pattern is given by:
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.
𝑉𝑆𝑊𝑅 =𝑉𝑚𝑎𝑥
𝑉𝑚𝑖𝑛=1 + Γ𝐿1 − Γ𝐿
𝑧 𝑑𝑚𝑎𝑥 =1 + Γ 𝑑𝑚𝑎𝑥
1 − Γ 𝑑𝑚𝑎𝑥=1 + Γ𝐿1 − Γ𝐿
= 𝑉𝑆𝑊𝑅
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 28
3. Smith Chart ApplicationsE. Given 𝒁 𝒅 , find 𝒀 𝒅
The normalized impedance and admittance are defined as:
Since:
The actual values are given by:
Example 6: Find YL given 𝑍𝐿 = 25 ± 𝑗100Ω, 𝑍0 = 50Ω.
𝑧 𝑑 =1 + Γ 𝑑
1 − Γ 𝑑𝑦 𝑑 =
1 − Γ 𝑑
1 + Γ 𝑑
Γ 𝑑 +𝜆
4= −Γ 𝑑 → 𝑧 𝑑 +
𝜆
4= 𝑦 𝑑
𝑍 𝑑 +𝜆
4= 𝑍0𝑧 𝑑 +
𝜆
4𝑌 𝑑 +
𝜆
4= 𝑌0𝑦 𝑑 +
𝜆
4=𝑦 𝑑 +
𝜆4
𝑍0
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 29
3. Smith Chart Applications
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 30
3. Smith Chart: Y Smith Chart The reflection coefficient is written as: 𝛤 =
𝑧 − 1
𝑧 + 1=
1𝑦 − 1
1𝑦 + 1
= −𝑦 − 1
𝑦 + 1
𝛤 =𝑧 − 1
𝑧 + 1
−𝛤 =𝑦 − 1
𝑦 + 1
: Z Smith Chart
: Y Smith Chart
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 31
3. Smith Chart: Y Smith Chart
𝛤 =𝑧 − 1
𝑧 + 1
−𝛤 =𝑦 − 1
𝑦 + 1
: Z Smith Chart
: Y Smith Chart
Since related impedance and
admittance are on opposite
sides of the same Smith
Chart, the imaginary parts
always have different sign.
Numerically we have:
𝑧 = 𝑟 + 𝑗𝑥 𝑦 = 𝑔 + 𝑗𝑏 =1
𝑧
𝑔 =𝑟
𝑟2 + 𝑥2
𝑏 = −𝑥
𝑟2 + 𝑥2
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 32
3. Smith Chart Applications
C110p
R50
L
22.5nH
C212p
Z
Example 7: Find impedance of a
complex circuit using Smith Chart where
𝑅0 = 50Ω and 𝜛 = 109 𝑟𝑎𝑑/𝑠.
1
1
0
1/1 2RC
R j Cz j
R
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 33
3. Smith Chart Applications
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 34
3. Smith Chart Applications
Example 8: A 50Ω lossless
T.L. of length 3.3𝜆 is
terminated by a load impedance
𝑍𝐿 = (25 + 𝑗50)Ω.
a. Find Γ.
b. Find VSWR.
c. Find dmax and dmin.
d. Find Zin of T.L.
e. Find Yin.
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 35
4. Impedance Matching
Impedance Matching
Maximum power transfer
ZL
Impedance
Matching
Network
ZS
What are Applications ?
T.L. Amplifier Design PA, LNA Component Design Equipment Interfaces
Using lump elements
Using transmission lines
ADS Smith Chart tool
Matching with Lumped Elements Single-Stub Matching Networks Quarter-wave Transformer
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 36
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.
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 37
4. Impedance MatchingA. Quarter wavelength Transformer Matching:
In case of complex impedance:
𝑍𝐿 = 40 Ω𝑍0 = 50
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 38
4. Impedance MatchingExample:
𝑍0 =60
𝜀ln
8ℎ
𝑊+𝑊
4ℎ𝑓𝑜𝑟 𝑊 ≤ ℎ
A simple but accurate equation for
Microstrip Characteristic Impedance:
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 39
4. Impedance MatchingB. 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 = −𝑏𝑑CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 40
4. Impedance MatchingB. 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.
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 41
4. Impedance MatchingC. 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.
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 42
More ExamplesExample 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.
𝒁𝒊𝒏 = (𝟗𝟓 − 𝒋𝟕𝟎)𝜴
𝒛𝑳 = 𝟏 + 𝒋𝟎. 𝟓
𝒚𝑨 = 𝟏. 𝟑𝟕 + 𝒋𝟎. 𝟒𝟓
𝟎. 𝟏𝟗𝟒𝝀
𝟎. 𝟑𝟗𝟒𝝀
𝒚𝑩 = 𝟑. 𝟎𝟒 + 𝒋𝟎. 𝟒𝟓
𝒛𝒊𝒏 = 𝟏. 𝟗 − 𝒋𝟏. 𝟒
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 43
More ExamplesExample 12: Use the Smith Chart to find 𝑍𝑖𝑛 of the 50Ω feedline shown in following
figure.
𝒁𝒊𝒏 = (𝟖𝟐. 𝟓 − 𝒋𝟖𝟗. 𝟓)𝜴
𝒛𝟏 = 𝟏 + 𝒋
𝒛𝟐 = 𝟏 − 𝒋
𝟎. 𝟒𝟏𝟐𝝀
𝟎. 𝟎𝟖𝟖𝝀
𝒚𝒊𝒏𝟏 = 𝟏. 𝟗𝟕 + 𝒋𝟏. 𝟎𝟐
𝒚𝒊𝒏𝟐 = 𝟏. 𝟗𝟕 − 𝒋𝟏. 𝟎𝟐
𝒚𝒋𝒖𝒏𝒄 = 𝟑. 𝟗𝟒
𝒛𝒊𝒏 = 𝟏. 𝟔𝟓 − 𝒋𝟏. 𝟕𝟗
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering 44
More ExamplesExample 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.
𝒅𝟏 = 𝟎. 𝟏𝟎𝟒𝝀, 𝒍𝟏 = 𝟎. 𝟏𝟕𝟑𝝀 𝒅𝟐 = 𝟎. 𝟑𝟏𝟒𝝀, 𝒍𝟐 = 𝟎. 𝟑𝟐𝟕𝝀
𝒛𝟏 = 𝟏. 𝟓 − 𝒋𝟎. 𝟒
𝟎. 𝟎𝟒𝟏𝝀
𝒚 = −𝒋𝟎. 𝟓𝟐
𝒛𝟏 = 𝟏. 𝟓 − 𝒋𝟎. 𝟒
𝟎. 𝟎𝟕𝟕𝝀
𝟎. 𝟑𝟐𝟕𝝀
𝒚 = 𝒋𝟎. 𝟓𝟐
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com
HCMUT / 2019Dung Trinh, PhD
Dept. of Telecoms Engineering
Q&A
45CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu d
uong
than
cong
. com