¡Descarga Análisis de ondas electromagnéticas en medios guiados: Transmisiones eléctricas y más Ejercicios en PDF de Electrónica solo en Docsity! Task 3 - Electromagnetic waves in guided media Individual work format EDUARD RICARDO CARRASQUILLA Rodriguez Group 203058_27 1101206091 UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA UNAD Teoría Electromagnética y Ondas 2022 16-01 Activity Answers: (write with your own words) 1. What do you understand by transmission line? A transmission line is a material structure of uniform geometry used to efficiently transport radiofrequency energy from one point to another, such as from one transmission equipment to another, from a transmitter to the antenna, among other applications. 2. Define the following electrical parameters of transmission lines: a. Input impedance 𝑍𝑖𝑛. — This The input impedance of an electrical network is the equivalent impedance "seen" by a power source connected to that network. If the source delivers a known value of voltage or current, such impedance can be calculated using Ohm's law. The input impedance is the Thevenin equivalent circuit of an electrical network, modeled by a combination of RL (resistance-inductance) or RC (resistance- capacitance), with equivalent values that would result in the same response as that of the network. It is also called Z11 in terms of Z- parameters. Roughly speaking, the exact definition depends on the field of study. b. Stationary wave ratio 𝑉𝑆𝑊𝑅: VSWR stands for Voltage Standing Wave Ratio, which refers specifically to the behavior of voltage (minimum and maximum) in a standing wave phenomenon between a transmission line and its end load. It is called in Spanish ROE Standing Wave Ratio, and it is implicit that it is the ratio (geometric ratio) between the maximum voltage and the existing minimum voltage. Exercises development Penetration depth of the wave in the conductor: 1 1 𝛿𝑝 = = √𝜋𝑓𝜎𝑐𝜇𝑐√𝜋 ∗ 037𝑥103𝐻𝑧 ∗ 4.3𝑥106𝑆𝑚/𝑚 ∗ 1.2566x10−6 T m/A 𝛿𝑝 = 12.618𝑥10−4𝑚 = 1261.802 𝜇𝑚 Having that in a coaxial line the parameters are classified according to the frequency: Considering the initial data: 𝑎 = 0.5𝑚𝑚 = 500 𝜇𝑚 𝑏 = 5𝑚𝑚 𝑡 = 37 𝜇𝑚 𝛿𝑝 > 𝑎, we have low media frequency, so: Resistance: 𝑅 = 1 𝜋𝜎𝑐 ( 1 𝑎2 + 1) 2𝑏𝑡 𝑅 = 1 𝜋∗4.3𝑥106𝑆𝑚/𝑚 ∗ ( 1 0.5𝑥10−4 𝑚 + 1 ) 2∗5𝑥10−2𝑚∗37𝑥10−5𝑚 𝑅 = 29.612 𝗇 Parallel conductance G: 𝐺 = 2𝜋𝜎𝑑 𝑏 ln ( 𝑎 ) Inductance L: 𝐺 = 2𝜋 ∗ 1𝑥10−10𝑆𝑚/𝑚 ln ( 5𝑥10 −2𝑚 ) 0.5𝑥10−4 𝑚 = 9.1𝑥10−11 𝑆 𝑚 𝑚 𝑎 𝐿 = 1.2566x10−6 T m/A (1 + ln ( 2𝜋 5𝑥10−2𝑚 −4 )) 𝐿 = 1.581𝑥10−6 𝐻 𝑚 0.5𝑥10 𝑚 Parallel capacitance C: 𝐶 = 2𝜋𝜀 𝑏 ln ( 𝑎 ) 𝜀 = 𝜀𝑟𝜀𝑜 = 2.1 ∗ 8.8542x10−12 𝐶2/N𝑚2 = 1.86𝑥10−11𝐶2/𝑁𝑚2 For point b: 𝐶 = 2𝜋 ∗ 2.1 ∗ 8.8542x10−12 𝐶2/N𝑚2 ln (5𝑥10−2𝑚 ) 0.5𝑥10−4 𝑚 = 1.69𝑥10−11 𝐹 𝑚 Propagation constant: 𝛾 = √(𝑅 + 𝑗𝜔𝐿)(𝐺 + 𝑗𝜔𝐶) 𝜔 = 2𝜋𝑓 = 2𝜋 ∗ 037𝑥103𝐻𝑧 = 23.247𝑥104 𝑟𝑎𝑑/𝑠 Ω 𝑟𝑎𝑑 𝐻 𝑆𝑚 𝑟𝑎𝑑 𝐹 𝛾 = √(29.612 + 𝑗 ∗ 23.247𝑥104 ∗ 1.581𝑥10−6 ) (9.1𝑥10−11 + 𝑗 ∗ 23.247𝑥104 ∗ 1.69𝑥10−11 ) 𝑚 𝑠 𝑚 𝑚 𝑠 𝑚 35 − 𝑗45 Ω + 𝑗50Ω ∗ 𝑇𝑎𝑛 ( ∗ 27 𝑚) 0.037 𝑚 Figure 2: Graphic representation of the transmission line. Exercise solution: Input impedance: 𝐿 = 27 𝑚 𝜆 = 037 𝑚𝑚 = 0.037 𝑚 2𝜋 𝑍𝑖𝑛 = 𝑍𝑜 𝑍𝐿 + 𝑗𝑍𝑜𝑇𝑎𝑛 ( 𝜆 𝐿) 2𝜋 𝑍𝑜 + 𝑗𝑍𝐿𝑇𝑎𝑛 ( 𝜆 𝐿) ( ) 2𝜋 𝑍𝑖𝑛 = 50Ω 0.037 𝑚 = 22.996 + 27.371𝑗Ω 50Ω + 𝑗(35 − 𝑗45)Ω ∗ 𝑇𝑎𝑛 ( 2𝜋 ∗ 27 𝑚) Reflection coefficient: 𝑍𝐿 − 𝑍𝑜 (35 − 𝑗45) Ω − 50Ω Γ = 𝑍𝐿 + 𝑍𝑜 = (35 − 𝑗45) Ω + 50Ω = 0.0810 − 𝑗0.486 VSWR: |Γ| = |0.0810 − 𝑗0.486| = 0.4931 arg(Γ) = −80.7° 𝑉𝑆𝑊𝑅 = 1 + | Γ| 1 − | Γ| 1 + 0.4931 = 1 − 0.4931 = 2.9 ll
Sciences, Technology
Al IZ
L=27
lamda = 0.047
Zo=50
Zi =35—i¡-45
= Das
ZI+iZo tg( 2 1)
Zin = lo 70 Mira
"OS oia D)
Universidad Na
Abierta y a Distancia
= 22006U614196461 + 27-37173722456941
¡ido
— zZl4 Zo
= 0.0810810810811 - 0.4864864864865
In
= 0.4031060619161
arg(r)
= -D0.53TOTTTOLOTAR
1+Ir
VSMR
1 rl
= 2463062564925
MODE 2: The results are shown for a Zo=500, ZL=(35-45i)0, L=25m and A=0.43m
Transmission Line
Features
Input impedance
Reflection
coefficient
VSWR
Characteristic impedance of the line
Load impedance
Physical length
Wavelength
58,139535| A
16.97155+0.38715i N
0.08108-0.48649¡
0.4932<-80.537870
2,94633
Electrical length
Input impedance
Reflection coefficient
Voltage Standing Wave Ratio
Interpretation: 𝑍𝑖𝑛 = 22.996 + 𝑗27.371 is the entrance impedance and is the ratio of total voltage to total current. Γ = 0.0810 − 𝑗0.486 is the reflection coefficient and describes the amplitude of the reflected wave relative to the incident wave. VSWR= 2.946 is called the standing wave ratio and is the ratio of the maximum voltage to the minimum voltage within the line. 3. Bearing in mind that Smith's letter is used to determine parameters of the transmission lines, use the "Smith 4.1" software to check the results obtained in point 2. a. Input impedance 𝑍𝑖𝑛. b. Reflection coefficient Γ. c. VSWR. Determining the electrical length: � � 𝑃 = � � 27 𝑚 = 0.037 𝑚 = 729 And entering the data in the software we obtain the parameter to compare with the ones in point 2:
