Transmission Lines
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Transcript of Transmission Lines
Department of Electrical Engineering
School of Electrical Engineering and Computer Science (SEECS)
National University of Sciences & Technology (NUST)
Fall 2015
EE342 Microwave Engineering
Instructor : Dr. M. Umar [email protected]
Transmission Line Equation
© Umar, 2015.
Previous Lecture Solution of Transmission line equation
Propagation constant, characteristic impedance
Lossless transmission line
This time Distortionless line
Input Impedance, SWR, Power
EE342 Microwave Engineering 2
© Umar, 2015.
Transmission-Line Equation
Bridges the gap between circuit theory and field analysis
Important for analysis of microwave circuits and devices
EE342 Microwave Engineering 3
© Umar, 2015.
Transmission-Line Equation
By applying KVL and KCL, we get two second order differential equations𝜕2𝑉𝑠𝜕𝑧2
− 𝛾2𝑉𝑠 = 0,𝜕2𝐼𝑠𝜕𝑧2
− 𝛾2𝐼𝑠 = 0
Where
𝛾 = 𝛼 + 𝑗𝛽 = (𝑅 + 𝑗𝜔𝐿)(𝐺 + 𝑗𝜔𝐶) is the propagation constant
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© Umar, 2015.
Transmission-Line Equation
The solution of the differential equation gives𝑉 𝑧 = 𝑉𝑜
+𝑒−𝛾𝑧 + 𝑉𝑜−𝑒𝛾𝑧
and
I 𝑧 = 𝐼𝑜+𝑒−𝛾𝑧 + 𝐼𝑜
−𝑒𝛾𝑧
The characteristic impedance is the ratio of positively traveling voltage wave to the current wave at any point on the line
𝑍𝑜 =𝑅 + 𝑗𝜔𝐿
𝐺 + 𝑗𝜔𝐶
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© Umar, 2015.
Lossless Line
For a lossless line, R = G = 0
Under such conditions, attenuation constant is zero
The characteristic impedance is real
𝛼 = 0 , 𝛽 = 𝜔 𝐿𝐶
𝑍𝑜 =𝐿
𝐶
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© Umar, 2015.
Distortionless Line
The line whose attenuation constant is not a function of frequency and phase constant is a linear function of frequency
The conditions for distortionless transmission line is:𝑅
𝐿=𝐺
𝐶 For such line, the propagation constant is :
𝛼 = 𝑅𝐺 , 𝛽 = 𝜔 𝐿𝐶
The characteristic impedance is :
𝑍𝑜 =𝐿
𝐶
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© Umar, 2015.
Summary
EE342 Microwave Engineering 8
© Umar, 2015.
Example
A transmission line operating at 500 MHz has 𝑍𝑜 = 80Ω, α =
0.04𝑁𝑝
𝑚, 𝛽 = 1.5
𝑟𝑎𝑑
𝑚. Find the line parameters R, L, C, G.
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© Umar, 2015.
Input Impedance, Standing Wave Ratio, Power Consider a transmission line connected to a load. The line extend
from z=0 at the generator to z= l at the load.
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© Umar, 2015.
Input Impedance, Standing Wave Ratio, Power The voltage and current waves on the line are :
𝑉 𝑧 = 𝑉𝑜+𝑒−𝛾𝑧 + 𝑉𝑜
−𝑒𝛾𝑧
I 𝑧 =𝑉𝑜+
𝑍𝑜𝑒−𝛾𝑧 −
𝑉𝑜−
𝑍𝑜𝑒𝛾𝑧
The conditions at the input and output are:𝑉 𝑧 = 0 = 𝑉𝑜 , 𝐼 𝑧 = 0 = 𝐼𝑜
𝑉 𝑧 = 𝑙 = 𝑉𝐿 , 𝐼 𝑧 = 𝑙 = 𝐼𝐿
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© Umar, 2015.
Input Impedance, Standing Wave Ratio, Power Input impedance at any point on the line is :
𝑍𝑖𝑛 =𝑉(𝑧)
𝐼(𝑧)
At the generator, the equation is :
𝑍𝑖𝑛 =𝑍𝑜(𝑉𝑜
+ + 𝑉𝑜−)
𝑉𝑜+ − 𝑉𝑜
−
Substituting for the load end and simplification
𝑍𝑖𝑛 = 𝑍𝑜𝑍𝐿 + 𝑍𝑜tanh(𝛾𝑙)
𝑍𝑜 + 𝑍𝐿tanh(𝛾𝑙)
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© Umar, 2015.
Input Impedance, Standing Wave Ratio, Power For a lossless line, the input impedance is :
𝑍𝑖𝑛 = 𝑍𝑜𝑍𝐿 + 𝑗𝑍𝑜tan(𝛽𝑙)
𝑍𝑜 + 𝑗𝑍𝐿tan(𝛽𝑙)
The voltage reflection coefficient (Γ) is the ratio of voltage reflection wave to the incident wave. At the load, it is :
Γ𝐿 =𝑉𝑜−𝑒𝛾𝑙
𝑉𝑜+𝑒−𝛾𝑙
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© Umar, 2015.
Input Impedance, Standing Wave Ratio, Power The voltage reflection coefficient (Γ) at any point on the line is given
as:
Γ(𝑧) =𝑉𝑜−
𝑉𝑜+ 𝑒
2𝛾𝑧
The standing wave ratio (SWR) is defined as:
𝑆𝑊𝑅 =1 + Γ𝐿1 − Γ𝐿
EE342 Microwave Engineering 14
© Umar, 2015.
Input Impedance, Standing Wave Ratio, Power The average input power at a distance ‘l’ from the load is
𝑃𝑎𝑣𝑔 =1
2𝑅𝑒 𝑉(𝑙)𝐼∗(𝑙)
Solving the above equation, we get power in terms of reflection coefficient at the load.
𝑃𝑎𝑣𝑔 =𝑉𝑜+ 2
2𝑍𝑜1 − Γ𝐿
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© Umar, 2015.
Special Case
Shorted Line If line is shorted , 𝑍𝐿 = 0
Open-Circuit Line If line is shorted , 𝑍𝐿 = ∞
Matched Line If line is shorted , 𝑍𝐿 = 𝑍0
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