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Fields and Waves I
Lecture 2Sine Waves on Transmission Lines
K. A. ConnorElectrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
4 September 2006 Fields and Waves I 2
These Slides Were Prepared by Prof. Kenneth A. Connor Using Original Materials Written Mostly by the Following:
Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic Institute, Troy, NYJ. Darryl Michael – GE Global Research Center, Niskayuna, NY Thomas P. Crowley – National Institute of Standards and Technology, Boulder, COSheppard J. Salon – ECSE Department, Rensselaer Polytechnic Institute, Troy, NYLale Ergene – ITU Informatics Institute, Istanbul, TurkeyJeffrey Braunstein – Chung-Ang University, Seoul, Korea
Materials from other sources are referenced where they are used.Those listed as Ulaby are figures from Ulaby’s textbook.
4 September 2006 Fields and Waves I 3
Overview
ReviewVoltages and Currents on Transmission LinesStanding WavesInput ImpedanceLossy Transmission LinesLow Loss Transmission Lines
Henry Farny Song of the Talking Wire
4 September 2006 Fields and Waves I 4
Transmission Line Representation
As 0z ⇒Δ limit
)z(V
I
)zz(V Δ+
tIL)z(V)zz(V∂∂
−=−+ Δ
zl Δ⋅zV
zV
tIl
∂∂
=ΔΔ
=∂∂⋅−
4 September 2006 Fields and Waves I 5
Transmission Line Representation
Similarly,tVc
zI
∂∂⋅−=
∂∂
2
2
2
2
tVlc)
zI(
tl)
tIl(
zzV
∂∂
=∂∂
∂∂
⋅−=∂∂⋅−
∂∂
=∂∂
Obtain the following PDE:2
2
2
2
tVlc
zV
∂∂
=∂∂
Solutions are: )uzt(f ±
These are functions that move with
velocity u
tIl
zV
∂∂⋅−=
∂∂looks like
4 September 2006 Fields and Waves I 6
Transmission Line Representation
Example:
Functions that move with velocity u
)zu
tcos( ωω ±
At t=0, )zu
cos( ω−
Wave moving to the right
At ωt =1 )zu
1cos( ω− ωt =0
ωt =1
zuω
4 September 2006 Fields and Waves I 7
Workspace – look at the general form of the solution
4 September 2006 Fields and Waves I 8
Some Numerical Experiments
PSpice can be used to do simple numerical experiments that demonstrate how transmission lines work
VVT1
0
R2
50
0
V1
FREQ = 1MegHzVAMPL = 10VOFF = 0
R1
50
Simple T-Line
4 September 2006 Fields and Waves I 9
PSpice
VVT1
0
R2
50
0
V1
FREQ = 1MegHzVAMPL = 10VOFF = 0
R1
50
Time
2.0us 2.5us 3.0us 3.5us 4.0us 4.5us 5.0usV(R1:2) V(T1:B+)
-5.0V
0V
5.0V
INPUT OUTPUT
t
4 September 2006 Fields and Waves I 10
Computer-Based Tools
When you use a program like PSpice, applets, or any handy tools available online … remain skeptical.Do not assume that the answers are correct. Apply crude plausibility checks.Know the assumptions and limitations of the tools you are using.Test all tools on problems you can solve other ways or with tools you have already tested.Use even sometimes incorrect tools as long as errors are recognized.
Pig from http://www.cincinnatiskeptics.org
4 September 2006 Fields and Waves I 11
PSpice Example
Let us return to the configuration shown above and simulate it using PSpiceList some conclusions from this exercise.• ?• ?• ?
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Sine Waves
The form of the wave solution
First check to see that these solutions have the properties we expect by plotting them using a tool like Matlab
( )A tu
z A t zcos cosω ω ω βm m⎛⎝⎜
⎞⎠⎟=
4 September 2006 Fields and Waves I 13
Sine Waves
The positive wave
u
cos cosω ω π πtu
z ft fu
z−⎛⎝⎜
⎞⎠⎟= −⎛
⎝⎜⎞⎠⎟
2 2
Apply frequency and wavelength analogy argument to show this is reasonable
4 September 2006 Fields and Waves I 14
Solutions to the Wave Equation
Thus, our sine wave is a solution to the voltage or current equation
if or
u = the speed of wave propagation = the speed of light
∂∂
∂∂
2
2
2
2
Vz
lc Vt
=
β ω ω= =u
lc ulc
=1
4 September 2006 Fields and Waves I 15
Sine Waves
Consider one other property. What is the distance required to change the phase of this expression by ? We just did this qualitatively.
This distance is called the wavelength or
cos cosω ω π πtu
z ft fu
z−⎛⎝⎜
⎞⎠⎟= −⎛
⎝⎜⎞⎠⎟
2 2
2π
β ω π πzu
z fu
z= = =2 2
βλ ω λ π λ π= = =u
fu
2 2 λ πβ
= =2 u
f
4 September 2006 Fields and Waves I 16
Sine Waves -- Summary
Solutions look like
β ω ω ω με πλ
= = = =u
lc 2
( )A t zcos ω βm
ω π π= =2 2f
Tu
lc= =
1 1με
λ πβ
= =2 u
fε ε ε= r o μ μ μ= r o
Figure from http://www.emc.maricopa.edu/
4 September 2006 Fields and Waves I 17
Phasor Notation
For ease of analysis (changes second order partial differential equation into a second order ordinary differential equation), we use phasor notation.
The term in the brackets is the phasor.
( ) { }( )f z t A t z Ae ej z j t( , ) cos Re= =ω β β ωm m
f z Ae j z( ) = m β
4 September 2006 Fields and Waves I 18
Phasor Notation
To convert to space-time form from the phasor form, multiply by and take the real part.
If A is complex
e j tω
( )f z t Ae e A t zj z j t( , ) Re( ) cos= =m mβ ω ω β
A Ae j A= θ
( )f z t Ae e e A t zj j z j tA
A( , ) Re( ) cos= = +θ β ω ω β θm m
4 September 2006 Fields and Waves I 19
Transmission Lines
Incident Wave
Reflected Wave
Mismatched load
Standing wave due to interference
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Standing Waves
Besser AssociatesNo Standing Wave
http://www.bessernet.com
4 September 2006 Fields and Waves I 21
Standing Waves
Besser AssociatesStanding Wave
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Standing Waves
Besser Associates
This may be wrong
We will see shortly
Standing Wave
4 September 2006 Fields and Waves I 23
Standing Waves
Besser Associates
This may be wrong
We will see shortly
Standing Wave
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Standing Waves
From Agilent
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Transmission Lines - Standing Wave Derivation
Phasor Form of the Wave Equation:
VclzV
tVcl
zV
22
2
2
2
2
2
⋅⋅⋅−=∂∂
⇒
∂∂⋅⋅=
∂∂
ω
where:
zjeVV ⋅⋅±⋅= βm
General Solution: zjzj eVeVV ⋅⋅+−⋅⋅−+ += ββ
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Workspace
4 September 2006 Fields and Waves I 27
Workspace
4 September 2006 Fields and Waves I 28
Transmission Lines - Standing Wave Derivation
zjzj eVeVV ⋅⋅+−⋅⋅−+ += ββ
Forward Wave
)ztcos( ⋅−⋅ βω
Backward Wave
)ztcos( ⋅+⋅ βω
Vmax occurs when Forward and Backward Waves are in Phase
Vmin occurs when Forward and Backward Waves are out of Phase
CONSTRUCTIVE INTERFERENCE
DESTRUCTIVE INTERFERENCE
TIME DOMAIN
4 September 2006 Fields and Waves I 29
Transmission Lines Formulas
Fields and Waves I Quiz Formula SheetIn the class notes
Note:
All are used in various handouts, texts, etc. There is no standard notation.
V V Vm++ += =
v z V e V ej z j z( ) = ++−
−+β β
i zV e V e
ZVZ
eVZ
ej z j z
o o
j z
o
j z( ) =−
= −+−
−+
+ − − +β β
β β
V V Vm−− −= =
4 September 2006 Fields and Waves I 30
RG58/U Cable
Assume 2 VP-P 1.5MHz sine wave is launched on such a line. Find and
Answers?
β ω ω ω με πλ
= = = =u
lc 2 λ
4 September 2006 Fields and Waves I 31
Reflection Coefficient Derivation
Define the Reflection Coefficient:
+− ⋅= mLm VV Γ
Maximum Amplitude when in Phase:−+ += mmmax VVV
)1(VV Lmmax Γ+⋅=∴ +
Similarly: )1(VV Lmmin Γ−⋅= +
Standing Wave Ratio (SWR) =L
L
min
max
11
VV
ΓΓ
−+
=
4 September 2006 Fields and Waves I 32
Transmission Lines - Standing Wave Derivation
Distance between Max and Min is λ/2
Forward Phase is = zj ⋅⋅− β
Backward Phase is = zj ⋅⋅+ β
Difference in Phase is = zj2 ⋅⋅⋅− β
Varies by 2π (distance between maxima)
Show this
πΔβ ⋅=⋅⋅∴ 2z2 22z λ
λππ
βπΔ =
⋅==
V ±Assume are real (will be complex if the load is complex)
4 September 2006 Fields and Waves I 33
Reflection Coefficient Derivation
Let z=0 at the LOAD
)1(
00
L
jjload
V
VV
eVeVV
Γ+⋅=
+=
⋅+⋅=⇒
+
−+
⋅⋅+−⋅⋅−+ ββ
Need a relationship between current and voltage:
tIl
zV
∂∂⋅−=
∂∂ Ilj
zV
⋅⋅⋅−=∂∂
⇒ ω
4 September 2006 Fields and Waves I 34
Reflection Coefficient Derivation
zV
lj1I
∂∂⋅
⋅⋅−=
ω
)(1 zjzj eVjeVjlj
⋅⋅+−⋅⋅−+ ⋅⋅⋅+⋅⋅⋅−⋅⋅⋅
−= ββ ββω
)( zjzj eVeVl
⋅⋅+−⋅⋅−+ ⋅−⋅⋅⋅
= ββ
ωβ
)( zjzj eVeVl
clI ⋅⋅+−⋅⋅−+ ⋅−⋅⋅⋅⋅⋅
=∴ ββ
ωω
zjzj e
cl
Ve
cl
V ⋅⋅+−
⋅⋅−+
⋅−⋅= ββ zjLzj e
cl
Ve
cl
V ⋅⋅++
⋅⋅−+
⋅Γ⋅
−⋅= ββ
4 September 2006 Fields and Waves I 35
Reflection Coefficient Derivation
At LOAD: LZIV=
Use derived terms of V and I at z=0 (position of the LOAD)
( ) LLL ZVV
cl
V
cl
V=⋅Γ+⋅
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛Γ⋅
− ++
−++
1
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
⋅=L
LL 1
1clZ
ΓΓ
OR0
0
ZZZZ
L
LL +
−=Γ
Note thatclZ0 =
4 September 2006 Fields and Waves I 36
Short Circuit Load
For ZL=0, we have ΓLL o
L o
o
o
Z ZZ Z
ZZ
=−+
=−+
= −00
1
( )v z V e V e V e ej zL
j z j z j z( ) = + = −+ − + + + − +β β β βΓ
e z j zj z+ = +β β βcos sine z j zj z− = −β β βcos sin
( )v z V j z( ) sin= − + 2 β
4 September 2006 Fields and Waves I 37
Short Circuit Load
Convert to space-time form
This is a standing wave
( ) ( )( )v z t v z e V j z ej t j t( , ) Re ( ) Re sin= = −+ω ωβ2
( )( ) ( )( )Re sin Re sin cos sin− = − −j z e z j z zj t2 2β β β βω
v z t V z t( , ) sin sin= +2 β ω
4 September 2006 Fields and Waves I 38
Short Circuit Load
What are the voltage maxima and minima?
Where are they?
The standing wave pattern is the envelope of this function.
v z t V z t( , ) sin sin= +2 β ω
4 September 2006 Fields and Waves I 39
Lumped Transmission Line
0
0
L1
1uH
L2
1uH
L3
1uH
L4
1uH
L5
1uH
L6
1uH
L7
1uH
L8
1uH
L9
1uH
L10
1uH
L11
1uH
L12
1uH
L13
1uH
L14
1uH
L15
1uH
L16
1uH
L17
1uH
L18
1uH
L19
1uH
L20
1uH
C1
390pF
C2
390pF
C3
390pF
C4
390pF
C5
390pF
C6
390pF
C7
390pF
C8
390pF
C9
390pF
C10
390pF
C11
390pF
C12
390pF
C13
390pF
C14
390pF
C15
390pF
C16
390pF
C17
390pF
C18
390pF
C19
390pF
C20
390pF
R1
50V1
R2
93
Function Generator
Load
1 2 3 4 5
15
15
5
20
10
4 September 2006 Fields and Waves I 40
Lumped Transmission Line
Input Not Shown
Both Outputs Shown
4 September 2006 Fields and Waves I 41
Lumped Transmission Line
In Out
1
15
0 5
Input is BNC Output is both BNC and Banana Plugs (for some loads)
4 September 2006 Fields and Waves I 42
Lumped Transmission Line Experiment
Treat the lumped version just like the reel of cable. (Connectors are opposite so you will need connector cables.) Monitor the output of the function generator on one channel Monitor the voltages on each node (one at a time) on the other channel. You can use just the signal (red) lead, since the ground (black) lead is connected through the other cables. Use the voltage cursors to obtain VP-P for each node. Record your values and plot with Excel, Matlab, etc.
4 September 2006 Fields and Waves I 43
Use PSpice to set up the standard transmission line, matched and notLook at the output for a variety of frequenciesSet up the lumped line in PSpice (more work) and repeatUse the lumped line model to show the standing wave patternWill there be any obvious differences between the physical and numerical experiments?
Lumped Transmission Line Numerical Experiment(Not required)
4 September 2006 Fields and Waves I 44
Workspace
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Workspace
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Workspace
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Workspace
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Workspace
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Workspace
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Workspace
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Workspace
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Workspace
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Workspace
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Workspace