lecture -- standing waves · 2020. 5. 11. · 6. the standing wave is stationary. 7. sin( z) and...
TRANSCRIPT
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Electromagnetics:
Electromagnetic Field Theory
Standing Waves
Lecture Outline
• Standing Waves
• Standing Wave Ratio (SWR)
Slide 2
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Slide 3
Standing Waves
Two Counter‐Propagating Waves (1 of 2)
Slide 4
Suppose we have two counter‐propagating waves of equal amplitude travelling in opposite directions.
Observations:
1. Things are boring until the waves overlap.
2. Large fluctuations in amplitude are observed.
3. Locations of the fluctuations are stationary.
4. Total field is zero at some points.
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Two Counter‐Propagating Waves (2 of 2)
Slide 5
New Observations:
1. Fluctuations are smaller.2. Fluctuations do not go
to zero.
Suppose we have two counter‐propagating waves that do not have equal amplitude travelling in opposite directions.
General Expressions for Forward and Backward Waves
Slide 6
An incident wave will reflected from an interface.On the reflection side, there will exist two counterpropagating waves.
1
1
i 0,i
0,ii
1
ˆ
ˆ
zx
zy
E z E e a
EH z e a
1
1
r 0,r
0,rr
1
ˆ
ˆ
zx
zy
E z E e a
EH z e a
Incident Wave
Reflected Wave
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Wave Incident on Metal (1 of 2)
Slide 7
To more easily understand what happens on the reflection side, let the wave be incident from a lossless dielectric (i.e. = 0) onto metal (i.e. = ).
In this case, the material parameters are
1 2 0
The reflection and transmission coefficients are
2 1
2 1
2
2 1
01
0
2 2 00
0
r
t
In this case, we get zero transmission and 100% reflection with a 180° phase shift.
Wave Incident on Metal (2 of 2)
Slide 8
The propagation constant 1 is
1 1 1
2 2
1 1 1 1 11
1 1
2 2
1 1 1 1 11 1 1
1 1
01 1 1 1 0
2 2
01 1 1 1
2 2
j
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Revised Expressions for Our Waves
Slide 9
Given the reflection coefficient r and phase constant , we can rewrite our wave expressions as
1
1
i 0,i
0,ii
1
ˆ
ˆ
j zx
j zy
E z E e a
EH z e a
1
1
r 0,i
0,ir
1
ˆ
ˆ
j zx
j zy
E z rE e a
rEH z e a
Incident Wave
Reflected Wave
1 10,i
1
ˆj z j zy
Ee e a
1 10,i ˆj z j z
xE e e a
Frequency‐domain Standing Waves (r = -1)
Slide 10
On the reflection side, the total electromagnetic field is the sum of both the incident and reflected wave.
The expressions in parentheses containing complex exponentials are the sine and cosine functions. The equations for the total field become
1 0,i 1
0,i1 1
1
ˆ2 sin
2ˆcos
x
y
E z j E z a
EH z z a
Recall r = -1
1 i rE z E z E z
1 10,i 0,iˆ ˆ j z j z
x xE e a E e a
1 i rH z H z H z
1 10,i 0,i
1 1
ˆ ˆj z j zy y
E Ee a e a
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0,i1
1
ˆ2 cos cos y
Ez t a
0,i 1 ˆ2 sin sin xE z t a
Time‐Domain Standing Waves (r = -1)
Slide 11
Converting our standing wave equations to the time‐domain, we get
1 0,i 1 ˆ, Re 2 sin j txE z t j E z a e
0,i 1 ˆ2 sin Re j txE z a je
0,i 1 ˆ2 sin Re cos sinxE z a j t t
0,i1 1
1
ˆ, Re 2 cos j ty
EH z t z a e
0,i1
1
ˆ2 cos Re j ty
Ez a e
0,i1
1
ˆ2 cos Re cos siny
Ez a t j t
Standing Waves When r = +1
Slide 12
1 1
1 1
1 i r 0,i 0,i 1
0,i 0,i1 i r 1
1 1
ˆ ˆ2 cos
ˆ ˆ2 sin
j z j zx x
j z j zy x
E z E z E z E e e a E z a
E EH z H z H z e e a j z a
In the frequency‐domain, we have
In the time‐domain, we have
1 0,i 1
0,i1 1
1
ˆ, 2 cos cos
ˆ, 2 sin sin
x
y
E z t E z t a
EH z t z t a
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Visualizing the Standing Waves (1 of 2)
Slide 13
1 0,i 1 ˆ, 2 sin sin xE z t E z t a
0,i1 1
1
ˆ, 2 cos cos y
EH z t z t a
We will let r = -1 represent the case where 1 > 2.
Observations:
1. 180° phase shift after reflection.2. Max E and min H occur at the same points.3. Min E and max H occur at the same points.4. E is minimum at the interface and H is maximum.5. Nodes occur a half‐wavelength apart.6. The standing wave is stationary.7. sin(z) and cos(z) terms describe the envelope
of the standing wave.
Visualizing the Standing Waves (2 of 2)
Slide 14
1 0,i 1 ˆ, 2 cos cos xE z t E z t a
0,i1 1
1
ˆ, 2 sin sin y
EH z t z t a
We will let r = +1 represent the case where 1 < 2.
Observations:
1. 180° phase shift after reflection.2. Max E and min H occur at the same points.3. Min E and max H occur at the same points.4. E is maximum at the interface and H is
minimum.5. Nodes occur a half‐wavelength apart.6. The standing wave is stationary.7. sin(z) and cos(z) terms describe the envelope
of the standing wave.
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More Rigorous Visualization (1 of 2)
Slide 15
More Rigorous Visualization (2 of 2)
Slide 16
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Slide 17
Standing Wave Ratio (SWR)
Definition of Standing Wave Ratio (SWR)
Slide 18
We wish to have a metric to quantify the severity of the standing wave.
To do this, we define the standing wave ratio (SWR) as the maximum electric field observed in the standing wave divided by the minimum electric field observed in the standing wave.
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Derivation of Standing Wave Ratio (SWR)
Slide 19
Let’s examine our expression for the electric field when we have counter propagating waves.
1 11 i r 0,i ˆz z
xE z E z E z E e re a
This expression has the following maximum and minimum.
1 0,i
1 0,i
max 1
min 1
E E r
E E r
Substituting these into our definition of SWR gives
1 0,i
0,i1
max 1SWR
1min
E E r
E rE
1
SWR1
r
r
Derivation in Terms of Magnetic Field
Slide 20
Let’s examine our expression for the magnetic field when we have counter propagating waves.
1 10,i1 i r ˆz z
y
EH z H z H z e re a
This expression has the following maximum and minimum.
0,i1
0,i1
max 1
min 1
EH r
EH r
Dividing these shows that we get the same expression for SWR
0,i
1
0,i1
1max 1
1min 1
ErH r
E rH r
max
min
1 SWR
1
H r
rH
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SWR in Decibel Scale
Slide 21
Very often the SWR is given on a decibel scale.
dB 10SWR 20log SWR
dBSWR 20SWR 10
Given the SWR in dB, we can calculate the SWR on a linear scale.
Usefulness of SWR
Slide 22
The standing wave ratio (SWR) is something that we can directly measure. Given the SWR, we can calculate the magnitude of the reflection coefficient.
SWR 1
SWR 1r
1SWR
1
SWR SWR 1
SWR SWR 1
SWR 1 SWR 1
SWR 1
SWR 1
r
r
r r
r r
r
r
Derivation
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Notes About the SWR
• Since 0 |r| 1, we conclude that 1 SWR .• SWR is very large when the reflection is very strong.• SWR = 1 (SWRdB = 0)• Zero standing wave• |r| = 0• No backward wave.
• SWR = (SWRdB = )• Does NOT imply infinite amplitude standing wave• Standing wave has a perfect null (amplitude goes to zero)• |r| = 1• Forward and backward waves have equal amplitude.
Slide 23
Example
Slide 24
Suppose we have a wave inside of a 50 medium that is incident onto a second medium with impedance 120 . What fraction of power is reflected? What is the standing wave ratio (SWR)?
SolutionThe reflection coefficient at the interface is
2 1
2 1
120 50 700.4118
120 50 170r
The SWR is
1 1 0.4118 1.4118SWR 2.4
1 1 0.4118 0.5882
r
r
The fraction of power reflected is the reflectance. 2 2
0.4118 0.1696 16.96%R r
dB 10 10SWR 20log SWR 20log 2.4 7.6 dB