voltage standing wave ratio pattern
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Transmission Lines
© Amanogawa, 2006 -–Digital Maestro Series 125
Standing Wave Patterns
In practical applications it is very convenient to plot the magnitudeof phasor voltage and phasor current along the transmission line.These are the standing wave patterns:
( )
( )0
(d) 1 (d)
(d) 1 (d)
V V
V I
Z
+
+
= ⋅ + Γ
= ⋅ − Γ
Loss - less line
( ))
( ))
d
d
0
(d) 1 d
(d) 1 d
V V e
V e I
Z
+ α
+ α
= ⋅
+ Γ
= ⋅ − ΓLossy line
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Transmission Lines
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The standing wave patterns provide the top envelopes that bound
the time-oscillations of voltage and current along the line. In other words, the standing wave patterns provide the maximum valuesthat voltage and current can ever establish at each location of thetransmission line for given load and generator, due to theinterference of incident and refelected wave.
The patterns present a succession of maxima and minima which
repeat in space with a period of length λ /2, due to constructive or
destructive interference between forward and reflected waves. The
patterns for a loss-less line are exactly periodic in space, repeatingwith a λ /2 period.
Again, note that although we talk about maxima and minima of thestanding wave pattern we are always examining a maximum of voltage or current that can be achieved at a transmission linelocation during any period of oscillation.
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Transmission Lines
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We limit now our discussion to the loss-less transmission line case
where the generalized reflection coefficient varies as
( ) ( ) ( )(d) exp 2 d exp exp 2 d R R j j j = Γ − β = Γ φ − β
Note that the magnitude of an exponential with imaginary argumentis always unity
( ) ( )exp exp 2 d 1 j j − β =
In a loss-less line it is always true that, for any line location,
(d) R= Γ
When d increases, moving from load to generator, the generalized
reflection coefficient on the complex plane moves clockwise on a
circle with radius |Γ R| and is identified by the angle φ - 2β d .
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The voltage standing wave pattern has a maximum at locations
where the generalized reflection coefficient is real and positive
( ) ( )
(d)
exp exp 2 d 1 2 d 2
R
j j n
Γ = Γ
φ − β = ⇒ φ− β = π
At these locations we have
( )max max
1 (d) 1
(d ) 1
R
RV V V +
+ Γ = + Γ
⇒ = = ⋅ + Γ
The phase angle φ - 2β d changes by an amount 2π, when moving
from one maximum to the next. This corresponds to a distance
between successive maxima of λ /2.
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The voltage standing wave pattern has a minimum at locations
where the generalized reflection coefficient is real and negative
( ) ( ) ( )
(d)
exp exp 2 d 1 2 d 2 1
R
j j n
Γ = − Γ
φ − β = − ⇒ φ− β = + π
At these locations we have
( )min min
1 (d) 1
(d ) 1
R
RV V V +
+ Γ = − Γ
⇒ = = ⋅ − Γ
Also when moving from one minimum to the next, the phase angle
φ - 2β d changes by an amount 2π. This again corresponds to a
distance between successive minima of λ /2.
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The voltage standing wave pattern provides immediate information
on the transmission line circuit
If the load is matched to the transmission line ( Z R = Z 0 ) the
voltage standing wave pattern is flat, with value | V+ |.
If the load is real and Z R > Z 0 , the voltage standing wave
pattern starts with a maximum at the load.
If the load is real and Z R < Z 0 , the voltage standing wave
pattern starts with a minimum at the load.
If the load is complex and Im(Z R ) > 0 (inductive reactance),
the voltage standing wave pattern initially increases whenmoving from load to generator and reaches a maximum first.
If the load is complex and Im(Z R ) < 0 (capacitive reactance),
the voltage standing wave pattern initially decreases whenmoving from load to generator and reaches a minimum first.
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Since in all possible cases
( )d 1≤
the voltage standing wave pattern
( )(d) 1 (d)V V += ⋅ + Γ
cannot exceed the value 2 | V+ | in a loss-less transmission line.
If the load is a short circuit, an open circuit, or a pure reactance,there is total reflection with
( )d 1=
since the load cannot consume any power. The voltage standingwave pattern in these cases is characterized by
max min2 0V V V += =and .
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The quantity 1 + Γ(d) is in general a complex number, that can be
constructed as a vector on the complex plane. The number 1 is
represented as 1 + j0 on the complex plane, and it is just a vector
with coordinates (1,0) positioned on the Real axis. The reflection
coefficient Γ(d) is a complex number such that |Γ(d)| ≤ 1.
Im
Re
1
Γ1+Γ
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We can use a geometric construction to visualize the behavior of
the voltage standing wave pattern
( )(d) 1 (d)V V += ⋅ + Γ
simply by looking at a vector plot of |(1 + Γ (d))| . |V +| is just a
scaling factor, fixed by the generator. For convenience, we placethe reference of the complex plane representing the reflection
coefficient in correspondence of the tip of the vector (1, 0).
1+ΓR φ
Im( Γ )
Re ( Γ )1
ΓR
Example: Loadwith inductivereactance
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( ) maxd 2 d 0Γ = φ − β =
( ) mind 2 dΓ = φ − β = −π
1+Γ(d)
1
ΓR
φ
Im( Γ )
Re ( Γ )
Maximum of voltage standing
wave pattern
Γ d
2 β dmax
1+Γ(d)
1
φ
Im( Γ )
Re ( Γ )
ΓR
Minimum of voltage standing
wave pattern
Γ(d)
2 β dmin
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The voltage standing wave ratio (VSWR) is an indicator of load
matching which is widely used in engineering applications
max
min
1
1
R
R
V VSWR
V
+ Γ= =
− Γ
When the load is perfectly matched to the transmission line
0 1 R VSWR= ⇒ =
When the load is a short circuit, an open circuit or a pure reactance
1 R VSWR= ⇒ →∞
We have the following useful relation
1
1 R
VSWR
VSWR
−Γ =
+
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Im Γ
Re Γ
The load reflection coefficientis in this part of the domain
1
Maxima and minima of the voltage standing wave pattern.
Load with inductive reactance
( ) ( ) 0
0
Im 0 Im Im 0 R R R
R
Z Z Z
Z Z
> ⇒ Γ = > +
The first maximum of the voltage standing wave pattern is closest
to the load, at location
( ) max maxd 2 d 0 d4
φ∠Γ = φ − β = ⇒ = λ
π
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Im(Γ)
Re(Γ)
The load reflection coefficient
is in this part of the domain 1
Load with capacitive reactance
( ) ( ) 0
0
Im 0 Im Im 0 R R R
R
Z Z Z
Z Z < ⇒ Γ = < +
The first minimum of the voltage standing wave pattern is closest tothe load, at location
( ) min mind 2 d d4
π − φ∠Γ = φ − β = π ⇒ = λ
π
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A measurement of the voltage standing wave pattern provides the
locations of the first voltage maximum and of the first voltageminimum with respect to the load.
The ratio of the voltage magnitude at these points gives directly thevoltage standing wave ratio (VSWR).
This information is sufficient to determine the load impedance Z R ,
if the characteristic impedance of the transmission line Z 0 is
known.
STEP 1: The VSWR provides the magnitude of the loadreflection coefficient
1
1 R
VSWR
VSWR
−Γ =+
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STEP 2: The distance from the load of the first maximum or
minimum gives the phase φ of the load reflection coefficient.
For an inductive reactance, a voltagemaximum is closest to the load and
max max4
2 d dφ = β =λ
For a capacitive reactance, a voltageminimum is closest to the load and
min min42 d dφ = −π + β = −π +λ
V
V
dmax 0
|V |
dmin 0
|V |V max
V min
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STEP 3: The load impedance is obtained by inverting the
expression for the reflection coefficient
( )
( )( )
00
0
exp
1 exp1 exp
R R R R
R R
R
Z Z j
Z Z
j Z Z j
−Γ = Γ φ = +
+ Γ φ⇒ =− Γ φ