Relationship to the reflection coefficient Standing wave ratio
incident wave (blue) reflected (red wave) out of phase @ short-circuited end of transmission line, creating net voltage (black) standing wave. Γ = −1, swr = ∞.
standing waves on transmission line, net voltage shown in different colors during 1 period of oscillation. incoming wave left (amplitude = 1) partially reflected (top bottom) Γ = 0.6, −0.333, , 0.8 ∠60°. resulting swr = 4, 2, 9.
the voltage component of standing wave in uniform transmission line consists of forward wave (with complex amplitude
v
f
{\displaystyle v_{f}}
) superimposed on reflected wave (with complex amplitude
v
r
{\displaystyle v_{r}}
).
a wave partly reflected when transmission line terminated other pure resistance equal characteristic impedance. reflection coefficient
Γ
{\displaystyle \gamma }
defined thus:
Γ
=
v
r
v
f
.
{\displaystyle \gamma ={\frac {v_{r}}{v_{f}}}.}
Γ
{\displaystyle \gamma }
complex number describes both magnitude , phase shift of reflection. simplest cases
Γ
{\displaystyle \gamma }
measured @ load are:
Γ
=
−
1
{\displaystyle \gamma =-1}
: complete negative reflection, when line short-circuited,
Γ
=
0
{\displaystyle \gamma =0}
: no reflection, when line matched,
Γ
=
+
1
{\displaystyle \gamma =+1}
: complete positive reflection, when line open-circuited.
the swr directly corresponds magnitude of
Γ
{\displaystyle \gamma }
.
at points along line forward , reflected waves interfere constructively, in phase, resulting amplitude
v
max
{\displaystyle v_{\text{max}}}
given sum of waves amplitudes:
|
v
max
|
=
|
v
f
|
+
|
v
r
|
=
|
v
f
|
+
|
Γ
v
f
|
=
(
1
+
|
Γ
|
)
|
v
f
|
.
{\displaystyle {\begin{aligned}|v_{\text{max}}|&=|v_{f}|+|v_{r}|\\&=|v_{f}|+|\gamma v_{f}|\\&=(1+|\gamma |)|v_{f}|.\end{aligned}}}
at other points, waves interfere 180° out of phase amplitudes partially cancelling:
|
v
min
|
=
|
v
f
|
−
|
v
r
|
=
|
v
f
|
−
|
Γ
v
f
|
=
(
1
−
|
Γ
|
)
|
v
f
|
.
{\displaystyle {\begin{aligned}|v_{\text{min}}|&=|v_{f}|-|v_{r}|\\&=|v_{f}|-|\gamma v_{f}|\\&=(1-|\gamma |)|v_{f}|.\end{aligned}}}
the voltage standing wave ratio then
vswr
=
|
v
max
|
|
v
min
|
=
1
+
|
Γ
|
1
−
|
Γ
|
.
{\displaystyle {\text{vswr}}={\frac {|v_{\text{max}}|}{|v_{\text{min}}|}}={\frac {1+|\gamma |}{1-|\gamma |}}.}
since magnitude of
Γ
{\displaystyle \gamma }
falls in range [0,1], swr greater or equal unity. note phase of vf , vr vary along transmission line in opposite directions each other. therefore, complex-valued reflection coefficient
Γ
{\displaystyle \gamma }
varies well, in phase. swr dependent on complex magnitude of
Γ
{\displaystyle \gamma }
, can seen swr measured @ point along transmission line (neglecting transmission line losses) obtains identical reading.
since power of forward , reflected waves proportional square of voltage components due each wave, swr can expressed in terms of forward , reflected power:
swr
=
1
+
p
r
/
p
f
1
−
p
r
/
p
f
.
{\displaystyle {\text{swr}}={\frac {1+{\sqrt {p_{r}/p_{f}}}}{1-{\sqrt {p_{r}/p_{f}}}}}.}
by sampling complex voltage , current @ point of insertion, swr meter able compute effective forward , reflected voltages on transmission line characteristic impedance swr meter has been designed. since forward , reflected power related square of forward , reflected voltages, swr meters display forward , reflected power.
in special case of load rl, purely resistive unequal characteristic impedance of transmission line z0, swr given ratio:
swr
=
(
r
l
z
0
)
±
1
{\displaystyle {\text{swr}}=\left({\frac {r_{\text{l}}}{z_{\text{0}}}}\right)^{\pm 1}}
with ±1 chosen obtain value greater unity.
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