SME
Engineering Electromagnetics
Transmission lines
Use this for “A 50 Ω line ends in a 100 Ω load. Find Γ and the VSWR.”
Diagram
Γ
=
RL − Z₀
RL + Z₀
=
100 − 50
100 + 50
=
0.3333
Vₘₐₓ
Vₘᵢₙ
=
1 + |Γ|
1 − |Γ|
=
1.333
0.6667
=
2,
the
VSWR
RL > Z₀ (Γ > 0): a maximum at the load; maxima repeat every λ/2.
This is an example. Type your own values to start a new problem.
Γ
=
RL (Ω)
−
Z₀ (Ω)
RL (Ω)
+
Z₀ (Ω)
VSWR
Standing-wave ratio
calculated
RL
Return loss
calculated · dB
share
Power reflected
calculated · %
Formulas
Γ
=
RL −
Z₀
RL +
Z₀
0.3333
=
100 − 50
100 + 50
VSWR
=
1 + |
Γ
|
1 − |
Γ
|
2
=
1 + |0.3333|
1 − |0.3333|
RL
=
−20
×
log₁₀(|
Γ
|)
9.5424 = −20 × log₁₀(|0.3333|)
share
=
100
×
Γ
2
11.1111
=
100
×
0.3333
2
Assumptions
•
A lossless line ending in a resistive load; complex loads are on the input-impedance page.
•
Γ = 0 means matched (RL = Z₀); Γ = +1 is an open end and −1 a short.
•
The sent and reflected waves add to a standing wave: Vₘₐₓ = 1 + |Γ| and Vₘᵢₙ = 1 − |Γ|, per |V⁺|.
Step-by-step
WE KNOW
Z₀ = 50 Ω, RL = 100 Ω
FIND
reflection coefficient (Γ), standing-wave ratio (VSWR), return loss (RL), power reflected (share)
Step 1 · Find reflection coefficient (Γ)
Use Γ = (RL − Z₀) ÷ (RL + Z₀)
The share of the voltage wave the load sends back: its mismatch over the sum.
Γ
=
100 − 50
100 + 50
Γ
=
50
150
Γ
=
0.333333
Step 2 · Find standing-wave ratio (VSWR)
Use VSWR = (1 + |Γ|) ÷ (1 − |Γ|)
The
standing
wave
swings
between
1
+
|
Γ
|
and
1
−
|
Γ
|;
the
VSWR
is
their
ratio.
VSWR
=
1 + |0.333333|
1 − |0.333333|
VSWR
=
1 + 0.3333
1 − 0.3333
VSWR
=
1.3333
0.6667
VSWR = 2
Step 3 · Find return loss (RL)
Use RL = −20 × log₁₀(|Γ|)
How far the reflected wave is below the sent one, in decibels of voltage.
RL = −20 × log₁₀(|0.333333|)
RL = −20 × log₁₀(0.3333)
RL = −20 × (−0.4772)
RL = 9.5424 dB
Step 4 · Find power reflected (share)
Use share = 100 × Γ²
Power
goes
as
voltage
squared,
so
Γ
2
of
the
power
comes
back;
times
100
for
percent.
share
=
100
×
0.333333
2
share = 100 × 0.1111
share = 11.1111%
Check
Put the numbers back in. Both sides should match.
0.333333
=
100 − 50
100 + 50
✓
2
=
1 + |0.333333|
1 − |0.333333|
✓
9.5424 = −20 × log₁₀(|0.333333|) ✓
11.1111
=
100
×
0.333333
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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