SME
Engineering Electromagnetics · Problem type
Quarter-wave transformer
Use this for “Match a 200 Ω load to a 50 Ω line at 1 GHz with a quarter-wave section. Find its impedance and length.”
Diagram
ZT
=
√
Z₀RL
=
√
50 × 200
ZT = 100 Ω
λ
=
v
f
=
0.2
m,
so
ℓ
=
λ
4
=
0.05
m
Zᵢₙ
=
ZT
2
RL
=
100
2
200
=
50
Ω
=
Z₀:
the
line
sees
a
match,
and
the
standing
wave
stays
inside
the
section
(|V|
at
the
load
2
×
|V⁺|).
This is an example. Type your own values to start a new problem.
ZT (Ω)
=
Z₀ (Ω)
×
RL (Ω)
λ (cm)
=
v (m/s)
f (GHz)
ℓ (cm)
=
λ (cm)
4
Formulas
ZT
=
√
Z₀
× RL
100
=
√
50 × 200
λ
=
v
f
0.2
=
2 ×
10
8
1 ×
10
9
ℓ
=
λ
4
0.05
=
0.2
4
Formulas are worked in Hz, m. The step-by-step shows the conversions.
Assumptions
•
Lossless lines and a resistive load; a section λ/4 long turns RL into ZT² ÷ RL at its input.
•
Choosing ZT = √(Z₀RL) makes that equal Z₀, so the main line sees no reflection.
•
The match holds only near the design frequency, where the section is a quarter of a wavelength.
•
λ is the wavelength on the section itself, from the signal speed v on it.
Step-by-step
WE KNOW
Z₀ = 50 Ω, RL = 200 Ω, f = 1 GHz, v = 2 × 10⁸ m/s
FIND
section impedance (ZT), wavelength (λ), section length (ℓ)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to Hz, m).
f = 1 GHz = 1,000,000,000 Hz (1 GHz = 10⁹ Hz)
Step 2 · Find section impedance (ZT)
Use ZT = √(Z₀ × RL)
The
section
must
turn
RL
into
Z₀
,
and
ZT
2
RL
=
Z₀
gives
the
geometric
mean.
ZT
=
√
50 × 200
ZT
=
√
10
,000
ZT = 100 Ω
Step 3 · Find wavelength (λ)
Use λ = v ÷ f
One
wavelength
passes
in
each
period
1
f
.
λ
=
2 ×
10
8
10
9
λ
=
0.2
m
Step 4 · Find section length (ℓ)
Use ℓ = λ ÷ 4
The section is a quarter of a wavelength long.
ℓ
=
0.2
4
ℓ
=
0.05
m
Step 5 · Convert the answers
Change each answer back to the unit you chose.
λ = 0.2 m = 20 cm (1 m = 100 cm)
ℓ = 0.05 m = 5 cm (1 m = 100 cm)
Check
Put the numbers back in. Both sides should match.
100
=
√
50 × 200
✓
0.2
=
2 ×
10
8
10
9
✓
0.05
=
0.2
4
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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