SME
Communication Systems
AM and FM
Use this for “A 100 W carrier is modulated to μ = 0.5. Find the total and sideband power.”
Diagram
Pₜ
=
Pc(1
+
μ
2
/2)
=
100
×
(1
+
0.5
2
2
)
=
113
W,
so
the
sidebands
carry
Psb
=
12.5
W
and
η
=
μ
2
2 +
μ
2
=
11.1%.
Each sideband is μ/2 = 0.25 of the carrier’s height, so the two hold μ²/2 = 0.125 of its power.
B = 2 × fₘ = 10 kHz: the sidebands sit at 995 and 1,005 kHz.
The carrier is drawn 11 cycles to each message cycle, far slower than it is.
This is an example. Type your own values to start a new problem.
Pₜ (W)
=
Pc (W)
×
1
+
μ
2
2
Psb (W)
=
Pc (W)
×
μ
2
2
η
Efficiency
calculated · %
fₘ
Message frequency
example · kHz
B
Bandwidth
calculated · kHz
fc
Carrier frequency
example · kHz
fL
Lower sideband
worked out (not typed) · kHz
fU
Upper sideband
worked out (not typed) · kHz
Formulas
Pₜ
=
Pc
×
(1
+
μ
2
2
)
112.5
=
100
×
(1
+
0.5
2
2
)
Psb
=
Pc
×
μ
2
2
12.5
=
100
×
0.5
2
2
η
=
100
×
μ
2
2 +
μ
2
11.1111
=
100
×
0.5
2
2 +
0.5
2
B
=
2
×
fₘ
10 = 2 × 5
fL = fc − fₘ
995 = 1,000 − 5
fU = fc + fₘ
1,005 = 1,000 + 5
Assumptions
•
Tone modulation: one message frequency fₘ, a cosine.
•
μ ≤ 1, so the envelope never reaches zero (no overmodulation).
•
The sidebands sit at fc ± fₘ, each μ ÷ 2 of the carrier’s amplitude.
Step-by-step
WE KNOW
Pc = 100 W, μ = 0.5, fₘ = 5 kHz, fc = 1,000 kHz
FIND
total power (Pₜ), sideband power (Psb), efficiency (η), bandwidth (B), lower sideband (fL), upper sideband (fU)
Step 1 · Find total power (Pₜ)
Use Pₜ = Pc × (1 + μ² ÷ 2)
Each
sideband
is
μ
2
of
the
carrier’s
amplitude,
so
the
two
add
μ
2
2
of
its
power.
Pₜ
=
100
×
(1
+
0.5
2
2
)
Pₜ
=
100
×
(1
+
0.25
2
)
Pₜ = 100 × (1 + 0.125)
Pₜ = 100 × 1.125
Pₜ = 112.5 W
Step 2 · Find sideband power (Psb)
Use Psb = Pc × μ² ÷ 2
What
the
sidebands
carry
is
the
total
less
the
carrier:
μ
2
2
of
the
carrier’s
power.
Psb
=
100
×
0.5
2
2
Psb
=
100
×
0.25
2
Psb
=
25
2
Psb = 12.5 W
Step 3 · Find efficiency (η)
Use η = 100 × μ² ÷ (2 + μ²)
The
share
of
the
power
in
the
sidebands,
where
the
message
is:
Psb
Pₜ
as
a
percent.
η
=
100
×
0.5
2
2 +
0.5
2
η
=
100
×
0.5
2
2 + 0.25
η
=
100
×
0.5
2
2.25
η
=
100
×
0.25
2.25
η
=
25
2.25
η
=
11.1111%
Step 4 · Find bandwidth (B)
Use B = 2 × fₘ
The sidebands reach fₘ either side of the carrier.
B
=
2
×
5
B
=
10
kHz
Step 5 · Find lower sideband (fL)
Use fL = fc − fₘ
The lower sideband sits fₘ below the carrier …
fL = 1,000 − 5
fL = 995 kHz
Step 6 · Find upper sideband (fU)
Use fU = fc + fₘ
… and the upper one fₘ above it.
fU = 1,000 + 5
fU = 1,005 kHz
Check
Put the numbers back in. Both sides should match.
112.5
=
100
×
(1
+
0.5
2
2
)
✓
12.5
=
100
×
0.5
2
2
✓
11.1111
=
100
×
0.5
2
2 +
0.5
2
✓
10 = 2 × 5 ✓
995 = 1,000 − 5 ✓
1,005 = 1,000 + 5 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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