SME
Electronics · Problem type
Cascaded stages in decibels
Use this for “Two stages with voltage gains of 20 and 50 are cascaded. Find the overall gain, as a ratio and in dB.”
Chart
Overall in dB: 26.0 + 34.0 = 60.0 dB
This is an example. Type your own values to start a new problem.
A
=
A₁
×
A₂
G (dB)
=
G₁ (dB)
+
G₂ (dB)
Formulas
G₁
=
20
×
log₁₀(
A₁
)
26.0206 = 20 × log₁₀(20)
G₂
=
20
×
log₁₀(
A₂
)
33.9794 = 20 × log₁₀(50)
G
=
20
×
log₁₀(
A
)
60 = 20 × log₁₀(1,000)
A
=
A₁
×
A₂
1,000 = 20 × 50
G
=
G₁
+
G₂
60 = 26.0206 + 33.9794
Assumptions
•
Each stage’s gain is measured with the next stage connected, so loading is already in it.
•
Gains are sizes (an inverting stage’s minus sign is left out); in decibels a voltage gain is G = 20 log₁₀ A.
Step-by-step
WE KNOW
A₁ = 20, A₂ = 50
FIND
stage 1 in dB (G₁), stage 2 in dB (G₂), overall gain (A), overall in dB (G)
Step 1 · Find stage 1 in dB (G₁)
Use G₁ = 20 × log₁₀(A₁)
A voltage ratio in decibels is 20 times its common log (stage 1).
G₁
=
20
×
log₁₀(20)
G₁
=
20
×
1.301
G₁
=
26.0206
dB
Step 2 · Find stage 2 in dB (G₂)
Use G₂ = 20 × log₁₀(A₂)
A voltage ratio in decibels is 20 times its common log (stage 2).
G₂
=
20
×
log₁₀(50)
G₂
=
20
×
1.699
G₂
=
33.9794
dB
Step 3 · Find overall gain (A)
Use A = A₁ × A₂
Stage 2 multiplies what stage 1 hands it, so gains in a chain multiply.
A
=
20
×
50
A
=
1,000
Step 4 · Find overall in dB (G)
Use G = G₁ + G₂
The log of a product is the sum of the logs, so decibels add.
G
=
26.0206
+
33.9794
G
=
60
dB
Check
Put the numbers back in. Both sides should match.
26.0206 = 20 × log₁₀(20) ✓
33.9794 = 20 × log₁₀(50) ✓
60 = 20 × log₁₀(1,000) ✓
1,000 = 20 × 50 ✓
60 = 26.0206 + 33.9794 ✓
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