SME
Electronics
Amplifiers
Use this for “A CE amplifier biased at IC = 1 mA has RC = RL = 4 kΩ. Find the voltage gain.”
Diagram
gₘ
=
IC
VT
=
1
mA
25.85
mV
=
38.68
mS
rπ
=
β
gₘ
=
100
38.68
mS
=
2.585
kΩ
Rₚ
=
RC·
RL
RC + RL
Rₚ
=
4
kΩ
×
4
kΩ
4 kΩ + 4 kΩ
=
2
kΩ
Aᵥ = −gₘ·Rₚ = −38.68 mS × 2 kΩ
Aᵥ = −77.37
Small signals: the base sees rπ, the collector a current source gₘ vπ into RC ∥ RL (emitter bypassed, rₒ ignored).
This is an example. Type your own values to start a new problem.
gₘ (mS)
=
IC (mA)
25.85
mV
Rₚ (kΩ)
=
RC (kΩ)
×
RL (kΩ)
RC (kΩ)
+
RL (kΩ)
Aᵥ
=
−
gₘ (mS)
×
Rₚ (kΩ)
β
Current gain
example
rπ
Input resistance
calculated
Formulas
gₘ
=
IC
0.02585
38.6847
=
1
0.02585
rπ
=
β
gₘ
2.585
=
100
38.6847
Rₚ
=
RC
×
RL
RC + RL
2
=
4
×
4
4 + 4
Aᵥ = −gₘ × Rₚ
−77.3694 = −38.6847 × 2
Assumptions
•
Thermal voltage VT = 25.85 mV (300 K); the signal at the base is small, a few mV.
•
The emitter resistor is bypassed by a capacitor, and the output resistance rₒ is ignored.
•
Units: mA ÷ V gives mS, β ÷ mS gives kΩ, and mS × kΩ is a plain ratio.
Step-by-step
WE KNOW
IC = 1 mA, β = 100, RC = 4 kΩ, RL = 4 kΩ
FIND
transconductance (gₘ), input resistance (rπ), collector load, RC ∥ RL (Rₚ), voltage gain (Aᵥ)
Step 1 · Find transconductance (gₘ)
Use gₘ = IC ÷ 0.02585
The collector current over the thermal voltage: how much IC moves per volt at the base.
gₘ
=
1
0.02585
gₘ = 38.6847 mS
Step 2 · Find input resistance (rπ)
Use rπ = β ÷ gₘ
The
base
current
is
the
collector
current
β
,
so
the
base
sees
β
times
1
gₘ
.
rπ
=
100
38.6847
rπ = 2.585 kΩ
Step 3 · Find collector load, RC ∥ RL (Rₚ)
Use Rₚ = RC × RL ÷ (RC + RL)
For the signal, the supply is ground, so RC and the load sit in parallel.
Rₚ
=
4
×
4
4 + 4
Rₚ
=
4
×
4
8
Rₚ
=
16
8
Rₚ = 2 kΩ
Step 4 · Find voltage gain (Aᵥ)
Use Aᵥ = −gₘ × Rₚ
The current gₘ vπ flows down through Rₚ, so the output swings the other way.
Aᵥ = −38.6847 × 2
Aᵥ = −77.3694
Check
Put the numbers back in. Both sides should match.
38.6847
=
1
0.02585
✓
2.585
=
100
38.6847
✓
2
=
4
×
4
4 + 4
✓
−77.3694 = −38.6847 × 2 ✓
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Common-source MOSFET amplifier
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