More on Output Resistance
Louis WY LIU
B111
Revisit the Concept of Emitter
Degeneration
• This topology will decrease the gain of the
amplifier but improve other aspects, such as
linearity, and input impedance.
Summary of the Basic Formula
Voltage Gain:
𝐴𝑣 =
−𝑔𝑚 𝑅𝐶
1+𝑔𝑚 𝑅𝐸
=
−𝑅𝐶
1
+𝑅𝐸
𝑔𝑚
Input impedance:
𝑣𝑖𝑛 =𝑖𝑏 𝑟𝜋 + 𝛽𝑖𝑏 𝑅𝐸
𝑣𝑖𝑛
𝑟𝑖𝑛 =
𝑖𝑏
=𝑟𝜋 + 𝛽𝑅𝐸 = 𝑟𝜋 (1 + 𝑔𝑚𝑅𝐸 )
Output impedance:
𝑟𝑜𝑢𝑡 = 𝑅𝐶
Degenerated CE Stage with VA<∞
• Emitter degeneration increases the output
impedance.
Use Thevinin conversion, we end up with the following
equivalent circuit.
𝑣𝑥 = 𝑖𝑥 𝑟𝑜 − 𝑔𝑚 𝑣𝜋 𝑟𝑜 − 𝑣𝜋
𝑣𝑥 = 𝑖𝑥 𝑟𝑜 + 𝑔𝑚 𝑟𝑜 𝑖𝑥 𝒓𝝅 // 𝑹𝑬 + 𝑖𝑥 𝒓𝝅 // 𝑹𝑬
𝑣𝑥
𝑟𝑜𝑢𝑡 =
= 𝑟𝑜 + (𝑔𝑚 𝑟𝑜 + 𝟏) 𝒓𝝅 // 𝑹𝑬
𝑖𝑥
𝑟𝑜𝑢𝑡 ≈ 𝑟𝑜 + 𝑔𝑚 𝑟𝑜 𝑟𝜋 // 𝑅𝐸
𝒓𝒐𝒖𝒕 = 𝒓𝒐 𝟏 + 𝒈𝒎 𝒓𝝅 // 𝑹𝑬
This relation depends on RE and the transistor only. RC is not
needed.
Use of the Output Resistance Formula
𝑅𝑜𝑢𝑡 = 𝑟𝑜 1 + 𝑔𝑚 𝑟𝜋 // 𝑅𝐸
Extreme Case 1:
If 𝑟𝜋 << RE, then 𝑅𝑜𝑢𝑡 ≈ 𝑟𝑜 1 + 𝑔𝑚 𝑟𝜋
Extreme Case 2:
If 𝑟𝜋 >> RE, then 𝑅𝑜𝑢𝑡 ≈ 𝑟𝑜 1 + 𝑔𝑚 𝑅𝐸
Degeneration by Another BJT
≡
• 𝑟𝑜𝑢𝑡 = 𝑟𝑜 1 + 𝑔𝑚 𝑟𝑜2 //𝑟𝜋1
This circuit is called “cascode”. It offers many
advantages.
Voltage Gain
Original Circuit
𝑣𝑖𝑛 = 𝑣𝜋 + 𝑖𝑥 𝑅𝐸
𝑣𝑖𝑛 − 𝑖𝑥 𝑅𝐸 = 𝑣𝜋
𝑣𝑖𝑛 +
𝑣𝑜𝑢𝑡
𝑅𝐸
𝑖𝑥
= 𝑣𝜋
(1)
Voltage Gain (con’t)
The approximated output network:
𝑣𝑜𝑢𝑡 = −𝑔𝑚 𝑣𝜋 𝑟𝑜
𝑅𝐶
𝑅𝐸 + 𝑅𝐶 + 𝑟𝑜
Voltage Gain (con’t)
Substituting (1):
𝑣𝑜𝑢𝑡 = −𝑔𝑚
𝑣𝑜𝑢𝑡
𝑣𝑖𝑛 +
𝑅𝐸
𝑖𝑥
𝑟𝑜
𝑅𝐶
𝑅𝐸 + 𝑅𝐶 + 𝑟𝑜
This yields:
𝑣𝑜𝑢𝑡
−𝑔𝑚 𝑅𝐶 𝑟𝑜
=
𝑣𝑖𝑛
𝑅𝐶 + 𝑅𝐸 + 𝑟𝑜 + 𝑔𝑚 𝑅𝐸 𝑟𝑜
Since 𝑔𝑚 𝑅𝐸 𝑟𝑜 >>𝑅𝐸 , we apply some approximation:
𝑣𝑜𝑢𝑡
−𝑔𝑚 𝑅𝐶 𝑟𝑜
≈
𝑣𝑖𝑛
𝑅𝐶 + 𝑟𝑜 1 + 𝑔𝑚 𝑅𝐸
Voltage Gain (Con’t)
Multiply the top and bottom with 1 + 𝑔𝑚 𝑅𝐸 :
𝑣𝑜𝑢𝑡
−𝑔𝑚 𝑅𝐶
≈
𝑣𝑖𝑛
𝑅𝐶 + 𝑟𝑜 1 + 𝑔𝑚 𝑅𝐸
𝑣𝑜𝑢𝑡
−𝑔𝑚
≈
𝑣𝑖𝑛
1 + 𝑔𝑚 𝑅𝐸
𝑟𝑜 (1 + 𝑔𝑚 𝑅𝐸 )
×
1 + 𝑔𝑚 𝑅𝐸
𝑅𝐶 //𝑟𝑜 (1 + 𝑔𝑚 𝑅𝐸 )
Where
𝑅𝐶 //𝑟𝑜 (1 + 𝑔𝑚 𝑅𝐸 ) is the combined output impedance, Rout.
−𝑔𝑚
is the transconductance, Gm.
1+𝑔𝑚 𝑅𝐸
New Theorem to Learn
Linear Theory of Circuit Network:
Voltage Gain, AV = Gm x Rout
Homework Q1
Consider the following circuit. IBIAS=1mA
1. Find the value of VBE . (ans: 0.836)
2. Is the BJT in the active mode? Why?
Find the small signal parameters of the BJT
under this bias condition. (ans: gm=38.5m,
r𝜋 = 2.6𝑘, 𝑟𝑜 = 10𝑘)
3. Derive the expression for voltage gain.
4. What is the numerical values of the voltage
gain and the output impedance in this
circuit? (-9.65, 9.58k)
5. Find the input impedance. (ans: 104.6k)