Learn about the stability factors for the stabilization techniques

1. COLLECTOR TO BASE BAIS
SELF BIAS
Dr.N.G.Praveena
Associate Professor/ECE
R.M.K. COLLEGE OF ENGINEERING
AND TECHNOLOGY

2. COLLECTOR-TO-BASE BIAS (DC BIAS
WITH VOLTAGE FEEDBACK)
IB + IC

3.  For dc analysis
ac input signal = 0
C1 and C2 – open circuit (∵f = 0)
IB + IC IB + IC

4. STEP 1: OBTAIN AN EXPRESSION FOR IB
Apply KVL to the base
𝑽𝑪𝑪 = 𝑰𝑩 + 𝑰𝑪 𝑹𝑪 + 𝑰𝑩𝑹𝑩 + 𝑽𝑩𝑬
𝑽𝑪𝑪= 𝑰𝑩𝑹𝑪 + 𝑰𝑪𝑹𝑪 + 𝑰𝑩𝑹𝑩 + 𝑽𝑩𝑬
𝑽𝑪𝑪 = (𝑹𝑩 + 𝑹𝑪 )𝑰𝑩 + 𝑰𝑪𝑹𝑪 + 𝑽𝑩𝑬
(𝑹𝑩+𝑹𝑪 )𝑰𝑩 = 𝑽𝑪𝑪 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑪
𝑰𝑩 =
𝑽𝑪𝑪 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑪
𝑹𝑩+𝑹𝑪
--- (1)
IB+IC

5. STEP 2: TO OBTIAN S , DIFFERENTIATE IB WITH RESPECT TO IC AND
SUBSITUTE IN THE STANDARD EQUATION OF S
Differentiate eqn (1) with respect to 𝑰𝑪
𝝏𝑰𝑩
𝝏𝑰𝑪
= -
𝑹𝑪
𝑹𝑪+ 𝑹𝑩
----------- (2)
We know that the standard equation of S is
S =
(1 + β)
1 – β ∂ 𝑰𝑩 / ∂ 𝑰𝑪
---(3)
Sub (2) in (3)
S =
(1 + β)
1 – β − 𝑹𝑪
𝑹𝑪+ 𝑹𝑩
S =
(1 + β)
1 + β 𝑹𝑪
𝑹𝑪+ 𝑹𝑩
𝑰𝑩 =
𝑽𝑪𝑪 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑪
𝑹𝑩+𝑹𝑪
--- (1)

6. STEP 3: TO OBTAIN S′, REPLACE IB INTERMS OF VBE IN THE
STANDARD EQUATION OF IC AND DIFFERENTIATE IT WITH RESPECT
TO VBE
The Standard Equation of Ic is
𝑰𝑪 = 𝜷𝑰𝑩 + (1+𝛃) 𝑰𝑪𝑶
Sub (1)
𝑰𝑪 = β (
𝑽𝑪𝑪 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑪
𝑹𝑩+𝑹𝑪
) (∵ 𝑰𝑪𝑶<<)
𝑰𝑪 (𝑹𝑩 + 𝑹𝑪 ) = β ( 𝑽𝑪𝑪− 𝑽𝑩𝑬 − 𝑰𝑪 𝑹𝑪)
𝑰𝑪 (𝑹𝑩 + 𝑹𝑪 ) + β 𝑰𝑪𝑹𝑪 = β ( 𝑽𝑪𝑪 − 𝑽𝑩𝑬)
𝑰𝑪 (𝑹𝑩 + 𝑹𝑪 + β 𝑹𝑪 ) = β ( 𝑽𝑪𝑪 − 𝑽𝑩𝑬)
𝑰𝑪 =
β( 𝑽𝑪𝑪 − 𝑽𝑩𝑬)
𝑹𝑩+𝑹𝑪 +β𝑹𝑪
𝐈𝐂 =
β(𝐕𝐂𝐂 − 𝐕𝐁𝐄)
𝐑𝐁+𝐑𝐂 (1 +β)
----------- (4)
𝑰𝑩 =
𝑽𝑪𝑪 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑪
𝑹𝑩+𝑹𝑪
--- (1)

7. 𝑰𝑪 =
𝜷(𝑽𝑪𝑪 − 𝑽𝑩𝑬)
𝑹𝑩+𝑹𝑪 (1 + 𝜷)
----------- (4)
Differentiate eqn (4) with respect to VBE
𝝏𝑰𝑪
𝝏𝑽𝑩𝑬
= -
𝜷
𝑹𝑩+𝑹𝑪 (𝟏+𝜷 )
∴ S′ =
𝝏𝑰𝑪
𝝏𝑽𝑩𝑬
= -
𝜷
𝑹𝑩+𝑹𝑪 (𝟏+𝜷 )

8. STEP 4 : TO OBTAIN S′′ DIFFERENTIATE THE EQUATION OBTAINED
IN STEP 3 WITH REPSPECT TO β
𝑰𝑪 =
𝜷(𝑽𝑪𝑪 − 𝑽𝑩𝑬)
𝑹𝑩+𝑹𝑪 (1 + 𝜷)
----------- (4)
Differentiate eqn (4) with respect to β
𝛛𝐈𝐂
𝛛𝛃
=
𝟏+ 𝛃 𝐑𝐂
+𝐑𝐁
𝐕𝐂𝐂
−𝐕𝑩𝑬
−𝛃(𝐕𝐂𝐂
−𝐕𝐁𝐄
)𝐑𝐂
𝟏+ 𝛃 𝐑𝐂
+𝐑𝐁
𝟐
=
𝐕𝐂𝐂
−𝐕𝑩𝑬
[ 𝟏+ 𝛃 𝐑𝐂
+𝐑𝐁
−𝛃𝐑𝐂)
𝟏+ 𝛃 𝐑𝐂
+𝐑𝐁
𝟐
=
𝐕𝐂𝐂
−𝐕𝑩𝑬
[𝐑𝐂
+𝐑𝐁
]
𝟏+ 𝛃 𝐑𝐂
+𝐑𝐁
𝟐
=
𝐕𝐂𝐂
−𝐕𝑩𝑬
[𝐑𝐂
+𝐑𝐁
]
[ 𝟏+ 𝛃 𝐑𝐂
+𝐑𝐁
][ 𝟏+ 𝛃 𝐑𝐂
+𝐑𝐁
]
S’’ =
𝛛𝐈𝐂
𝛛𝛃
=
𝑰𝑪(𝑹𝑩+𝑹𝑪)
𝜷[ 𝟏+𝜷 𝑹𝑪
+𝑹𝑩]
[using eqn 4]

9. Advantages
 Simple method as it requires only one resistance
RB.
 Provides some stabilization of the operating point
Disadvantages
 Does not provide good stabilization because
stability factor is fairly high
 Negative feedback which reduces the gain of the
amplifier

10. VOLTAGE DIVIDER BIAS
• R1 and R2 are connected across the supply
voltage VCC and provide biasing.
• RE provides stabilization

11.  For dc analysis
ac input signal = 0
C1 and C2 – open circuit (∵f = 0)

12. Equivalent circuit
𝑽𝑻𝑯 = 𝑽𝑪𝑪
𝑹𝟐
𝑹𝟏+ 𝑹𝟐
RB =

13. STEP 1:OBTAIN AN EXPRESSION
FOR 𝑰𝑩
 Apply KVL to the base
𝑽𝑻𝑯 = 𝑰𝑩𝑹𝑩 + 𝑽𝑩𝑬 + 𝑰𝑬𝑹𝑬
= 𝑰𝑩𝑹𝑩 + 𝑽𝑩𝑬 + (𝑰𝑩 +𝑰𝑪)𝑹𝑬
𝑰𝑩𝑹𝑩 + 𝑰𝑩𝑹𝑬= 𝑽𝑻𝑯 - 𝑽𝑩𝑬 - 𝑰𝑪𝑹𝑬
𝑰𝑩(𝑹𝑩 + 𝑹𝑬)= 𝑽𝑻𝑯 - 𝑽𝑩𝑬 - 𝑰𝑪𝑹𝑬
𝑰𝑩 =
𝑽𝑻𝑯 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑬
𝑹𝑩 + 𝑹𝑬
------(1)

14. STEP 2: TO OBTIAN S , DIFFERENTIATE IB
WITH RESPECT TO IC AND SUBSITUTE IN
THE STANDARD EQUATION OF S
𝑰𝑩=
𝑽𝑻𝑯 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑬
𝑹𝑩 + 𝑹𝑬
------(1)
Differentiate eqn (1) with respect to Ic
𝝏𝑰𝑩
𝝏𝑰𝑪
= -
𝑹𝑬
𝑹𝑩 + 𝑹𝑬
------(2)
We know that the standard equation of S is
S =
(1 + β)
1 – β ∂ 𝐈𝐁 / ∂ 𝐈𝐂
---(3)
Sub (2) in (3)
S =
(1 + β)
1 – β − 𝐑𝐄
𝐑𝐄+ 𝐑𝐁
=
(1 + β)
1 + β 𝐑𝐄
𝐑𝐄+ 𝐑𝐁
By rearranging the above expression we get
S =
(1 + β)
1 + β 𝐑𝐄
𝐑𝐄+ 𝐑𝐁
≈ 𝟏 (∴
𝑹𝑩
𝑹𝑬
<< 1 )

15. 𝑰𝑩 =
𝑽𝑻𝑯 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑬
𝑹𝑩 + 𝑹𝑬
------(1)
The Standard Equation of Ic is
𝑰𝑪 = β𝑰𝑩 + (1+β) 𝑰𝑪𝑶
Sub (1)
𝑰𝑪 = β
𝑽𝑻𝑯 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑬
𝑹𝑩 + 𝑹𝑬
(∵
𝑰𝑪𝑶 << )
𝑰𝑪(𝑹𝑩 + 𝑹𝑬 ) = β (𝑽𝑻𝑯 − 𝑽𝑩𝑬 − 𝑰𝑪𝑹𝑬)
𝑰𝑪(𝑹𝑩 + 𝑹𝑬 + β 𝑹𝑬 ) = β (𝑽𝑻𝑯 − 𝑽𝑩𝑬 )
𝑰𝑪 =
β (𝑽𝑻𝑯 − 𝑽𝑩𝑬 )
𝑹𝑩 + 𝑹𝑬 (1+ β )
---------(4)
STEP 3: TO OBTAIN S′, REPLACE IB INTERMS OF
VBE IN THE STANDARD EQUATION OF IC AND
DIFFERENTIATE IT WITH RESPECT TO VBE

16. 𝑰𝑪 =
β (𝑽𝑻𝑯 − 𝑽𝑩𝑬 )
𝑹𝑩 + 𝑹𝑬 (1+ β )
---------(4)
Differentiate eqn (4) with respect to 𝑽𝑩𝑬
𝝏𝑰𝑪
𝝏𝑽𝑩𝑬
= -
𝜷
𝑹𝑩+𝑹𝑬 (𝟏+𝜷 )
∴ S′ =
𝝏𝑰𝑪
𝝏𝑽𝑩𝑬
= -
𝜷
𝑹𝑩+𝑹𝑬 (𝟏+𝜷 )

17. STEP 4 : TO OBTAIN S′′ DIFFERENTIATE
THE EQUATION OBTAINED IN STEP 3
WITH REPSPECT TO β
 Differentiate eqn (4) with respect to β
𝛛𝐈𝐂
𝛛𝛃
=
𝟏+ 𝛃 𝐑𝑬
+𝐑𝐁
𝐕𝑻𝑯
−𝐕𝑩𝑬
−𝛃(𝐕𝑻𝑯
−𝐕𝐁𝐄
)𝐑𝑬
𝟏+ 𝛃 𝐑𝑬
+𝐑𝐁
=
𝐕𝑻𝑯
−𝐕𝑩𝑬
[ 𝟏+ 𝛃 𝐑𝑬
+𝐑𝐁
−𝛃𝐑𝑬
)
𝟏+ 𝛃 𝐑𝑬
+𝐑𝐁
𝟐
=
𝐕𝑻𝑯
−𝐕𝑩𝑬
[𝐑𝑬
+𝐑𝐁
]
𝟏+ 𝛃 𝐑𝑬
+𝐑𝐁
𝟐
=
𝐕𝑻𝑯
−𝐕𝑩𝑬
[𝐑𝑬
+𝐑𝐁
]
[ 𝟏+ 𝛃 𝐑𝑬
+𝐑𝐁
][ 𝟏+ 𝛃 𝐑𝑬
+𝐑𝐁
]
S’’ =
𝛛𝐈𝐂
𝛛𝛃
=
𝑰𝑪
(𝐑𝑬
+𝐑𝐁
)
𝜷[ 𝟏+𝜷 𝐑𝑬
+𝐑𝐁
]
[using eqn 4]
𝑰𝑪 =
β (𝑽𝑻𝑯 − 𝑽𝑩𝑬 )
𝑹𝑩 + 𝑹𝑬 (1+ β )
---------(4)

18. Advantages
 Provides good stability.