Bias Stability of BJT Amplifiers
Earlier we claimed that the four-resistor bias circuit for BJT amplifiers was
remarkably stable. The following circuit shows its implementation for an NPN BJT
common emitter amplifier:
We now demonstrate the bias stability of this circuit by using the bias equivalent
circuit that we developed earlier for this circuit:
This circuit is clearly equivalent to the following:
2
If we use the following Thevenin equivalent circuits,
we can redraw the original circuit as follows:
3
After these Thevenin transformations, we can easily use mesh analysis with
mesh currents I b and I c to solve for the collector current, I c . Writing KVL around
the base loop gives:
Rb1 Vcc

Rb1  Rb 2
 Rb
 r  I b  Vbe* 
 Ib
 I c  Re
Writing KVL around the collector loop gives:
Vcc
Ib
I c*
1

  Rc 

 Ic - 
G
G
G

 Ib
 I c  Re
We can collect terms in the two equations above to get the following linear
algebraic equations for I b and I c :
 Rb
 r  Re  I b 
 Re  I c

Rb1 Vcc
- Vbe*
Rb1  Rb 2
I c*

1



R
I

R


R
I

V

e c
cc
 e
 b
 c
G
G
G



We can use Cramer's rule to solve this pair of equations for I c (actually I cq , but
we leave off the subscript q for simplicity in notation):
4


I*  
   Rb1 Vcc
Ic  =  R b + r + R e  Vcc  c  -  Re - Vve* 
 
G 
G   Rb1  Rb 2


where
 
 Rb
R
1


 r  Re   Rc 
 Re  - Re 2   e
G
G


For any bias circuit, the approximate change in collector current, I c , caused by
small changes in the transistor parameters is given by the first order terms in a
multivariable Taylor's series expansion:
I 
  Ic  *
 I 
 I 
I 
I c   c*  hI c*  
Vbe   c  G   c     c  r
* 
 G 
  
 r 
  Ic 
  Vbe 
It usually makes more sense to look at the percentage change of Ic:
  I c  I c*
  I c  Vbe*   I c  G   I c  
I c
  I  r
 



  c




*
* 
Ic
  G  Ic
  r  Ic
   I c
  Ic  Ic
  Vbe  I c
We can easily rewrite this result as follows:
  I I *  I *
  I c Vbe*  Vbe*   I c G  G
I c
  c* c  *c  
 * 

*
Ic
  Ic Ic  Ic
  Vbe I c  Vbe
  G Ic  G
  I      I c r  r
  c



   Ic  
  r Ic  r
The advantage of this form is that the percentage variation of the collector
current, I c / I c , is given as a function of the percentage variations of the
transistor parameters, I c* , Vbe* , G ,  and r . The coefficients in parentheses are
dimensionless numbers known as sensitivity coefficients. If the sensitivity
coefficient that corresponds to a certain parameter is much smaller than one,
then the collector current for the transistor in that circuit will be relatively stable
against variations in that particular parameter. A larger sensitivity coefficient
5
means that the collector current may vary considerably as that transistor
parameter varies because of changes in temperature or because of the statistical
distribution of values for that parameter among various transistors with the same
type number.
Because the operating point, Vceq , I cq  , must lie on the load line, note that Vceq is
stable if I cq is stable. We conclude, therefore, that the operating point for a
transistor that resides in a particular bias circuit is stable if the sensitivity
coefficients for each parameter that can vary are much less than one.
We demonstrate the stability of the four resistor bias scheme by showing that the
sensitivity coefficients for the various parameters are small compared to one. We
begin with the sensitivity coefficient that corresponds to I c* . It helps to note that
Cramer's delta does not depend on I c* and hence can be treated as a constant in
calculating the partial derivative in the sensitivity coefficient for I c* :
 Ic
1 1

   Rb  r  Re 
*
 Ic
 G

1
G
R
1


 Re  - Re 2   e
 Rc 
G
G


 Rb
 Rb
 r  Re 
 r  Re 
This expression is complicated enough that it is hard to see what is going on. We
can use some of the biasing conditions that we imposed during our bias design
procedure to simplify considerably this expression, and others that we are about
to encounter. First, we recall the expressions for Re and r  hie  rbb '  rb ' e :
Re 
VRe
2

I Re
I cq
r  0.026

I cq
6
Clearly,
r

Re
r
 Re
or
Also,
Rb  Rb1 
27
I cq
Thus,
Rb
27 I cq

 13.5
Re
I cq 2
This means
Rb
Re
and that
Rb
13.5

 Re

1
so that
Rb
 Re
Also, usually,
r
 I cq
 0.026
1
Re
I cq 2
7
because usually   100 if the BJT is to be of any interest. Thus, r and Re are
typically of the same order of magnitude.
In addition,
Vceq I cq
Vceq
Rc


 1
Re
I cq 2
2
Also, usually, the BJT I c vs Vce curves in the active region are much flatter than
the load line. That is, G is usually small enough so that:
1
G
Rc
Let's summarize some useful results:
 Re
r  hie  rbb '  rb ' e
 Re
Rb
1
G
Rc
Re
Re
r ~ Re
I c*
Ic
I c   Ib
Recall that the last two results come from our discussion of linear models for
BJTs.
By using these simplifying relationships, we find that our earlier equation
8
1
G
R
1


 Re  - Re 2   e
 Rc 
G
G


 Rb
 Ic

 I c*
 Rb
 r  Re 
 r  Re 
simplifies to
Rb
G
 Ic

 I c*
Because I c*
Rb
R
  e
G
G

1
R
1   e
Rb
 1
I c , then
  I c I c* 


*
  Ic Ic 
1
That is, the sensitivity coefficient corresponding to I c* is small, so we conclude
that the quiescent point for this bias circuit is not sensitive to variations in I c* .
Let's next calculate the sensitivity coefficient that corresponds to Vbe* .
 Ic
1 
 

 Re 
*
 Vbe
 
G

 Rb
 r  Re 
 

 Re 
G

R
1


 Re  - Re 2   e
 Rc 
G
G


Using some of the approximations summarized above, we find:
 Ic

 Vbe*

-

G
Rb
 Re

G
G
-
Rb   Re
9
The minus sign indicates that an increase in Vbe* results in a decrease in the
collector current. Because we are mainly interested in the size of the sensitivity
coefficient rather than its sign, we can drop the negative sign for simplicity in
manipulation. Thus the sensitivity coefficient for Vbe* becomes:
  I c Vbe* 
Vbe*



 
*
Rb   Re I c
  Vbe I c 
Vbe*
 Rb   Re  Ib
or
  I c Vbe* 
Vbe*
Vbe*
Vbe*





*
 Re Ib
I c Re
Ve
  Vbe I c 
If we choose Ve  2V (as we did in our biasing procedure) and since Vbe*  0.7V ,
then
Vbe*
1

Ve
3
and
  I c Vbe* 
1

 
*
3
  Vbe I c 
We conclude that this bias circuit is relatively insensitive to changes in Vbe* when
we follow our biasing procedure, although we would have preferred that this
sensitivity coefficient be smaller. Fortunately, Vbe* is mainly determined by the
semiconductor material from which the BJT is made and consequently varies
little from transistor to transistor and varies only slightly with temperature. Thus,
this circuit is stable in practice even though this sensitivity coefficient is larger
than we might wish.
Things get a little more complicated when we calculate the sensitivity coefficients
for G and  because Cramer's delta is a function of these parameters and
10
hence can no longer be treated as constant with respect to differentiation. Our
strategy to simplify the calculation is to simplify I c before we differentiate by
discarding (according to the approximations that we developed earlier) certain
small terms. We must be careful, however, not to throw away even small terms
that depend on G or  because the derivative of even small terms can be large.
Recall:
 Rb
Ic 

I* 
 r  Re  Vcc  c  G

 Rb  r  Re   Rc 



 Re 
G

1

 Re  G

 Rb1 Vcc
*
 R  R - Vbe 
b2
 b1

R
Re 2   e
G
After simplifying, we find:


I* 
  Rb1 Vcc
Rb Vcc  c  
- Vbe* 

G
G  Rb1  Rb 2


Ic 
Rb
 Re

G
G
Note that in the approximations, no small terms containing  or G have been
discarded. Simplifying, we find:
 R V

Rb GVcc  I c*     b1 cc
- Vbe* 
 Rb1  Rb 2

Ic 
Rb   Re
or
 Rb1 Vcc

- Vbe* 
*



Rb GVcc  I c 
R  Rb 2

Ic 
  b1
 Re
Re
To calculate the sensitivity coefficient for G , we first calculate
11
 Ic
R V
 b cc
 G
 Re
Thus, the sensitivity coefficient becomes
  Ic G 
Rb Vcc G
Vcc
 GRb

 
 Re I c
 Ve
  G Ic 
But GRb ~ 1 and Vcc /  Ve
1 so that
  Ic G 

 << 1
  G Ic 
We therefore conclude that this bias circuit is stable with respect to variations in
G.
To calculate the sensitivity coefficient for  , we first calculate
*
 Ic
Rb GVcc  I c 
 
2
Re
The minus sign indicates that an increase in  results in a decrease in the
collector current. Because we are mainly interested in the size of the sensitivity
coefficient rather than its sign, we can drop the negative sign for simplicity in
manipulation. Thus the sensitivity coefficient for  becomes:
Rb GVcc  I c* 
 
Ic 
 Re I c
  Ic

 
 
  Ic


 
or
But
Rb GVcc  I c*


Ic 
 Re
Ic
12
Rb
 Re
1
and
GVcc  I c*
Ic
1
so that clearly
  Ic

 
 

Ic 
1
We conclude, therefore, that this bias circuit maintains a stable operating point
despite any variations in  due to changes in temperature or statistical
variations from one BJT to another.
To calculate the sensitivity coefficient for the parameter r , we use logarithmic
differentiation. That is, we take the logarithm of both sides of the equation for the
collector current before we differentiate it. (Logarithmic differentiation sometimes
simplifies the process of taking the derivative of complicated expressions.)

ln I c  ln  Rb  r  Re 

 

I c* 
   Rb1 Vcc

V

- Vce*  

 cc
 -  Re G
G   Rb1  Rb 2


 
R 

1


- ln  Rb  r  Re   Rc 
 Re  - Re 2   e 
G
G



Thus,
13
Vcc 
1  Ic

Ic  r
I c*
G


I c* 
  R V
 Rb  r  Re  Vcc   -  Re -   b1 cc - Vbe* 
G
G   Rb1  Rb 2



1


 Re 
 Rc 
G


R
1


 Re  - Re 2   e
 Rb  r  Re   Rc 
G
G


We can simplify this expression by using the approximations developed for our
biasing procedure for this circuit:
1  Ic

Ic  r
I c*
1
G
G
R
 Re

Ic * 
  Rb1 Vcc
b



- Vbe* 


G
G
G
G  Rb1  Rb 2

Vcc 

Rb Vcc

To find the sensitivity coefficient, we multiply this result by r :
  Ic r 

 
  r Ic 
r
r

Rb   Re
 Rb1 Vcc

- Vbe* 

 Rb1  Rb 2


Rb 
I c*
G

Vcc 


G
Because
Rb1 Vcc
- Vbe  0
Rb1  Rb 2
then
  Ic r 
r
r


 
Rb
Rb   Re
  r Ic 
1
We can therefore conclude that this circuit is stable with respect to variations in
r . Furthermore, we have now demonstrated that this four-resistor bias circuit,
14
with our design rules of thumb, maintains a stable operating point despite
temperature changes and even despite substituting a transistor with different
parameters because the operating point is not sensitive to any of the five
transistor parameters in our linear equivalent circuit for the transistor.