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:

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If we use the following Thevenin equivalent circuits,

we can redraw the original circuit as follows:

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After these Thevenin transformations, we can easily use mesh analysis with mesh currents bI and cI to solve for the collector current, cI . Writing KVL around

the base loop gives:

( ) ( )*1

1 2

b ccb b be b c e

b b

R V R r I V I I RR R

= + + + ++

Writing KVL around the collector loop gives:

( )*1 - - b c

cc c c b c eI IV R I I I R

G G Gβ = + + +

We can collect terms in the two equations above to get the following linear algebraic equations for bI and cI :

( ) ( ) *1

1 2

-

b ccb e b e c be

b b

R VR r R I R I VR R

+ + + =+

*

1 -

c

e b c e c ccIR I R R I V

G G Gβ + + + = +

We can use Cramer's rule to solve this pair of equations for cI (actually cqI , but

we leave off the subscript q for simplicity in notation):

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( )*

*1c b e

1 2

I = R + r + R - - -

c b cccc e ve

b b

I R VV R VG G R R

β ∆ + +

where

( ) 21 - eb e c e e

RR r R R R RG G

β ∆ = + + + + +

For any bias circuit, the approximate change in collector current, cI∆ , caused by

small changes in the transistor parameters is given by the first order terms in a multivariable Taylor's series expansion:

* ** *

c c c c cc c be

c be

I I I I II h I V G rI V G r

ββ

∂ ∂ ∂ ∂ ∂∆ = ∆ + ∆ + ∆ + ∆ + ∆ ∂ ∂ ∂ ∂ ∂

It usually makes more sense to look at the percentage change of Ic:

* *

* *

c c c c be c c c

c c c be c c c c

I I I I V I I IG rI I I V I G I I r I

ββ

∆ ∂ ∆ ∂ ∆ ∂ ∂ ∂∆ ∆ ∆ = + + + + ∂ ∂ ∂ ∂ ∂

We can easily rewrite this result as follows:

* * * *

* * * *

c c c c c be be c

c c c c be c be c

I I I I I V V I G GI I I I V I V G I G

∆ ∂ ∆ ∂ ∆ ∂ ∆= + + ∂ ∂ ∂

c c

c c

I I r rI r I rβ β

β β ∂ ∂∆ ∆

+ + ∂ ∂

The advantage of this form is that the percentage variation of the collector current, /c cI I∆ , is given as a function of the percentage variations of the

transistor parameters, *cI , *

beV , G , β 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

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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, ( ),ceq cqV I , must lie on the load line, note that ceqV is

stable if cqI 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 *

cI . It helps to note that

Cramer's delta does not depend on *cI and hence can be treated as a constant in

calculating the partial derivative in the sensitivity coefficient for *cI :

( )*

1 1

cb e

c

I R r RI G

∂ = + + ∂ ∆

( )

( ) 2

1

1 -

b e

eb e c e e

R r RG

RR r R R R RG G

β

+ +=

+ + + + +

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 eR and ' 'ie bb b er h r r= = + :

2 Ree

Re cq

VRI I

= =

0.026 cq

rIβ

≈

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Clearly,

er Rβ

or

er Rβ

Also,

127 b b

cq

R RI

< =

Thus,

27 13.5 2

cqb

e cq

IRR I

< =

This means

b eR R

and that

13.5 1

b

e

RRβ β

<

so that

b eR Rβ

Also, usually,

0.026 1 2

cq

e cq

IrR I

β= ≥

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because usually 100β ≥ if the BJT is to be of any interest. Thus, r and eR are

typically of the same order of magnitude.

In addition,

1 2 2

ceq cq ceqc

e cq

V I VRR I

= = >>

Also, usually, the BJT cI vs ceV curves in the active region are much flatter than

the load line. That is, G is usually small enough so that:

1 cRG

Let's summarize some useful results:

' ' e ie bb b eR r h r rβ = = +

e b eR R Rβ

1 c eR RG

~ er R

* c cI I

c bI Iβ≥

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

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( )

( )*

2

1 1 -

b ec

ecb e c e e

R r RI GRI R r R R R R

G Gβ

+ +∂=

∂ + + + + +

simplifies to

*

1 1 1

b

c

b e ec

b

RI G

R R RIG G R

β β

∂≈ = <

∂ + +

Because *c cI I , then

*

*

1

c c

c c

I II I

∂ ∂

That is, the sensitivity coefficient corresponding to *cI is small, so we conclude

that the quiescent point for this bias circuit is not sensitive to variations in *cI .

Let's next calculate the sensitivity coefficient that corresponds to *beV .

*

1 -

ce

be

I RV G

β∂ = ∂ ∆

( ) 2

-

1 -

e

eb e c e e

RG

RR r R R R RG G

β

β

=

+ + + + +

Using some of the approximations summarized above, we find:

*

-

c

b ebe

I GR RVG G

β

β∂

≈∂ +

- b eR R

ββ

=+

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The minus sign indicates that an increase in *beV 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 *

beV becomes:

( )* * *

*

c be be be

b ec b e c b e b

I V V VV I R R I R R I

ββ β

∂= ≤ ∂ + +

or

* * * *

*

c be be be be

be c e b c e e

I V V V VV I R I I R Vβ

∂≤ ≈ ≈ ∂

If we choose 2eV V= (as we did in our biasing procedure) and since * 0.7beV V≈ ,

then

* 1 3

be

e

VV

≈

and

*

*

1 3

c be

be c

I VV I

∂≤ ∂

We conclude that this bias circuit is relatively insensitive to changes in *beV when

we follow our biasing procedure, although we would have preferred that this sensitivity coefficient be smaller. Fortunately, *

beV 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 β because Cramer's delta is a function of these parameters and

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hence can no longer be treated as constant with respect to differentiation. Our strategy to simplify the calculation is to simplify cI 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 β because the derivative of even small terms can be large. Recall:

( )

( )

**1

1 2

2

- - -

1 -

c b ccb e cc e be

b bc

eb e c e e

I R VR r R V R VG G R R

IRR r R R R R

G G

β

β

+ + + + = + + + + +

After simplifying, we find:

**1

1 2

-

c b ccb cc be

b bc

b e

I R VR V VG G R R

I R RG G

β

β

+ + + ≈

+

Note that in the approximations, no small terms containing β or G have been discarded. Simplifying, we find:

* *1

1 2

-

b ccb cc c be

b bc

b e

R VR GV I VR R

IR R

β

β

+ + + ≈

+

or

*1*

1 2

-

b ccbe

b cc c b bc

e e

R V VR GV I R RI

R Rβ

+ + ≈ +

To calculate the sensitivity coefficient for G , we first calculate

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c b cc

e

I R VG Rβ

∂≈

∂

Thus, the sensitivity coefficient becomes

c b cc ccb

c e c e

I R V VG G GRG I R I Vβ β

∂≈ = ∂

But ~ 1bGR and / 1cc eV Vβ so that

∂∂

I G

G I

<< 1c

c

We therefore conclude that this bias circuit is stable with respect to variations in G .

To calculate the sensitivity coefficient for β , we first calculate

*

2

- cc cc b

e

GV II RRβ β

+∂ ≈∂

The minus sign indicates that an increase in β 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 β becomes:

*

b cc cc

c e c

R GV III R Iβ

β β

+ ∂ ≈ ∂

or

*

c b cc c

c e c

I R GV II R Iβ

β β ∂ +

≈ ∂

But

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1

b

e

RRβ

and

* 1cc c

c

GV II+

so that clearly

1

c

c

IIβ

β ∂ ∂

We conclude, therefore, that this bias circuit maintains a stable operating point despite any variations in β 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.)

( )

( )

**1

1 2

2

ln ln - - -

1- ln -

c b ccc b e cc e ce

b b

eb e c e e

I R VI R r R V R VG G R R

RR r R R R RG G

β

β

= + + + + + + + + +

Thus,

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( )

( )

*

**1

1 2

2

1 - - -

1 -

1 -

ccc

c

c c b ccb e cc e be

b b

c e

eb e c e e

IVI GI r I R VR r R V R V

G G R R

R RG

RR r R R R RG G

β

β

+∂=

∂ + + + + + + + + + + +

We can simplify this expression by using the approximations developed for our biasing procedure for this circuit:

*

*1

1 2

1 1 - * -

ccc

c

b ec c b ccb cc be

b b

IVI G GR RI r I R VR V V G GG G R R

ββ

+∂≈

∂ ++ + +

To find the sensitivity coefficient, we multiply this result by r :

*1

1 2*

-

c

b ccc b ebe

b bb

ccc

I r r rR Vr I R RV

R RRIG VG

ββ

∂≈ + ∂ +

+ + +

Because

1

1 2

- 0

b ccbe

b b

R V VR R

>+

then

1

c

c b b e

I r r rr I R R Rβ

∂< + ∂ +

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,

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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.

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