Saturday 13 September 2014

Voltage divider bias - Stability factor 'S'(Stabilization in voltage divider circuit)


Various biasing circuits are designed for amplifiers(a refreshment is provided here). One which is familiar to you is Voltage divider bias, also known as Self bias. Here, we are discussing about various stability factors in self bias.
First, consider diagram for voltage divider bias :
fig 1
This is the commonly used biasing circuit as it provides good bias stability. It is connected in Common Emitter configuration. In order to connect as an amplifier, Emitter-base junction is to be
forward biased and Emitter-collector junction to be reverse biased. R1-R2 network provides the necessary base voltage. The emitter resistance RE provides stabilization. Voltage drop across RE ensures reverse bias across Emitter junction.

 In order to understand how stabilization is obtained in potential divider circuit,  lets do the analysis and derive equations for stability factors.

In amplifiers, stabilization means keeping the operating point fixed against variations in temperature and transistor parameters. Both VCE and Ic together decides the operating point. Equation for VCE is   VCE = Vcc - Ic(Rc + RE ). This does not contains β term and so VCE is independent of variations in β. So we need to bother only about variations in operating point due to changes in Ic

Stabilization in voltage divider circuit

Take only input side of fig 1 into consideration as shown below(inside green line) and find Thevenin's equivalent. This will reduce complexity of analysis.
fig 2

 Redraw the input side, we will get
fig 3
Consider Thevenin equivalent model of input side( left side of Blue line).First find out RTH and then ETH.
fig 4
In order to calculate RTH , replace voltage source(Vcc) by a short circuit in fig 3. Then look into the circuit as shown in figure 4. We can see that R1 is parallel to R2. RTH =R1 ll R2.
Then ETH is the voltage across R2 in fig 3.
fig 5
{E_{TH}} = \frac{{{R_2}Vcc}}{{{R_1} + {R_2}}}

In fig 3, replace the section for which we are taking Thevenin's (left side of blue line) with Thevenin equivalent resistance RTH and voltage ETH obtained.

fig 6
With this simplified circuit, lets do some analysis about its stability. For that find an equation for Ic.
For that, first write an equation for IB.

           Apply KVL,     ETH - IBRTH  - VBE - IERE = 0  ----(1)     re-writing,
                                                       
      {I_B} = \frac{{{E_{TH}} - {V_{BE}}}}{{{R_{TH}} - (1 + \beta ){R_E}}}       -----------(2)                  

We have a general equation between IB and IE  

      {I_B} = \frac{{{I_E}}}{{1 + \beta }}      -----------(3)       Using equ. (2) and (3) we can write an equ. for IE .

    {I_E} = \frac{{{E_{TH}} - {V_{BE}}}}{{\frac{{{R_{TH}}}}{{1 + \beta }} + {R_E}}} \simeq {I_c}     -------(4)     ( Because IE and Ic are nearly equal)

Stability Factors

Next we are going to find out equations of Stability factors for voltage divider circuit. In this post, i am including the first stability factor 'S' only. Sv and S β are described in other posts.
Just recollect the definition: Stability factor (S) is defined as the rate of change of collector current(Ic) with respect to the reverse saturation current(Ico), keeping β and VBE  constant.




{\rm{S = }}\frac{{dIc}}{{dIco}}  at constant β and VBE .     
Consider fig 6 for simplicity. Take euq (1) and modified as follows.

           ETH = IBRTH  + VBE + (IB+ Ic)RE  ----(5)

Consider equ for stability factor S, from previous post 'Stability factors & Biasing circuits'.

 i) Stability factor (S)

\therefore S = \frac{{(1 + \beta )}}{{1 - \beta \frac{{dIB}}{{dIc}}}}     Here we have a term dIB/dIc. We will get an equ for substituting for this if                                       we differentiate equ(5) with respect to Ic(take VBE and ETH as a constants).
From (5), 
                0 = \frac{{d{I_B}}}{{d{I_c}}}{R_{TH}} + \frac{{d{I_B}}}{{d{I_c}}}{R_E} + {R_E}   re-arranging we will get,
             
           \frac{{d{I_B}}}{{d{I_c}}} = \frac{{ - {R_E}}}{{{R_{TH}} + {R_E}}}    --------(6)    Put this in equ for 'S'.

              S = \frac{{1 + \beta }}{{1 + \beta \frac{{{R_E}}}{{{R_{TH}} + {R_E}}}}}    -------(7)   take denominator and re-arrange as

              S = \frac{{1 + \beta }}{{1 + \beta \frac{1}{{\left( {({R_{TH}}/{R_E}) + 1} \right)}}}}    -------  multiply denominator and numerator by (RTH / RE) +1


             S = \frac{{[1 + \beta ][({R_{TH}}/{R_E}) + 1]}}{{({R_{TH}}/{R_E}) + 1 + \beta }}   ------------ (8)

Value of 'S' depends on two factors of the equation, (RTH / RE) and β('1' is very less compared to them).
          Minimum value of  'S' is '1' for small values of (RTH / RE).
           Maximum value of 'S' is 1+β for large  (RTH / RE).

## When  (RTH/RE)˂˂1.    [1+ β][(RTH/RE)+1]  →     1+ β       and   1+(RTH/RE)+β   →   1+ β  
     When  (RTH/RE)˃˃ 1 .      [1+ β][(RTH/RE)+1]  →   [1+ β](RTH/RE)  and  1+(RTH/RE)+β   →   (RTH/RE)  ##


In the previous post 'Stability factors & Biasing circuits'  it is given that for proper stability of an amplifier, 'S' should be small. For voltage divider circuit, we obtained that 'S ' is small for small values of (RTH / RE). So RTH should be small and RE should be large. We can conclude that value of 'S' is determined by R1,R2 and RE  resistors.
While designing a voltage divider circuit, for good stability against variations in Ic with respect to Ico (see definition of 'S'), we should select small values R1 and R2  and high value RE .

In this post,the stability factor 'S' for the voltage divider circuit is discussed. To get an idea about other stability factors for voltage divider circuit - visit Sv and S β .           
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