Easy Analog Circuit: 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 R_E provides stabilization. Voltage drop across R_E 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 V_CE and Ic together decides the operating point. Equation for V_CE is V_CE = Vcc - Ic(Rc + R_E ). This does not contains β term and so V_CE 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 R_TH and then E_TH.

fig 4

In order to calculate R_TH , 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. R_TH =R1 ll R2.

Then E_TH 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 R_TH and voltage E_TH 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 I_B.

           Apply KVL,     E_TH - I_BR_TH - V_BE- I_ER_E= 0 ----(1) re-writing,

    ${I_B} = \frac{{{E_{TH}} - {V_{BE}}}}{{{R_{TH}} - (1 + \beta ){R_E}}}$    -----------(2)

We have a general equation between I_Band I_E

_{-----------(3)}Using equ. (2) and (3) we can write an equ. for I_E .

    ${I_E} = \frac{{{E_{TH}} - {V_{BE}}}}{{\frac{{{R_{TH}}}}{{1 + \beta }} + {R_E}}} \simeq {I_c}$ -------(4) ( Because I_E 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 _βare 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.

E_TH = I_BR_TH + V_BE+ (I_B+ I_c)R_E ----(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 V_BE and E_THas 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 (R_TH/ R_E) +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, (R_TH/ R_E) and β('1' is very less compared to them).
Minimum value of 'S' is '1' for small values of (R_TH/ R_E).
Maximum value of 'S' is 1+β for large (R_TH/ R_E).

## When (R_TH/R_E)˂˂1. [1+ β][(R_TH/R_E)+1] → 1+ β and 1+(R_TH/R_E)+β → 1+ β
When (R_TH/R_E)˃˃ 1 . [1+ β][(R_TH/R_E)+1] → [1+ β](R_TH/R_E) and 1+(R_TH/R_E)+β → (R_TH/R_E) ##

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 (R_TH/ R_E). So R_TH should be small and R_E should be large. We can conclude that value of 'S' is determined by R₁,R₂ and R_E 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 R_E .

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.

Easy Analog Circuit

Saturday 13 September 2014

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

Stabilization in voltage divider circuit

Stability Factors

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