Easy Analog Circuit: Voltage divider circuit

Myself Stability factor S_β. My address is $S\beta = \frac{{dIc}}{{d\beta }}$ .
To view my history visit Stability factors & Biasing circuits

Request you to visit my Seniors also- Voltage divider bias - Stability factor 'S'

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Voltage divider circuit - stability factor (Sv)

Referring to Thevenin's equivalent of Voltage divider circuit(fig 1)- ( details are given in Voltage divider bias - Stability factor 'S' )- equation for Thevenins voltage is given below.

fig 1

${E_{TH}} = {I_B}{R_{TH}} + {V_{BE}} + {I_B}{R_E} + {I_{_C}}{R_E}$ -------(1) then,

${E_{TH}} = {V_{BE}} + {I_{_C}}{R_E} + {I_B}\left( {{R_E} + {R_{TH}}} \right)$ substitute for I_B

${E_{TH}} = {V_{BE}} + {I_{_C}}{R_E} + \frac{{{I_C}}}{\beta }\left( {{R_E} + {R_{TH}}} \right)$ re-arrange

$({E_{TH}} - {V_{BE}})\beta = {I_{_C}}{R_E}\beta + Ic\left( {{R_E} + {R_{TH}}} \right)$ differentiate this equation with respect to β

${E_{TH}} - {V_{BE}} = {R_E}\beta \frac{{d{I_{_C}}}}{{d\beta }} + {I_C}{R_E} + \frac{{dIc}}{{d\beta }}\left( {{R_E} + {R_{TH}}} \right)$ we have dIc/dβ = S_β

${E_{TH}} - {V_{BE}} = {R_E}\beta {S_\beta } + {I_C}{R_E} + {S_\beta }\left( {{R_E} + {R_{TH}}} \right)$ re-arrange

${E_{TH}} - {V_{BE}} = {S_\beta }({R_E}\beta + {R_E} + {R_{TH}}) + {I_C}{R_E}$

${S_\beta } = \frac{{{E_{TH}} - {V_{BE}} - {I_C}{R_E}}}{{{R_E}\beta + {R_E} + {R_{TH}}}}$ -------(2) Numerator of this equation can be substituted using equ(1).

From equation (1) we will get, ${E_{TH}} - {V_{BE}} - {I_{_C}}{R_E} = {I_B}{R_{TH}} + {I_B}{R_E}$ Substitute in (2).

${S_\beta } = \frac{{{I_B}{R_{TH}} + {I_B}{R_E}}}{{{R_E}\beta + {R_E} + {R_{TH}}}}$

${S_\beta } = \frac{{{I_B}({R_E} + {R_{TH}})}}{{{R_E}\beta + {R_E} + {R_{TH}}}}$ -------(3) divide numerator and denominator with R_TH+R_{E .}

${S_\beta } = \frac{{{I_B}}}{{1 + \beta \frac{{{R_E}}}{{{R_E} + {R_{TH}}}}}}$ -----------(4) Smelling some similarity with equation of 'S'. Verify equ(7).

Now we can follow conclusion for 'S' and also can substitute 'S' in the above equation (4).

${S_\beta } = \frac{{S{I_B}}}{{1 + \beta }}$ Substitute for I_Bin terms of Ic.

${S_\beta } = \frac{{SIc}}{{\beta (1 + \beta )}}$

Like 'S', S_βalso depends on values of R_TH and R_E.

For visiting other stability factors for voltage divider circuit - S and Sv .

Easy Analog Circuit

Saturday 20 September 2014

Voltage divider circuit - stability factor (Sβ)

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