Analog Integrated Circuits (Jieh Tsorng Wu)

Then js can be expressed as 2 js where g wb qni wb pb

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Unformatted text preview: eVBE /UT − eVBC /UT where 2 JS ≡ BJT qni WB pb dx 0 Dn 2-11 Analog ICs; Jieh-Tsorng Wu Gummel Number (G) Dn is a weak function of x . Then, JS can be expressed as 2 JS = where G≡ WB qni WB pb dx 0 Dn pb(x )dx ≈ 0 2 qni D n = G WB NA(x )dx 0 • The Gummel number, G , is simply the dopant concentration per unit cross-sectional area of the base. • For a uniform base region, NA(x ) = NA, then G = WB NA. BJT 2-12 Analog ICs; Jieh-Tsorng Wu Base Transport Current The total minority carrier transport current across the base is 2 IT = JN × A = IS eVBE /UT − eVBC /UT where IS = JS × A = qni D n G ×A The transport current can be separated into forward and reverse components as IT = IS eVBE /UT − 1 − IS eVBC /UT − 1 = ICF + IE R • If VBE > 0 and VBC < 0, the device is biased in the forward-active region, IT = IS eVBE /UT • If VBE < 0 and VBC > 0, the device is biased in the inverse-active region, IT = IS eVBC /UT • If VBE > 0 and VBC > 0, the device is biased in the saturation region. BJT 2-13 Analog ICs; Jieh-Tsorng Wu Base Current In the forward-active region IB = IBB + IBE • IBB is due to the recombination of holes and electrons in the base. • IBE is due to the injection of holes from the base into the emitter. Define Qe as the minority carrier charge in the base region Qe = qA 2 WB nb(x )dx 0 or ni V /U 1 1 Qe = qAWB nb(0) = qAWB e BE T 2 2 NA IBB is related to Qe by the lifetime of minority carriers in the base, τb IBB BJT 2 Qe 1 qAWB ni = = · eVBE /UT τb 2 τb NA 2-14 Analog ICs; Jieh-Tsorng Wu Base Current IBE depends on the gradient of minority carriers (holes) in the emitter. • For a “long-base” emitter (all minority carriers recombine in the quasi-neutral region) with a diffusion length Lp IBE = qADp Lp peoeVBE /UT = qADp n2 i Lp ND eVBE /UT ND = Emitter Doner Density • For a “short-base” emitter (all recombination at the contact) with emitter width WE , WE simply replaces Lp in the expression for IBE . The total base current in the forward-active region is 2 qADp n2 1 qAWB ni i eVBE /UT + IB = 2 τB NA Lp ND • In modern narrow-base transistors IBE BJT IBB . 2-15 Analog ICs; Jieh-Tsorng Wu Forward Current Gain In the forward-active region, the forward current gain is βF ≡ IC IB 1 = 2 WB DWN 2τb Dn + Dp L B N A n P D The emitter current is IE = −(IC + IB ) = − IC + where IC =− αF IC βF IC βF 1 = = αF ≡ − = IE βF + 1 1 + 1 β F αT = 1 2 WB 1 + 2τ 1 2 WB 1 + 2τ γ= b Dn + 1 ≈ αT · γ DWN 1 + Dp L B N A n B Dn Dp WB NA Dn LP ND P D • αT is called the base transport factor, and γ is called the emitter injection efficiency. BJT 2-16 Analog ICs; Jieh-Tsorng Wu BJT DC Large-Signal Model in Forward-Active Region IB IB IC B C B IC VBE C VBE(on) IE IE E E IS V /U IB = e BE T βF IC = βF IB • The voltage on the emitter junction can be approximated by a constant VBE (on) . ◦ • VBE (on) is usually 0.6 V to 0.8 V, and has a temperature coefficient of −2 mV/ C. BJT 2-17 Analog ICs; Jieh-Tsorng Wu Dependence of βF on Operating Condition • At high currents, due to high-level injection IC → IS eVBE /(2UT ) • At low currents, due to recombination in the B-E depletion region IB → IS eVBE /(2UT ) BJT 2-18 Analog ICs; Jieh-Tsorng Wu Collector Voltage Effects In the forward-active region, an increase ∆VCE in VCE results in an increase in the collector depletion layer width, thereby reducing WB by ∆WB , and increasing IC . 2 IC = IS eVBE /UT = A ∂IC ∂VCE BJT qni Dn G eVBE /UT 2 = −A qni Dn G2 e VBE /UT 2-19 G = Gummel number IC d G dG · =− · d VCE G d VCE Analog ICs; Jieh-Tsorng Wu Collector Voltage Effects For a uniform-base transistor G = WB NA and ∂IC ∂VCE IC d WB =− · WB d VCE • d WB /d VCE is typically a weak function of VCE for a reverse biased collector junction and is often assumed to be constant. The Early voltage, VA, is given by VA = IC ∂IC /∂VCE = −WB 1 d WB /d VCE The influence of changes in VCE on IC can thus be represented as IC = IS e VBE /UT 1+ VCE VA • Typical values of VA are 15–100 V. BJT 2-20 Analog ICs; Jieh-Tsorng Wu Base Transport Model C IC IS /βR IT B IE IS /βF E IT = IS eVBE /UT − eVBC /UT IC = IT − BJT IS IS IE = −IT − eVBC /UT − 1 eVBE /UT − 1 βR βF IS IS IB = eVBE /UT − 1 + eVBC /UT − 1 βF βR 2-21 Analog ICs; Jieh-Tsorng Wu Ebers-Moll Model Recalling IT = IS eVBE /UT − eVBC /UT IC = IT − IS eVBC /UT − 1 IE = −IT − IS eVBE /UT − 1 βR βF SPICE uses the base transport model with the equations rewritten as: IC = IS e VBE /UT − 1 − IS IE = −IS 1 1+ βF e VBE /UT 1 1+ βR e VBC /UT − 1 −IS e VBC /UT − 1 = IS e VBE /UT IS −1 − eVBC /UT − 1 αR IS −1 =− eVBE /UT − 1 −IS eVBC /UT − 1 αF • Note that, in the classical Ebers-Moll model, parameters IE S and ICS are defined such that αF IE S = αR ICS = IS BJT 2-22 Analog ICs; Jieh-Tsorng Wu Leakage Current VBE /UT In the forward-active region, e IC ≈ IS e VBE /UT 1 and e IS + αR VBC...
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