Sedra-Smith 4.11 Basic Single-Stage BJT Amplifier Configurations Transistor Parameters IE & IC & IB = dc emitter & collector & base current α = IC/ IE β = IC/ IB α = β/(β + 1) re = VT/IE gm = IC/VT β = gmrπ rπ = (β + 1) re r o = VA/IC

VT = 25 mV at 290 K VA = Early Voltage Rb = Thevenin Equivalent Resistance seen looking out from base to ground May include bias resistors and output resistance of previous stage or signal source internal resistance (usually denoted as Rs) Rc = Thevenin Equivalent Resistance seen looking out from collector to ground

May include bias resistors and input resistance of next stage or load resistor (usually denoted as RL)

Re = Thevenin Equivalent Resistance seen looking out from emitter to ground May include bias resistors and either output resistance of previous stage or

signal source internal resistance (usually denoted as Rs) or input resistance of next stage or load resistor (usually denoted as RL)

RinQ = Thevenin Equivalent Resistance seen looking into BJT signal input terminal e. g. base for Common-Emitter or Common-Collector or

emitter for Common-Base RoutQ = Thevenin Equiv. Resistance seen looking into BJT signal output terminal e. g. collector for Common-Emitter or Common-Base or

emitter for Common-Collector

The Common Emitter (CE) Stage See Fig. 4.43 on p. 283. Re = 0 Rb = Rs Rc = RC Signal Input Terminal is Base and Signal Output Terminal is Collector In Section 4.11 it is assumed that rx = 0 Note that the equations in Sedra-Smith for the CE amplifier are in terms of rπ and not re. Input Resistance Ri = RinQ = rπ (4.59) Output Resistance Ro = RC// RoutQ = RC// ro where RoutQ = ro (4.67) vo/vs = vc/vb x vb/vs vc/vb = vc/vπ = - gm(RC// ro) (4.61) vb/vs = Ri /( Rs + Ri) = rπ/( Rs + rπ) (4.60) vo/vs = [- gm(RC// ro)] x [rπ/( Rs + rπ)] = -[β(RC// ro)] ÷ (Rs + rπ) (4.62) For rx > 0 See Fig. 4.70 on p. 323 Input Resistance Ri = RinQ = rπ + rx vc/vb = vc/vπ x vπ /vb = [- gm(RC// ro)] x [rπ/( rx + rπ)] = -β(RC// ro)/( rx + rπ) vb/vs = Ri /( Rs + Ri) = (rπ + rx)/( Rs + rx + rπ) vc/vs = -β(RC// ro)/( Rs + rx + rπ)

The Common Emitter (CE) Stage with a Resistance in the Emitter: Fig. 4.44 Re > 0 Rb = Rs Rc = RC Assume rx = 0 Signal Input Terminal is Base and Signal Output Terminal is Collector Output Resistance Ro = RC// RoutQ where RoutQ = ro[1 + (βRe)/(Re + rπ + Rs)] Derivation of RoutQ to be posted. The equation for RoutQ also includes a term Re //( rπ + Rs) that was neglected because it is small. Note that RoutQ can be much greater than ro. For that reason ro is ignored by Sedra-Smith in Fig. 4.44 (c) to simply the following calculations: Note that the equations in Sedra-Smith for the CE stage with a resistance in the emitter are in terms of re and not rπ. Equations will be presented in terms of both re and rπ in alternating order. The equation number in Sedra-Smith is given when available. For rx = 0 & ro = ∞ Input Resistance Ri = RinQ = (β + 1)(re + Re) (4.70)

Ri = RinQ = rπ + (β + 1)Re vo/vs = vc/vb x vb/vs

For rx = 0 & ro = ∞

vc/vb = -(αRC) ÷ (re + Re) (4.73) vc/vb = -[β/(β + 1) x RC] ÷ [re + Re] = -βRC ÷ [rπ +(β + 1)Re]

vb/vs = Ri /( Rs + Ri) For rx = 0 & ro = ∞ vb/vs = [(β + 1)(re + Re)] ÷ [Rs + (β + 1)(re + Re)] vb/vs = [rπ + (β + 1)Re] ÷ [Rs + rπ + (β + 1)Re] vo/vs = -βRC ÷ [Rs + (β + 1)( re + Re)] (4.75) vo/vs = -βRC ÷ [Rs + rπ +(β + 1)Re] If rx > 0 & ro = ∞

Ri = RinQ = rx + (β + 1)(re + Re)

Ri = RinQ = rx + rπ + (β + 1)Re

vc/vb = -βRC ÷ [rx + (β + 1)(re + Re)] vc/vb = -βRC ÷ [rx + rπ + (β + 1)Re] vb/vs = [rx + (β + 1)(re + Re)] ÷ [Rs + (β + 1)(re + Re)] vb/vs = [rx + rπ + (β + 1)Re] ÷ [Rs + rπ + (β + 1)Re] vo/vs = -βRC ÷ [Rs + rx + (β + 1)( re + Re)] vo/vs = -βRC ÷ [Rs + rx + rπ +(β + 1)Re]

The Common-Base (CB) Stage See Sedra-Smith Fig. 4.45 on p. 289 Rb = 0 Re = Rs Rc = RC Assume for rx = 0 Signal Input Terminal is Emitter and Signal Output Terminal is Collector Output Resistance Ro = RC// RoutQ where RoutQ = ro[1 + (βRs)/(Rs + rπ)] Note: a term Rs//rπ in the equation for RoutQ was neglected Note also that RoutQ can be much greater than ro. For that reason ro is ignored by Sedra-Smith in Fig. 4.45 (b) to simply the following calculations. Note that the equations in Sedra-Smith for the CB are in terms of re and not rπ. Equations will be presented in terms of both re and rπ in alternating order. The equation number in Sedra-Smith is given when available. For rx = 0 & ro = ∞

Input Resistance Ri = RinQ = re (4.77) Input Resistance Ri = RinQ = rπ /(β + 1)

vo/vs = vc/ve x ve/vs

vc/ve = (-αieRC)/(-ie re ) = αRC/re vc/ve = β RC/((β + 1)re ) = β RC/rπ

vb/vs = Ri /( Rs + Ri) vb/vs = re /( Rs + re) vb/vs = [rπ /(β + 1)] ÷ [ Rs + rπ /(β + 1)] = rπ ÷ [(β + 1)Rs + rπ]

vo/vs = [αRC/re] ÷ [Rs + re] (4.78) vo/vs =[β RC] ÷ [(β + 1)Rs + rπ]

For rx > 0 RoutQ = ro[1 + βRs/(Rs + rπ + rx)]

For rx > 0 & ro = ∞ Input Resistance Ri = RinQ = rx/(β + 1) + re

Input Resistance Ri = RinQ = (rx + rπ) /(β + 1) vo/vs = vc/ve x ve/vs

vc/ve = (-αieRC)/(-ie re ) = αRC/[re + rx/(β + 1)]

vc/ve = [β/(β + 1)] x RC ÷ [ re + rx/(β + 1)] = β RC/(rπ+ rx) ve/vs = Ri /( Rs + Ri)

ve/vs = [rx/(β + 1) + re] ÷ [ Rs + rx/(β + 1) + re]

vb/vs = [(rx + rπ) /(β + 1)] ÷ [ Rs +(rx + rπ) /(β + 1)] = (rx + rπ) ÷ [(β + 1)Rs + rx + rπ] vo/vs = [αRC] ÷ [ Rs + rx/(β + 1) + re]

vo/vs =[β RC] ÷ [(β + 1)Rs + rx+ rπ]

The Common-Collector (CC) Stage See Sedra-Smith Fig. 4.46 on p. 291 Rc = 0 Rb = Rs Re = RL Signal Input Terminal is Base and Signal Output Terminal is Emitter In Section 4.11 it is assumed that rx = 0 Note that the equations in Sedra-Smith for the CC are in terms of re and not rπ. Equations will be presented in terms of both re and rπ in alternating order. The equation number in Sedra-Smith is given when available. For rx = 0 & ro < ∞ Input Resistance Ri = RinQ = (β + 1)(re + RL// ro) (4.81) Input Resistance Ri = RinQ = rπ + (β + 1)x(RL// ro) For ro >> RL, Ri = RinQ = rπ + (β + 1)RL

Output Resistance Ro = RoutQ = ro//[ re + Rs/(β + 1)] (4.87) Output Resistance Ro = RoutQ = ro//[(rπ + Rs)/(β + 1)]

For ro >> (rπ + Rs)/(β + 1)

Ro = RoutQ = re + Rs/(β + 1) (4.88) Ro = RoutQ = (rπ + Rs)/(β + 1)

vo/vs = ve/vb x vb/vs ve/vb = [RL// ro] ÷ [re + RL// ro] (4.84) or ve/vb = [(β + 1)x(RL// ro)] ÷ [rπ + (β + 1)x(RL// ro)]

vb/vs = Ri /( Rs + Ri) vb/vs = [(β + 1)(re + RL// ro)] ÷ [Rs + (β + 1)(re + RL// ro)] (4.83) or vb/vs = [rπ + (β + 1)x(RL// ro)] ÷ [Rs + rπ + (β + 1)x(RL// ro)] vo/vs = ve/vs = [(β + 1)x(RL// ro)] ÷ [Rs + (β + 1)(re + RL// ro)] (4.85) also vo/vs = ve/vs = (RL// ro) ÷ [Rs/(β + 1) + re + (RL// ro)] (4.86) or vo/vs = ve/vs = [(β + 1)x(RL// ro)] ÷ [Rs + rπ + (β + 1)x(RL// ro)] If rx > 0 Input Resistance Ri = RinQ = rx + (β + 1)(re + RL// ro) or Input Resistance Ri = RinQ = rx + rπ + (β + 1)x(RL// ro) ve/vb = [RL// ro] ÷ [rx/(β + 1) + re + RL// ro] or ve/vb = [(β + 1)x(RL// ro)] ÷ [rx + rπ + (β + 1)x(RL// ro)] vb/vs = Ri /( Rs + Ri) = [rx + (β + 1)(re + RL// ro)] ÷ [Rs + rx + (β + 1)(re + RL// ro)] or vb/vs = [rx + rπ + (β + 1)x(RL// ro)] ÷ [Rs + rx + rπ + (β + 1)x(RL// ro)] vo/vs = ve/vs = (β + 1) x [RL// ro] ÷ [Rs + rx + (β + 1)(re + RL// ro)] Also vo/vs = ve/vs = (RL// ro) ÷ [(Rs + rx) /(β + 1) + re + (RL// ro)] or vo/vs = ve/vs = [(β + 1)x(RL// ro)] ÷ [Rs + rx + rπ + (β + 1)x(RL// ro)] For ro >> RL replace (RL// ro) by RL

Basic Single-Stage BJT Amplifier Equations: Hybrid-pi Model (rπ) Rb = Thevenin Equivalent Resistance seen looking out from base to ground Rc = Thevenin Equivalent Resistance seen looking out from collector to ground Re = Thevenin Equivalent Resistance seen looking out from emitter to ground IE & IC & IB = dc emitter & collector & base current α = IC/ IE β = IC/ IB α = β/(β + 1) re = VT/IE gm = IC/VT β = gmrπ rπ = (β + 1) re r o = VA/IC

VT = 25 mV at 290 K VA = Early Voltage CE CE Re > 0 CB CC

Input base base emitter base Output collector collector collector emitter

Re = 0 Re > 0 Rb = 0 Rc = 0 Rc = RC Rc = RC Rc = RC Re = RL

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Basic Single-Stage BJT Amplifier Equations: Tee Model (re) Rb = Thevenin Equivalent Resistance seen looking out from base to ground Rc = Thevenin Equivalent Resistance seen looking out from collector to ground Re = Thevenin Equivalent Resistance seen looking out from emitter to ground IE & IC & IB = dc emitter & collector & base current α = IC/ IE β = IC/ IB α = β/(β + 1) re = VT/IE gm = IC/VT β = gmrπ rπ = (β + 1) re r o = VA/IC VT = 25 mV at 290 K VA = Early Voltage CE CE Re > 0 CB CC

Input base base emitter base Output collector collector collector emitter

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CE Re = 0 CE Re > 0 CB CC Ro ro ro[1 +

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ro[1 + (βRs)/(Rs + rπ)]

ro//[(rπ + Rs)/(β + 1)]

Ri rπ + rx rx + rπ + (β + 1)Re

(rx + rπ) /(β + 1)

rx + (β + 1)(re + RL// ro)

vo/vi -β(RC// ro) ÷

( rx + rπ)

vc/vb = -βRC ÷ [rx + rπ + (β + 1)Re]

β RC/(rπ+ rx) [(β + 1)x(RL// ro)] ÷ [rx + rπ + (β + 1)x(RL// ro)]

vi/vs Ri /( Rs + Ri) Ri /( Rs + Ri) Ri /( Rs + Ri) Ri /( Rs + Ri) vo/vs (vo/vi) x (vi/vs) (vo/vi) x (vi/vs) (vo/vi) x (vi/vs) (vo/vi) x (vi/vs)