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 = g m rπ
VT = 25 mV at 290 K
rπ = (β + 1) re
r o = VA/IC
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
Re = 0
Rb = Rs
See Fig. 4.43 on p. 283.
Rc = R C
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 = R C
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)]
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]
(4.75)
The Common-Base (CB) Stage
See Sedra-Smith Fig. 4.45 on p. 289
Rb = 0
Assume for rx = 0
Re = Rs
Rc = R C
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
Rc = 0
Rb = Rs
See Sedra-Smith Fig. 4.46 on p. 291
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]
or ve/vb = [(β + 1)x(RL// ro)] ÷ [rπ + (β + 1)x(RL// ro)]
(4.84)
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 = g m rπ
rπ = (β + 1) re
r o = VA/IC
VT = 25 mV at 290 K
VA = Early Voltage
Input
Output
CE
base
collector
Re = 0
Rc = R C
Rb = Rs
CE Re > 0
base
collector
Re > 0
Rc = R C
Rb = Rs

β Re 
ro 1 +

+
+
R
r
R


e
s
π


CB
emitter
collector
Rb = 0
Rc = RC
Re = Rs

β Rs
ro 1 +

 R s + rπ
CC
base
emitter
Rc = 0
Re = R L
Rb = Rs




 Rs + rx + rπ 
ro // 

 β+ 1 
Ro
ro
Ri
rπ + r x
rx + rπ + (β + 1)Re
vo
vi
− β(RC //ro )
rx + rπ
− βRC
rx + rπ+ ( β+ 1)Re
β RC
rx + rπ
vi
vs
Ri
Ri + Rs
Ri
Ri + Rs
Ri
Ri + Rs
Ri
Ri + Rs
vo
vs
vo vi
x
vi vs
vo vi
x
vi vs
vo vi
x
vi vs
vo vi
x
vi vs
 rx + rπ 


 β+ 1 
rx + r π +
(β+1)(RL// ro)
(β+ 1)(RL//ro)
rx+ rπ+ (β+ 1)( RL//ro)
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 = g m rπ
rπ = (β + 1) re
r o = VA/IC
VT = 25 mV at 290 K
VA = Early Voltage
CE
base
collector
Re = 0
Rc = R C
Rb = Rs
CE Re > 0
base
collector
Re > 0
Rc = R C
Rb = Rs
CB
emitter
collector
Rb = 0
Rc = RC
Re = Rs
Ro
ro


β Re
ro 1 +

 Re + (β + 1)re + Rs 


β Rs
ro 1 +

 R s + ( β + 1) re 


Ri
rx + (β + 1)re
rx + (β + 1)(re +Re)
vo
vi
− β(RC //ro )
rx + (β + 1) re
− βRC
rx + (β+ 1)(re + Re )
β RC
rx + (β + 1)re
vi
vs
Ri
Ri + Rs
Ri
Ri + Rs
Ri
Ri + Rs
Ri
Ri + Rs
vo
vs
vo vi
x
vi vs
vo vi
x
vi vs
vo vi
x
vi vs
vo vi
x
vi vs
Input
Output
 rx 

+ re
β + 1 
CC
base
emitter
Rc = 0
Re = R L
Rb = Rs
R + r

ro //  s x + re 
 β+ 1

rx +
(β+1)( re +RL// ro)
(β+ 1)(RL//ro)
rx+ (β+ 1)( re+ RL//ro)
Ro
CE Re = 0
ro
Ri
rπ + r x
vo/vi
vi/vs
vo/vs
CE Re > 0
ro[1 +
(βRe)/(Re + rπ
+ Rs)]
rx + rπ + (β +
1)Re
vc/vb = -βRC ÷
[rx + rπ + (β +
1)Re]
CB
ro[1 +
(βRs)/(Rs +
rπ)]
(rx + rπ) /(β +
1)
β RC/(rπ+ rx)
CC
ro//[(rπ +
Rs)/(β + 1)]
rx + (β + 1)(re
+ RL// ro)
[(β + 1)x(RL//
-β(RC// ro)
ro)] ÷ [rx + rπ
÷
+ (β + 1)x(RL//
( rx + rπ)
ro)]
Ri /( Rs + Ri) Ri /( Rs + Ri)
Ri /( Rs + Ri)
Ri /( Rs + Ri)
(vo/vi) x (vi/vs) (vo/vi) x (vi/vs) (vo/vi) x (vi/vs) (vo/vi) x (vi/vs)