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Section 5.6
Small Signal Model
& Analysis
Quiz No 1 DE 27 (CE)
06-03-07
State the purpose and four steps (each) that
are to be taken for carrying out
(a)DC Analysis
(b) Small signal Analysis
The operation of the transistor as an
amplifier.
Conceptual circuit with the signal source eliminated .
(vbe =0)
DC Analysis
Signal source eliminated
vbe 0
I C ISe
IE
IC
IB
IC
VBE
VT
IE
IB
1
VC VCE VCC - I C R C
Active Mode Verification
V >V -0.4 V
The collector Current & Trans-conductance
vBE VBE vbe
iC I S e
iC I C e
v BE
VT
ISe
VBE
VT
e
vbe
VT
vbe
VT
vbe VT
vbe
iC I C 1
V
T
Valid for vbe 10m V
Sm all Signal
Approxim at
ion
The collector Current & Trans-conductance
IC
iC I C vbe
VT
DC Bias
Signal Component
IC
ic vbe
VT
ic g m vbe
IC
g m transconduc tance
DirectlyProportion
al to I C
VT
For vbe<< VT, the transistor behaves as a
voltage-controlled current device
The trans-conductance of the controlled source is gm
Output resistance is infinity
Linear operation of the transistor under the smallsignal condition:
Base Current & Input Resistance at the Base
IC
iC I C vbe
VT
iC
iC g m vbe
IC
IC
iB
vbe
VT
iB I B ib
IB
IC
I v
g v
ib C be m be
VT
r input resist ancebetween Base & Emit ter
Lookingint o t hebase
v
r be
ib
V
r
T
gm I B
ib g m
vbe
iC ib
vbe
r
ib
gm
ie
g m vbe
vbe
re
ie
gm
r 1re
Emitter Current & Input Resistance @ Emitter
iE
iC
IC
iC
iE I E ie
ie
iC
IC
I
vbe E vbe
VT
VT
re Small resistancebetween base & Emittter,
lookinginto theEmitterKnown as emitt erresistance
re
vbe VT
1
ie
I E gm gm
vbe ie re
vbe ib r ie re 1i re
r
iC
re 1re
ib
ib 1ie
rπ 1re
Voltage Gain
vC VCC RC iC
VCC RC I C ic
VCC RC I C RC ic
vC VC RC ic
vc RC ic g m RC vbe
vc
IC
Av
g m RC RC
vbe
VT
I C RC
VRC
Av g m RC
VT
VT
DC-AC Models
Large Signal Model
Small Signal Model
The amplifier circuit
Figure 5.51 Two slightly different versions of the simplified
hybrid- model for the small-signal operation of the BJT.
Figure 5.52 Two slightly different versions of what is
known as the T model of the BJT.
Small Signal Analysis
• Coupling Capacitors
– Couples the input signal vi to the emitter while
blocks the DC signals
– Don’t let dc biasing established by VCC &VEE be
disturbed. when vi is connected
– Capacitor is of very large value –infinite, acts
as short circuit at signal frequency of interest.
Application (Steps) : Small Signal Model
•
Suppress ac independent sources
–
–
–
•
ac Voltage Sources be short circuited
ac Current Sources be open circuited
Capacitors be Open circuited
Determine DC operating Point IC
Active Mode Verification
VBE > 0.7 V
VC> VB-0.4 V
I C ISe
IB
VBE
VT
IC
IE IB IC
Calculate
gm
IC
VT
ic g m vbe ib ie
r
re
gm
gm
Small Signal Analysis
•
Suppress DC independent sources
–
–
–
DC Voltage Sources be short circuited
DC Current Sources be open circuited
Capacitors be short circuited
•
Replace BJT with small signal Model
•
Analyze the resulting circuit of find voltage gain & input/output resistance
VT
IB
VT
IE
The Early Effect
• In real world
– (a) Collector current does show some
dependence on collector voltage
– (b) Characteristics are not perfectly
horizontal line
iC vCE
Figure 5.19 (a) Conceptual circuit for measuring the iC –vCE
characteristics of the BJT. (b) The iC –vCE characteristics of a
practical BJT.
The Early Effect
vCE
iC I S e 1
vA
Nonzero slopeindicatesthatoutput resistanceis not infinite
v BE
vT
iC
and defined as ro
vCE vBE Cons tan at
1
v A vCE
o
IC
I C & vCE are the valuesat operatingpoint
vA
o '
IC
I 'C I S e
v BE
vT
Figure 5.58 The hybrid-pi small-signal model, in
its two versions, with the resistance ro included.
The hybrid- small-signal model, with the resistance ro
included.
VA VCE VA
ro
IC
IC
VBE & I B and VCE & I C are DC bias values
vo -gm vbe ro || RC T husgain is slightly reduced
ro can be neglectedif ro 10RC
Problem 5.130
• Find the common-emitter amplifier shown in Fig.
P5.130, Let VCC =9V, R1 = 27kΩ, R2 = 15kΩ, RE
= 1.2kΩ, and Rc = 2.2kΩ. The transistor has β =
100 and VA = 100 V. Calculate the dc bias
current IE. If the amplifier operates between a
source for which Rsig = 10 kΩ and a load of 2kΩ
replace the transistor with its hybrid-Π model,
and find the value of Rm, the voltage gain vo
v sig
and the current gain io
ii
Figure P5.130
Figure P5.130
DC Analysis
Suppress the AC (independent Sources)
Short Circuit Voltage Sources
Open Circuit the Capacitors
Calculate DC Node Voltages & Loop Currents
Figure P5.130
9v
DC Analysis
IC
2.2k
IE
1.2k
IR
VBB
RB
vBE
β = 100 , α = 0.99
VA = 100V
IE = ?, Rin = ?, overall gain vo/vsig, io/i1
VCC 15k
VBB
3.21V
27 15k
15 27
RB
9.64k
15 27
Solution P5.130
• DC Values
IE
9v
VBB VBE
R
RE B
1
IC
IR
2.2k
1.92 mA
9.64 KΩ
3.21 0.7
1.94m A
9.64
1.2
101
I C 0.99 1.94 1.92m A
IE
RB
v BE
3.21 V
1.94 mA
IE
1.2k
Solution P5.130
• Check for Mode
9v
VB VBE VE 0.7 1.941.2 2.33V
IC
IR
VC VCC RC I C 9 2.2 1.92 4.776V
1.92 mA
9.64 KΩ
RB
v BE
VC VB 0.4 4.776 2.33 2.446V
3.21 V
ACTIVE MODE VCB > - 0.4 V
2.2k
1.94 mA
IE
1.2k
Solution P5.130
• Small Signal Model
v sig
10k
27k
15k
r
vbe
vo
g m vbe
ro
RC
RL
E
IC = 1.92 mA
VT = 25 mV
β = 100 , α = 0.99
VA = 100 V
gm
I C 0.991.94
76.8m A/ V
VT
0.025
100
r
1.3k
g m 76.8
ro
VA
52.1k
IC
Ri RB || r 1.15k
Ro ro || RC 2.11k
Solution P5.130
v sig
10k
27k
15k
r
vbe
vo
g m vbe
ro
E
vo vi vo
Av
vs vs vi
vi
Ri
1.15
vS Rsig Ri 10 1.15
Av
io
vo
-gm ro || RC || RL
vi
Ri
g m ro || RC || RL 8.13V / V
Rsig Ri
vo
RL
ii
vS
Rsig Rin
Ai
io
v Rsig Ri
o
45.3 A / A
ii
vS
RL
RC
RL
CE with pi Model
v sig
10k
27k
15k
r
vbe
E
vo
g m vbe
ro
RC
RL
CE with ‘T’ Model
Comparison ‘pi’ Vs ‘T’ Model
v sig
10k
27k
15k
r
vbe
E
vo
g m vbe
ro
RC
RL
Single Stage BJT Amplifier
• Three Configurations
– Common Emitter (CE)
• Common Emitter (CE) with Emitter Resistance
– Common Base (CB)
– Common Collector (CC)
Figure 5.60 (a) A common-emitter amplifier using
the structure of Fig. 5.59.
Amplifiers Configurations
Common Emitter
Amplifiers Configurations Common Emitter
DC Analysis
Suppress Independent ac Source
Voltage source ----- Short Cct
Current Sources --- Open
Capacitors ---- Open Cct
Redraw the Circuit
VC
Analysis
VB
IE=I
IC=αIE
IB=(β+1)IE
VC=VCC-ICRC
VB=-IBRB
VE=VB-VBE
VE
gm=Ic/VT
rл=β/gm
re=α/gm
Amplifiers Configurations Common Emitter
Small Signal Analysis
Suppress Independent DC Source
Voltage source ----- Short Cct
Current Sources --- Open
Capacitors ---- Short Cct
Redraw the Circuit by replacing BJT
With pi Model
Analysis
gm=Ic/VT
rл=β/gm
re=α/gm
Find Rin, Rout, Voltage Gain vo/vi
Common Emitter
+
vbe
-
Av
vo vo vbe
vi vbe vsig
vbe
RB || R
vsig Rsig RB || R
Rin=RB||rл
Rout=RC|RL
vbe
g m RL || RC
vsig
Short Circuit Current Gain Ais
Ais = ios/ii
ios=-gmvbe
vbe=vi=iiRin
Ais=-gmRin
RB || R
vo
g m RL || RC
vi
Rsig RB || R
Summary : CE
Ri RB || r ~ r
Input Resistance
Low to moderate typically a few kilohms
Output Resistance
Ro ro || RC ~ RC
Output Resistance is relatively low
Open Circuit Voltage Gain
Av
Ri
g m Ro || RL ~ g m ro || RC
Rsig Ri
Voltage gain of a few hundred
Short Circuit Current Gain
Ai
io
g v
m be g m r
vbe
ii
Rin
Current gain equal to β
Figure 5.61 (a) A common-emitter amplifier with an
emitter resistance Re.
Quiz No 2 (DE 27 CE)
13-03-2007
Redraw the circuit for DC
analysis
Redraw the circuit for Small
Signal pi model analysis
Figure 5.61 (a) A common-emitter amplifier with an
emitter resistance Re.
Figure 5.61 (a) A common-emitter amplifier with an
emitter resistance Re.
Small Signal Analysis : CE with Emitter Resistance
Input Resistance
Multiplication by a factor (1+β) is known as the
Resistance Reflection Rule.
Analysis
Small Signal Analysis : CE with Emitter Resistance
Voltage Gain
Voltage gain is lower than that of CE
because of the additional term (1+β)Re
Small Signal Analysis : CE with Emitter Resistance
Current Gain
Small Signal Analysis : CE with Emitter Resistance
Summary
Re introduces negative feedback gives it the name
emitter degenerative resistance
Comparison ‘T’ Vs ‘pi’ Model
A common-base amplifier
A common-base amplifier with its T model.
Small Signal Analysis : CB
CB has low input resistance
CB is non-inverting amplifier
Summary : CB
• Very Low input resistance
Rin=re
• Short Circuit Current Gain is nearly unity
• Open circuit Voltage Gain is equal to CE and is positive
gm RC
• Relatively high output resistance (Rc) same as CE
• Excellent high frequency performance
• As short circuit current gain is unity Current Buffer, it accept
an input signal current at a low input resistance and delivers
equal current at a very high output resistance.
An emitter-follower circuit : Common Collector
Non-unilateral Amplifier
Input Resistance depends upon RL
Output Resistance depends upon Rsig
Common Collector An emitter-follower circuit : T model
An equivalent circuit of the Emitter Follower - CC
An equivalent circuit of the Emitter Follower - CC
Overall Voltage Gain is less than unity:
RB>>Rsig, (β+1)(re+(ro||RL))>>(Rsig||RL)
The voltage at the emitter (vo) follows very closely the voltage at the input
thus give the circuit the name Emitter Follower
The emitter follower : Reflecting resistance into emitter
For RB>> Rsig & ro >> RL
Gain approaches Unity when Rsig/(1+β)<<RL89
Short Circuit Current Gain = 1+β
Common Collector : Output Resistance
Output Resistance is low
Summary : Common Collector
Non-unilateral Amplifier
Input Resistance depends upon RL
Output Resistance depends upon Rsig
High Input Resistance
Low out Resistance
Voltage Gain ≈ unity
Relatively Large Current = 1+β
An equivalent circuit of the emitter follower
BJT Configurations
Common Emitter
Common Emitter with Emitter Resistance
Common Base : Current Buffer
Common Collector : Voltage Follower
Summary & Comparison
Comparison of Transistor
Configurationsж
Quantity
Common
Emitter
(CE)
Common
Common
Collector
Base
(CC)
(CB)
AI
Current Gain
High (-50)
High (50)
Low (0.98)
AV
Voltage Gain
High (-136)
Low (0.99)
High (1.4)
Ri
Input Resistance
Medium (1 kΩ)
High (154 kΩ)
Low (21 Ω)
Ro
Output
Resistance
High (∞)
Low (80 Ω)
High (∞)
ж r = 1.1 kΩ, β = 50, R = R = 3kΩ
e
L
s
Problem 5.135
• The amplifier of Fig. P5.135 consists of two identical
common-emitter amplifier connected in cascade.
Observe that the input resistance of the second stage,
Rin2, constitutes the load resistance of the first stage.
– For Vcc = 15V, R1 = 100kΩ , R2 = 47kΩ , RE = 3.9kΩ , Rc = 6.8kΩ
, and β = 1000, determine the dc collector current and dc
collector voltage of each transistor.
– Draw the small-signal equivalent circuit of the entire amplifier
and give the values of all its components. Neglect ro1 and ro2
– Find Rin1 and vb1/vsig for Rsig = 5 kΩ
– Find Rin2 and vb2/vb1.
– For RL = 2kΩ , find vo /vb2
– Find the overall voltage gain vo /vsig
Figure P5.135
Solution P5-135
DC Analysis
Suppress the AC (independent Sources)
Short Circuit Voltage Sources
Open Circuit the Capacitors
Calculate DC Node Voltages & Loop Currents
•β=100, α=0.99
Solution P5-135
DC Analysis
47
VBB
15 4.8V
100 47
47 100
RB
32k
147
VBB VBE
IE
0.97m A
RB
RE
1
I C I E 0.99 0.97 0.96m A
VC VCC RC I C 15 0.96 6.8k 8.5V
gm
IC
3.8mA / V
VT
Small Signal Model
Suppress the DC (independent Sources)
Short Circuit Voltage Sources
Open Circuit Current Sources
Short Circuit the Capacitors
Draw the Small Signal Model
Small Signal Model
RS
ig
vb1
r 1
vi
Rin1
vo
C
RC1
g m vi 2
R B1
g m vi1
Rin 2
r 2
RC 2
RL
Small Signal Model
vb1
RS
v sig
r 1
vi
RC1
g m vi 2
R B1
r 2
RC 2
g m vi1
Rin1
Rin 2
RB1 RB 2 32k
r 1 r 2
vo
C
gm
2.6k
RC1 RC 2 6.8k
ro1 ro 2
Rin1 Rin 2 RB1 || r 2.4k
vi1 Rin1
0.32V / V
vsig Rin1 RS
vi 2 g m vi1 RC1 || Rin2
vi 2
68.1V / V
vi1
vo g m vi 2 RC 2 || RL
vo
59.3V / V
vi 2
vo vi1 vi 2 vo
. .
1292V / V
vS vS vi1 vi 2
RL
Figure P5.141 Common Base
For the circuit shown, Assume β=100
(a) Find the input resistance Rin
(b) Find the voltage gain vo/vsig
Figure P5.141 (Common Base)
DC Analysis
Suppress the AC (independent Sources)
Short Circuit Voltage Sources
Open Circuit the Capacitors
Calculate DC Node Voltages & Loop Currents
Figure P5.141 (Common Base)
DC Analysis
Calculate DC Node Voltages & Loop Currents
β =100
I = IB +IC=IE=0.33 mA
re
gm
IC
VT
VT
IE
VT
25
75
I E 0.33
IB
IE
IC
Figure P5.141 (Common Base)
Small Signal Analysis
Suppress the DC (independent Sources)
Short Circuit Voltage Sources
Open Circuit Current Sources
Short Circuit the Capacitors
Draw the Small Signal Model
Small Signal Analysis
C
Rin re 75
vo
v
i v
e e o
vsig vsig ve ie
ve
re
0.5V / V
vsig re Rsig
ie
1 1
re
75
ve
ic=αie
B
ie
ve
E
vo
( R B ||R L ) 0.99(100k || 1.5k)
ie
vo
9V / V
vsig
Rin
vo
Figure P5.143 Common Collector ( Emitter follower)
For the circuit shown, Assume β=40
(a) Find IE,VE,& VB
(b) Find the input resistance Rin
(c) Find the voltage gain vo/vsig
Figure P5.143 Common Collector ( Emitter follower)
Suppress the AC (independent Sources)
Short Circuit Voltage Sources
Open Circuit the Capacitors
Calculate DC Node Voltages & Loop Currents
Figure P5.143 Common Collector ( Emitter follower)
Calculate DC Node Voltages & Loop Currents
β=40
IE
VCC VBE
2.41mA
RB
RE
1
VE I E RE 2.41V
VB VE VBE 3.11V
VT
re
10.37
IE
Figure P5.143 Common Collector ( Emitter follower)
Small Signal Analysis
Suppress the DC (independent Sources)
Short Circuit Voltage Sources
Open Circuit Current Sources
Short Circuit the Capacitors
Draw the Small Signal Model
Small Signal Model P5-143
Small Signal Model P5-143
Ri RB || re ( 1)(RE || RL ) 17.3
vo
vb vo
vsig vsig vb
vb
vb
Ri
vsig Rsig Ri
vo
vo
Rc || RL
vb re ( Rc || Rl )
vo
Ri
Rc || RL
0.621 V / V
vsig Rsig Ri re ( Rc || Rl )
(β+1)ib
Ri
Figure P5.144
Problem
Problem
Problem 5.147
• For the circuit in Fig P5.147, called a bootstrapped follower:
– Find the dc emitter current and gm, re, and rΠ Use β =
100.
– Replace the BJT with its T model (neglecting ro), and
analyze the circuit to determine the input resistance
Rin and the voltage gain vo/vsig.
– Repeat (b) for the case when capacitor CB is open –
circuited. Compare the results with those obtained in
(b) to find the advantages of bootstrapping.
Boot-Strapped Follower
P5-147
Figure P5.147
DC Analysis
Suppress the AC (independent Sources)
Short Circuit Voltage Sources
Open Circuit the Capacitors
Calculate DC Node Voltages & Loop Currents
Figure P5.147
Calculate DC Node Voltages & Loop Currents
DC Analysis
9V
10k
10k
VE
4.5V
2k
Solution 5-147
9V
DC Analysis
10k
10k
VBB 4.5V
RB 10k
V VBE
I E BB
R
RC B
1
4.5 0.7
IE
1.73mA
10 10
2k
101
I C I E 1.71mA
VE
4.5V
2k
I C I E
gm
68.5mA / V
VT VT
re
r
VT
25
0.99
14.5
I C 1.73 g m 68.5
gm
1.46k
Figure P5.147
Small Signal Model
Suppress the DC (independent Sources)
Short Circuit Voltage Sources
Open Circuit Current Sources
Short Circuit the Capacitors
Draw the Small Signal Model
Figure P5.147
Small Signal Model
C
αie
B
re
E
Solution 5-147
C
B
αie
E
i
B
E
ie
Rin
αie
C
i
B
Solution 5-147
E
ie
vo
αie
C
vb
i
i (1 )ie
Rin
Rin
vb
vb
(14.48 1667) (1681.48)
vb
i (1 0.99)ie
(168148)
v
R r
Rin b E e ( 1)( RE re ) 168.148k
i
1
ie
v0 v0 vb RE Rin
0.93V / V
vsig vb vsig RE re Rsig Rin
Solution 5-147
Without Boot-Strap Capacitor
Rin
Rib
vb RE re
( 1)( RE re ) 203k
i
1
R (20k)
Rin ib
18.12k
Rib (20k)
Rib
v0 v0 vb RE Rin
0.64V / V
vsig vb vsig RE re Rsig Rin
The value of overall voltage gain and Rin obtained by using Bootstrap
capacitor is higher than cct ,without Bootstrapping
Bootstrapping is used to avoid loading of the input cct and to have higher gain.
Comparison of Transistor Configurations
Quantity
Common
Emitter
(CE)
Common
Common
Collector
Base
(CC)
(CB)
AI
Current Gain
High
High
Low
AV
Voltage Gain
High
Low
High
Ri
Input Resistance
Medium
High
Low
Ro
Out Resistance
High
Low
High