BJT Small-Signal Analysis
Contents
Common-Emitter fixed-bias configuration Voltage divider bias CE Emitter bias Emitter-follower configuration Common-base configuration Collector-feedback configuration Hybrid equivalent circuit and model
3
Typical amplifier operation.
RB
RC
Q1
VCCVB(ac)
IB(ac)
VCE(ac)
IC(ac)
4
A generic dc load line.
IC
VCE
(sat)CC
CC
VI
R
(off )CE CCV V
CC CEC
C
V VI
R
5
RB
RC2 k
Q1
+12 V
VCE2 4 6 8 10 12
2
4
6
8
IC
IC(sat)
VCE(off)
Plot the dc load line for the circuit shown in Fig.
6
Plot the dc load line for the circuit shown in Fig. Then, find the values of VCE for IC = 1, 2, 5
mA respectively.
RB
RC1 k
Q1
+10 V
VCE2 4 6 8 10
2
4
6
8
IC
10IC (mA) VCE (V)
1 9
2 8
5 5
CE CC C CV V I R
7
Optimum Q-point with amplifier operation
βC BI I
CE CC C CV V I R
VCEIB = 0 A
IB = 10 A
IB = 20 A
IB = 30 A
IB = 40 A
IB = 50 A
IC
Q-Point
VCCVCC/2
IC(sat)
IC(sat)/2
IB
8
Base bias (fixed bias).
CC BEB
B
V VI
R
βC BI I
CE CC C CV V I R
RC
RB
+0.7 V
IC
IB
IE
Input
Output
VBE
VCC
Q1
= dc current gain = hFE
9
Example
RC2 k
RB360 k
+0.7 V
IC
IB
IEVBE
+8 V
hFE = 100
0.7V 8V 0.7V
360kΩ
20.28μA
CCB
B
VI
R
100 20.28μA
2.028mAC FE BI h I
8V 2.028mA 2kΩ
3.94V
CE CC C CV V I R
The circuit is midpoint biased.
10
Example
Construct the dc load line for the circuit shown in Fig, and plot the Q-point from the values obtained in Example Determine whether the circuit is midpoint biased.
VCE (V)2 4 6 8 10
1
2
3
4
IC (mA)
Q
(sat )
8V4mA
2kΩCC
CC
VI
R
off 8VCCCEV V
11
Example (Q-point shift.)
The transistor in Fig. 7.12 has values of hFE = 100 when T =
25 °C and hFE = 150 when T = 100 °C. Determine the Q-point values of IC and VCE at both of these temperatures.
RC2 k
RB360 k
+0.7 V
IC
IB
IEVBE
+8 V
hFE = 100 (T = 25C)hFE = 150 (T = 100C)
Temp(°C) IB (A) IC (mA) VCE (V)
25 20.28 2.028 3.94
100 20.28 3.04 1.92
12
Base bias characteristics. (1)
RC
RB
+0.7 V
IC
IB
IE
Input
Output
VBE
VCC
Q1 Advantage: Circuit simplicity.
Disadvantage: Q-point shift with temp.
Applications: Switching circuits only.
Circuit recognition: A single resistor (RB) between the base terminal and VCC. No emitter resistor.
13
Base bias characteristics. (2)
RC
RB
+0.7 V
IC
IB
IE
Input
Output
VBE
VCC
Q1
(sat )
(off )
CCC
C
CE CC
VI
R
V V
Load line equations:
Q-point equations:
CC BEB
B
C FE B
CE CC C C
V VI
R
I h I
V V I R
14
Voltage divider bias. (1)
R1
R2 RE
RC
+VCC
Input
Output
I1
I2 IE
IB
IC
Assume that I2 > 10IB.
2
1 2B CC
RV V
R R
0.7VE BV V
EE
E
VI
R
Assume that ICQ IE (or hFE >> 1). Then
CEQ CC CQ C EV V I R R
15
Which value of hFE do I use?
Transistor specification sheet may list any combination of the following hFE: max. hFE,
min. hFE, or typ. hFE. Use typical value if there is one. Otherwise, use
(ave) (min) (max)FE FE FEh h h
16
Stability of Voltage DividerBias Circuit
The Q-point of voltage divider bias circuit is less dependent on hFE than that of the base bias (fixed
bias).
For example, if IE is exactly 10 mA, the range of hFE is 100 to 300. Then
10mAAt 100, 100μA and 9.90mA
1 101E
FE B CQ E BFE
Ih I I I I
h
10mAAt 300, 33μA and 9.97mA
1 301E
FE B CQ E BFE
Ih I I I I
h
ICQ hardly changes over the entire range of hFE.
17
Load line for voltage divider bias circuit.
2 4 6 8 10 12
5
10
15
20
25
IC (mA)
VCE (V)
(sat )
10V20mA
260Ω+240ΩCC
CC E
VI
R R
(off ) 10VCE CCV V
Circuit values are from Example 7.9.
18
Emitter bias.
RC
RE
RB
IC
IE
IBQ1
Input
Output
+VCC
-VEE
0.7V
1EE
BB FE E
VI
R h R
19
Load Line forEmitter-Bias Circuit
(sat )
( )CC EE CC EEC
C E C E
V V V VI
R R R R
( )CE off CC EE CC EEV V V V V
VCE
IC
IC(sat)
VCE(off)
20
Emitter-bias characteristics
RC
RE
RB
IC
IE
IBQ1
Input
Output
+VCC
-VEE
Circuit recognition: A split (dual-polairty) power supply and the base resistor is connected to ground.
Advantage: The circuit Q-point values are stable against changes in hFE.
Disadvantage: Requires the use of dual-polarity power supply.
Applications: Used primarily to bias linear amplifiers.
21
Collector-feedback bias.
RB
RC
+VCC
IC
IE
IB
CC C B C B B BEV I I R I R V
( 1)CC BE
BFE C B
V VI
h R R
CQ FE BI h I
1CEQ CC FE B C
CC CQ C
V V h I R
V I R
22
Circuit Stability ofCollector-Feedback Bias
RB
RC
+VCC
IC
IE
IB
hFE increases
IC increases (if IB is the same)
VCE decreases
IB decreases
IC does not increase that much.
Good Stability. Less dependent on hFE and temperature.
23
Collector-FeedbackCharacteristics (1)
RB
RC
+VCC
IC
IE
IB
Circuit recognition: The base resistor is connected between the base and the collector terminals of the transistor.
Advantage: A simple circuit with relatively stable Q-point.
Disadvantage: Relatively poor ac characteristics.
Applications: Used primarily to bias linear amplifiers.
24
Emitter-feedback bias.
RB RC
+VCC
RE
IB
IE
IC
1CC BE
BB FE E
V VI
R h R
CQ FE BI h I
CEQ CC C C E E
CC CQ C E
V V I R I R
V I R R
1E FE BI h I
25
Circuit Stability ofEmitter-Feedback Bias
hFE increases
IC increases (if IB is the same)
VE increases
IB decreases
IC does not increase that much.
IC is less dependent on hFE and temperature.
RB RC
+VCC
RE
IB
IE
IC
26
Emitter-FeedbackCharacteristics (1)
Circuit recognition: Similar to voltage divider bias with R2 missing (or base bias with RE added).
Advantage: A simple circuit with relatively stable Q-point.
Disadvantage: Requires more components than collector-feedback bias.
Applications: Used primarily to bias linear amplifiers.
RB RC
+VCC
RE
IB
IE
IC
27
Emitter-FeedbackCharacteristics (2)
RB RC
+VCC
RE
IB
IE
IC
Q-point relationships:
( 1)CC BE
BB FE E
V VI
R h R
CQ FE BI h I
CEQ CC CQ C EV V I R R
• re transistor model – employs a diode and controlled current source to duplicate the behavior of a transistor in the region of interest.
• The re and hybrid models will be used to analyze small-signal AC analysis of standard transistor network configurations.
Ex: Common-base, common-emitter and common-collector configurations.
• The network analyzed represent the majority of those appearing in practice today.
BJT Small Signal Analysis
AC equivalent of a network is obtained by:
1. Setting all DC sources to zero
2. Replacing all capacitors by s/c equiv.
3. Redraw the network in more convenient and logical form
Common-Emitter (CE) Fixed-Bias Configuration
The input (Vi) is applied to the base and the output (Vo) is from the collector.
The Common-Emitter is characterized as having high input impedance and low output impedance with a high voltage and current gain.
Removing DC effects of VCC and Capacitors
Common-Emitter (CE) Fixed-Bias Configuration
re Model
Determine , re, and ro: and ro: look in the specification sheet for the transistor or test the transistor using a curve tracer.re: calculate re using dc analysis:
Ee I
26mVr
Common-Emitter (CE) Fixed-Bias Configuration
36
The Norton Equivalent Circuit
• Get the Norton Equivalent Circuit from the Thevenin by Source Transformation.
Impedance Calculations
Input Impedance: Output Impedance:
eBi r||RZ
eB ei r10RrZ
Or||RZ Co
c o 10roZ RRc
Common-Emitter (CE) Fixed-Bias Configuration
Gain Calculations
Voltage Gain (Av):
Current Gain (Ai):
Current Gain from Voltage Gain:
e
oC
i
ov r
)r||(R
V
VA
Coe
Cv 10Rrr
RA
)r)(RR(r
rR
I
IA
eBCo
oB
i
oi
eBCoi r10R ,10RrA
C
ivi R
ZAA
Common-Emitter (CE) Fixed-Bias Configuration
Voltage Gain
e
CvCo
e
oC
eb
oCbv
eb i
oCbO
i
Ov
r
RA 10Ror r if
r
)r||(R
βrI
)r||(RβIA
βrIV
)r||(RβIV
V
VA
Common-Emitter (CE) Fixed-Bias Configuration
Current gain
C
ivi
Bo
Bo
i
oi
eBCo
eBCo
Bo
i
oi
eB
B
Co
o
i
b
b
o
i
oi
eB
B
i
b
eB
iBb
Co
o
b
o
Co
boo
R
ZAA
ooequation t thisusecan or we
βRr
βRr
I
IA
,βr10R and 10R r if
βrRRr
βRr
I
IA
βrR
R
Rr
βr
I
I
I
I
I
IA
βrR
R
I
I and
βrR
IRI
Rr
βr
I
I and
Rr
βIrI
circuitsoutput andinput the toruledivider -current
theapplyingby determined isgain current The
Common-Emitter (CE) Fixed-Bias Configuration
Phase Relationship
The phase relationship between input and output is 180 degrees. The negative sign used in the voltage gain formulas indicates the inversion.
Common-Emitter (CE) Fixed-Bias Configuration
CE – Voltage-Divider Bias Configuration
re Model
You still need to determine , re, and ro.
CE – Voltage-Divider Bias Configuration
Impedance Calculations
Input Impedance: Output Impedance:
21
2121
RR
RRR||RR
er||RZi
oC r||RZo
C C 10RroRZo
CE – Voltage-Divider Bias Configuration
Gain Calculations
Voltage Gain (Av):
Current Gain (Ai):
Current Gain from Voltage Gain:
e
oC
i
ov r
r||R
V
VA
Coe
C
i
ov 10Rrr
R
V
VA
)rR)(R(r
rR
I
IA
eCo
o
i
oi
Coei
oi 10RrrR
Rβ
I
IA
eCoi
oi r10R ,10RrI
IA
C
ivi R
ZAA
CE – Voltage-Divider Bias Configuration
Voltage Gain
e
C vCo
e
oC v
oCe
io
e
ib
oCbO
r
RA 10Ror r if
r
)r ||(RA
)r ||(Rβr
VβV
βr
VI
)r ||)(RI (βV
CE – Voltage-Divider Bias Configuration
Current gain
e
eo
o
i
oi
Co
eCo
o
i
oi
B21
βrR'
βR'
βrR'r
rβR'
I
IA
,R10rfor
βrR'Rr
rβR'
I
IA
RR||RR'
format. same thehave
gain willcurrent for theequation the,R' the
for except ion,configurat bias-fixedemitter -
common that similar to so isnetwork thesince
CE – Voltage-Divider Bias Configuration
C
iVi
i
oi
i
oi
e
R
ZAA
optionan as
I
IA
R'
βR'
I
IA
,r10R' if And
CE – Voltage-Divider Bias Configuration
Phase Relationship
A CE amplifier configuration will always have a phase relationship between input and output is 180 degrees. This is independent of the DC bias.
CE – Voltage-Divider Bias Configuration
CE Emitter-Bias Configuration
Unbypassed RE
re Model
Again you need to determine , re.
CE Emitter-Bias Configuration
Impedance Calculations
Input Impedance: Output Impedance:
Eeb 1)R(rZ
)R(rZ Eeb
eE Eb rRRZ
bBi Z||RZ Co RZ
CE Emitter-Bias Configuration
Phase RelationshipA CE amplifier configuration will always have a phase relationship between input and output is 180 degrees. This is independent of the DC bias.
CE Emitter-Bias Configuration
Bypassed RE
This is the same circuit as the CE fixed-bias configuration and therefore can be solved using the same re model.
CE Emitter-Bias Configuration
Emitter-Follower Configuration
You may recognize this as the Common-Collector configuration. Indeed they are the same circuit. Note the input is on the base and the output is from the emitter.
re Model
You still need to determine and re.
Emitter-Follower Configuration
Impedance Calculations
Input Impedance:
bBi Z||RZ
Eeb 1)R(rZ
)R(rZ Eeb
Eb RZ
Emitter-Follower Configuration
Calculation for the current Ie
Ee
ie
eee
Ee
i
Ee
ie
b
b
ibe
b
ib
Rr
VI
rβ
βr1)β(
βr and
β1)β(but R1)β(
βrV
1)Rβ(βr
1)Vβ(I
gives for Z gsubtitutin
Z
V1)β(1)Iβ(I
Z
VI
Emitter-Follower Configuration
Impedance Calculations (cont’d)Output Impedance:
eEo r||RZ eE
eo rRrZ
Ee
ie Rr
VI
ionconfiguratfollower emitter for the impedenceoutput theDefining
Emitter-Follower Configuration
Gain CalculationsVoltage Gain (Av):
Current Gain (Ai):
Current Gain from Voltage Gain:
eE
E
i
ov rR
R
V
VA
EeEeEi
ov RrR ,rR 1
V
VA
bB
Bi ZR
RA
E
ivi R
ZAA
Emitter-Follower Configuration
Voltage gain
1V
VA
RrR
,ran greater thmuch usually R
rR
R
V
VA
rR
VRV
i
ov
EeE
eE
eE
E
i
ov
eE
iEo
Emitter-Follower Configuration
Current Gain
E
ivi
bB
Bi
bB
B
i
b
b
o
i
oi
b
o
beo
bB
B
i
b
bB
iBb
R
ZAAor
ZR
RA
,)1( since
ZR
R)1(
I
I
I
I
I
IA
)1(I
I
I)1(II
ZR
R
I
I
ZR
IRI
Emitter-Follower Configuration
Phase RelationshipA CC amplifier or Emitter Follower configuration has no phase shift between input and output.
Vo
Emitter-Follower Configuration
Common-Base (CB) Configuration
The input (Vi) is applied to the emitter and the output (Vo) is from the collector.
The Common-Base is characterized as having low input impedance and high output impedance with a current gain less than 1 and a very high voltage gain.
re Model
You will need to determine and re.
Common-Base (CB) Configuration
Impedance Calculations
Input Impedance: Output Impedance:
eEi r||RZ Co RZ
Common-Base (CB) Configuration
Gain Calculations
Voltage Gain (Av):
Current Gain (Ai):
e
C
e
C
i
ov r
R
r
R
V
VA
1I
IA
i
oi
Common-Base (CB) Configuration
Voltage & Current gain
e
C
e
C
i
oV
Ce
io
e
ie
Ce
CcCoo
r
R
r
Rα
V
VA
Rr
VαV
r
VI
RαI
)RI(RIV
1I
IA
III
II
i
oi
ieo
ie
Common-Base (CB) Configuration
Phase Relationship
A CB amplifier configuration has no phase shift between input and output.
Vo
Common-Base (CB) Configuration
Approximate Hybrid Equivalent Circuit
The h-parameters can be derived from the re model:
hie = re hib = rehfe = hfb = -hoe = 1/ro
The h-parameters are also found in the specification sheet for the transistor.
Hybrid equivalent model re equivalent model
Approximate Common-Emitter Equivalent Circuit
Hybrid equivalent model re equivalent model
Approximate Common-Base Equivalent Circuit
Chapter 7: BJT Transistor ModelingChapter 7: BJT Transistor Modeling
84
Disadvantages
• Re model– Fails to account the output impedance level of device
and feedback effect from output to input
• Hybrid equivalent model– Limited to specified operating condition in order to
obtain accurate result
85
VS
VCC
C1
C2
C3
+
-
Vo
RS
Vi
+
-RE
RCR1
R2
VS
+
-
Vo
RS
Vi
+
-
RCR1
R2
•I/p coupling capacitor s/c• Large values• Block DC and pass AC signal • Bypass
capacitor s/c•Large values
DC supply “0” potential
Voltage-divider configuration under AC analysis
Redraw the voltage-divider
configuration after removing dc supply and
insert s/c for the capacitors
• O/p coupling capacitor s/c• Large values• Block DC and pass AC signal
86
VS
RSR2 R1
Rc
Transistor small-signal ac
equivalent cct
Vo
Zi
Ii
Zo
Io
Vi
+ +
- -
B
E
C
Redrawn for small-signal AC analysis
Modeling of BJT begin
HERE!
VS
+
-
Vo
RS
Vi
+
-
RCR1
R2
87
AC bias analysis :
1) Kill all DC sources
2) Coupling and Bypass capacitors are short cct. The effect of there capacitors is to set a lower cut-off frequency for the cct.
3) Inspect the cct (replace BJTs with its small signal model:re or hybrid).
4) Solve for voltage and current transfer function, i/o and o/p impedances.
88
IMPORTANT PARAMETERS
• Input impedance, Zi
• Output impedance, Zo
• Voltage gain, Av
• Current gain, Ai
Input Impedance, Zi(few ohms M)
The input impedance of an amplifier is the value as a load when connecting a single source to the I/p of terminal of the amplifier.
89
VS Two-portsystem
Vi
Rsense
IiZi
+
-
Determining Zi
+
-
sense
isi
R
VVI
i
ii
I
VZ
Two port system-determining input impedance Zi
• The input impedance of transistor can be approximately determined using dc biasing because it doesn’t simply change when the magnitude of applied ac signal is change.
90
Demonstrating the impact of Zi
VS=10mVTwo-portsystem
Vi
Rsource
Zi
+
-
+
-1.2 kΩ
600Ω
mV6.6600k2.1
)m10(k2.1
RZ
VZV
Ω600R impedance, sourceWith
system the toapplied 10mV Full
0ΩR source, Ideal
sourcei
sii
source
source
91
Example 6.1: For the system of Fig. Below, determine the level of input impedance
VS=2mV Two-portsystem
Vi=1.2mV
RsenseZi
+
-
+
-
1 k Ω
A8.0k1
m8.0
k1
m2.1m2
R
VVI
sense
isi
:Solution
k5.18.0
m2.1
I
VZ
i
ii
92
Output Impedance, Zo (few ohms 2M)
The output impedance of an amplifier is determined at the output terminals looking back into the system with the applied signal set to zero.
Two-portsystem
Rsource
Vs=0V
Rsense
V
+
-
+
-
Io
Zo
Vo
Determining Zo
sense
oo
R
VVI
o
oo
I
VZ
cctopen become ZRZ oLo RLZo=Ro
Iamplifier
IRo
IL
RoL
Lo
II
RRFor
93
Voltage Gain, AV
• DC biasing operate the transistor as an amplifier. Amplifier is a system that having the gain behavior. • The amplifier can amplify current, voltage and power.• It’s the ratio of circuit’s output to circuit’s input.• The small-signal AC voltage gain can be determined by:
i
ov
V
VA
94
VS AvNLVi
Rsource
Zi
+
-
+
-Vo
+
-
Determining the no load voltage gain
By referring the network below the analysis are:
cct)(open ΩRi
oLvNL
V
VA
load no
vNLARZ
Z
V
VA
:resistance sourcewith
si
i
s
ovs
95
Current Gain, Ai
• This characteristic can be determined by:
i
oi
I
IA
BJTamplifier
Vi
Zi
+
-
Vo
+
-
Ii
RL
Determining the loaded current gain
Io
L
ivi
R
ZAA
Li
io
ii
Lo
RV
ZV
Z/V
R/V
L
oo
RV
I
96
re TRANSISTOR MODEL
• employs a diode and controlled current source to duplicate the behavior of a transistor.• BJT amplifiers are referred to as current-controlled devices.
Common-Base Configuration
Common-base BJT transistorre modelre equivalent cct.
97
E
BB
C
Common-base BJT transistor - pnp
Ic Ie
e
b b
c
ec I αI
IcIe
re model for the pnp common-baseconfiguration
e
b b
c
ec I αI
IcIe
common-base re equivalent cct
re
current emitter
of level DC the isII
26mVr E
E(dc)
e
isolation part,Zi=re
e
b b
c
A0Ic
IcIe=0A
Determining Zo for common-base
reVs=0V
Zo
Therefore, the input impedance, Zi = re
that less than 50Ω.
For the output impedance, it will be as follows;
98
The common-base characteristics
99
e
b b
c ec I αI Ie
re
Defining Av=Vo/Vi for the common-base configuration
BJT common-basetransistor amplifier
Vi Vo
+
-
+
-
Zi
oZ RL
Io
LeLcLoo RIRIRIV
e
L
e
Lv
r
R
r
RA
gain, Voltage
eeiei rIZIV
ee
Lev
rI
RI
Vi
VoA
100
1A
gain,Current
i
e
e
e
c
i
oi
I
I
I
I
I
IA
e
b b
c ec I αI Ie
re
Defining Ai=Io/Ii for the common-base configuration
BJT common-basetransistor amplifier
Vi Vo
+
-
+
-
ZioZ RL
Io
105
Common-Emitter Configuration
Common-emitter BJT transistorre modelre equivalent cct.Still remain controlled-current source (conducted between collector and base terminal)Diode conducted between base and emitter terminal
Input Output
Base & Emitter terminal Collector & Emitter terminal
106
common-emitter BJT transistor
EE
B
C
Ib
Icbc I I
c
e e
b
Ic
Ib
re model npn common-emitter configuration
bc I I
c
e e
b
Ic
Ii=Ib
Determining Zi using re equivalent model
re
Ie+
-
Vbe
+
-
Vi
(1) Ii
ViZi
gives (1) into subtitute
and IbreIereVbeVi
b
eb
b
be
I
rI
I
VZi
erZi 7k~6 to hundred between ranges iZ
107
The output graph
108
bI
c
e
bIi=Ib
re model for the C-E transistor configuration
rero
e
0AbI
c
e
bIi=Ib
rero
e
Vs=0V
= 0A
oZ
impedance)high cct,(open ΩZ
the thusignored is r if
rZ
o
o
oo
Output impedance Zo
109
e
b b
cbco I II Ii=Ib
re
Determining voltage and current gain for the common-emitter amplifier
BJT common-emittertransistor amplifier
Vi Vo
+
-
+
-
oZ RL
Io
ei rZ
e
Lv
r
RA
Ib
Ib
Ib
Ic
Ii
IoA
gain,Current
i
LbLcLoo RIRIRIV
ebiii rIZIV
eb
Lb
i
ov
rI
RI
V
VA
gain, Voltage
iA
110
Hybrid Equivalent Model
• re model is sensitive to the dc level of operation that result input resistance vary with the dc operating point
• Hybrid model parameter are defined at an operating point that may or may not reflect the actual operating point of the amplifier
111
Hybrid Equivalent Model
The hybrid parameters: hie, hre, hfe, hoe are developed and used to model the transistor. These parameters can be found in a specification sheet for a transistor.
112
Determination of parameter
0VVo
i12
0VVi
i11
o12i11i
o
o
VV
h
IV
h
VhIhV
0AIo
o22
0VVo
i21
o
o22i21O
o
o
VI
h
II
h
, 0VV Solving
VhIhI
H22 is a conductance!
113
General h-Parameters for any Transistor Configuration
hi = input resistancehr = reverse transfer voltage ratio (Vi/Vo)hf = forward transfer current ratio (Io/Ii)ho = output conductance
114
Common emitter hybrid equivalent circuit
115
Common base hybrid equivalent circuit
116
Simplified General h-Parameter Model
The model can be simplified based on these approximations:
hr 0 therefore hrVo = 0 and ho (high resistance on the output)
Simplified
117
Common-Emitter re vs. h-Parameter Model
hie = rehfe = hoe = 1/ro
118
Common-Emitter h-Parameters
[Formula 7.28]
[Formula 7.29]
acfe
eie
h
rh
119
Common-Base re vs. h-Parameter Model
hib = rehfb = -
120
Common-Base h-Parameters
1fb
eib
h
rh