Download 2011. Lecture 2

Lecture 2: Introduction to electronic analog circuits 361-1-3661
1
2. Elementary Electronic Circuits with
a BJT Transistor
© Eugene Paperno, 2008
Our main aim in the two next lectures is to build all the
possible practical circuits (amplifiers) by using a BJT
transistor and a resistor. (We use the resistor to translate the
output current of the circuit into voltage; otherwise the circuit
will not be able to provide a voltage gain.) We then analyze
and compare the circuits' small-signals gains to understand for
what applications they can be suitable. We are particularly
interested in the applications where there is a need to amplify
power and dc signals.
In this lecture, we develop all the models for the transistors
− as we did this for the diode − and then will build and analyze
− with the help of these small-signal models − all the possible
single-transistor amplifiers.
C
p
n
B
B
p
n
n
E
p
E
Injection
Extraction
W
W'
n++
p+
n Bo e
E
VB ' E / V
n
*
T
*
iE
iB
2.1. BJT transistor: symbol, physical structure,
analytical model, and graphical characteristics
The symbols of the npn and pnp BJT transistors and the
physical structure of the npn transistor are given in Fig. 1. We
will analyze in the lectures only npn transistors. The only
difference between the npn and pnp transistors is in their static
states: the static state of the pnp transistors is reverse to that of
the npn ones because of their opposite structures. There will be
no difference in the small-signal behavior and models. The
circuits analyzed in home exercises, the lab, and the exam will
comprise both npn and pnp transistors.
In analog circuits, the operating point of transistors is
usually defined in active (linear) region, where the emitter
junction is forward biased and the collector junction is reverse
biased. Thus, the emitter injects the electrons into the base,
and the collector collects them. The amount of the injected
electrons is controlled by the emitter-base voltage, vBE (or
base-to-emitter current, iB). The collector collects almost all
the electrons from the base if its potential is sufficiently high:
is greater or equal to that of the base. The base is very thin and
the electrons prefer entering the collector − even its potential
equals that of the base − and not the base, because the
resistance that they see looking into the base is much greater
than that they see looking into the collector.
To define the operating point of the transistor in active
region, we ground the emitter and bias the transistor junctions
with a current and voltage source as shown in Fig. 1. A single
transistor circuit (with no other components, except
independent sources) with grounded emitter is called the
common-emitter configuration. Although we develop all the
models of the transistor for the common-emitter
C
iC
iR
pCo
nBo
pEo
B'
B
vCE
VCE >VBE + 180 mV
rB
iB
C
e180/26 = 1015.44
V'CE >VCE
VCE=VBE
I'B= 0.5 IB; I'C= 0.5 IC
* VCE = 0
Fig. 1. Symbol of the n-p-n and p-n-p BJT transistors and the physical
structure of the npn transistor. Note that for a fixed iB, v'BE is also fixed.
configuration, they can also be used (see the Appendix) for
any transistor in a circuit, no matter which terminal of the
transistor is grounded (if at all).
Analytical model: transistor equations
Let us first write the equations for the transistor current
based on the concentrations of the minor charge carriers in
Fig. 1:
iC  Dn q
nBoe v BE / VT
ABE  iR
w
iC  i R
.
n
 Dn q Bo ABE e v BE / VT  I CS e v BE / VT
w


I CS
(1)
Lecture 2: Introduction to electronic analog circuits 361-1-3661
iC
VCE
IC
2
iC
Q
iC
VCE
IC
Q
IB
IC
Q
hoe
gm
hfe
0.5IB
vCE
0.5 IC
0.5 hoe
vCE
I"B I'B
IB
iB
VBE
iB
1/ro
vBE
 VA
VCE
0.7 V
I"B
V'CE
vCE
vBE
VCE
1/hie
Q
Q
IB
VBE
IB
hre
vCE
VBE
iE
vBE
VCE
V'CE
vCE
VCE
iC
Q
IE
iB
1/re
vCE
C
B
vCE
iB
vBE
VBE
iE
E
vBE
Fig. 2. Common-emitter characteristics of an npn transistor.
iB 
D p q ABE pEo
L pE




i BS
n
Dn q Bo ABE e v BE / VT
iC
w
F  
D p q ABE pEo v BE / VT
iB
e
L pE
( e v BE / VT  1)
i B  i BS
.
(4)
(2)
 I BS e v BE / VT

iE  iC  iB  I CS e v BE / VT  I BS e v BE / VT
D p pEow
(3)
F 
I ES
We now can define the static current gains
 1
n Bo  p Eo ; L pE W
and
.
 ( I CS  I BS ) e v BE / VT  I ES e v BE / VT

Dn nBo L pE
iC
iC
i /i

 C B
i E i B  iC 1  iC / i B
.

F
1 F
1
 F 1
(5)
Lecture 2: Introduction to electronic analog circuits 361-1-3661
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Note that according to (4), the transistor iC-iB characteristic
should be a linear one (see Fig. 2), of course, provided that F
is constant (in a real transistor, F depends on iB, but we will
neglect this in our theory). It is also apparent from (1)-(3) that
the iC-vBE, iB-vBE, and iE-vBE characteristics are exponential.
Since according to (1), the collector current is a function of
the base width, w, and w decreases with increasing vCE, the
transistor output characteristics have a slope that is
proportional to IC. (This is unlike the Ebers-Moll model, where
the transistor output characteristics are horizontal.) Indeed,
 iC
w
Q
n


  Dn q Bo ABE e v BE / VT 
w



w
Having all the needed transistor characteristics, we can
define the small-signal gains as the slopes of the characteristic
at their operating points.
The small-signal current gains
ic
ib
h fe 
f 
ic
ie

Q , v ce  0
ic
ib  ic
 F ,
(7)
Q , vce 0

Q , v ce  0
ic / ib
1  ic / ib
Q , v ce  0
.
Q

1
n
1
 Dn q Bo ABE eVBE / VT  I C
W
W
W
  iC 
Small-signal parameters
w
IC .
W
h fe

1  h fe
F
 F
1  F
The small-signal conductance and resistance of the emitter
(6)
1
i
 e
re vbe

v ce  0
I ES eVBE / VT
I
 E
VT
VT
.
For
w
 vCE

:
W
VCE
 iC 
(8)
 vCE
I C   iC  I C .
VCE
Due to the linear dependence of the slope of the output
characteristics on IC, their extrapolations meet at the one and
the same point on the vCE axis, so-called Early voltage, VA.
When vCE increases, the base width w decreases, and the
base resistance, rB, increases. Therefore, the static VBE voltage
should increase for the same static bias current IB (see the iBvBE and vBE-vCE characteristics in Fig. 2). As a result, the iB-vBE,
characteristic decreases a bit with increasing VCE. Since
decreasing w causes much more substantial increase in iC and
iE than in vBE, the iC-vBE and iE-vBE characteristics increase with
increasing vCE.
The effect associated with the change (modulation) of the
base width by the collector voltage, vCE, and with the
corresponding behavior of the transistor characteristics is
called Early effect.
re 
VT
IE
(9)
 26 
300 K, I E  1 mA
The small-signal (mutual) conductance gain
gm 
ic
vbe

Q , v ce  0
I CS eVBE / VT
I
 C,
VT
VT
.
gm 
ic
vbe

 f ie
v ce  0
vb' e

v ce  0
(10)
f
re
The small-signal input conductance and resistance
1
i
 b
hie vbe


ic / h fe
v ce  0
ie
(1  h fe )v'be
hie  (1  h fe )re
vbe

v ce  0

v ce  0
 f ie / h fe
vbe
v ce  0
1
;
(1  h fe )re
300 K, I E  1 mA, h fe 100
. (11)
 2.6 k
The small-signal output conductance and resistance ("r-out",
not "r-zero")
Lecture 2: Introduction to electronic analog circuits 361-1-3661
4
IC
i'c
C
hfeib
gmv'be
ic
IB
IC
i"c
B
IB
rb
B'
i"C
VCE
ib
ib
re
hrevce
ro
hrevce
IB
E
i"C
i'C
iC
hfeib
gmv'be
i"C
ib
E
VBE+ v'be
IE
B
Fig. 3. "Large"-signal equivalent circuit (model) for the transistor. Note that
another VCE source is added to cancel the effect of the static collector-toemitter voltage, VCE, on the current through ro. Thus, only the small-signal
collector-to-emitter voltage, vce, generates the small-signal current through ro,
which is in accordance with the Early effect. Note also that alternating the
polarity of the vs source causes the corresponding alternating the polarity of
the hfeib source.
1
i
 c
roe vce

Q , ib  0
VA  VCE
IC
rb
B'
ib
C
B
(1+hfe) re
v'BE
hrevce
IC
VA  VCE
1+hfe
hfe
hrevce
vce
re
ro
(b)
E
rb
. (12)
ro 
(a)
vce
i'c
IB
hoe 
ro
re
V
ib
hfeib
gmv'be
C
B'
v'BE
IC
ib
iC
vce
VCE
IB
B
rb
i'C
 100 k
i'C
B'
ib
v'BE
C
i"C
hie
I C  1 mA, V A 100 V VCE
iC
hfeib
gmv'be
hrevce
vce
ro
(c)
And finally, the small-signal reverse-voltage gain
hre 
vbe
vce
.
(13)
E
Q , ib  0
i'C
B
iC
C
"Large"-signal model for the transistor
To develop a "large"-signal model (see Fig. 3) for the
transistor, we first replace the base-emitter diode with the
"large"-signal model of the diode, add the IB dependent
source (this completes the static signal translation), and then
add the hfeib, or what is the same gm v'be dependent source to
represent the effect of v'be on ic, add ro together with an
additional independent voltage source VCE to represent the
effect of vce on ic, and finally add the hrevce source to represent
the effect of vce on v'be. Note that we add another VCE source to
cancel the effect of the static collector-to-emitter voltage, VCE,
on the current through ro. Only the small-signal collector-toemitter voltage, vce, should generate the small-signal current
through ro, which is in accordance with the Early effect.
ib
i"C
vBE
hfeib
gmvbe
hie
ro
vce
(d)
E
Fig. 4. Small-signal equivalent circuits (models) for the transistor. (a) T smallsignal model of the BJT transistor, (b) separating the input and output loops
of the T model by applying the Miller theorem, (c) hybrid- small-signal
model, (d) simplified hybrid- model with the hrevce source and rb neglected.
Lecture 2: Introduction to electronic analog circuits 361-1-3661
5
i
ki
(1+k) Z
vin
Electronic circuit
V
1+k
k
i
ki
Z
Z
vo
V
vin
vo
V
vCE
Fig. A2. Miller's theorem (for voltages).
iB
Fig. A1. Transistor in an arbitrary electronic circuit connected to equivalent
signal sources. According to the substitution theorem, a branch of the
network that is not coupled to other branches can be replaced by an
equivalent independent current or voltage source without affecting any other
branch current or branch voltage. To apply the substitution theorem, the
network has to have a unique solution for all its branch currents and branch
voltages. The network does not have to be linear.
source. We can omit these two components because they do
not affect the hfeib source and, therefore, do not affect the
model output voltage and current: vce and ic.
Neglecting the hrevce source (the typical value of hre is very
small, about 10-3), we obtained in Fig. 4(d) a simplified
hybrid- model. We will use this model in all our further
analysis.
Either the T or  models can be used in a small-signal
analysis to replace a transistor in an electronic circuit.
Naturally, all the small-signal parameters of the models should
be found in advance as a function of the transistor operating
point.
APPENDIX
Small-signal model for the transistor
Note that the circuit in Fig. 3 is a linear one. Hence, to
obtain a small-signal model for the transistor [see Fig. 4(a)],
we simply suppress all the static sources in Fig. 3. The circuit
in Fig 4(a) is called the T small-signal model of the BJT
transistor.
The T model can be simplified by separating its input and
output loops [see Fig. 4(b)] by applying the Miller theorem for
voltages (see the Appendix). Such a separation provides us
with so-called hybrid- small-signal model shown in Fig 4(c).
Note that in Fig. 4(b) we short-circuited the resistor and the
voltage source that are connected in series with the hfeib
Fig A1 illustrates that the effect of the electronic circuit on a
transistor can be modeled with two independent sources.
Fig. A2 illustrates the Miller theorem for voltages: the input
and output loops of a T network can be separated without
changing the states of the network ports if the values of the
impedances Zin and Zo are increased to compensate for the
reduction of the currents through them relative to the current in
the impedance Z of the original T network.
REFERENCES
[1]
[2]
J. Millman and C. C. Halkias, Integrated electronics, McGraw-Hill.
A. S. Sedra and K. C.Smith, Microelectronic circuits.