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Chapter 3 The Bipolar Junction Transistor 1
CHAPTER 3
The Bipolar Junction Transistor
3.1. Transistor Structure. Operating Modes
A Bipolar Junction Transistor (BJT) is a semiconductor device formed by two pn junctions. Therefore, it will have three alternating regions. It can be either a narrow n type region placed between two p type layers (forming a pnp transistor), or a thin p-
type region between two n- type layers (representing a npn transistor). Fig. 3.1 shows the construction and symbols of the two kinds of BJTs. The three terminals of the transistor are called emitter (E), base (B) and collector (C), respectively. The arrow at
the emitter indicates the conventional direction of the current through the device.
The name BJT comes from the fact that the transistor operates with two type of
charge carriers, electrons and holes, in the same time. The first BJT was invented in 1947 at Bell Labs by William Shockley, Walter Brattain and John Bardeen. Their
invention was awarded with the Nobel Prize in 1956.
Fig 3.1 BJT construction and symbols: a) pnp transistor (left); b) npn transistor
(right)
p
p+
n
C
B
E
n
n+
p
C
B
E
B - Base
E - Emitter
C - Collector
C
B
E iE
iC
iB
vEB
vEC
C
B
E iE
iC
iB
vBE
vCE
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The key fact in BJT manufacturing consists in making the middle layer (the base) as
thin as possible. Because of this feature, the transistor functionality differs from two diodes placed back to back. Normally, the base - emitter (BE) junction is forward biased, while the base – collector (BC) junction is reverse biased (fig. 3.2). Also, the
emitter has a higher concentration of impurities than the other two layers. This is marked by a p+ or n+ sign in fig. 3.1 and fig 3.2.
Since the BE junction is forward biased, the emitter electrons diffuse into the base.
Their flow produce the emitter current iE. To be noticed that the conventional current direction is opposite to the electron flow direction.
+
-
+
-
VBC
VBE
C
B
E
+ + + + + + + + - - - - - - - - - -
n
+ + + + + + + +
- - - - - - - - - -
p
n+
iE
iC
iB
C
B
E iE
iC
iB
vBE
vCE
Electron
Hole
Depletion region
Electron flow
Fig. 3.2 BJT operating principle illustrated for a npn type transistor
Chapter 3 The Bipolar Junction Transistor 3
A few of the emitter electrons recombine with the holes found into the base, thus
forming a very small current called the base current, iB. However, most of the emitter electrons cross over into the BC depletion region, where the strong electric field found here leads them directly into the collector, thus creating the collector current iC.
Practically 99% of the emitter current reaches the collector and only 1% flows into the base. This happens because the base layer is very thin. (If it were not so, the majority of the emitter electrons would recombine with the base holes, never reaching the
collector). The operation of a pnp –type transistor is analogous to that of a npn type, with the role of the charge carriers reversed.
From the above discussions, the following equations can be written for describing
the transistor operation:
0
;
CEBC
CBE
Iii
iii
(3.1)
where is called current amplification factor (current gain). It is a dimensionless number, typically in the range of 100 to 800. In the literature it is also denoted with
hFE. The term ICE0 is the leakage current produced by the minority charge carriers. It can be neglected in practice. Since the base current iB is hundreds of times smaller than the collector current iC, it can be neglected also in (3.1):
.
;
BC
CE
ii
ii
(3.2)
Fig. 3.3 presents the typical physical appearance of a transistor.
As can be noticed, the most important
feature of the transistor is the fact that it can be used to control a large current to pass between the emitter and collector, by
the means of a much smaller current (iB). The device can be compared to a faucet, where the flow of the water can be
controlled by opening /closing a control knob. In the same way, a voltage or current applied to the transistor base may
allow much or less current to pass between the emitter and the collector. At the limit, the device can be completely closed, when iC = 0 and the transistor is blocked. This operation mode is called cut – off regime and
corresponds to the case when both junctions are reverse-biased. On the other hand, the device can be made maximum opened when the collector current is maximum. This
mode is called saturation, and corresponds to the case when both junctions are forward biased. Therefore the transistor can be used as a switch when operating in one of these two modes (cut-off / saturation). Between the two limits, the device works as an
Fig 3.3 BJT common packages
E C
B
E C
B
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amplifier. The current gain is a measure of the transistor effectiveness as an amplifying device.
In many applications the transistor can be connected using one of the following three
possible configurations: common emitter (CE), common base (CB) or common collector (CC), depending on which terminal is
common between the input and the output (fig. 3.4). Among them, the most used configuration is the CE connection.
It can be noticed (equation (3.1) and fig.3.1) that two currents and two voltages are
sufficient to specify the transistor operation. (The third one can be computed from the other two.)
Also the transistor operation can be described by its i-v characteristics. For the
CE connection, they are defined as follows:
a) The input characteristic relates the input
current iB with the input voltage vBE, when the output voltage vCE remains constant:
.constvvii CEBEBB (3.3)
b) The output characteristic relates the
output current iC with the output voltage vCE, with the condition that the input current iB is kept constant:
.constivii BCECC (3.4)
c) Finally, the transfer characteristic is
defined by the variation of the output current iC versus the input voltage vBE, when the output voltage vCE is maintained constant:
.constvvii CEBECC (3.5)
If the constant parameter in the above equations is varied, a family of characteristics can be drawn, each curve corresponding to a certain parameter. The common emitter i-v qualitative characteristics are depicted in Fig. 3.5.
The most important of all these are the output characteristics, since they can be used to describe the transistor behavior. In the (iC, vCE) plane four main regions can be
defined (fig. 3.5c), corresponding to four different operating modes of the BJT. The features of each region are summarized below:
C
B
E
iE
iC
iB vBE
vCE
C
B
E
iE vEB iC vCB
iB
vEC
C
B
E
vBC iB
iC
iE
Fig. 3.4 BJT main connections: a) CE (top); b) CB (middle); c) CC (bottom)
Chapter 3 The Bipolar Junction Transistor 5
a) The cut-off region is characterized by the fact that both junctions are reverse
biased, therefore iB 0, and consequently iC 0. In conclusion no current flows through the transistor.
b) The saturation region corresponds to the case when both junctions are forward
biased. The collector current iC reaches its maximum. Also the collector emitter voltage
vCE is smaller than the base– emitter voltage, vBE (typical VCE,sat 0.2 V)
IB1
Saturation
vCE
0
iC
VCE,Max VCE,Sat
ICMax Breakdown
Cut-off
Active Region
IB2
IB3
IB4
IB=0
0
iB
0.6V vBE
VCE1 VCE2
VCE2 > VCE1
vCE = const.
0
iC
0.6V vBE
vCE = const.
Fig. 3.5 Typical i-v characteristics for a npn BJT operating in the CE mode: a) input curves (top left hand corner); b) transfer characteristic (top right-hand corner); c) output characteristic
(bottom)
ICE0
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c) The active region has the base– emitter junction forward biased, while the base –
collector junction is reverse biased. In this case the transistor acts as a linear amplifier. The collector current iC can be controlled by the base current iB, according to (3.2). Also the collector – emitter voltage has to be such that VBE < VCE < VCC, where VCC is
the value of the voltage supply.
d) The breakdown region corresponds to the situation when iC and vCE exceed the
specifications given in the transistor data sheet. Above these values the transistor is damaged.
Table 3.1 summarizes the main transistor parameters that can be found in a data sheet, for BC 108, a general purpose npn transistor.
Table 3.1. Typical ratings for BC 108, a common npn transistor, at room temperature (25ºC)
Symbol Parameter BC 108
ICE0 Collector cut - off current 15 A
VCE0 Max Maximum value of collector-emitter voltage with the base terminal
left open circuit (IB = 0) 20 V
VCE,sat Collector emitter saturation voltage (at IC = 100 mV) 0.2 V
IC Max Collector current maximum value 100 mA PD Maximum total power dissipation 300 mW
hFE min Minimum DC current gain at IC = 2mA 110
3.2. BJT Operating Point
The operating point (Q point or bias point) of the BJT is defined as the DC component pair of the collector current IC
Q and the collector – emitter voltage VCEQ:
Q
CE
Q
C
V
IQ
(3.6)
Graphically, the operating point is situated at the intersection of the load line and the output i-v characteristic of the transistor (fig. 3.6). It can be noticed that when the
base current IB decreases, the corresponding Q point approaches the cut – off region. Conversely, as IB increases, it falls near the saturation region. Therefore, when designing a transistor amplifier, the Q point has to be chosen as much as possible in the
middle of the active region.
A practical diagram for biasing the BJT is shown in fig. 3.7. It is called the self-
bias circuit and has several advantages compared to other types of biasing diagrams. First of all, it is better from a practical perspective, since it doesn’t require two voltage sources. Secondly, the Q point can be stabilized so it doesn’t depend on the current
gain, which may vary in a wide range (e.g. 100 … 800) from transistor to transistor. This can be done by choosing proper values of the base resistors, so that the base current IB is much smaller than the divider current ID. Typically it is required that
Chapter 3 The Bipolar Junction Transistor 7
BD II 10 (3.7)
Generally, the algorithm summarized in
fig. 3.8 can be used for performing the DC analysis of a circuit containing transistors working in the active region.
In the first step of this algorithm all capacitors can be replaced with open
circuits, since their reactance
CX C
1 (3.8)
is practically infinite at zero frequency
(DC regime). Also, the signal sources are being replaced by short-circuits. For example, the circuit shown in fig. 3.7
represents already the DC equivalent diagram of a single transistor amplifier (the AC varying source, the load and the coupling capacitors were not presented yet.).
In the second step, on the resulted diagram, it is important to mark the transistor currents and voltages with their corresponding conventional directions (as shown in fig. 3.1).
Fig. 3.6 Graphical interpretation of the BJT operating point
Saturation
vCE 0
iC
Q
Load line
Cut-off
Active Region
IB
VCEQ
ICQ
RC
RE RB2
VCC +15V
VC
VB
RB1
100k
50k 3k
ICQ
5k
VE
IEQ
ID
IBQ
VCEQ
VBE0
Fig. 3.7 Typical circuit for biasing the BJT
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Next, the transistor equations can be
written in their simplified form (3.2). Also, the base – emitter voltage can be assumed equal with the offset voltage of
forward biased pn junction:
VVBE 6.00 (3.9)
The equations given by the external circuit can be obtained by applying
Kirchhoff Laws, Ohm’s Law, the voltage divider rule, the Thevenin theorem, etc.
In the current example (fig. 3.7), after applying the Thevenin theorem between the transistor terminal base and the
ground, a simplified diagram like the one shown in fig. 3.9can be obtained, where:
kRRR
VVRR
RV
BBBB
CC
BB
BBB
3.33||
5
21
21
2
Next, by applying Kirchhoff second law, around the left loop, and taking into
account the transistor equations (3.2) the collector current can be derived immediately:
EBB
BEBBQ
CRR
VVI
0 (3.10)
It can be noticed that generally ICQ
depends on the current gain . In practice this relation is undesirable,
due to the large range variation of this parameter. In turn this may cause a significant variation of the Q point,
which is required to be as stable as possible inside the active region.
This inconvenient can be avoided if
EBB RR (3.11)
which is satisfied in the current
example for 100.
1. Plot the DC equivalent circuit
2. Mark transistor currents and
voltages
3. Write the transistor equations
(3.2)
4. Write the external
circuit equations
5. Solve to find the
unknown quantities
Fig. 3.8 General algorithm for DC analysis
of transistor circuits
RC
RE
RBB
VCC +15V
VC
3k
ICQ
5k
VE
IEQ
IBQ
VCEQ
VBE0
VBB +
-
VCC +
-
Fig. 3.9 Equivalent diagram obtained by applyin
the Thevenin Theorem for the circuit in fig. 3.7
Chapter 3 The Bipolar Junction Transistor 9
Then the bias point collector current can be approximated as:
mAR
VVI
E
BEBBQ
C 47.10
(3.12)
The collector-emitter voltage, VCEQ can be drawn from the second Kirchhoff law
written for the loop on the right:
VIRRVV Q
CECCC
Q
CE 24.3 (3.13)
Thus it can be concluded that the operating point in the current example is ICQ =
1.47 mA and VCEQ = 3.24 V.
The same result can be obtained directly, if neglecting the base current IBQ relative
to the divider current ID. The left branch of the circuit in fig.3.7 can be approximated with a voltage divider. Then the base voltage VB can be computed easily by applying
the voltage divider rule:
VVRR
RV CC
BB
BB 5
21
2
(3.14)
Next the emitter voltage and current can be obtained immediately from the bottom-left loop:
mAR
VII
VVVV
E
EQ
C
Q
E
BEBE
47.1
4.40
(3.15)
The collector- emitter voltage VCEQ is computed identically with (3.13).
The initial assumption (3.7) can be next verified by comparing the two currents:
mAR
VVI
B
BCCD 1.0
1
(3.16)
If the minimum value of the current gain = 100,
D
Q
CQ
B ImAI
I 014.0
Finally it can be concluded that the initial assumption was true.
The load line equation can also be drawn from the second Kirchhoff law written for the loop on the right, similar to (3.13:
CECECCC viRRV (3.17)
This line can be plotted in the( iC, vCE) plane by finding the intercepts with the axes (fig.3.10):
(3.18)
mARR
Viv
VVvi
EC
CCCCE
CCCEC
875.10
;150
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It can also be noticed in fig.3.10 that the
position of the Q point is indeed in the active region, thus the transistor works as an
amplifier, as desired. However, when a signal source is applied at the input, the transistor currents and voltages will have variations
around their DC values. If the input signal amplitude is increased, the total collector current iC or the collector – emitter voltage
vCE may extend beyond the active region boundaries reaching into the saturation or into the cut-off domains. In this case, the
signal at the amplifier’s output may be distorted (fig. 3.11). Thus, if a sine wave is applied at the input, the voltage at the
amplifier’s output may present flattened peaks. The decomposition in Fourier series of this waveform leads to additional superior
Fig. 3.10 The load line and the Q point
Saturation
vCE[V] 0
iC[mA]
Q
Load line
Cut-off
Active Region
VCEQ
ICQ
VCC
VCC
RC+RE
15
1.875
3.24
1.47
Vce
Qopt
VCEQ
opt
ICQ
opt
7.5
0.94
ic
Ic
Fig 3.11 Signal distortion example
Input signal vi(t)
Output signal vO (t)
time
vi(t), vO(t)
Vi
t[s]
0
t[s]
-Vi
t[s]
Chapter 3 The Bipolar Junction Transistor 11
harmonics. For example in audio applications the listening quality may be seriously
affected.
In order to obtain a maximum gain from the amplifier and in the same time no
output signal distortions, it is a good practice to choose the transistor’s bias point as much as possible in the middle of the load line segment from the active region (fig.3.10). This particular operating point is called optimum, Qopt. In the current
example,
VV
V
mARR
VI
QCCQ
CE
EC
CCQ
C
opt
5.72
94.02
(3.19)
Example 1
In the above circuit (fig. 3.7), often the inverse problem might arise: for example to find the resistances RB1 and RB2, such that the transistor works in its optimum Q point.
It is assumed that the total value of the base resistors remains the same i.e. 150 k. In this case the analysis starts from the “tail” to the “head”. First, the emitter voltage VE is found:
VRIV E
Q
CE 8.2394.0 (3.20)
The base voltage VB can be computed next, from the bottom-left loop:
VVVV BEEB 4.36.08.20 (3.21)
Requiring the condition (3.7), thus assuming IB >> ID, then it follows that:
mARR
VI
BB
CCD 1.0
150
15
21
(3.22)
Still at the same time,
kI
VR
D
BB 34
1.0
4.32 (3.23)
and therefore,
kRB 116341501 (3.24)
If the value of the total base resistance RB1 + RB2 is not given, then the divider
current ID has to be chosen, according to (3.7) considering the worst conditions. For
example assuming the minimum value of the current gain factor min = 100, it follows that
mAI
I
Q
Copt
BMax 0094.0100
94.0
min
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and therefore the divider current can be chosen ten times higher:
mAII BMaxD 094.010
Consequently,
k6.159094.0
15
I
VRR
D
CC2B1B (3.25)
The rest of the problem can be solved in a similar manner, by following the same
equations (3.23 – 3.24) and choosing a proper RB1 + RB2 value, according to (3.25).
3.3. BJT Large Signal Model
The large signal model describes the behavior of the transistor in the presence of relatively large base and collector currents. It includes all three basic operating modes highlighted previously, in the output characteristic plane (fig. 3.6).
Thus, in the cut-off region, both junctions are reverse biased. The collector current is given by the small leakage current
00 CEC II (3.26)
Therefore, it acts virtually as an open circuit and can be replaced by the corresponding circuit shown in fig. 3.12 a, where ICE0 denotes a very small current source.
In the active mode the BE junction is forward biased, therefore it can be modeled by
a DC voltage source VBE = 0.6 V, placed between the base and the emitter. At the same time, the BC junction is reverse biased, so there is an open circuit between these two
terminals. In this case the base current is amplified by the gain factor at the collector
according to (3.2), and thus IC can be modeled by a current source IB, as shown in fig. 3.12 b.
Finally, in the saturation regime both junctions are forward biased. Again the BE
junction can be replaced by a DC voltage source VBE = 0.6 V as in the previous case. Also, the voltage drop between the collector and the emitter is very small (typically less than 0.2 V). Thus an additional DC voltage source VCE,sat can be placed between
these two terminals to specify this property (fig. 3.12 c). (Evidently, since VBE > VCE,sat the BC junction still remains forward biased).
Example 2
In some microcomputer applications it is required to turn on an off an LED from
one of its digital output ports. A transistor has to be employed to drive the LED device
as shown in fig. 3.13 a. It is known that VBE0 = 0.6V; VCEsat = 0.2V; = 65. The LED is turned on at VLED = 1.4V and ILED > 15mA. Also its maximum dissipated power is PMax = 100mW. On the other hand, the microcomputer output resistance is RB=1k and its
output voltage levels corresponding to the off and on states are VOFF = 0V and VON = 5V, respectively. Also it can provide an output current no more than 5 mA.
Chapter 3 The Bipolar Junction Transistor 13
The transistor large signal model can be used to find the collector resistance RC for
this application. It is also required to check if the power dissipated by the LED doesn’t exceed the maximum limit.
When the microcomputer voltage is V1 = VOFF = 0, the transistor is obviously in its
cut – off region since the BE junction is reverse biased. Thus IB = 0, and therefore IC 0 also, so the LED is off (fig. 3.13 b).
On the other hand, when the microcomputer voltage is V1 = VON = 5V, the transistor has to be biased in its saturation region. It follows that the device can be replaced by DC voltage sources: VBE0 = 0.6V between the base and the emitter and VCEsat = 0.2V
between the collector and the emitter, respectively (fig. 3.13 c).
The Kirchhoff second law written for the loop on the right gives:
C
satCELEDCC
CI
VVVR
,
Fig. 3.12 The large signal models of the BJT: a) Cut – off mode – bottom; b) Active mode - top right hand corner; c) Saturation – top left hand corner.
Saturation
vCE
0
iC
Cut-off
Active Region
IB
+
- + -
IC
C
B
E
VCE, sat
VBE IB
+ -
C
B
E
IC=IB
VBE
C
B
E
ICE0 0
VCE,Sat
ICE0
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At the same time,
mAII LEDC 15
Therefore, it follows that
22015
2.04.15CR
If for example the chosen collector resistance
is RC = 110 , then the collector current will be
mAIC 3011.0
2.04.15
In this last case, the power dissipated by the LED
mWmWVIP LEDCLED 100424.130
doesn’t exceed the maximum power limit PMax
It can be noticed that the transistor in this example works indeed in the saturation
mode. Evidently, the collector voltage VC = VCE,sat = 0.2V is smaller than the base voltage VB= VBE0 = 0.6V, and thus the BC junction is forward biased (like the BE
junction). From the left loop it can be drawn that
IB = 0
RB
1k VE
VBE0 VI=0
IC
VLED
RC
5V +VCC
VC
B ICE0 0
Fig. 3.13 b) BJT replaced by its cut-off model (left); c) BJT replaced by its saturation model (right)
IB
RB
1k VE
VBE0
VI=5V
IC
VLED
RC
5V +VCC
VC
+
- + -
IC
VCE, sat VB
Fig. 3.13 a) LED driver circuit
IC
IB
VLED
RC
5V +VCC
RB
VC
1k VE
VCE
VBE0
V1
Chapter 3 The Bipolar Junction Transistor 15
mAR
VVI
B
BEONB 4.4
1
6.050
However, it can be seen that equation (3.2b) cannot be applied here anymore since it will lead to a different result:
mAII BC 4184.495
3.4. BJT Small Signal Model
The small signal models of the BJT are based on the fact that the transistor characteristics can be assumed linear in the neighborhood of the operating point. These
models can be applied when the amplitudes of voltages and currents are much smaller than the values from the Q point. Typically, the small signal condition is given by
CmVVV thbe 25/25 (3.27)
This condition is usually satisfied in practice (e.g. amplifiers used to magnify low
level voltages acquired by various sensors).
There are known many models which approximate the dynamic behavior of a BJT
such as the natural model (also known as the Giacoletto model), the hybrid parameter (h-parameter) model, etc.
In the following discussion, the hybrid parameter model is used. According to this representation the
transistor is regarded as a quadripole (a two port device). The base and emitter terminals represent
the input port, denoted 11’ (fig. 3.14). In the same manner, the collector and emitter terminals form
the output port, 22’. Obviously, in this representation the emitter is common between the two ports.
The transistor is thus modeled as a “black box” with its properties
defined by a matrix of parameters, specified also in the device data sheets:
2
1
2221
1211
2
1
v
i
hh
hh
i
v
ee
ee (3.28)
where the index e denotes the fact that the transistor has the common emitter connection. Equivalently, it can be written that:
C
B
E
i2 = iC
i1= iB
v1 = vBE
v2 = vCE 1
1’
2
2’
Fig. 3.14 Transistor modeled as a quadripole (two – port device)
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2221212
2121111
vhihi
vhihv
ee
ee (3.29)
Thus the first parameter h11e is defined as the input impedance when the output voltage variation around the Q point is zero:
01
111
2
v
d
ei
vh (3.30)
Knowing that
CBCEBE iiiivvvv 2121 ;;; (3.31)
it follows that graphically, the h11e parameter represents the slope to the input
characteristic, taken in the Q point (fig.3.15), since evidently it can also be written as:
constvb
bee
CE
I
Vh
11 (3.32)
where Vbe, Ib denote the small signal amplitudes of voltage and current.
Sometimes h11e is also denoted hie, to indicate an input parameter. Typically it
ranges within a few kilo ohms, h11e = 2k … 4 k.
Fig. 3.15 Graphical interpretation of the h11e parameter
vBE 0
iB
VBEQ
Q
IBQ
vbe
t
Vbe
tg h11e
BJT input characteristic for vCE = VCE
Q
The Load line
Linear approximation
around the Q point
ib
t Ib
Chapter 3 The Bipolar Junction Transistor 17
The second parameter denoted in (3.28) with h12e is defined as the input to output
voltage ratio, thus is a dimensionless quantity. It represents the reverse voltage gain factor when the input is open:
02
112
1
i
d
ev
vh (3.33)
In terms of transistor voltages and currents, equation (3.33) can be written as:
.
12
constice
bee
B
V
Vh
(3.34)
Fig. 3.16 presents the graphical interpretation of this parameter. In the literature it is also denoted hre (reverse parameter). Typically,
4
12 10eh
but in practice it is usually neglected.
Another h- parameter, denoted h21e in (3.28) is
the current gain factor computed when the output voltage variation is zero:
01
221
2
v
d
ei
ih (3.35)
It is also identical with:
.
21
constvb
ce
CE
I
Ih
(3.36)
This parameter represents the current gain of the BJT and is approximately equal to
the transistor parameter described by (3.1) – (3.2). In the literature it is also denoted
with hfe (forward parameter). The graphical interpretation of h21e is shown in fig.3.17a.
Finally, the last parameter in (3.28) is h22e. It is defined as the output admittance
when the output is kept open:
02
222
1
i
d
ev
ih (3.37)
Clearly it can also be expressed in terms of transistor voltages and currents as:
.
22
constice
ce
B
V
Ih
(3.38)
0
iB
Q
vBE
IBQ
VBEQ
iB = const.
2Vbe
VCEQ
2Vce
Fig. 3.16 Graphical interpretation
of the h12e parameter
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Its graphical interpretation is depicted in fig. 3.17 b. This parameter indicates that
the output characteristic is not exactly flat. Precisely, h22e represents the slope of the output characteristic with the horizontal axis, in the Q point. It has units of
conductance (i.e. Siemens). Typically, h22e 10-5 S, or h22e-1 100 k. In many
applications this parameter can be neglected. In the literature this h22e is also denoted
hoe (specifying an output parameter).
Assembling the previous definitions, the BJT small signal model for medium
frequencies can be derived and is presented in fig. 3.18.
vCE 0
iC
Q IBQ
VCEQ
ICQ
2Ic
2Vce
tg h22e
vCE 0
iC
Q IBQ
VCEQ
ICQ
2Ic IB
Q + Ib
IBQ - Ib
Fig. 3.17 Graphical interpretations of the BJT small signal h parameters:
a) h21e (vCE = VCEQ = const.) – top; b) h21e (iB = IB
Q = const.) – bottom.
Chapter 3 The Bipolar Junction Transistor 19
3.5. The Transconductance
Another useful small signal parameter is the transfer conductance or simply the
transconductance. It is defined as the local slope of the transfer characteristic (fig. 3.19) and is denoted with gm:
intintint poQbe
b
b
c
poQbe
c
poQBE
Cd
mV
I
I
I
V
I
v
ig
(3.39)
Therefore the transconductance can be expressed in mA/V. Taking into consideration
the equations (3.32) and (3.36), it follows immediately that gm can also be written as:
e
em
h
hg
11
21 . (3.40)
On the other hand, it can be shown that:
CmVq
kTV
V
Ig th
th
Q
Cm 25/25; (3.41)
C B
E
i2 = Ic i1= Ib
v1 = Vbe v2 = Vce
1
1’
2
2’
~
E
h11e
h12eVce
1
h22e
h21eIb
Neglected parameter
C B
E
Ic Ib
Vbe Vce
1
1’
2
2’ E
h11e h21eIb
Fig. 3.18 BJT h-parameter small signal model for medium scale
frequencies: a) Complete diagram (top); b) Simplified model (bottom)
________________________________________________________________
Thus, assuming the room ambient temperature (25ºC), in most applications the
transconductance can be computed as:
Q
Cm Ig 40 (3.42)
It can be noticed that the collector current in the BJT h- parameter model (fig. 3.18) can also be computed in terms of the transconductance:
bembmebec VgIghIhI 1121 (3.43)
3.6. Using the h- Parameter Model in AC Circuit Analysis
Fig. 3.20 a presents a typical voltage amplifier using a single BJT - common emitter connection. It can be observed that this circuit contains two voltage sources. The DC voltage source VCC is required for biasing the transistor in the active region so
that it can work as an amplifier. On the other hand, the AC source vg provides the variable signal to be amplified.
Usually it is more convenient to analyze such circuits in two steps, applying the superposition theorem. The DC analysis can be performed first, by considering inactive the AC source vg. The DC components of voltage/ currents can be computed for
example by following the general algorithm described in section 3.2, fig. 3.8. The AC analysis can be done next, in order to determine the small signal components which are superimposed over their DC values. Typically, the quantities which are usually
tg gm
BJT transfer characteristic for vCE = VCE
Q
The Load line
Linear approximation
around the Q point
vBE 0
iC
VBEQ
Q
ICQ
vbe
t
Vbe
ic
t Ic
Fig. 3.19 Graphical interpretation of the BJT transconductance, gm
Chapter 3 The Bipolar Junction Transistor 21
required when performing the AC analysis are the voltage gain, Av and the input /
output circuit resistances Ri and Ro, respectively, as defined below:
.;
;
0
gVo
od
o
i
id
i
i
od
v
I
VR
I
VR
V
VA
(3.44)
This amplifier can also be referred as a two port circuit (fig. 3.20 b) for which the quantities defined in (3.44) are sufficient to describe its functionality.
VBE0
RC RB1
RL
RB2
VCC +15V
+1600 5V
Vo
vO Vi
vi 3k
ICQ
5k
50k50k
~
Rg
Vg
600
IC
Q
CE
C1
C2
VCEQ
VB
VC
VE
ICQ
ID
+
-
RE
~
Vg +
-
-
RL Vo vO Vi
vi 50k
50k
Rg
~
600
600
vg ~ AvVi
+
Ro
-
Ri
Ii Io
Fig. 3.20 Voltage amplifier. a) Typical single transistor diagram
(top); b) Two-port equivalent circuit (bottom)
Ii
________________________________________________________________
In the above diagram depicted in fig 3.20a, the capacitors are chosen such that at
the working frequency (usually a few kilohertz), their reactance can be neglected,
compared to the circuit resistances. For example, if the capacitor value is chosen 1 F its reactance at 5 kHz will not exceed 100 ohms.
321010514.32
1
2
1163fCC
X C
Like in the DC case, an algorithm can be developed for performing the AC analysis too. The steps of this procedure are being summarized in fig. 3.21.
Thus in the first step, the AC equivalent circuit has to be plotted, taking into account that all capacitors can now be replaced by
shortcircuits (wires). In addition to this, the DC voltage source VCC is passivated and can also be replaced by a shortcircuit between its terminals.
The AC equivalent circuit for the amplifier presented in fig. 3.20a is shown in fig. 3.22a.
Next, the transistor can be replaced by its simplified h-parameter small-signal model discussed in section 3.4, fig 3.18b. The resulted
diagram for the current example is shown in fig. 3.22b.
The currents and voltages will be marked on the new circuit usually using small letters (for variations), or capital letters with small
subscripts (for amplitudes). The transistor currents/ voltages will respect their conventional directions.
The BJT the small signal parameters h11e and gm can be found with the help of the equations
(3.40) and (3.42), and the value of the collector current from the Q point, computed in the DC analysis stage.
Next, linear circuit equations can be written for the AC diagram using the Kirchhoff laws,
Ohm’s law, the voltage divider rule, etc.
Finally, the AC required quantities such as
the voltage gain, the input and output resistances, can be computed from their definitions (3.44) and the equations written in
the previous step.
Fig. 3.21 General algorithm for AC analysis of transistor circuits
1. Plot the AC
equivalent circuit
2. Replace the
transistor by its small signal model
3. Mark voltages
and currents
4. Compute the
transistor small-signal parameters
5. Write circuit
equations
6. Solve to find the
AC quantities
Chapter 3 The Bipolar Junction Transistor 23
Example 3
For the amplifier shown in fig. 3.20a, knowing that the transistor has min = 50 and VBE0 = 0.6 V the first question is to find the base resistances RB1 and RB2 such that a
maximum voltage gain is obtained and in the same time the output signal is free of distortions.
In this situation, the voltage, current and power gains are to be computed, together with the amplifier’s input and output resistances, given the fact that h21e = 200.
Finally, the value of the coupling capacitors C1 and C2 are required, that will permit
the transistor to operate in the audio range.
For this example, the load line equation, the optimum Q point and the emitter and
base voltages are given by (3.17) – (3.21), as shown previously in section 3.2. The base resistors RB1 and RB2 can be computed in a similar way as in Example 1. Thus, for
min= 50, the current flowing through the base voltage divider can be taken
mAI
IIQ
CBMaxD 19.01010
min
Therefore, the base resistances can be chosen accordingly:
kI
VR
kI
VVR
D
BB
D
BCCB
1819.0
4.3
6119.0
4.315
2
1
In order to perform the AC analysis of the amplifier, the corresponding AC
equivalent circuit is derived (fig. 3.22), by following the steps specified in the general algorithm described previously (fig. 3.21).
Fig. 3.22 AC equivalent circuit for the amplifier in fig. 3.21
C
B
E
Ic
Ib
~ Vg
+ -
-
Vi vi
Rg
~
600600
vg
Ii
Vo vi
RB2
61 k
vg
RC RL
5 k
vg
50 k
vg
T
Ri,T Ri
RB1
18 k
vg
Io
________________________________________________________________
Here Ri,T denotes the transistor input resistance, defined by
b
id
TiI
VR , (3.45)
Fig. 3.23 presents also the AC circuit with the transistor replaced by its equivalent
small signal h-parameter simplified model (fig. 3.18).
In this equivalent diagram
kRRR BBB 14||21
(3.46)
It can be noticed that the input transistor resistance can be derived from Ohm’s Law:
e
b
iTi hI
VR 11, (3.47)
and the parameter h11e can be computed from (3.40) - (3.42):
kI
h
g
hh
Q
C
e
m
ee 7.4
94.040
200
40
2121
11
Consequently, the input amplifier resistance will be in this case
TiTiB
i
ii RkkkRRI
VR ,, 4.37.4||14|| (3.48)
The voltage gain can be written as a product of three fractions
i
b
b
c
c
o
i
ov
V
I
I
I
I
V
V
VA
where each term can be computed easily. Thus,
Fig. 3.23 AC diagram for the amplifier in fig. 3.21 with the transistor replaced by its small signal model
Ib
~ Vg
+
-
-
Vi vi
Rg
~
600600
vg
Ii
Vo vi
RB
14 k
vg
RC||RL
4.5k
vg
Ri,T Ri
C B
E
Ic Ib
Vbe Vce
E
h11e h21eIb
(gmVbe)
Chapter 3 The Bipolar Junction Transistor 25
.1695.494.040||
1||
1||
11
21
,
21
LCm
e
eLC
Ti
eLC
i
b
b
c
c
ov
RRg
hhRR
RhRR
V
I
I
I
I
VA
(3.49)
The minus sign in the voltage gain expression (3.49) indicates the fact that the phase shift between the output and the input voltage is 180º.
Also, noticing that the input signal generator resistance Rg and the input amplifier equivalent resistance Ri form a voltage divider, the voltage gain relative to the voltage
source Avg, can be written as:
1436.04.3
4.3169
gi
iv
g
i
i
o
g
ovg
RR
RA
V
V
V
V
V
VA (3.50)
The output resistance of the transistor Ro,T is defined as the equivalent resistance seen at the output of the device, when the input is short circuit to the ground while an external source is connected at the output (fig. 3.24). In this case the base current Ib=0
and consequently Ic=0. It follows therefore that :
0
,
gVc
od
ToI
VR (3.51)
Thus, the output resistance of the amplifier will be:
kRRRI
VR CToC
Vo
od
o
g
5|| ,
0
(3.52)
In a similar manner, the current gain Ai of the amplifier can be put in the form:
Fig. 3.24 Circuit configuration for computing the output resistances
C
B
E
Ic
Ib=0
Vi=0
vi
Rg
~
600600
vg
Vo vi RB
14 k
vg
RC RL
5 k
vg
50 k
vg
T
Ro,T Ro
Io
________________________________________________________________
i
b
b
c
c
o
i
oi
I
I
I
I
I
I
I
IA (3.53)
The first and the last fractions in (3.53) can be computed by applying the current divider rule, it follows that:
13,
21
TiB
Be
LC
Ci
RR
Rh
RR
RA
Finally, the power gain Ap will be given by:
1859
iv
ii
oo
i
op AA
IV
IV
P
PA
In conclusion, the transistor in this example amplifies both voltage and current, and thus amplifies in power.
The capacitors C1 and C2 in the amplifier diagram in fig.3.20a are designed to perform the coupling of the AC source and the load to the remainder of the circuit (For this reason they are called coupling capacitors). Basically, they have to provide
separate paths for the DC and AC components in the circuit. Evidently, they act as open circuits for the DC voltages and currents, allowing the transistor to be biased in the active region, so it may work as an amplifier. On the other hand, their impedances
have to be negligible at the working frequencies, so they can act as short circuits, thus connecting the AC source and the load to the amplifier. The emitter capacitor CE (called emitter bypass capacitor or decoupling capacitor) serves a similar purpose, i.e.
to provide a stable DC bias point and on the other hand, to increase the amplifier gain.
The audio frequency domain is considered between 20 Hz and 20 kHz. It follows
that the capacitive reactance of C1 has to be much smaller than the equivalent resistance “seen” by the capacitor between its terminals.
ig
L
C RRCf
X 12
11
(3.54)
even at the lowest frequency fL = 20 Hz. Thus, C1 >> 2F. For example it can be
chosen 22F, a standard value. In a similar way,
Lo
L
C RRCf
X 22
12
(3.55)
Therefore, C2 >> 0.14 F, so its value can be chosen for example 2F.
Finally, the emitter bypass capacitor “sees” across its terminals two resistances connected in parallel: RE and the equivalent resistance of the transistor between its
emitter and ground, Ri,TE (fig. 3.25)
1
||||
21
11
21
11
,
e
gBe
beb
bgBbe
e
eTEi
h
RRh
IhI
IRRIh
I
VR (3.56)
Chapter 3 The Bipolar Junction Transistor 27
It can be noticed in (3.56) that Rg << RB and thus Rg || RB Rg . Therefore,
121
11
,
e
ge
TEih
RhR (3.57)
After replacing the corresponding values, Ri,TE 26, which is a small resistance. The bypass emitter capacitor has to be chosen such that
TEiE
EL
CE RRCf
X ,||2
1
(3.58)
or
F
RRfC
TEiEL
E
300||2
1
,
The capacity for this case can be chosen at least ten times higher than the computed
value, i.e. 3000F, which is a high capacity. Yet in practice it would be reasonable to
select a somewhat a smaller value (e.g. 1000 F).
C
B
E
Ie
Ve
Ic
Ib RC||RL
Rg||Ri
C
B
E
Ie
Ve
Ic
Ib
RC||RL
Rg||Ri
E
Ib Vbe h11e
Fig. 3.25 Circuit configuration for computing equivalent resistance
in the emitter of the BJT