Tài liệu Electronics and Circuit Analysis using MATLAB doc

IJA
SS
=
(12.8)

where

A
is the area of the emitter and

J
S
is the transport saturation current density, and it can be
further expressed as

J
qD n
Q
S
ni
B
=
2
(12.9)

where


D
n
is the average effective electron diffusion constant

n
i
is the intrinsic carrier concentration in silicon (
n
i
= 1.45 x
10
10
atoms / cm
3
at 300
o
K)

Q
B
is the number of doping atoms in the base per unit area.



The dc equivalent circuit of the BJT is based upon the Ebers-Moll model.
The model is shown in Figure 12.2. The current sources
α
RR
I
indicate the
interaction between the base-emitter and base-collector junctions due to the
narrow base region.

In the case of a pnp transistor, the directions of the diodes in Figure 12.2 are
reversed. In addition, the voltage polarities of Equations (12.1) and (12.2) are
reversed. The resulting Ebers-Moll equations for pnp transistors are

II
V
V
EES
EB
T
=






−






exp 1
−






−






α
RCS
CB
T
I
V
V
exp 1
(12.10)

II
V
V
CFES
EB
T
=−






−






α
exp 1
+






−






I
V
V
CS
CB
T
exp 1
(12.11)


© 1999 CRC Press LLC

© 1999 CRC Press LLC

α
I
C
I
E
I
R
I
F
R
I
F
R
I
F
V
BC
V
BE
I
B
α
+
+
-
-


Figure 12.2 Ebers-Moll Static Model for an NPN transistor
(Injection Version)


The voltages at the base-emitter and base-collector junctions will define the
regions of operation. The four regions of operations are forward-active,
reverse-active, saturation and cut-off. Figure 12.3 shows the regions of
operation based on the polarities of the base-emitter and base collector
junctions.


Forward-Active Region

The forward-active region corresponds to forward biasing the emitter-base
junction and reverse biasing the base-collector junction. It is the normal
operational region of transistors employed for amplifications. If
V
BE
> 0.5 V
and
V
BC

< 0.3V, then equations (12.1) to (12.4) and (12.6) can be rewritten
as

II
V
V
CS
BE
T
=






exp
(12.12)


© 1999 CRC Press LLC

© 1999 CRC Press LLC


I
IV
V
E
S
F
BE
T
=−






α
exp
(12.13)

From Figure 12.1,

()
III
BCE
=− +
(12.14)

Substituting Equations (12.12) and (12.13) into (12.14), we have

()
II
V
V
BS
F
F
BE
T
=
−






1
α
α
exp
(12.15)


=






IV
V
S
F
BE
T
β
exp
(12.16)

where

β
F
=
large signal forward current gain of common-emitter
configuration

β
F
=
α
α
F
F
1
−
(12.17)

From Equations (12.12) and (12.16), we have

II
CFB
=
β
(12.18)

We can also define,
β
R
, the large signal reverse current gain of the common-
emitter configuration as

β
α
α
R
R
R
=
−
1
(12.19)




© 1999 CRC Press LLC

© 1999 CRC Press LLC

reverse bias
cut-off
forward bias
reverse-active
V
BC
V
BE
forward-active
reverse bias forward bias
saturation


Figure 12.3 Regions of Operation for a BJT as Defined by the Bias
of
V
BE
and
V
BC



Reverse-Active Region

The reverse-active region corresponds to reverse biasing the emitter-base
junction and forward biasing the base-collector junction. The Ebers-Moll
model in the reverse-active region (
V
BC

> 0.5V and
V
BE

< 0.3V) simplifies to

II
V
V
ES
BC
T
=






(12.20)

I
IV
V
B
S
R
BC
T
=






β
exp
(12.21)

Thus,
II
ERB
=
β
(12.22)

The reverse-active region is seldom used.



© 1999 CRC Press LLC

© 1999 CRC Press LLC

Saturation and Cut-off Regions

The saturation region corresponds to forward biasing both base-emitter and
base-collector junctions. A switching transistor will be in the saturation region
when the device is in the conducting or “ON” state.

The cut-off region corresponds to reverse biasing the base-emitter and base-
collector junctions. The collector and base currents are very small compared
to those that flow when transistors are in the active-forward and saturation
regions. In most applications, it is adequate to assume that
III
CBE
===
0
when a BJT is in the cut-off region. A switching
transistor will be in the cut-off region when the device is not conducting or in
the “OFF” state.


Example 12.1

Assume that a BJT has an emitter area of 5.0 mil
2
,
β
F
=
120,

β
R
=
03.

transport current density,
J
S
=
−
210
10
*

µ
Amil
/
2
and
T
= 300
o
K. Plot
I
E
versus
V
BE
for
V
BC
= -1V. Assume 0 <
V
BE
< 0.7 V.

Solution

From Equations (12.1), (12.2) and (12.4), we can write the following
MATLAB program.

MATLAB Script

%Input characteristics of a BJT
diary ex12_1.dat
diary on
k=1.381e-23; temp=300; q=1.602e-19;
cur_den=2e-10; area=5.0; beta_f=120; beta_r=0.3;
vt=k*temp/q; is=cur_den*area;
alpha_f=beta_f/(1+beta_f);
alpha_r = beta_r/(1+beta_r);
ies=is/alpha_f;
vbe=0.3:0.01:0.65;
ics=is/alpha_r;
m=length(vbe)
for i = 1:m
ifr(i) = ies*exp((vbe(i)/vt)-1);


© 1999 CRC Press LLC

© 1999 CRC Press LLC
ir1(i) = ics*exp((-1.0/vt)-1);
ie1(i) = abs(-ifr(i) + alpha_r*ir1(i));
end
plot(vbe,ie1)
title('Input characteristics')
xlabel('Base-emitter voltage, V')
ylabel('Emitter current, A')

Figure 12.4 shows the input characteristics.



Figure 12.4 Input Characteristics of a Bipolar Junction Transistor


Experimental studies indicate that the collector current of the BJT in the
forward-active region increases linearly with the voltage between the collector-
emitter V
CE
. Equation 12.12 can be modified as

II
V
V
V
V
CS
BE
T
CE
AF
≅






+






exp 1
(12.23)

where
V
AF
is a constant dependent on the fabrication process.


© 1999 CRC Press LLC

© 1999 CRC Press LLC

Example 12.2

For an npn transistor with emitter area of 5.5 mil
2
,
α
F
=
098.,

α
R
=
035.,

VV
AF
=
250
and transport current density is
20 10
9
.
x
−

µ
Amil
/
2
. Use
MATLAB to plot the output characteristic for
V
BE
= 0.65 V. Neglect the
effect of
V
AF
on the output current
I
C
. Assume a temperature of 300
o
K.


Solution

MATLAB Script

%output characteristic of an npn transistor
%
diary ex12_2.dat
k=1.381e-23; temp=300; q=1.602e-19;
cur_den=2.0e-15; area=5.5; alpha_f=0.98;
alpha_r=0.35; vt=k*temp/q; is=cur_den*area;
ies=is/alpha_f; ics=is/alpha_r;
vbe= [0.65];
vce=[0 0.07 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 4 6];
n=length(vbe);
m=length(vce);
for i=1:n
for j=1:m
ifr(i,j)= ies*exp((vbe(i)/vt) - 1);
vbc(j) = vbe(i) - vce(j);
ir(i,j) = ics*exp((vbc(j)/vt) - 1);
ic(i,j) = alpha_f*ifr(i,j) - ir(i,j);
end
end
ic1 = ic(1,:);
plot(vce, ic1,'w')
title('Output Characteristic')
xlabel('Collector-emitter Voltage, V')
ylabel('Collector current, A')
text(3,3.1e-4, 'Vbe = 0.65 V')
axis([0,6,0,4e-4])


Figure 12.5 shows the output characteristic.

© 1999 CRC Press LLC

© 1999 CRC Press LLC


Figure 12.5 Output Characteristic on an NPN Transistor



12.2 BIASING BJT DISCRETE CIRCUITS

12.2.1 Self-bias circuit

One of the most frequently used biasing circuits for discrete transistor circuits
is the self-bias of the emitter-bias circuit shown in Figure 12.6.


V
CC
R
BI
R
C
R
E
R
B2
C
E


(a)

© 1999 CRC Press LLC

© 1999 CRC Press LLC



-
V
CC
R
C
R
E
R
BB
V
BB
V
CE
I
E
I
B
I
C
+
+
-


(b)

Figure 12.6 (a) Self-Bias Circuit (b) DC Equivalent Circuit of (a)


The emitter resistance,
R
E
, provides stabilization of the bias point. If
V
BB

and
R
B
are the Thevenin equivalent parameters for the base bias circuit, then

V
VR
RR
BB
CC B
BB
=
+
2
12
(12.24)

RRR
BBB
=
12
(12.25)

Using Kirchoff’s Voltage Law for the base circuit, we have

VIRVIR
BB B B BE E E
=++
(12.26)

Using Equation (12.18) and Figure 12.6b, we have

()
IIII I I
EBCBFB F B
=+=+ = +
ββ
1
(12.27)

Substituting Equations (12.18) and (12.27) into (12.26), we have


© 1999 CRC Press LLC

© 1999 CRC Press LLC
()
I
VV
RR
B
BB BE
BF E
=
−
++
β
1
(12.28)
or


()
I
VV
R
R
C
BB BE
B
F
F
F
E
=
−
+
+
β
β
β
1
(12.29)

Applying KVL at the output loop of Figure 12.6b gives


VVIRIR
CE CC C C E E
=− −
(12.30)


=− +






VIR
R
CC C C
E
F
α
(12.31)



12.2.2 Bias stability

Equation (12.30) gives the parameters that influence the bias current
I
C
. The
voltage
V
BB
depends on the supply voltage
V
CC
. In some cases,
V
CC
would
vary with
I
C
, but by using a stabilized voltage supply we can ignore the
changes in
V
CC
, and hence
V
BB
.
The changes in the resistances
R
BB
and
R
E
are negligible. There is a variation of
β
F
with respect to changes in
I
C
.
A typical plot of
β
F
versus
I
C
is shown in Figure 12.7.


B
f
________
B
f
max
0.5
1
I
C

Figure 12.7 Normalized plot of
β
F
as a Function of Collector
Current

© 1999 CRC Press LLC

© 1999 CRC Press LLC

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