FRANKFURT UNIVERSITY OF APPLIED SCIENCES
Faculty of Computer Science and Engineering
Electronics
Academic Year 2017/2018
Prof. Dr.-Ing. G. Zimmer
Contents
1
2
3
Semiconductor Basics
1.1 Band theory of solids . . . . . . . .
1.2 Intrinsic conductivity . . . . . . . .
1.3 Doping . . . . . . . . . . . . . . .
1.4 Diffusion currents in semiconductors
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Semiconductor diode and applications
2.1 The pn-junction . . . . . . . . . . . . . . . . .
2.1.1 pn-junction with zero bias . . . . . . .
2.1.2 pn-junction with bias . . . . . . . . . .
2.1.3 Small-signal model of a pn-junction . .
2.1.4 Spice model of a semiconductor diode .
2.1.5 Different types of semiconductor diodes
2.2 Diode applications . . . . . . . . . . . . . . .
2.2.1 Diode as rectifier . . . . . . . . . . . .
2.2.2 Voltage multiplier . . . . . . . . . . . .
2.2.3 Zener diode as voltage regulator . . . .
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The bipolar junction transistor and applications
3.1 The bipolar junction transistor . . . . . . . . . . . . .
3.1.1 Structure and operation principles of a npn BJT
3.1.2 Static input and output characteristics of a BJT
3.1.3 Simple small signal BJT model . . . . . . . .
3.1.4 Advanced small signal BJT model . . . . . . .
3.1.5 SPICE model of a BJT . . . . . . . . . . . . .
3.2 Small signal amplifier . . . . . . . . . . . . . . . . . .
3.2.1 BJT biasing . . . . . . . . . . . . . . . . . . .
3.2.2 Common-emitter amplifier . . . . . . . . . . .
3.2.3 Common-collector amplifier . . . . . . . . . .
3.3 Integrated circuit techniques . . . . . . . . . . . . . .
3.3.1 The differential amplifier . . . . . . . . . . . .
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10
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17
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34
36
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41
41
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44
47
50
50
52
52
55
61
65
65
3.3.2
3.3.3
3.3.4
3.3.5
Current Sources . . . . . . . .
Active Load . . . . . . . . . .
Level-Shifting Circuits . . . .
Complementary Output Stage
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71
72
73
74
Chapter 1
Semiconductor Basics
Important elements of electric communications systems are devices capable of
amplifying the weak received electrical signals making further signal processing possible. Up to the sixties in most purchasable receivers vacuum tubes were
used for this purpose. In most pratical applications vacuum tubes were replaced
by transistors after the bipolar transistor was invented by Bardeen and Brattain in
1948 and the theoretical prediction of the planar bipolar transistor by Schockley in
1949. Compared to vacuum tubes transistors have an almost infinite lifetime and
it is possible to combine a large amount of transistors to form integrated electronic
circuit with a very high functionality. To understand the operation principles of
semiconductor devices, in the first section their physical basics will be summarized, while in the following sections the devices and their equivalent circuits are
discussed. Today the most important semiconducting material is silicon (Si). In
contrast to metal the conductivity of a semiductor is quite low but raises with increasing temperature. To understand this strange physical behaviour we will first
discuss the atomic structure of a semiconductor and we will have a look on the
band theory of solids.
1.1
Band theory of solids
In the framework of Maxwells theory the influence of bodies is described by
scalare values like the conductivity κ, the permittivity ε and the permeability µ.
But they are not subjects of the theory itself. To explain their physical base solid
states physics was introduced, which has its own base in atom physics.
Since the beginning of the nineteenth century most physicists agree that matter
is composed out of atoms, introduced by the greek philosopher Demokrit. Rutherford, an English physicist showed with experiments that an atom consists of a very
small positive nucleus (diameter ≈ 10−13 to 10−12 cm) carrying positive charges
2
surrounded by the same amount of negative charges, called electrons, so that the
atom itself is neutral. In this classical picture electrons circle around the positive
nucleus like the planets circle around the sun.
According to Maxwells theory the classical picture of an atom cannot be right,
since an electron moving around the nucleus of an atom would losse its energy by
emitting an electromagnetic wave, thus losing its energy and dropping into the
nucleus. It was Niels Bohr, a Danish physicist, who postulated that an atom does
not behave like a classical object, being able to exchange arbitrary amounts of
electromagnetic energy with its environment, but only in mulitples of an energy
unit ∆W , already introduced by the German physicist Max Planck, to describe the
black-body radiation
∆W = h f
(1.1)
In equation 1.1 h = 6.624 10−34 Ws2 stands for Plancks action quatum or Plancks
constant while f describes the frequency of the radiated electromagnetic wave.
According to Niels Bohr an electron bound to the nucleus of an atom can only
occupy certain levels of total energy, as shown in Fig. 1.1a. If an electron does
W 6
x
W 6
-
W3
W2
a)
W1
b)
Figure 1.1: Energy levels of an atom and electronic band structure of a crystal
lattice
not occupy its lowest energy level, it can drop from the energy level W j to the
lower energy level Wi by emitting an electromagnetic wave of frequency
f =
1
(W j − Wi )
h
One says the electron changed its quantum state. If we put a large amount of atoms
together we can in principle form a crystal. Due to the Pauli exclusion principle,
different electrons may not exist in the same quamtum state. That is the reason
why in a crystal the single energy levels of an atom will split up into closely spaced
energy levels forming a so-called electronic band structure as shown in Fig. 1.1b.
In principle all energy levels within a band may be occupied by electrons, while
no electrons may exist at energy levels between the bands. If we cool down a
3
crystal to an absolute temperature of T ≈ 0K, all atoms of the crystal will exist
at their ground states and all energy levels within the electronic band structure
will be occupied up to a certain level. This level is called Fermi level WF . If
the temperature is increased, energy levels above the Fermi level may also be
occupied by electrons. The propability p(W ) that a certain energy level W is
occupied by an electron is given by the so called Fermi-Dirac distribution [].
p(W ) =
1
W −WF
exp(
)+1
kB T
with kB = 1, 38 10−23 J/K
(1.2)
(Boltzmanns constant)
Considering the band with the highest energy one can distinguish between two
different cases:
1. Electrons do not occupy all energy levels within the band. As a result there
will exist free energy states slightly above the states already occupied. If an
electric field is applied, electrons are being accelerated by the field, enhancing their kinetic energy and thus reaching higher energy levels. Electrons
will move thru the crystal due to the electric field, resulting in an electric
current. This scenario describes the situation within metals as shown in Fig.
1.2a.
2. At the absolute temperature T = 0K all lower energy bands are totally occupied. The occupied band with the highest energy level is called valence
band. Normally there will exist a further energy band above the valence
band, called conduction band. The energy difference between the highest
possible energy state in the valence band and the lowest energy state in the
conduction band is called the band gap ∆W of the crystal. If we have ∆W <
5eV one speaks of a semiconductor while for ∆W > 5eV we speak of an
isolator. In Fig. 1.2 the band structure of the different materials is shown.
Since the band structure shows the energy of the negative electrons inside a crystal
the product of the electrostatic potential function Φe and the elementary charge e
is up to an arbritray constant equal to the band energy. Thus the following relation
holds true:
1
1
(1.3)
Φe = − WL +C1 = − WV +C2
e
e
1.2
Intrinsic conductivity
The semiconductor silicon is a group IV element of the periodic table, thus it
possesses four valence electrons and forms a face-centered diamond cubic crystal
4
W
6
0
W
W
6
0
6
0
WL
WL
WL
WV
WV
WV
a) metal
b) semiconductor
c) isolator
Figure 1.2: Band structure of a metal, semicondcutor and isolator
structure. In the ideal crystal each atom forms covalent bondings with its four
neighbours, as schematically illustrated in Fig. 1.3a, which shows a plane model
of the crystal. At the absolute temperature T = 0K, all valence electrons are
trapped in covalent bondings. Thus considering the band structure, the valence
band is totally occupied, while the conduction band is totally empty as shown in
Fig. 1.3b. Hence there do not exist free charges inside the crystal and it is an
j
j
j
j
W 6
j
j
j
j
WC
j
j
j
j
j
j
j
j
WV
a)
j
Si-atom
b)
covalent bonding
Figure 1.3: Plane model of Si-crystal and band structure at T = 0K
isolator. If the temperature is enhanced the atoms of the crystal will perform a
vibration around their mean location. With increasing temperature the thermal
movement of the single atoms can become so strong that single covalent bondings
will break. Now the valence electron will no longer be trapped to the bonding but
can almost freely move within the crystal. This situation is sketched in Fig. 1.4a.
In the picture of the band structure the thermal energy of the atom has moved an
electron from the valence to the conduction band. If an electric field is applied to
5
j
j
j
j
j j
u
j
j
j j
u
j
j
?
j
j
?
j
W 6
6
Si-atom
u
j
electron
u
u
WV
e
e
E
b)
a)
j
WC
e
hole
Figure 1.4: Plane model of Si-crystal and band structure at T > 0K
the crystal the free electrons will move against the field direction. But not only
the free electrons will move, the electrons trapped in a bonding will move too.
Since one bonding electron is missing, other valence electrons may replace the
missing electron resulting in a movement of the missing electron in the field direction. The missing electron thus behaves like a positive charge and is called a
hole. The thermal induced breaking of a bonding thus results in the creation of an
electron-hole pair in the picture of the band structure. Beside the thermal creation
of electron-hole pairs there exists a process called recombination. In this process
a free electron will be trapped again in a covalent bonding, which is equivalent
to the annihilation of an electron-hole pair. In the thermal equilibrium both processes are in balance and for a given temperature we will have a certain density of
electrons n and holes p in the crystal.
To calculate their values one not only has to take into account the Fermi-Dirac
distribution, but also the function D(W ) which describes the density of states in
the crystal. If one approximates the Fermi-Dirac distribution by the Boltzmann
distribution one finds the following equations describing the electron and hole
density inside a crystal []:
n = NC exp(−
2πm∗e kB T 3/2
WC −WF
) mit NC = 2(
)
kB T
h2
(1.4)
2πm∗p kB T 3/2
WF −WV
) mit NV = 2(
)
(1.5)
kB T
h2
Where m∗e denotes the effective electron mass in the conduction band and m∗p
the effective hole mass in the valence band. This correction has to be done to
reflect the difference between a free particle and an almost free particle in the
p = NV exp(−
6
periodic potential inside a cyrstal. The values NC and NV are called effective
density of states in the conduction band respectively valence band. Table 1.1
gives some examples for the effective masses of electrons and holes for different
semiconductors. As already discussed earlier, the electron and hole density are
Semiconductor m∗e /me
Si
0,33
Ge
0,22
GaAs
0,067
InP
0,078
m∗p /me
0,56
0,33
0,48
0,64
Table 1.1: Effective masses of electrons and holes for different semiconductors []
equal in an ideal semiconductor. This opens the opportunity to define the so-called
intrinsic charge density ni of a semiconductor by:
ni =
√
np
(1.6)
With the help of the equations 1.4 and 1.5 and ∆W = WC −WV we find:
ni =
√
∆W
NL NV exp(−
)
2kB T
(1.7)
Example: Intrinsic charge density
Germanium: ∆W = 0,63 eV, Silicon: ∆W = 1,14 eV, T = 300K
ni Ge ≈ 1, 8 1013 1 3
cm
9
ni Si ≈ 2, 6 10 1 3
cm
The examples show that we have a much lower intrinsic charge density in silicon at
the same temperature, due to its larger band gap. If we expose the semiconductor
to an electric field the electrons as well as the holes will move with different mean
velocities thru the crystal lattice. This effect is described by the electron mobility
µe and the hole mobility µ p . Table 1.2 gives the mobility of electrons and holes for
different crystals. With the help of the mobility of electrons and holes and their
densities one can formulate the law describing the conductivity of a semiconductor
[].
κ = e(µn n + µ p p)
(1.8)
Example: Intrinsic conductivity of germanium and silicon at T = 300 K
7
Crystal Electron Holes
Si
1300
500
Ge
4500 3500
GaAs
8800
400
InSb
77000
750
InAs
33000
460
InP
4600
150
Table 1.2: Mobility of electrons and holes for different crystals in cm2 /Vs []
κi Ge ≈ 2.3 10−2 S/cm
κi Si ≈ 7.5 10−7 S/cm
For example copper at the same temperature has a conductivity of κCu ≈ 5.9 105 S/cm
which is by a factor of 107 higher than the conductivity of geramium.
1.3
Doping
The property of semiconductors that makes them most useful for constructing
electronic devices is that their conductivity may easily be modified by introducing
impurities into their crystal lattice. The process of adding controlled impurities to
j
z
u
j
j
W 6
j
j
j
j
WC
j
j
z
j
j
j
j
z
u
u
donor
level
u
h
u
h
u
h
WF
WV
a)
z donor atom
u
b)
free electron
Figure 1.5: Plane lattice and band structure of a n-condcutor
a semiconductor is known as doping. The amount of impurity, or dopant, added
to an intrinsic semiconductor can variegate its level of conductivity in a wide
8
range. Most useful doping materials are atoms of group 5 of the periodic table of
elements like phosphor (P), arsenic (As) and antimony (Sb) and atoms of group
3 like boron (B), aluminium (AL) and indium (In). To clearyfy the influence of
doping we will have a look on Fig.1.5. Again Fig. 1.5 shows a plane model of the
Si lattice. But in contrast to an ideal Si lattice some of the Si atoms are replaced
by atoms having five valence electrons. To build up the crystal lattice only four
valence electrons are needed, thus the fifth electron is only weakly bounded to the
impurity atom. So only very little thermal energy is needed to free the electron. In
the picture of the band structure each impurity atom will contribute its fifth valence
electron to the conduction band. If we use ND to denote the volume density of the
donator atoms, this will resut in
n ≈ ND
Hence with the help of the donator atoms, we can influence the density of the free
electrons in the semiconductor, which is according to equation 1.4 equivalent to a
shift of the Fermi-level
NC
(1.9)
WF ≈ WL − kB T ln( )
ND
Since the product np = n2i only depends on the band gap of the semiconductor we
have,
n2
n2
p = i ≈ i
n
ND
while the conductivity of the n-conductor is essentially given by
κ ≈ eµn ND
(1.10)
In a n-doped semiconductor the elctrons are called majority carrier while the holes
are called minority carrier. If we use doping atoms out of group 3 of the periodic
table of elements, one valence electron is missing. Due to thermal vibrations
this missing bonding can easily move from one atom to the other as shown in
Fig. 1.6. But as already introduced, a missing bonding electron is called a hole in
semiconductor theory. If we use NA to denote the volume density of the impurities,
each so-called acceptor atom will contribute a free hole to the valence band and
we have,
p ≈ NA
and with the help of equation 1.5 we can find the shift of the Fermi-level
NV
(1.11)
WF ≈ WV + kB T ln( )
NA
In a p-doped semiconductor the holes are the majority carriers while the electrons
are the minority carriers. For the conductivity of a p-type semiconductor we find:
κ ≈ eµ p NA
9
(1.12)
j
z
j
j
W 6
j
WC
e
j
j
j
j
j
z
j
z
j
j
j
e
e
atom
e
h
e
h
e
WF
WV
b)
a)
z acceptor
h
e
acceptor
level
free hole
Figure 1.6: Plane lattice and band structure of a p-condcutor
1.4
Diffusion currents in semiconductors
In contrast to a metal diffusion currents may play an important roll within semiconductors, due to the effect that there might exist electrons and holes within the
same volume, forming an electric neutral carrier concentration. To explain the
process of diffusion we have a look at Fig. 1.7. It shows a plane section of a crysr r r r r
r
r
r r r r r r r
r
r
r
r
r
r
r r r
r r r
r r r r r r
r
r
r r r
rr r r r r r
r
r
x-
Figure 1.7: Diffusion process within a crystal lattice
tal lattice in which a uniform concentration drop in the x direction exists, which
is represented by a different amount of particles within a certain region. If we
assume that due to thermal motion one half of the particles moves to the right and
the other half moves to the left, we get a net particle flow in the direction of the
concentration drop. So, diffusion does not need external forces to act on a group
of particles, but is just driven by their thermal energy. If we define with J pD (x)
the one dimensional diffusion current density of the holes and with JnD (x) the diffusion current density of the electrons, the diffusion current is described by the
following equations:
J pD (x) = −eD p
dp
dx
JnD (x) = eDn
10
dn
dx
(1.13)
The positive sign in the equation of the electron current density reflects the defintion of the positive technical current direction, which is contrary to the movement
of the electrons. The constants D p and Dn are called diffusion coefficients and
they are related to the mobility of the carriers by Einsteins relation [?, ]
Dp = µp
kB T
e
Dn = µn
kB T
e
(1.14)
Thus the total current density of the holes J p and of the electrons in a semiconductor is composed of the drift current due to an electric field and the diffusion
current.
dp
J p = eµ p pE − eD p
(1.15)
dx
dn
Jn = eµn nE + eDn
(1.16)
dx
11
Chapter 2
Semiconductor diode and
applications
2.1
The pn-junction
The simplest semiconductor component fabricated from both n-type and p-type
material is the semiconductor diode, a two-terminal device which, ideally, permits
conduction with one polarity of applied voltage and completely blocks conduction
when the voltage is reversed. For the mathematical despriction of a pn-junction we
will assume that changes in the crystal structure only occur in the x-direction while
the structure is homogenous in the y- and z-direction. As a result all considered
properties will only be functions of the x-coordinate.
2.1.1 pn-junction with zero bias
To understand the physical behaviour of a pn-junction we will first consider the
junction being separated by an ideal, fictive, infinite thin membrane as shown in
Fig. 2.1. In the n-region will exist a huge amount of free electrons, moving arbitrarily thru that region due to their thermal energy. There will also exist the same
amount of positive donator atoms being fixed in the crystal lattice. In the adjacend
p-region we formally have the same situation but now the holes play the role of the
electrons and the donators are replaced by fixed negative acceptor atoms. If we assume the fictive membrane to be removed, due to the difference in concentration,
the free holes of the p-region will diffuse into the n-region, while the free electrons
of the n-region will diffuse in the p-region and a recombination of electron-hole
pairs will occur. As a result a transition region will be established between the pand n-region, were only the fixed acceptor and donator atoms exist but essentially
no free carrier. As a further consequence an internal electric field will be built up,
12
e
e
e
e
e
e
a)
e
e
e
u
e
e
u
e
e
e
e
e
e
e
e
e
b)
e
u
u
u
u
u
free hole
free electron
u
u
E
u
u
u
e
u
u
u
u
u
u
u
u
u
u
fixed acceptor
fixed donator
Figure 2.1: pn-junction with and without a fictive membran
canceling the diffusion process of the free carriers and also resulting in a potential
difference between the end faces of the crystal. This potential difference is called
diffusion or build-in voltage UD and is given by the following equation:
UD = Φe (∞) − Φe (−∞)
(2.1)
To calculate the hole distribution p(x) we use equation 1.15 and consider the fact
that the diffusion process has stopped (JP = 0) and that one can deduce the electric
field by the gradient of the potential function, which is related to the valence band
energy WV (x) via equation 1.3:
kB T
dWV (x)
d p(x)
= p(x)
dx
dx
The last differential equation can be solved by separation of the variables, while
the neccessary constant can be deduced from the boundary condition p(x → −∞) =
NA . Thus we get for the distribution of the holes:
)
(
WV (−∞) −WV (x)
(2.2)
p(x) = NA exp −
kB T
In an analog manner we get for the distribution of the electrons using the boundary
condition n(x → ∞) = ND :
(
)
WC (x) −WL (∞)
n(x) = ND exp −
(2.3)
kB T
13
To get a unique relation between the potential function Φe (x) and the band energies one uses the condition Φe (x → −∞) = 0. As a result we get the following
relation between the potential function and the band energies of the valence and
the conduction band:
1
1
Φe (x) = − [WV (x) −WV (−∞)] = − [WC (x) −WC (−∞)]
e
e
(2.4)
With the help of the last equations the hole and the electron distribution may be
expressed by the potential function and the diffusion voltage:
(
)
Φe (x)
p(x) = NA exp −e k T
B
(
)
(2.5)
UD − Φe (x)
n(x) = ND exp −e
k T
B
The last two equations in combination with equation 1.7, may be used to deduce an
expression for the diffusion voltage without knowledge of the potential function
Φe (x):
NA ND
kB T
ln( 2 )
(2.6)
UD =
e
ni
Example: Diffusion voltage of a pn-junction in silicon
NA = ND = 1015 cm−3 , T = 300K
)
(
(1015 )2
= 660 mV
UD ≈ 25.9 mV ln
(2.6 109 )2
According to equation 2.2 and 2.3 the decline of the electron and hole distribuρ(x) 6
−w p
eNA
eND
-
wn
x
Figure 2.2: Charge distribution of an abrupt pn-junction
tion follows an exponential function. To calculate the potential function one can
approximate the carrier distributions in the transition region by a step function
14
according Fig. 2.2. This approximation is called abrupt pn-junction [2] and considers a constant negative charge distribution NA in the region −w p < x < 0 and
a constant positive charge distribution ND in the region 0 < x < wn . Since the
pn-junction is electrically neutral, the following equation must hold true:
ND wn = NA w p
(2.7)
To calculate the internal electric field und potential function of an abrupt pnjunction we use the one-dimensional divergence theorem of the electrical field,
which results in the following differential equations for the electric field:
dE
e
= − NA
dx
ε
− wp < x < 0
for
(2.8)
dE
e
= ND
for
0 < x < wn
(2.9)
dx
ε
The last equations can be integrated easily and one finds the following functional
dependence taking into account that the electric field may only exist in the region
−w p < x < wn
E(x) = −
eNA
(x + w p )
ε
for
− wp < x < 0
(2.10)
eND
(wn − x)
for
0 < x < wn
(2.11)
ε
According to the above equations the value of the electric field will first fall linear
reaching its negative maximum at x = 0 and then will rise also linear to reach zero
again at x = wn . The negative sign of the electric field reflects the fact that it is
directed in the negative x-direction, as already shown in Fig. 2.1b. The potential
function can again be evaluated by integration, while the integration constants
must be chosen to reflect the following boundary conditions Φe (−w p ) = 0 and
Φe (x = 0− ) = Φe (x = 0+ ).
E(x) = −
Φe (x) =
eNA
(x + w p )2
2ε
for
− wp ≤ x ≤ 0
(2.12)
eND
(wn (w p + 2x) − x2 ) for 0 ≤ x ≤ wn
(2.13)
2ε
In Fig 2.3 the functional dependence of the electric field and the potential function
of an abrupt pn-junction is shown. With the help of the last equation and equation
2.7 the values w p and wn of the depletion zone may be evaluated.
Φe (x) =
15
−w p
@
@
E(x)
6
wn
-
x
@
@
@
−Emax
Φe (x) 6
6
UD
?
-
x
Figure 2.3: Electric field and potential function of an abrupt pn-junction without
bias
√
wp =
√
wn =
ND
2ε U
e D NA (NA + ND )
NA
2ε U
e D ND (NA + ND )
As a result we get for the total width of the depletion zone:
√
2ε NA + ND
w =
UD
(2.14)
e NA ND
According to Fig. 2.3 the electric field reaches its highest absolute value Emax at
the coordinate x = 0. Since the potential function is in the one-dimensional case
the integral of the electric field, the easiest way to calculate its value is to evaluate
the area under the graph of the function:
1
UD = (w p + wn )Emax
2
Using equation 2.14 we find for the maximum of the electric field:
√
2UD
2eUD NA ND
Emax =
=
w
ε NA + ND
(2.15)
Fig. 2.4 shows the energy band model of a pn-junction at zero bias. Due to the
locally fixed acceptor and donator atoms an internal electric field is created within
the depletion area, which results in a potential difference between the p- and nconductor called diffusion voltage or build-in voltage UD and in a band bending.
16
W 6
p-conductor
PP
P
n-conductor
u
PPX
z
X
PX
P
P
P
e e e e e e e e e e PP
X
yP
XPeP
PP
P
P
?
eUD
uuuuuuuuu
6
WC
WF
WV
-
−w p
wn
x
Figure 2.4: Energy band model of a pn-junction at zero bias
The influence of the electric field on thermally excited electrons can easily be
illustrated with the help of the band bending. If a thermally excited electron tries
to jump over the potential barrier it behaves like a sphere on a hill, which rolls
to the bottom again. In contrast holes act quite different. They behave more like
balloons in a water basin, they always bob up to the highest energy value in the
valence band, as illustrated in Fig. 2.4.
2.1.2
pn-junction with bias
With the help of the energy band diagrams shown in Fig. 2.5 in a first step we
now want to discuss qualtively the operation principles of a pn-junction, if a bias
is applied. According to Fig. 2.5 a bias voltage is applied to the pn-junction with
a direction opposed to the internal electric field. Hence it will lower the potential
barrier between the p- and n-conductor. Due to their thermal energy now electrons
of the n-conductor as well as holes of the p-conductor are able to surmount the
potential barrier and will diffuse into the p- as well as into the n-conductor. Being
minority carrier in these regions they will recombine and as a result a current
will flow in the direction of the applied voltage. If we change the direction of
the applied voltage, the internal electric field will be enhanced, resulting in an
enhanced potential barrier. As a result the thermal energy neither of the holes nor
of the electrons is high enough to surmount the barrier. So, in principle no carrier
exchange between the two regions of the pn-junction will take place. Only due to
the intrinsic conductivity there will be a small amount of reverse current flow.
Analysis
After the qualitative discussion of the operation principles we will now describe
the process taking place in more mathematical depth. To deduce the mathematical
17
U
HH
s
s
W 6
e(UD −U)
uuu X
?
XXX
Xy
""
X
XXX
XXu u u u u u u u u u
X
WC
6
6
WF
?
e - eeeeeeeX
eX
XX
XX
XX
XX
X
zX e e e
WV
a)
-
−w p
W
HH
x
wn
6
6
H
HHu
e(UD −U)
HH
j
H
HH u u u u u u u u u u u
?
H
hhh
hhhh
hh
e e e e e e e e e eH
HH
He
H
YH
HH
H
HH
b)
−w p
WC
WF
WV
-
wn
x
Figure 2.5: Energy band model a) forward bias b) reverse bias
description we will use the following basic assumptions:
• The voltage drop along the regions of the p- and n-conductor is neglected
and it is assumed that it only takes place along the depletion zone of the
pn-junction.
• The current due to the minority carrier can solely be described as diffusion
current.
• In the depletion zone no recombination takes place. As a result the total
current thru the diode can be described as the diffusion current of the minority carrier at the boundaries of the depletion zone at each side of the
pn-junction.
We will start our analysis by considering the density of holes p(x) in the nconductor. The concentration of the electrons n(x) in the p-conductor can be
18
deduced in an equivalent way. Starting from equation 2.5 we get for the hole
density at x = wn in dependence of the applied voltage U:
p(+wn ) = NA exp(
−e(UD −U)
eU
) = pno exp( )
kB T
kT
(2.16)
In the last expression the constant pno denotes the hole density in the undisturbed
n-region (x → ∞). According to equation 2.16 the hole concentration will rise
with U > 0 and decay for U > 0. To calculate the hole distribution in the n-region
we use the rate equation ??, extended by the divergence term of the currents [2]:
∂p
1 ∂J p p − pno
= −
−
∂t
e ∂x
τ
In the stationary case ( ∂ = 0) this expression reduces to:
∂t
dJ p
p − pno
= −e
(2.17)
dx
τ
According to our assumption the current J p is solely a diffusion current due to the
minority carrier and we get for the region wn < x < ∞ the following differential
equation:
1
1
d2 p
=
(p
−
p
)
=
(p − pno )
(2.18)
no
D pτ
dx2
L2p
√
In the last term the constant L p = D p τ was introduced, it posseses the dimension
of a length and hence denotes the mean length along which a minority carrier can
diffuse in its lifetime τ before it will recombinate. The solution of the above
differential equation has to reflect that for x = wn the hole density is given by
equation 2.16 and hence we find as solution:
p(x) = (p(wn ) − pno ) exp(−
x − wn
) + pno
Lp
(2.19)
According to equation 2.19 the denisty of the minority carrier in the n-region is
governed by an decaying exponential function. Using equation 1.15 we find the
following diffusion current density at x = wn :
(
)
Dp
eU
J p (wn ) = e pno exp(
)−1
(2.20)
Lp
kB T
In an equivalent way one can also deduce an expression for the diffusion current
Jn (w p ) and since we assume that there will be no recombination in the depletion
zone we get for the total current thru a pn-junction:
(
(
)
)
Dp
eU
Dn
2
J = Js exp(
)−1
with Js = e ni
+
(2.21)
kB T
L p ND Ln NA
19
According to equation 2.21 the current density thru the pn-junction will rise exponentially for positive voltages U, while it will decay for negative values. In the
limit it will reach a value of Js , hence this value is called reverse biased saturation current density. If we multiply the current density with the area A pn of the
pn-junction we get the static I-U characteristic of an ideal pn-junction.
(
)
U
kB T
I = Is exp( ) − 1
with Is = A pn Js and UT =
(2.22)
UT
e
In equation 2.22 the constant UT was introduced, which is called thermal voltage.
At room temperature (T = 300 K) it shows a vaule of approximately 26 mV. Since
the voltage drop along the p- and c-conductor was neglected, equation 2.22 is
only valid for small currents. The ohmic behaviour for these regions can in a first
step be approximated by a resistor Rs . As a result the ideal pn-junction is only
Figure 2.6: Static I-U-characteristic of an ideal pn-junction with Is = 10nA
controlled by the reduced voltage U − Rs I. To demonstrate the influence of this
resistor in Fig. 2.6 the static I-U-characteristic of an ideal pn-junction with Is =
10nA and of the same diode with Rs = 1Ω is shown.
20
2.1.3 Small-signal model of a pn-junction
The equations deduced in the preceding section describe the behaviour of a pnjunction only for almost static time functions. To get an idea of its dynamic behaviour it is useful to study small signal exitation at a given operation point. In
principle we will study the circuit given in Fig. 2.7, where a DC current source
I is used to setup a certain operation point and a sinusoidal current source i(t) is
used to realize the small signal exitation.
r
I 6
i(t) 6
A
D
r
Figure 2.7: Small signal exitation of a pn-junction
Dynamic resistance rD
According to Fig. 2.7 we assume the pn-junction to be operated in a given operation point (I, U). Due to the sinusoidal current source with amplitude Iˆ there
will also exist a sinusoidal voltage across the pn-junction with amplitude Û. One
speaks of small signal exitation as long as the following relations hold true:
Iˆ ≪ I
and Û ≪ U
For a first order approximation, we will describe the current voltage characteristic
by its slope at the operation point. Hence for the amplitudes of the sinusoidal time
functions the following relation holds true:
Û ≈
1 ˆ
dU ˆ
I =
I = rD Iˆ
dI
dI
dU
In the last equation the dynamic resistance rD of a pn-junction was introduced.
Assuming the pn-junction is forward biased, we get the following expression for
the dynamic resistance using equation 2.22:
rD =
21
UT
I
(2.23)
Example: Dynamic resistance of a pn-junction at an operation point of I = 10
mA.
According to equation 2.23 we find
rD =
25, 9 mV
≈ 2, 6 Ω
10 mA
Diffusion capacitance cD
As already discussed in section 2.1.2 a forward biased pn-junction will store minority carrier in the n- as well as in the p-region. So each change in voltage ∆u
at a given operation point will also result in a change for stored minority carrier.
To calculate the stored minority carrier in the n-region we use equation 2.19 and
perform an integration over the n-region:
)
∫ ∞(
x − wn
Q(U) = eA pn
) dx
(p(wn ) − pno ) exp(−
Lp
wn
For a differential change of the applied voltage ∆u we can write:
∆q ≈
eAD L p pno
U
dQ(U)
exp( ) ∆u
∆u =
dU
UT
UT
(2.24)
Equation 2.24 can be used to define the diffusion capacitance cD of a pn-junction.
cD =
eAD L p pno
U
τ
τ
∆q
=
exp( ) =
I =
∆u
UT
UT
UT
rD
(2.25)
Example: Diffusion capacitance of pn-junction at an operation point of I = 1 mA.
In silicon diodes the minority carriers have a lifetime of τ ≈ 2.5 10−3 s
cD =
2, 5 ms
1mA ≈ 97 µF
25, 9 mV
According to the last example, the diffusion capacitance shows relatively high
values. Since the dynamic resistance and the diffusion capacitance are essentially
connected in parallel, the storage of the minority carrier in the p- and n-regions
inhibits the technical usage of the dynamic resistance at higher frequencies of an
ordinary pn-junction diode, since it is short circuited by the capacitance.
22
Junction capacitance cJ
To deduce an expression for the junction capacitance we have a look at Fig. 2.3,
which shows the electric field distribution inside the depletion zone of the pnjunction. If the applied voltage is changed with time also the electric field will
change, resulting in a displacement current density. To find an expression of its
value we start with equation 2.15 and assume that the total voltage u pn (t) is given
by the sum of a DC voltage Uo and a time varying voltage ∆u(t). Hence we get
for the electric field in the pn-junction
√
2e NA ND
1
E(t, x = 0) = −
(UD −Uo ) (1 −
∆u(t))
(2.26)
ε NA + ND
2(UD −Uo )
and for the displacement current density
√
dE(t, x = 0)
NA ND
1
d∆u(t)
Jv = ε
= eε
dt
NA + ND 2(UD −Uo ) dt
(2.27)
Since we assume a homogenous distribution across the cross-section of the pnjunction, the total displacement current can be calculated by multiplication with
the area A pn of the pn-junction. According to the definition of a capacitance the
factor in front of the time differential of the voltage must be the expression for the
junction capacitance.
√
eε
NA ND
(2.28)
cJ = A pn
NA + ND 2(UD −Uo )
To describe the small signal frequency response of a real semiconductor diode in
s
s LS
CP
RS
s
rD
cJ
s
s
s
Figure 2.8: Small signal equivalent circuit of a real semiconductor diode
Fig. 2.8 its equivalent circuit is given. Besides the elements already discussed two
further elements are included. This is a series inductance LS accounting for wire
bonds and a parallel capacitance CP reflecting the influence of the packaging.
23
2.1.4 Spice model of a semiconductor diode
In the preceding section we discussed the behaviour of an ideal pn-junction. As an
electric two terminal device it is called semiconductor diode. Since all electronic
devices exhibit strong nonlinearities the behaviour of an electronic circuit can only
be analysed by using sophisticated simulation tools. Most of todays commercial
available tools are based on a simulator called SPICE Simulation programm with
Integrated Circuit Emphasis which was developed at the University of Berkley [].
Even though we already discussed several effects and parameters of an ideal pnjunction a real diode needs even more parameters to describe its real behaviour.
In the following section we will give a short introduction to the equation used
to describe a real diode in the Spice simulation tool, while the denotation of the
parameters a summarised in Table ?? at the end of this section. Fig. 2.9 shows
the equivalent circuit that is used in SPICE. The total time dependent current iD (t)
s
?iD
RS
uD
s
?ID
A
? s
s
CJ
CD
s
s
Figure 2.9: Spice model of a semiconductor diode
thru the diode is calculated using the following equation:
iD = ID + CD
duD
duD
+ CJ
dt
dt
(2.29)
Static diode current ID
Forward biased, the static diode current ID is equal to the current of an ideal pnjunction already given in equation 2.22, but with a further parameter N included,
called emission coefficient.
[
(
]
)
uD
IDi = IS(T ) exp
−1
(2.30)
N ·UT
24
Here the temperature dependence of the saturation current IS(T ) is given by the
following expression:
(
T
IS(T ) = IS ·
T0
)(XT I/N)
(
EG(1 − T0 /T )
· exp
N kB T0
)
(2.31)
If a diode is reverse biased, experiments show that the real reverse current is higher
than that predicted by equation 2.30. To account for this effect an additional
current IDc of a so-called correction diode is added:
IDc
[
(
= ISR exp
uD
NR ·UT
)
] [(
]M/2
uD )2
−1 · 1−
+ 0, 005
VJ
(2.32)
If the reverse voltage of the diode is further enhanced reverse breakdown occurs
which is modeled by an exponential function:
)
(
−uD − BV
(2.33)
ID = −IBV exp
NBV ·UT
Dynamic diode current
To account for the dynamic behaviour of a real diode expressions for the junction capacitance and diffusion capacitance have to be considered. According to
equation 2.28 the junction capacitance varies proportional to the square root of the
applied reverse voltage. For a real diode this expression is slightly modified
(
uD )−M
CJ = CJO 1 −
VJ
(2.34)
If a diode is forward biased the lifetime of the miniority carrier of the junction has
to be considered. In its implementation SPICE uses also equation 2.25 already
discussed earlier.
diD
TT
=
(2.35)
CD = T T
duD
rD
In the following table the essential SPICE parameters used to specify a real diode
are summarized
IS
N
ISR
NR
BV
IBV
saturation current
emission coefficient
saturation current of correction diode
emission coefficient of ISR
reverse breakdown voltage
current at break-down voltage
25
NBV
RS
TT
CJ0
VJ
M
FC
EG
XT I
KF
AF
coefficient of IBV
series resistance
minority carrier life time
zero-bias junction capacitance
junction potential
grading coefficient
coefficient for forward-bias depletion capacitance formula
activation energy
temperature exponent of IS
flicker noise coefficient
flicker noise exponent
To include different diodes into SPICE ordinary ASCII-files are used as shown in
the following example.
Example: SPICE diode data sets
*----------------------------------------------------------.MODEL BAT68
D(IS=8N RS=2 N=1.05 XTI=1.8 EG=.68
+ CJO=.77P M=.075 VJ=.1 FC=.5 BV=8 IBV=1U TT=25P)
*----------------------------------------------------------.MODEL BA592 D (IS=185F RS=.15 N=1.305 BV=70 IBV=.1N
+ CJO=1.17P VJ=.12 M=.096 TT=125N)
*----------------------------------------------------------.MODEL BAS116 D(
+ AF= 1.00E+00
BV= 7.50E+01
CJO= 1.83E-12 EG= 1.11E+00
+ FC= 5.00E-01
IBV= 1.00E-04 IS= 1.48E-13
KF= 0.00E+00
+ M= 2.62E-01
N= 1.33E+00
RS= 8.48E-01
TT= 8.66E-09
+ VJ= 3.44E-01
XTI= 3.00E+00)
*-----------------------------------------------------------
2.1.5
Different types of semiconductor diodes
There were developed different types of junction diodes by emphasizing different
physical aspects for example by geometric scaling, by changing doping levels or
by the use of different semiductor materials. In the following section we will give
a short overview of the diodes most often used in electronics.
26
Zener diodes
The ordinary junction diode will be destroyed, if a reverse voltage is applied, extending their maximum reverse voltage and breakdown occurs. Zener diodes in
this sense are special diodes that will not be destroyed when the breakdown occurs. Furthermore it is possible to controll the breakdown voltage or Zener voltage
of the diode very precisley. Fig. 2.10 shows the current voltage characteristic of
an ideal Zener diode, which will be conducting as soon as the applied reverse voltID
−UZ0
6
-
UD
Figure 2.10: I-U characteristic of a ideal Zener diode
age exceeds the Zener voltage UZ0 . In practical applications these diodes are used
to stabilize a voltage to a certain level.
Schottky diode
From a historical point of view not the pn-junction but the crystal detector was
the first electronic device already used at the end of the 18th century. In principle
it consists of thin sharpened metal wire pressed against a crystal, thus forming a
metal to semiconductor contact. Today this kind of diode can also be constructed
using semiconductor technology and is called Schottky diode. But in contrast to
a pn-junction no minority carrier is essential for the nonlinear behaviour and they
tend to show a much lower junction capacitance. Thus they can be used up to very
high frequencies as mixers and detectors [].
Varactor diodes
As already discussed in section 2.1.3 if reverse biased, each junction diode shows a
certain capacitive value, that depends on the applied reverse voltage. Furthermore
the value of capacitance and its voltage dependence can be controlled using certain
doping profiles. Thus varactor diodes can be used to replace a capacitor, with the
advantage of being adjustable by an applied voltage. One of the main practical
application are their use in voltage controlled oscillators.
27
Photo detector
If a pn-junction is reverse biased only a small reverse current exists, due to thermal
creation of electron hole pairs within the depletion region. But if the pn-junction
is exposed to light and the photon energy is high enough to surmount the band gap
energy of the semiconductor ∆Wg they can create electron hole pairs. This process
is called absorption. If no external voltage is applied, the photodiode operates in
the mode of a solar cell, converting optical into electrical energy. If the diode is
reverse baised, it operates in the mode of a photo detector and can be used to sense
light. In this case the reverse current, called photo current I ph , is proportional to
the incident optical power Popt and the proportional constant is called responsivity
Rsp of the photo detector.
I ph = Rsp Popt
(2.36)
Light emitting diodes (LEDs)
The fundamental physical principle LEDs are based on is called spontaneous
emission []. If an electron of the conduction band recombines with a hole of
the valence band, the energy may be emitted as photon of a certain wavelength or
frequency, depending on the bandgap ∆Wg of the semiconductor.
λ =
C0 h
∆Wg
f =
∆Wg
h
(2.37)
But this process may only take place in certain semiconductors, called direct bandgap semiconductor. Unfortunatly silicon is no direct-band gap semiconductor.
So more sophisticated materials like GaAs have to be used. All LEDs produce
incoherent, narrow-band light.
Laser diodes
In a crude approximation a laser diode is a LED-like structure with an additional
optical resonator, formed by the endfaces of the semiconductor crystal itself. Due
to this resonator the bandwith of the light due to spontaneous emission is reduced
and stimulated emission takes place resulting in light with a high coherence [].
Laser diodes are commonly used in optical storage devices and for high speed
optical communication.
28
2.2
Diode applications
2.2.1 Diode as rectifier
In the previous sections we discussed intensively how to describe and model the
electrical behaviour of a semiconductor diode. For the basic understanding of
diode applications such as a rectifier circuit these models are even far to complicated. So here we will introduce the simplest possible model of a diode. From Fig.
2.6 we know that a semiconductor diode has a very strong nonlinear behaviour.
Essentially there will be no current flow, if it is reverse biased, but arbitrarilly
high currents if it is biased in the froward direction. In Fig. 2.11 the static I-UID
6
ideal diode→
← diode with threshold voltage Uth
-
Uth
UD
Figure 2.11: Diode modeled as a voltage sensitive switch
characteristic of an ideal diode is given. Essentially an ideal diode will behave like
a voltage sensitive switch. If the voltage UD across the diode is negative the diode
will show an infinite resistance, thus it behaves like an open switch. If on the other
hand the voltage across the diode is positive, it shows a very low resistance or the
switch is closed. Specially for discussion of the following basic diode applications this model is appropriate for their principle understanding. Especially, when
dealing with small voltages, the model with a certain threshold voltage Uth can be
used, also shown in Fig. 2.11. For normal silicon diodes the value of the threshold
voltage lies in the range from 0.6 V to 0.7V. The slight differences in behaviour
of real diodes can be examind using simulation tools.
Almost in all electronic equipment DC voltages of different values are needed
for their operation. Since the electric power distribution system uses AC voltages
of 230 V nominal they usually have to be transformed to a lower level and converted to DC. This process is called rectification and in most practical application
this is done with the help of semiconductor diodes. In the following sections we
will discuss different circuits that are used for rectification.
29
Half-wave rectifier
uD
r
uS
H
H
-
r
RL
?
uR
?
r
r
Figure 2.12: Circuit schematic of a half-wave rectifier
Fig. 2.12 shows the circuit schematic of a half-wave rectifier. It consists of an
alternating source delivering a sinusoidal voltage uS (t), a diode and a load resistance RL . It should be noted that the nominal output voltage UN of a transformer
is the effective value of the sinusoidal time √
function, so one always has to remember, that the amplitude Û is by a factor of 2 higher than the nominal value UN .
To understand the behaviour of the circuit we introduce the voltage uD (t) across
the diode and the voltage uR (t) across the load resistance. According to KVL the
following equation holds true:
−uS (t) + uD (t) + uR (t) = 0
Since the diode is essential for the operation of the circuit we first have a look on
the voltage across the diode
uD (t) = uS (t) − uR (t) = uS (t) − RL iD (t)
(2.38)
Starting with a positive half cycle all voltages are zero and so is the diode current
iD (t). If now the source voltage becomes positive, the diode voltage becomes
positive too and according to our model the diode will switch into its on state.
As a result the source voltage will drop across the load and the voltage across the
diode will essentially be zero. If now the negative half cycle will start, at first
again the diode current will be zero and as a result the voltage across the diode
will become negative. According to our model the diode will now switch into
its off state. No current iD (t) will exist and thus there will be no voltage drop
across the resistor, but the whole voltage of the source will drop across the diode.
As an example Fig. 2.13 shows the time function across the resistor as result
of a simulation with SPICE. The amplitude of the sinusoidal voltage source was
chosen to be 5V, with a load resistance of 500Ω and the diode BA592. Essentially
it shows the half-wave of the exiting voltage source, but there is a remarkable
30
Figure 2.13: Output voltage of a half-wave rectifier
difference. While the model of an ideal diode would propose an amplitude of
5V, the simulation shows that there will be a voltage drop of about 0.8V across
the diode at the peak voltage of the half cycle. For practical applications it is
neccessary to choose an appropriate diode for the application. Thus one has to
consider certain maximum ratings of a diode, which are usally given in their data
sheet. Two crucial parameters are the maximum reverse voltage URmax and the
maximum forward current IFmax . In case of a half-wave rectifier we must fulfill
the following conditions:
URmax > Û =
√
2UN
and
IFmax >
Û
RL
(2.39)
Of course the voltage shown in Fig. 2.13 is not yet a DC voltage but still a periodic
time function, with a DC part given by the following equation.
√
2
Û
=
UN
(2.40)
UDC =
π
π
To further smooth the ouput voltage a capacitor may be used as shown in Fig.
2.14. Fig. ?? shows the simulation results of the same half-wave rectifier where
according to Fig. 2.14 a capacitor of 100µF was included for smoothing, also
shown is the time function without a capacitor. So even if there seemed to be only
31
Figure 2.14: Half-wave rectifier with smoothing capacitor
a little change in the circuit due to the capacitor there is an significant change in
the maxium ratings the diode now has to withstand. At first we will have a look on
the simple equation for the diode voltage 2.38. In the limit of high load resistances
RL the maximum voltage will become nearly equal to the amplitude Û of the AC
voltage. So, according to equation 2.38 the maximum reverse voltage may reach a
value of 2Û, thus the following condition must be fulfilled, in case of a half-wave
rectifier with smoothing capacitor.
URmax > 2Û
(2.41)
But not only the diode must withstand a two times higher reverse voltage, but also
the maximum possible forward current is significantly changed due to the capacitor. This is because at swichting time a capacitor behaves like a short circuit.
Thus, if the rectifier is not switched on at a zero crossing, but at a certain positive
voltage value of the alternating source, the maximum forward current is only limited by internal resistances and can reach fairly hight values. To circumvent this
problem, it is sometimes neccessary to include a resistor in series to the diode to
limit the maximum possible forward current. Even though the circuit of a halfwave rectifier is very simple, it is also very inefficient for power transfer, since
only one half-cycle is used.
32
Full-wave rectifier
The circuit that allows us to use every half-wave of a cycle is called full-wave
rectifier. Fig. 2.15 shows its circuit schematic. To realise the two equal voltage
sources, in pratice a transformer is used whose secondary winding is split into two
with a center tap connected to the ground. In principle the upper and lower part
r
uS (t)
H
H
r
uD1 (t)
uR (t)
RL
?
r
?
uS (t) ?
uD2 (t)
r
H
H
Figure 2.15: Circuit schematic of a full-wave rectifier
of the circuit each work like a half-wave rectifier, but if the anode of diode one is
positive, due to the grounding of the sources, the anode of diode two is negative
and vice versa. Since now each half-wave will be rectified, we get for the DC part
of the voltage:
√
2
Û
UN
(2.42)
UDC = 2 = 2
π
π
while the maximum reverse voltage will reach a value of 2Û and thus the following condition must be fullfilled.
URmax > 2Û
(2.43)
One disadvantage of this kind of full-wave rectifier is the costly transformer, due
to its center tap. To over-come this a so-called bridge rectifier as shown in Fig.
2.16 may be used. With the help of this circuit the costly transformer is omitted by
the expense of two further diodes. During the positive half cycle D1 and D4 will
be conducting, while diodes D2 and D3 are reverse biased. Thus the current will
flow in the direction of diode D1 thru the resistor RL . If the polarity of the cycle
changes, now diodes D3 and D2 are conducting, while diodes D1 and D4 are reverse biased. Now the current will flow in the direction of D3 thru the resistor, but
this direction is identical to that during the positive half cycle. Thus independent
of the polarity of the half cycle, the current will always flow in the same direction
thru the load resistance. Fig. 2.17 shows the simulation results, again using the
33
D2
H
H
uS (t)
?
r
D1
H
H
r
r
D4
H
H
r
RL
-
D3
H
H
uR (t)
Figure 2.16: Circuit schematic of a bridge rectifier
diode BA592 and a 500Ω load resistance. Comparing the maximum amplitude,
with the simulation given in Fig. 2.13 shows, that in the case of the bridge recticfier the peak voltage is further reduced, since in the rectification process two
diodes are involved always. Of course as in the case of the half-wave rectifier,
also in the case of the bridge rectifier a smoothing capacitor may be connected in
parallel to the load resistance. The result using a cpacitor of 100µF parallel to the
load resistance is also shown in Fig. 2.17.
Series and parallel connection of diodes
Under certain circumstances there may exist a neccessity to use diodes that for
example cannot withstand the occuring reverse voltage or forward current. In the
first case diodes can be connected in series to reach the necessary reverse voltage
capability as shown in Fig. 2.18a, but with the help of two parallel resistors it must
be assured that the voltage will drop equally across the diode to compensate for
differences in their saturation currents. To enhance the forward current capability,
two diodes may be connected in parallel, as shown in Fig.2.18b. But here series
resistors have to be used to compensate for differences in current distribution.
2.2.2
Voltage multiplier
Before we will discuss the circuit of a voltage multiplier according to Greinacher,
we will again have a look on the simple circuit of a half-wave rectifier shown in
Fig. 2.19 where the positions of the capacitor and diode are changed with respect
to the ground and compared to the circuit of Fig. 2.14. Using KVL we get for the
ouput voltage uo (t) of the circuit
uo (t) = uS (t) + uC (t)
34
Figure 2.17: Output voltage of a brigde rectifier without and with smoothing capacitor
Rs
r
r
H
H
RP
r
r
H
H
r
r
r
Rp
r
Rs
a)
b)
H
H
r
r
H
H
Figure 2.18: Combined diodes to enhance reverse voltage or forward current capability
Fig. 2.20 shows the simulation result for the time function uo (t) according to the
circuit of Fig. 2.19. As source voltage uS (t) a sinusoidal time function with a 5 V
amplitude was used. Roughly spoken the output voltage uo (t) shows a maximum
amplitude of approximately 10 V, which is two times the source amplitude, because the capacitor is charged to 5 V. Of course the output voltage may be used
as an input of a further half-wave rectifier as shown in Fig. 2.21a. The principle
of the voltage doubler shown in Fig. 2.21a was extended by Greinacher to reach
even higher voltage levels by adding further stages, as shown in Fig 2.21b. In
principle the voltages of the capacitors in the lower line will add up to the final
voltage level U0 . The time function of a two stage voltage multiplier ist given
in Fig. 2.22. According to our crude approximation, with two stages we should
reach a voltage level of 20 V. As the simulation shows we only reach a value of
35
C
uC (t)
uS (t) A
?
uo (t)
?
r
Figure 2.19: Half-wave rectifier with interchanged capacitor and diode
Figure 2.20: Output voltage uo (t)
approximately 17 V. If we would assume a voltage drop of approximately 0.7V
across each diode, this would sum up to a value of 2.8V, which may essentially
explain the difference.
2.2.3
Zener diode as voltage regulator
In the circuit shown in Fig. 2.23 a zener diode is used to stabilize the output
voltage U0 to the Zener voltage of the diode. To describe the performance of a
Zener diode usually the current and voltage directions given in Fig. 2.23a are
used. These results in the I-U characteristic of a Zener diode given in Fig. 2.23b.
In contrast to the very sophisticated models that can be used with SPICE, we will
36
r
uS (t)
?
H
H
r
r
r
uS (t)
Uo
A
r
?
r
?
r
r
r
r
r
A
A
A
A
r
r
r
r
r
U0
Figure 2.21: Voltage doubler and multiplier circuits
Figure 2.22: Output voltage of a two stage voltage multiplier
restrict our considerations to idealized Zener diodes. As shown in Fig 2.23b we
will describe the diode by its Zener voltage UZ0 and a resistance rZ , which will
become zero in the limit of an ideal Zener diode. Of course the circuit of Fig.
2.23 is only able to stabilize the output voltage to the Zener voltage as long as
the relation Ui > UZ0 holds true. One crucial parameter of a Zener diode is its
maximum possible dissipation power Pmax , which will limit the maximum current
IZmax thru the diode and we have:
Izmax ≈
Pmax
UZ0
(2.44)
But for proper operation at least a certain minimum current IZmin must flow thru
the diode. To deduce an expression for the series resistor we use the loop equation
37
r
Ui
Rs
r
?IZ
A
?
r
IZ 6
Io
r
r -
UZ
?
Uo
?
r UZ0
-
UZ
Figure 2.23: Simple circuit to regulate the ouput voltage
of the circuit and solve it for the resistor:
Rs =
Ui − Uo
Io + IZ
(2.45)
In the practical operation of the circuit there are two extreme cases possible:
• The input voltage reaches its minimum value Uimin while the maximum output current Iomax is drawn. Under these circumstances it has to be sure that
IZ must not fall below IZmin , thus resulting in an upper limit for the series
resistor.
Uimin − Uo
Rs <
(2.46)
Iomax + IZmin
• The input voltage reaches its maximum value Uimax while only a minimal
value of output current is drawn Iomin . Under these circumstances it has to
be sure that IZ must not exceed its maximum value IZmax , thus defining a
lower limit for the series resistor.
Rs >
Uimax − Uo
Iomin + IZmax
(2.47)
Only if the two inequalities are both fulfilled, the circuit according to Fig. 2.23 is
realisable with the chosen Zener diode. To compare different circuits to stabilize
the output voltage we define the following stability factor S
S =
dUi /Ui
∆Ui /Ui
≈
∆Uo /Uo
dUo /Uo
(2.48)
In principle the last form of equation 2.48 allows us to deduce expressions for the
stability factor using small signal approximations. Of course, if we would assume
an ideal Zener diode with rZ = 0, the stability factor S would become infinte since
a variation of the input voltage would not result in a variation of the ouput voltage
38
at all. If we now, in a first order approximation consider the Zener diode to have
a non zero rZ , a change in the input voltage Ui will also result in a change of the
output voltage Uo . According to the circuit schematic of Fig. 2.23a the following
equations are valid:
Ui = Rs I + rZ IZ + UZ0
and Uo = rZ IZ + UZ0
If there is a change in the input voltage dUi there will also be a change in the
current I and the current IZ , so we have,
dUi = Rs dI + rZ dIZ
and there will also be a change in the output voltage
dUo = rZ dIZ
So we find for the ratio dUi /dUo :
Rs dI + rZ dIZ
Rs dI
dUi
=
≈
dUo
rz dIZ
rZ dIZ
for
Rs ≫ rZ
In a first order approximation we can neglect a current change due to the change
of the output voltage and we have dI = dIz and we get:
dUi
Rs
=
dUo
rz
So we get as final result for the stability factor of the circuit according to Fig.
2.23a:
Rs Uo
(2.49)
S ≈
rz Ui
Example: The current thru a load may vary between 0 mA and 100 mA, while
the voltage should be kept stable at 15 V and the input voltage may vary between
27 V and 33 V. A diode with IZmax = 200 mA and IZmin = 20 mA is used. Find the
value of Rs and the stability factor.
According to the equations 2.46 and 2.47 we get for the series resistance the following relations,
Rs < 100Ω and Rs > 90Ω
so the ratings of the Zener diode are sufficient and the series resistor may be
chosen to be RS = 95Ω. From the data sheet of the Zener diode one finds the
maximum dynamic resistance rZ to be 7Ω, so we get for the stability factor
S =
95 15
≈ 6.8
7 30
39
Stabilization of low voltages
Usually Zener diodes are built for breakdown voltages above 3 V. So, if one has
to stabilize an output voltage below this value one has to use an other circuit. One
possible simple circuit is shwon in Fig. 2.24. Here the series connection of diodes
Rs
r
r
r
A
Ui
Uo
?
r
A
r
?
r
Figure 2.24: Circuit to stabilize low voltages
is used to stabilize the output voltage. Roughly spoken each diode needs a voltage
of approximately 0.6 V to become conducting. So the output voltage is a multiple
of this value.
40
Chapter 3
The bipolar junction transistor and
applications
3.1
The bipolar junction transistor
We will now discuss the bipolar junction transistor (BJT), which started the age of
electronics. Since its invention in 1948 a lot of different electronic devices have
been realized capable to amplify weak electric signals. Even though today the
most commonly used transistor is the field effect transistor, we will start our discussion with the BJT since its operation principles are based on the the behaviour
of a pn-junction, we already discussed.
3.1.1 Structure and operation principles of a npn BJT
Fig. 3.1a shows the simplified physical layout and 3.1b the circuit schematic of
a npn BJT. It consists of a highly n-doped conductor called emitter (E), followed
by a thin p-doped zone, called base. The adjanced zone is called collector, which
is again formed by a n-doped conductor. In Fig. 3.1c an example of a cross
sectional view of a npn-BJT is given, which is realized with the help of SBCtechnique (Standard Buried Collector ) [?], [?] inside an integrated circuit. The
realisation process starts with weak p-conducting silicon crystal. With the help
of gas phase epitaxy a weakly doped n-conductting layer is formed, realizing the
collector (NDC ≈ 1015 cm−3 ). With the help of the p-zones on both sides the
single transistor is isolated to the adjanced ones. With the help of an oxidation
process a silicon oxid layer is formed, in which a window defining the base is
etched. In the following diffusion process the base is formed using Bor atoms
with a concentration of approximately NAB ≈ 1017 cm−3 . In a further oxidation
and etching process the window for the emitter is formed and finally with the help
41
a)
E s
N
P
N
s
C
s
E s
p+
B
b)
E s
sC
@
@
I
@
B
s
Bs
+
n %
++
n
p
sC
n
+
p
&
p-Silizium
c)
Figure 3.1: a) simplified physical layout, b) circuit schematic and c) cross sectional view of a npn-BJT
of a diffusion process a donator concentration of approximately NDE ≈ 1022 cm−3
is realized in the emitter zone, leaving a thin p-conduction layer, which forms the
base of the transistor. To discuss the principle of operation of a BJT we have a
look on Fig. 3.2. In the upper part a simple cross sectional view of the different
layers of the npn BJT ist given. Since the volatge UBE > 0 the E-B junction is
forward biased and since the voltage UBC < 0 the B-C junction is reverse biased.
Also sketched are widths of the depletion zone of the two junctions. Since the
emitter is highly doped the depletion zone of the E-B junction extends wider into
the base and since the base is normally higher doped than the collector, here the
depletion zone extends wider into the collector. In the lower part of Fig. 3.2 the
energy band diagram under typical basing conditions is shown. Under these conditions the emitter-base-diode is forward biased and thermally excited electrons
are able to surmount the potential barrier to the base, in which they will diffuse.
Since they are minority carriers some of them will recombine and result in a base
current. But if the diffusion length is much longer than the thickness of the base,
the majority of electrons entering the base from the emitter will diffuse thru the
base and enter the depletion zone between base and collector. Since this diode is
based in reverse direction there will exist an electric field, accelerating the electrons into the collector. Hence creation of an emitter base current will result also
in an emitter collector current. Thus with the current thru the emitter base diode
the current from the emitter to the collector may be controlled. This is essentially the principle of operation of a npn BJT. To reach this state of operation the
following conditions must be met:
• The current thru the emitter base diode must be essentially an electron current. According to equation 2.21 this is only valid for highly doped emitters.
• The majority of electrons entering the base are only capable to reach the
42
B
UBE > 0
UBC < 0
s
?
s
?
s
E
C
W
* u- u u u u u u
PP
PP
P
6
uuuu
Q
c
Qs
cQ
c
H
?
ee c
c
HH
H
c
c
c
cc u u u u u -
WC
HH
WF
cc
WV
-
x
Figure 3.2: npn BJT forward baised E-B junction reversed baised B-C junction
collector if the diffusion length Ln inside the base is longer than the base
thickness dB .
• The reverse current of the base collector diode has to be negligible small.
In Fig. 3.3 the current distribution inside a BJT is shown qualitatively. The directions of the currents IE , IB and IC where chosen to give the technical current
directions, which is opposed to the movement of the electrons. The thinner arrows
denote the unwanted hole currents between the emitter and the base as well as the
reverse current of the base collector diode. As a result of Fig. 3.3 it is clear that
the collector current is proportional to the emitter current.
IC = α0 IE
(3.1)
The parameter α0 of the last equation is called static current gain in a common
base circuit, despite the fact that due to the recombination of electrons in the base
its value is always lower than one (αo ≈ 0,9 · · · 0,999). Since the BJT is a node
43
IE
@
@
s
E
B
C
-
-
IC
s
s
6IB
Figure 3.3: Current distribution in a npn BJT under typical operation conditions
we can apply Kirschhoffs current law:
IE = IB + IC
If we use the last equation to give the collector current as function of the base
current we will get:
α0
IB = β0 IB
(3.2)
IC =
1 − α0
The parameter of equation 3.2 is denoted as static current gain in a common emitter circuit. Depending on the transistor β0 can reach values between approximately 30 and 500.
3.1.2
Static input and output characteristics of a BJT
According to the arragement of the layers a transistor can be represented by two
diodes which are connected at their p-layer. Such a circuit would of course not
act as a transistor because the anode of the emitter base diode is also the anode of
the base collector diode in a physical sense, but not only in an electrcical sense as
modeled by the equivalent circuit. To account for the transistor effect, according
to [?], a current controlled current source has to be included parallel to the base
collector diode, which represent the electron current from the emitter to the collector of a real transistor. As a result we get the equivalent circuit of a transistor
given in Fig. 3.4 under typicall operation condictions, describing its static behaviuor. The currents ISE and ICE representing the saturation currents of the emitter
base and base collector diode, while the resistances of the semiconductor layers
are neglected.
)
(
UEB
)−1
(3.3)
IE = ISE exp(−
UT
44
IE
H
H
s
s
H
H
s
αo IE
IC
s
s UEB
UCB
?
s
?
s
s
Figure 3.4: Simplified equivalent circuit according to Ebers-Moll under typical
operation conditions
(
)
UCB
IC = αo IE − ISC exp(−
)−1
(3.4)
UT
As shown in Fig. 3.5 the equivalent circuit of Fig. 3.4 can also be given in a
common emitter configuration. With the help of equations 3.3 and 3.4 we will
H
H
s
s
#
#
I
s B-
@
@
R
UBE
?
s
s
UCE
?
s
UBE
?
s
IC
s s
α0 IE s A
I
?E
s
UCE
?
s
Figure 3.5: BJT in common emitter configuration and its Ebers-Moll-model
now deduce an expression for the input characteristic IB = f (UBE ,UCE ) and for
the output characteristic IC = f (UCE ,UBE ). Staring point is again KCL for the
BJT as a whole:
)
(
UCB
)−1
IB = IE − IC = (1 − α0 )IE + ISC exp(−
UT
Accounting the different definitions of the voltages UEB = −UBE and UCB =
UCE −UBE we get for the base current:
(
)
(
)
UBE
UCE −UBE
IB = (1 − α0 )ISE exp(
) − 1 + ISC exp(−
)−1
UT
UT
For typical conditions of operation we have UCE ≫ UBE and the last term may be
neglected, resulting in the following expression for the base current:
)
)
(
(
UBE
UBE
) − 1 = ISB exp(
)−1
(3.5)
IB = (1 − α0 )ISE exp(
UT
UT
45
The form of the last equation is equivalent to that of a normal pn-junction, which
has a saturation current of ISB = (1 − α0 )ISE . Thus the input characteristic of a
transistor is equivalent to that of a pn-junction already shown in Fig. 2.6. With
the help of equation 3.4 we get as expression for the collector current IC ,
(
)
)
(
UBE
UCE −UBE
IC = α0 ISE exp(
) − 1 − ISC exp(−
)−1
(3.6)
UT
UT
or with reference to the saturation current ISB
(
)
(
)
UBE
UCE −UBE
α0
ISB exp(
) − 1 − ISC exp(−
)−1
IC =
1 − α0
UT
UT
(3.7)
Example: Theoretical output characteristic of a BJT
As example Fig. 3.6 shows the output characteristic of a BJT according to equa-
Figure 3.6: Theoretical output characteristic of a BJT with UBE as parameter
tion 3.6, where the following values were used for its calculation: ISE = ISC =
1 nA, UT = 43 mV, α0 = 0, 999. Comparing the theoretical predicted output characteristic shown if Fig.3.6 with that of a real BJT shows that the current IC of a
real BJT rises with rising voltage UCE . Is effect is called Early-effect [?].
The first term of equation 3.7 can be identified as current gain β0 of the common emitter configuration:
α0
(3.8)
β0 =
1 − α0
46
while the second term describes nothing else but the dependence of the base current IB on the voltage UBE .
(
)
UBE
IB (UBE ) = ISB exp(
)−1
(3.9)
UT
With the help of the introduced parameters equation 3.7 can be rewritten as
(
)
UCE −UBE
IC = β0 IB (UBE ) − ISC exp(−
)−1
(3.10)
UT
Since under normal operation conditions the base collector diode is based in reverse direction, the last term of equation 3.10 can be neglected, which reduces this
equation to
IC ≈ β0 IB (UBE )
(3.11)
describing essentially the behaviour of a BJT in a common emitter configuration.
Equation 3.9 and equation 3.11 can be used to setup a large signal model of a BJT
operating in common emitter configuration, as shown in Fig. 3.7 It consists of the
β
0 IB r C
B r -IB
A
r
rE
Figure 3.7: Large signal model of a BJT under normal operation conditions
base emitter diode carrying the current IB and an ideal current controlled current
source being responsible for the collector current.
3.1.3
Simple small signal BJT model
One of the applications of transistors is the amplification of small signals. To use
a transistor for amplification one has to operate the transistor under certain DC
conditions UCE , IC , called biasing. In principle one uses voltage or current sources
to establish the DC conditions under which the transistor shows the described
transistor effect. Fig. 3.8 shows a BJT circuit in common emitter configuration.
The DC voltage sources UBE and UCE are chosen to establish the typical operation
conditions of the transistor, emitter base diode forward biased and base collector
diode biased in reverse direction. In the input circuit an additional sinusoidal
47
voltage source uBE (t) is used, with an amplitude ÛBE fulfilling the small signal
condition ÛBE ≪ UBE . In the ouput circuit a load resistance RL is included to allow
an alternating voltage uCE (t) to exist, also fulfilling the small signal condition
ÛCE ≪ UCE . A mathematical exact analysis of the circuit shown in Fig. 3.8 is
us (t)
UBE
RL
@
R
@
?
?
r
U0
?
Figure 3.8: Common emitter circuit of a BJT with small signal excitation
very complicated due to the nonlinear behaviour of the equations 3.9 and 3.10
and only possible using advanced simulation tools. Since we are at the moment
only interested in the small signal behaviour we linearize equations 3.9 and 3.11
in the vicinity of the point of operation and thus establish a small signal equivalent
circuit of the BJT at the point of operation. In mathematical sense we will perform
a Taylor approximation at the opertaion point.
∆IB =
UBE
| IBE |
ISB
exp(−
)∆UBE ≈
∆UBE
UT
UT
UT
(3.12)
In analogy to the pn-junction one can introduce the dynamic resistance rBE of the
base emitter diode.
UT
rBE =
(3.13)
| IB |
Also linearizing equation 3.11 yields the equivalent circuit shown in Fig. 3.9,
where the resistor rCE was additionaly included to account for the Early effect, already mentioned above. Since the equivalent circuit was deduced from equations
3.9 and 3.11 describing the static behaviour of the transistor, it is only valid for
low frequencies.
h-parameter
To describe the small signal behaviour of BJT at low frequency also the h-parameters
are used []. In this case these parameters are real and they describe the influence
48
βiB
iB
uBE (t)
?
r -
rBE
r
r
r
r
r CE r
r
RL uCE (t)
?
r
Figure 3.9: Simple common emitter small signal equivalent circuit of a BJT
of the output voltage uCE (t) and input current iB (t) on the input voltage uBE (t)
and the output current in a more formalized way given by the following equation:
uBE = hie iB + hre uCE
(3.14)
iC = h f e iB + hoe uCE
From equation 3.14 it becomes clear that the single parameters are defined according to the following equations:
hie = uiBE |uCE =0
B
BE |
hre = uuCE
iB =0
i
h f e = iC |uCE =0
B
C |
hoe = uiCE
iB =0
short circuit input resistance
open circuit reverse voltage ratio
(3.15)
short circuit forward current gain
open circuit output conductance
Since the first equation of 3.14 defines the small signal voltage uBE it may be
interpreted to be the result of Kirchhoff’s voltage law at the input of the transistor
while iC of the second equation is the result of Kirchhoff’s current law at its output.
Using these interpretations one finds the equivalent circuit shown in Fig. 3.10.
Comparing the equivalent circuit of Fig. 3.10 with that already given in Fig. 3.9
iB
hie
h f e iB
s
uBE
?
s
iC
?
s s
hoe
hre uCE
?
s
s
uCE
s
?
s
Figure 3.10: Small signal equivalent circuit according to the h-parameters
reveals the following equivalences:
hie = rBE
hfe = β
49
hoe = 1/rCE
(3.16)
On the other hand, the parameter hre finds no equivalent element in Fig. 3.9,
because it was deduced from the static behaviour and the parameter hre decribes
the feeback of an alternating voltage at the output on the input voltage, which of
course cannot be deduced using static equations. Nevertheless we will use the
parameters given in Fig. 3.9 for a first order analysis of small signal amplifier,
because of their physical significance. If a more detailed analysis is needed this
can today be done using advanced simulation tools. Nevertheless h-parameters
are often specified in data sheets of single transistors for a certain operation point.
3.1.4
Advanced small signal BJT model
If we want to have a more accurate equivalent circuit describing the behaviour also
up to higher frequencies, we have at least to account for the capacitve behaviour
of all encountered pn-junctions. According to Fig. 3.7 we must consider the
capacitance cBE of the base emitter diode, which essentially will reflect minority
storing, due to the diffusion process. Even if we neglected the reverse biased base
collector diode, to reach the simplified equivalent circuit of Fig. 3.7, its junction
capacitance cBc has to be considered. A further effect we also did not consider up
RBB′
cB′C
r
r
r
?
I
B
r
β0 I B ?
U BE
?
r
cB′ E
r
r
r
rCE
rB′ E
r
r
r
U CE
?
r
Figure 3.11: Advanced common emitter small signal equivalent circuit of a BJT
to now is the bulk semiconductor material between the base and the active base
emitter junction, giving rise to the resistor RBB′ . Of course all considered voltages
or currents are now given in phasor notation. The extended equivalent circuit of a
BJT shown in Fig. 3.11 is referred to as full hybrid π model.
3.1.5
SPICE model of a BJT
Fig. 3.12 shows the so-called Gummel-Poon-model of a npn BJT as it is implemented in the spice circuit simulator. As in the model already given in Fig. 3.4
also in the Gummel-Poon model a transistor consists essentially of the base emitter diode DBE and the base collector diode DBC and current controlled current
sources iF and iR . In contrast to the Ebers-Moll model they are controlled by the
base currents iBE and iBC . The current voltage characteristic of this diode is described by the exponential law already used to describe a real pn-junction (2.30).
50
r
Kollektor
RC
r
uBC
Basis
6
RBB
r
r
CsBC
r
CdBC
r
CsBE
uBE
?
r
CdBE
r
r
r
A
DLC A DBC
r
r
A
DLE A DBE
r
r
r
r
?iF
6iR
r
r
RE
r
Emitter
Figure 3.12: Gummel-Poon-model of a npn BJT
The diodes DLE and DLC are incorporated to describe leakage currents which have
no influence on the controlled current sources, and the capacitances C jBE , CdBE ,
C jBC , CdBC account for the junction und diffusion capacitance of the doides DBE
and DBC . In the following table the SPICE data set of the BJT BFP420 is given:
SPICE data set:
**********************************************************
.MODEL BFP420 NPN(
+ IS = 17.7E-18
RB = 9.47
CJC = 380E-15 BF = 117
+ IRB = 0.5E-3
VJC = 1.0
NF = 0.98
RBM = 5.47
+ MJC = 0.5
VAF = 45
RE = 0.948
XCJC = 0.18
+ IKF = 0.15
RC = 4.4
TR = 5.0E-9
ISE = 4.5E-12
+ CJE = 130E-15
CJS = 0
NE = 2.31
VJE = 1.0
+ VJS = 0.8
BR = 1.0
MJE = 0.5
MJS = 0.33
+ NR = 1.0
TF = 9.6E-12 XTB = 0
VAR = 1000
+ XTF = 0.457
EG = 1.16
IKR = 1000
VTF = 0.413
+ XTI = 3.0
ISC = 0
ITF = 41E-3
FC = 0.78
+ NC = 2.0
PTF = 0 )
51
************************************************************
3.2
Small signal amplifier
In order to use a BJT as small signal amplifier at first certain DC values UCE and
IC have to be established ensuring that the base emitter diode is forward biased,
the base collector diode is reversed biased and the transistor effect can take place.
For single transistors a certain point of operation is often recommended on its data
sheet. Under these circumtances small signals may be applied and the transistor
can be used as an amplifier.
3.2.1 BJT biasing
At first we now want to discuss different possiblities to reach this point of operation. For calculation we use the equivalent circuit of the BJT shown in Fig. 3.7,
were we assume that the base emitter diode may be modeled by an ideal diode
with a fixed threshold voltage UBE according to Fig. 2.11 and the BJT has a constant current gain β0 .
Constant base current biasing
One of the simplest possible circuit to reach a certain point of operation is shown
in Fig. 3.13. Since the whole circuit is driven by the DC voltage source U0 , so we
r
r
RC
RB
UBE ?
r
r
@
@
R
r
r
U0
UCE
?
?
r
Figure 3.13: Constant base current circuit
get for the collector resistance RC :
RC =
U0 −UCE
IC
52
(3.17)
Due to the given current gain β0 of the transistor the base current IB is also known
and we can setup an equation for the resistor RB .
RB =
U0 −UBE
β0
IC
(3.18)
Usually the value of the DC voltage U0 is much higher than the voltage UBE and
further-more the base current IB is very low, so the value of RB becomes very high
and essentially it behaves like a current source. So the base current is essentially
constant and the temperature dependence to the operation points is only due to the
temperature dependence of β0 (T ). A very crucial drawback of the circuit is that
the value of RB depends directly on the value of β0 , which usually shows a high
variation for single transistors.
A circuit with which one can over-come the considered problems is shown in
Fig. 3.14. For the collector resistor we get:
r
RB
IB ?
RC
r
?IC
U0
UCE
UBE
?
r
@
R ?
@
r
?
r
Figure 3.14: Constant base current circuit with feed back
RC =
U0 −UCE
IC (1 + 1/β0 )
(3.19)
And the equation for the resistor RB reads:
RB =
UCE −UBE
β0
IC
(3.20)
Although the equation for RB does not exhibit an obvious decrease in β0 dependence, the feedback does tend to stabilize the operation point. For exmaple, if
the transitor has a much higher β0 than the nominal value used to calculate RB ,
the collector current will be higher and the collector voltage lower than the design values. With the lower collector voltage there is less voltage across RB and
therefore less base current, thus at least partially compensating for the higher β0 .
53
Conversely, if β0 is lower than the nominal value, the collector current is less than
the design value giving a greater voltage across RB and hence results in a higher
base current, again partially compensating for the low β0 .
Biasing using a base voltage divider
Fig. 3.15a shows a further possiblity to bias the transistor to a certain operation
point UCE and IC . In contrast to the preceeding circuit now the voltage UB is
r
r
RC
R1
r
IT ?
R2
I
-B
UB
?
IC ?
@
R
@
UCE
U0
IE ? ?
RE
UE
r
?
?
r
Figure 3.15: Base voltage divider with current feedback
essentially kept constant. Without the resistor RE this circuit would be very problematic, because of the self heating of the transistor. But with the resistor RE a
feedback due to the current IE is introduced. For example, assume the current
IE will increase due to a rise of temperature. Since the voltage UB is fixed, the
voltage UBE driving the base current IB will be reduced and hence the current IE
is reduced again. Typically the voltage UE across RE is chosen to be in the range
of 1 V to 3 V. For the voltage devider represented by the resistors R1 and R2 to act
as a voltage source the transverse current IT has to be large compared to the base
current IB . Here typically a factor of 10 is chosen. As for the preceeding circuit,
we assume to know the transistor parameter UBE and β0 . From the given voltage
UE we first can calculate the value of the resistor RE .
RE ≈
UE
IC
(3.21)
With the help of KVL in the output circuit, we find for the resistor RC .
RC =
U0 −UCE −UE
IC
54
(3.22)
The voltage UB can also be calculated as a result of KVL in the lower base circuit,
UB = UBE + UE
and if we assume the transverse current IT to be 10 times the base current, we find
for the resistor R2 ,
UB
UBE + UE
R2 =
=
(3.23)
IT
10 IC /β0
and finally we get for the resistor R1
R1 =
U0 − UB
U0 − UBE − UE
=
IT + IB
11 IC /β0
(3.24)
Example: Biasing of the transistor BC548C
Data: U0 = 12V, UCE = 5V, IC = 2mA, β0 = 400, UBE = 0.65V and UE = 2V
RE =
2
Ω = 1kΩ
2 10−3
12 − 5 − 2
Ω = 2.5kΩ
2 10−3
As a result the voltage UCE will become:
RC =
chosen RC = 2.7kΩ
UCE = U0 − RC IC − UE = 4.6V
0.65 + 2
Ω = 53kΩ chosen R2 = 47kΩ
10 · 5 10−6
As a result the transverse current will become IT = 56µA wich is approximately
11 times the base current, so we get for the resistor R1
R2 =
R1 =
3.2.2
12 − 2.65
≈ 156kΩ chosen
12 · 5 10−6
R1 = 150kΩ
Common-emitter amplifier
One of the most commonly used configurations to realize a small signal amplifier
using a BJT is the common-emitter amplifier shown in Fig. 3.17. It is called
common-emitter configuration, because the emitter of the transistor belongs to
the input as well as to the output of the circuit. In this circuit the operation point
of the transistor is realized using a base voltage divider. The signal of the source
55
r
RS
r
C1
r
r
@
@
R
R2
C2
RL
r
R1
RC
CE
RE
r
r
r
r
r C∞
r
6
U0
Figure 3.16: Common-emitter amplifier
is AC coupled with the help of capacitor C1 to the base to the transistor. This must
be done to not disturb the operation point by the internal resistance of the signal
source. For the same reason the capacitor C2 is used to AC couple the output
signal to the load resistance RL . As already mentioned in section 3.2.1 the resistor
RE will result in a current feedback, which will reduce the overall amplification
of the circuit, thus the capacitor CE is used to allow the AC current to by pass the
resistor. Even so the DC supply voltage source U0 has a zero internal resistance
theoretical on a practical circuit board the additional capacitor C∞ has to be used
to realised the AC short circuit on the circuit board.
AC equivalent circuit
Of course all currents and voltages at the transistors are a superposition of DC and
AC values. Since we already discussed how to realise a certain operation point the
DC values are no longer of interest and the transistor may be replaced by its small
signal equivalent circuit shown in Fig. 3.9. Further-more we want to assume
IB
RS
US
?
r
-
r
R2
r
r -
β IB ?
U BE
R1
r
rBE
?
r
r
rin
r
RC
U CE
r
r
r
RL
?
r
rout
Figure 3.17: AC equivalent circuit of a common-emitter amplifier
56
that the capacitors C1 , C2 and CE are sufficiently high-valued to have negligble
reactance at the frequencies of interest. With the help of these considerations,
we can setup the AC equivalent circuit of the common-emitter amplifier shown
in Fig. 3.17, where the small signal output resistance rCE of the transistor is also
neglected.
Small signal voltage gain
Normally a small signal amplifier is used to amplify weak input signals. Of course
one is interested in the value of the output signal, if a certain input signal is applied. The ratio of these two voltages is called voltage gain v which usually is
a function of the frequency. Using the circuit shown in Fig. 3.17 we define the
voltage gain of this stage to be:
vu =
U CE
U BE
(3.25)
Introducing a resistor R′L
RC RL
RC + RL
being equal to the parallel connection of RC and RL , we can write for the output
voltage U CE
U CE = −β R′L I B
R′L =
According to Fig. 3.17 we have the following relation between the base current
I B und the base voltage U BE ,
U
I B = BE
rBE
and hence we get for the output voltage,
U CE = −β R′L
U BE
rBE
and finally for the voltage gain of the common-emitter stage:
vu =
U CE
R′
= − Lβ
U BE
rBE
(3.26)
Input and ouput resistance
According to the equivalent circuit shown in Fig. 3.17 the input resistance rin of
the stage is given by the parallel connection of R1 , R2 and rBE ,
rin =
1
1
1
1
+
+
R1 R2 rBE
57
(3.27)
while the ouput resistance rout is equal to the collector resistance RC .
rout = RC
(3.28)
Choosing the values of the capacitors
All capacitors included in the common-emitter amplifier circuit shown in Fig. 3.17
will contribute to its highpass character. So the amplifier will only properly work
above a certain lower frequeny limit we call flow . Instead of a rigorous treatment
of the equivalent circuit incorporating all capacitors we will discuss the influence
of each capacitor separately in a more heuristic manner.
Capacitor CE The purpose of this capacitor is to reduce the influence of the
resistor RE on the AC gain of the amplifier. Since it is connected in parallel to
the resistor, with increasing frequency the total reactance will become smaller
and smaller and for high frequencies there will exist no current feedback at all.
If already for the lower frequency limit flow there shall be no remarkable current
feedback, the following relation must hold true,
1
2π flowCE
≪ RE
which gives a lower limit how to chose the value of CE .
CE ≫
1
2π flow RE
(3.29)
Capacitor C1 If the capacitor CE is chosen high enough, the resistors RS , rin
and the capacitor C1 will form a first order high-pass filter. Its corner frequency
fc being given by:
1
fc =
2πC1 (RS + rin )
Choosing the corner frequency equal to the lower frequency limit flow gives a
lower limit for the capacitor C1
C1 ≥
1
2π flow (RS + rin )
(3.30)
Capacitor C2 The capacitor C2 in the ouput circuit is equivalent to the capacitor
C1 in the input circuit. Thus it will also form a first order high-pass filter with
58
the resistors RC and RL , if the capacitor CE is chosen high enough. So we get an
equivalent lower limit for its value analog to the limit of capacitor C1 .
C2 ≥
1
2π flow (RC + RL )
(3.31)
Example: Common-emitter amplifier with the transistor BC548C
As an example we will calculate the gain to be expected from a common-emitter
amplifier using the transistor BC548C with a biasing according to the last section.
We will choose the load resistance RL to be equal to RC and we assume a source
with internal resistance RS of 50Ω. According to equation 3.13 we get for the
dynamic resistance rBE ,
rBE =
0.026
Ω = 5.2kΩ
0.002/400
and hence for the expected voltage gain
v = −1.35kΩ
400
≈ −104
5.2kΩ
and for input resistance:
rin =
1
kΩ = 4.54kΩ
1
1
1
+
+
47 150 5.2
If we want the amplifier to work above a lower frequency limit of flow = 50 Hz
we have to choose the capacitor CE according to equation 3.29:
CE ≫
1
F = 3.18 µF
2π 50 · 103
chosen CE = 300µF
Using the equations 3.30 and 3.31 we find for the cpacitors C1
C1 ≥
1
F = 0.69µF chosen C1 = 1µF
2π 50 · 4.59 103
and C2
1
F = 0.59µF chosen C2 = 1µF
2π 50 · 5.4 103
To verify the prediction above of the gain and the values of the chosen capacitors
Fig. 3.18 shows the simulated frequency response of the common-emitter amplifier with the transistor BC548C. The source amplitude was set to be 1mV. With
C2 ≥
59
Figure 3.18: Simulated frequency response of the common-emitter small signal
amplifier
the help of the marker function we can read out an output amplitude of almost
40dBmV, which corresponds to an absolute value of a small signal gain of 100,
which is very close to the predicted value. The 3 dB cut-off frequency of the highpass can be read out to be approximately 110Hz, which is considerably above the
envisaged lower frequency limit of 50 Hz. But the lower frequency limit can now
easily be adjusted by increasing the values of the coupling capacitors C1 and C2 . It
is interesting to notice the limitation of the gain at higher frequencies predicted by
the simulation. Of course the SPICE model of the transistor is far more elaborated
than the simple small signal model of Fig. 3.17 and the internal capacitances of
the transistor will limit the frequency range of operation and result in a low-pass
characteristic for high frequencies with a 3 dB cut-off frequency of approximately
14 MHz. If one liked to use the ampflier as the input stage of an audio amplifier,
one should reduce the cut-off frequency to lower values using additional capacitors in the circuit.
Summarizing the properties we can say that a common-emitter state shows
• a high voltage gain
• a high current gain
60
• a high power gain
• a moderate input resistance approximately equal to rBE
• a moderate output resistance approximately equal to RC
3.2.3
Common-collector amplifier
A further possibility to realize a small signal amplifier with the help of a BJT is
the common-colllector circuit shown in Fig. 3.19. It is called common-collector
RS
r
C1
r
r
@
R
@
R2
C2
r
r
RE
R1
r
C∞
RL
r
r
r
r
r
6U
0
Figure 3.19: Common-collector amplifier
amplifier, since the collector of the BJT belongs to the input as well as to the
output circuit of the amplifier. Also the term emitter-follower is used. To establish
the operation point again a biasing base voltage divider is used, and established
with the help of the resistors R1 , R2 and RE . In contrast to the common-emitter
amplifier the resistor RC is missing. The AC input signal is coupled to the base
using the capacitor C1 while the capacitor C2 is used to couple the output signal to
the load resistance RL .
AC equivalent circuit
As in the case of the common-emitter amplifier we will assume that the capacitors C1 and C2 are sufficiently high-valued to have negligible reactance at the frequencies of interest. With the help of these considerations, we can setup the AC
equivalent circuit of the common-collector amplifier shown in Fig. 3.20, where
the small signal output resistance rCE of the transistor is also neglected.
61
RS
US
?
r
-
I B rBE
r
R2
r
IE
r
r
R1 U
BC
rin
r
r
RE
U EC
βI B 6
r
?
r
r
-
?
r
RL
r
rini
Figure 3.20: AC equivalent circuit of a common-collector amplifier
Input resistance
We will start our analysis calculating the input resistance rini of the intrinsic common collector stage first excluding the contribution of the resistors R1 and R2 . As
a first step we introduce the resistor R′L which reflects the parallel connection of
the resistors RE and RL .
1
R′L =
1
1
+
RE RL
Using KVL for the input circuit we find,
U BC = [rBE + (β + 1)R′L ]I B
and hence we get for the intrinsic input resistance of a common-collector stage:
rini =
U BC
= rBE + (β + 1)R′L
IB
(3.32)
Comparing rini of a common-collector stage with the intrinsic input resistance rini
of the common-emitter stage, which is equal to rBE , shows that it is considerable
larger, since the total load resistance R′L is multiplied by the factor β and added to
rBE , according to equation 3.32. So the resistors of the base voltage divider may
not be neglected calculating the input resistance of the total stage.
rin =
1
1
1
1
+
+
rini R1 R2
Small signal voltage and current gain
The voltage gain of the common-collector amplifier is defined by:
vu =
U EC
U BC
62
(3.33)
According to the equivalent circuit shown in Fig. 3.20 we get for the output voltage U EC ,
U EC = R′L I E
while we get for the current I E and hence for the current gain
I E = (β + 1)I B
I E = (β + 1)
→
vi =
IE
= (β + 1)
IB
(3.34)
U BC
β+1
=
U
rini
(β + 1)R′L + rBE BC
So we find for the voltage gain of the common-collector amplifier
(β + 1)R′L
< 1
(β + 1)R′L + rBE
vu =
(3.35)
which is always lower than one.
Output resistance
To deduce an expression for the output resistance of the common-collector stage
we again consider Fig. 3.20 and introduce the resistor R′ S which accounts for the
parallel connection of the resistors RS , R1 and R2 , assuming the voltage source to
be turned off.
1
R′ S =
1
1
1
+
+
RS R1 R2
According to Fig. 3.20 the output resistance routi is given by:
routi = −
U EC
IE
A voltage UEC applied at the output will result in the following base and emitter
current:
U
U
→ I E = −(β + 1) ′ EC
I B = − ′ EC
R S + rBE
R S + rBE
as a result we get or the output resistance:
routi =
R′ S + rBE
β+1
63
(3.36)
Choosing the values of the capacitors
All capacitors included in the common-collector amplifier circuit shown in Fig.
3.19 will contribute to its highpass character. So the amplifier will only work
properly above a certain lower frequency limit we call flow . Instead of a rigorous treatment of the equivalent circuit incorporating all capacitors we again will
discuss the influence of each capacitor separately in a more heuristic manner.
Capacitor C1 Again the resistors RS , rin and the capacitor C1 will form a first
order high-pass. Its corner frequency fc being given by:
fc =
1
2πC1 (RS + rin )
Choosing the corner frequency equal to the lower frequency limit flow gives a
lower limit for the capacitor C1
C1 ≥
1
2π flow (RS + rin )
(3.37)
Capacitor C2 The capacitor C2 in the ouput circuit is equivalent to the capacitor
C1 in the input circuit. Thus it will also form a first order high-pass filter with the
resistors rout and RL . So we get an equivalent lower limit for its value in analogy
to the limit of capacitor C1 .
C2 ≥
1
2π flow (rout + RL )
(3.38)
Summarizing the properties we can say that a common-collector state shows
• a voltage gain which is lower than one
• a high current gain
• a moderate power gain
• a high input resistance equal to rBE + (β + 1)R′L
′
• a low output resistance of R S + rBE
β+1
64
3.3
Integrated circuit techniques
In designing integrated circuits the economic rules of discrete component circuit
design are reversed. Active devices are inexpensive since they can be realized
on a much smaller area than resistors or capacitors. Hence, every effort must be
made to minimize the total resistance in a circuit and certainly replace passive
components with transistors or diodes wherever possible. In the following section
we will start with the conventional design of the operational amplifier and we will
discuss different possibilities to make it a circuit with almost no passive elements.
3.3.1 The differential amplifier
One serve disadvantage of the amplifier circuits discussed in the previous section
is that single stages have to be coupled using capacitors which is inpratical for
integrated circuit technology. Another drawback is that also due to the coupling it
is not possible to amplify voltages of very low frequencies down to DC. To overcome these problems the circuit shown in Fig. 3.21 can be used, which is called
differential amplifier. Essentially it consists of two identical transistors coupled
r
r
RC
r
r
Up ?
r
U0
RC
UTo
r r
U1o
@
@
R
-
r
U2o
r?
?
r
r
IE1 @
@
r
?
r
Un
IE2
I0 ?
r −U0
Figure 3.21: Differential amplifier
by an ideal current source of constant current I0 . In contrast to conventional amplifiers a positive as well as a negative supply voltage is used and two voltages
U p and Un are used as input signal. To get an ouput signal to equal load resistors
RC are used in the collector branch of each transistor. Due to the symmetry of the
circuit an output voltage UTo will only exist, if there is a difference between the
65
input voltages. Thus it is convenient first to define the difference voltage Ud by:
Ud = U p − Un
(3.39)
It is interesting to point out that according to KVL the difference voltage Ud is
also equal to the difference of the base emitter voltages of the two transistors.
Ud = UBE1 − UBE2
(3.40)
To further analyse the behaviour of the circuit especially in dependence of an
applied difference voltage, we start with the node equation at the current source,
which forces the sum of the two currents IE1 and IE2 to be constant.
IE1 + IE2 = I0
(3.41)
Of course the emitter current of each transistor may also be expressed according
to equation 3.3 as a function of its base emitter voltage,
)
(
)
(
UBE2
UBE1
IE2 = IS exp
IE1 = IS exp
UT
UT
assuming identical transistors at the same temperature. Using equation 3.41 we
get
[
(
)
(
)]
UBE1
UBE2
I0 = IS exp
+ exp
UT
UT
and the last expression can be rearranged to:
(
)[
(
)]
UBE1
UBE2 −UBE1
I0 = IS exp
1 + exp
UT
UT
According to 3.3 the first term of the last equation is equal to the emitter current of
the first transistor and in the last term, the difference of the base emitter voltages
may be expressed by the difference voltage Ud . As a result we get for the emitter
current IE1 :
I0
(
)
IE1 =
Ud
1 + exp −
UT
By analogy one can deduce the equivalent equation for the emitter current IE2 .
IE2 =
I0
( )
Ud
1 + exp
UT
66
Since we have IC1 = α IE1 and IC2 = α IE2 we get for both collector currents:
IC1 =
αI0
(
)
Ud
1 + exp −
UT
and IC2 =
αI0
( )
Ud
1 + exp
UT
(3.42)
Fig. 3.22 shows the normalised collector currents as a function of the applied
difference voltage Ud . According to Fig. 3.22 for a zero difference voltage the
1
0.9
I
I
C2
C1
0.8
0.6
IC/(α I0)
→
0.7
0.5
0.4
0.3
0.2
0.1
0
−5
−4
−3
−2
−1
0
Ud/UT
→
1
2
3
4
5
Figure 3.22: Normalised collector currents as function of the applied difference
voltage
differential amplifier is in balance and the total current I0 is equally distributed
on both transistors. For a positive difference voltage transistor 1 becomes more
conductive carrying more current, while the conduction of transistor 2 is reduced.
Similarly for a negative difference voltage the current I0 is steered more towards
67
transistor 2 while now the conduction of transistor 1 is reduced.
To deduce an expression for the voltage UTo we use KVL and apply it to the
top mesh of the circuit shown Fig. 3.21:
UTo = RC (IC1 − IC2 )
With the help of equation 3.42 we can rewrite the equation above as:



UTo = RC α I0 


1
1
(
)−
( )
Ud
Ud 
1 + exp −
1 + exp
UT
UT
Using the definitions of the functions cosh(x) and sinh(x), the last expression can
be manipulated to:
( )
Ud
sinh
UT
( )
(3.43)
UTo = RC α I0
Ud
1 + cosh
UT
Fig. 3.23 shows the normalised output voltage as function of the difference voltage Ud . For a difference input voltage Ud < 2UT ≈ 50mV we have an almost
linear dependence of the output voltage on Ud . But as the difference voltage increases the ouput voltage will reach a saturation value of ≈ RC I0 . Of course this
value may not become larger than the supply voltage U0 , which can be used to get
a relation between the current I0 and the value of RC .
RC I0 < U0
→
RC <
U0
I0
Small Signal Gain
As already mentioned above, for low values of the difference input voltage Ud
the differential amplifier behaves linear and can be described by the differential
voltage gain vd . Approximation of equation 3.43 yields:
UTo ≈
RC α I0
Ud
2UT
→
vd =
RC α I0
RC I0
≈
2UT
2UT
(3.44)
Using the deduced relation above for RC we find
vd <
U0
2UT
68
(3.45)
1
0.8
0.6
→
0.4
Uout/(RCα I0)
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−8
−6
−4
−2
0
U /U
d
T
→
2
4
6
8
Figure 3.23: Normalised output voltage as function of the difference voltage
With the last equation we found an upper limit for the attainable differential voltage gain of a single stage BJT differential amplifier with a resistive load. It is
interesting to notice that this limit is essentially independent of the used BJT as
long as its current gain β is sufficiently high and that it depends strongly on the
used level of supply voltage.
Realizing a Simple Current Source
The simplest way to realize the current source of the differential amplifier is shown
in Fig. 3.24. Here the ideal current source is just replaced by the resistor R0 . To
understand why this resistor acts like a current source, one should again consider
Fig. 3.23 which showed that already a very small differential voltage Ud is sufficient to drive the output voltage into the region of saturation. On the other hand
this implies that the voltage level of the base of the transistors is very close to
69
r
r
RC
r
r
Up ?
RC
UTo
r r
U1o
@
@
R
r
U0
U2o
r?
-
r
?
r
r
@
@
IE1
r
?
r
Un
IE2
R0
I0 ?
r −U0
Figure 3.24: Differential amplifier with a resitor used as current source
the ground level and as a result the voltage drop across the current source will
be equal to the supply voltage U0 . Applying these considerations to the circuit
shown in Fig. 3.24 gives the following approximation between the current I0 and
the resistor R0 .
U0
(3.46)
I0 ≈
R0
Common Mode Rejection
A differential amplifier should only respond to the difference between the input
signals Ud . But of course the input signal U p and Un can also show a common
mode voltage Uc , defined as:
Uc =
1
(U p + Un )
2
(3.47)
As a consequence there may also exist a common mode gain vcm , defined by
vcm =
∆Uout
∆Uc
(3.48)
Considering the circuit according to Fig. 3.24 an increase in common mode voltage ∆Uc will result in a change of the current thru the resistor R0 :
∆I0 =
70
∆Uc
R0
Assuming an operation near the balance point of the amplifier, both collector currents will increase
∆I0
∆Uc
=
2
2 R0
Now one must distinguish between two cases:
∆Ic =
1. In a double-ended output defined by the voltage UTo , given that the circuit
is truly symmetrical, both collector currents will alter by the same amount
and the differential ouput voltage will still be zero and no common-mode
gain will occur.
2. In a single-ended mode, defined by the output voltages U1o or U2o there is
an ouput response and for the common-mode gain we will get:
|vcm | =
∆U1o
Rc
=
∆Uc
2 R0
(3.49)
A useful figure of merit for the differencing performance of the differential amplifier may be defined as the ratio of differential voltage gain vd to common-mode
gain vcm . This is called the common-mode rejection ratio (CMRR). With the help
of equations 3.44 and 3.49.
CMRR =
3.3.2
2UT
vcm
=
vd
R0 I0
(3.50)
Current Sources
Current sources are very important electronic circuits providing for example biasing functions such as the current source of a differential amplifier. According
to the simple large signal equivalent circuit shown in Fig. 3.7 a BJT essentially
acts like a current source, with the advantage of the current being adjustable by
the base current. Thus in principle the constant base voltage circuit with current
feedback shown in Fig. 3.15 can also be used to realise the current source of a
differential amplifier as shown in Fig. 3.25a, which can be used to replace the
resistor. Of course the voltage UCE of the BJT may not drop below approximately
0.7 V, which is the onset of saturation. The advantage of the BJT as current source,
is its higher ouput resistance, when compared to the single resistor. But with the
expense of using three additional resistors, which will be a severe drawback for
the realisation as integrated circuit. A circuit to overcome these problems is shown
in Fig. 3.25b. It is called current mirror. In contrast to the previous circuit here
only one resistor is needed to define the current I0 . A further advantage of the
integrated version of the circuit is that both BJTs may be realised in close vicinity,
71
R1
r
r
r
r
? I0
?I1
R
UCE
@
R
@ ?
r
R2
@
@
RE
r
r
r
r
r
2IB r I0
?
?
r
@
R
@
r
-U0
r
-U0
Figure 3.25: Single BJT as current source and current mirror
so that process variations will not cause severe differences between the transistors
and on the other hand they will be closely related in temperature so that thermal
tracking will take place. If we assume strictly equal transistors with a static current gain β0 , we will have the following relation between the currents I0 and I1 ,
which shows that for high static current gain both currents are almost identical.
I0 =
3.3.3
β0
I1
β0 + 2
(3.51)
Active Load
A further step in reducing the number of passive components in a differential
r
R
@
T 3@
r
r
r
T1
@
@
R
@
@
r
U0
T4
r
r
@
T 2@
r
r
r
Figure 3.26: Differential amplifier with current mirror as active load
amplifier is to replace the resistors RC by a pnp current mirror as shown in Fig.
72
3.26, which is called active load. This load circuit is very economical in area
since T1 provides the current I1 for the current mirror and no resistors are used.
By using an active load, a high-impedance ouput load can be realized without
excessively large resitors and hence large power-supply voltage. As a result, for a
given power-supply voltage, a larger voltage gain can be achieved using an active
load than would be possible, if a resistor would be used as load. For example, if a
50 kΩ load would be used with a bias current of 1 mA, a resistive-load approach
would require a power-supply voltage of at least 50 V. An active load makes use
of the possibilities of a transitor to create simultaneously a large bias current and
a large small-signal output resistance rCE .
3.3.4
Level-Shifting Circuits
Despite using circuit techniques which avoid the use of high-valued decoupling
capacitors, there still remains the problem of coupling one circuit to another. Of
course direct coupling between amplifier stages removes the requirement for coupling capacitors but it is difficult because of the different DC-levels. So DC levelshifting circuits must be introduced. In principle the DC level may be shifted by
using an ideal voltage source and due to its zero internal resistance no AC-signal
attentuation would occur. In section 2.2 we already studied how to use normal
diodes to stabilize an ouput voltage. Figure 3.27a shows the principle layout of
r
r
r
A
r
r n
r
A
uin
r
?
r
r
R1
r
uin
R2
r
uout
?
r
?
r
@
R
@
r
r
r
r
uout
?
r
Figure 3.27: Level-shifting circuit with diodes and an amplified diode
a circuit to shift the DC level by an integer number times the threshold voltage
Uth ≈ 0.7 V of a diode. Using KVL we find for the ouput voltage of the first
circuit:
uout (t) ≈ uin (t) − nUth
To deduce an expression for the second circuit we must first consider the circuit
consisting of the two resistors and the transistor. Assuming a transistor with a high
73
current gain β0 the current IT thru the voltage divider R1 and R2 will not change
and we will have:
→
UR2 = UBE = R2 IT
UR1 = R1 IT =
R1
UBE
R2
So, we get for the total voltage across the voltage divider and hence across the
transistor UCE
)
(
R1
UBE
(3.52)
UCE = 1 +
R2
Which can be adjusted to be a non-integer fraction of the diode voltage UBE and
is thus called amplified diode. With the help of the last equation we find for the
output voltage of the circuit with amplified diode:
(
)
R1
uout (t) = uin (t) − 1 +
UBE
(3.53)
R2
3.3.5
Complementary Output Stage
In Fig. 3.28 the circuit schematic of a complementary emitter-follower output
stage is shown. In contrast to the input stages of an operational amplifier the output
U0
r
Uin
r
r
@
R
@
r
Uout
-
RL
@
R
@
r
−U0
Figure 3.28: Complementary emitter-follower output stage
stage must be able to drive other circuits and so the main emphasis does not lie on
the voltage gain of the stage but on the current the stage is able to deliver to the
circuits to be driven. As already discussed in section 3.2.2 an emitter follower has
a voltage gain of nearly one but a high current gain and provides a small output
resistance. In contrast to the conventional emitter-follower, the transistors shown
74
in Fig. 3.28 will only amplify on half of the sine wave. So the npn transistor will
only conduct for a positive signal, while the pnp transistor will only contact for a
negative signal.
75
Bibliography
[1] Jackson J.D.: Classical Electrodynamics, John Wiley & Sons, New York,
1999.
[2] Sze, S.M.: Physics of Semiconductor Devices, John Wiley & Sons, 1981.
[3] John, D; Martin, K.:Analog Integrated Circuit Design, JoŽhn Wiley & Sons,
1996.
[4] Vladimirescu, A.: The Spice Book, John Wiley& Sons Inc., 1993.
[5] Kittel, C.: Introduction to Solid State Physics, John Wiley & Sons, 2004.
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