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Fachhochschule Frankfurt am Main

University of Applied Sciences

Faculty of Computer Science and Engineering

ElectronicsAcademic Year 2011/2012

Prof. Dr.-Ing. G. Zimmer


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Contents

1 Semiconductor Basics 2

1.1 Band theory of solids . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Intrinsic conductivity . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Diffusion currents in semiconductors . . . . . . . . . . . . . . . . 10

2 Semiconductor diode and applications 12

2.1 The pn-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 pn-junction with zero bias . . . . . . . . . . . . . . . . . 12

2.1.2 pn-junction with bias . . . . . . . . . . . . . . . . . . . . 17

2.1.3 Small-signal model of a pn-junction . . . . . . . . . . . . 21

2.1.4 Spice model of a semiconductor diode . . . . . . . . . . . 242.1.5 Different types of semiconductor diodes . . . . . . . . . . 26

2.2 Diode applications . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.1 Diode as rectifier . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2 Voltage multiplier . . . . . . . . . . . . . . . . . . . . . . 34

2.2.3 Zener diode as voltage regulator . . . . . . . . . . . . . . 36

3 The bipolar junction transistor and applications 40

3.1 The bipolar junction transistor . . . . . . . . . . . . . . . . . . . 40

3.1.1 Structure and operation principles of a npn BJT . . . . . . 40

3.1.2 Static input and output characteristics of a BJT . . . . . . 43

3.1.3 Simple small signal BJT model . . . . . . . . . . . . . . 463.1.4 Advanced small signal BJT model . . . . . . . . . . . . . 49

3.1.5 SPICE model of a BJT . . . . . . . . . . . . . . . . . . . 49

3.2 Small signal amplifier . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 BJT biasing . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.2 Common-emitter amplifier . . . . . . . . . . . . . . . . . 54

3.2.3 Common-collector amplifier . . . . . . . . . . . . . . . . 60

3.3 Integrated circuit techniques . . . . . . . . . . . . . . . . . . . . 64

3.3.1 The differential amplifier . . . . . . . . . . . . . . . . . . 64

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3.3.2 Current Sources . . . . . . . . . . . . . . . . . . . . . . . 70

3.3.3 Active Load . . . . . . . . . . . . . . . . . . . . . . . . . 713.3.4 Level-Shifting Circuits . . . . . . . . . . . . . . . . . . . 71

3.3.5 Complementary Output Stage . . . . . . . . . . . . . . . 73

4 Field-Effect Transistors and their Applications 74

4.1 Junction Field-Effect Transistor . . . . . . . . . . . . . . . . . . 74

4.1.1 Cross-Section and Static IU-Characteristic of a JFET . . . 74

4.1.2 Small Signal Equivalent Circuit of a JFET . . . . . . . . . 78

4.1.3 SPICE Model of a JFET . . . . . . . . . . . . . . . . . . 80

4.1.4 Common-Source small Signal Amplifier with a JFET . . . 81

4.2 Metal-Oxide Semiconductor Field-Effect Transistor (MOSFET) . 864.2.1 N-Channel MOSFET . . . . . . . . . . . . . . . . . . . . 86

4.2.2 P-Channel MOSFET . . . . . . . . . . . . . . . . . . . . 87

4.2.3 Static Characteristics of a n-Channel Enhancement MOS-

FET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.4 Small Signal Equivalent Circuit of a n-Channel MOSFET 90

4.2.5 SPICE Model of a MOSFET . . . . . . . . . . . . . . . . 91

4.2.6 Common-Source Small Signal Amplifier with a n-Channel

MOSFET . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Operational Amplifiers and their Applications 95

5.1 Basic Linear Model of an Operational Amplifier . . . . . . . . . . 95

5.2 Basic Linear Op-Amp Circuits . . . . . . . . . . . . . . . . . . . 97

5.2.1 Inverting Amplifier . . . . . . . . . . . . . . . . . . . . . 97

5.2.2 Inverting summing amplifier . . . . . . . . . . . . . . . . 98

5.2.3 Non-Inverting Amplifier . . . . . . . . . . . . . . . . . . 99

5.2.4 Inverting Integrator . . . . . . . . . . . . . . . . . . . . . 100

5.2.5 Inverting Differentiator . . . . . . . . . . . . . . . . . . . 102

5.2.6 First-Order Low-Pass Filter . . . . . . . . . . . . . . . . 104

5.2.7 Second-Order Low-Pass Filter . . . . . . . . . . . . . . . 105

5.3 Basic Non-Linear Op-Amp Circuits . . . . . . . . . . . . . . . . 109

5.3.1 Op-Amp as Comparator . . . . . . . . . . . . . . . . . . 1095.3.2 Schmitt Trigger Realised with an Op-Amp . . . . . . . . 110

5.4 Digital-to-Analog Converter (DAC) . . . . . . . . . . . . . . . . 113

5.4.1 Op-Amp Summer as DAC . . . . . . . . . . . . . . . . . 114

5.4.2 DAC with R-2R-Ladder Network . . . . . . . . . . . . . 115

5.5 Analog-to-Digital Converter (DAC) . . . . . . . . . . . . . . . . 118

5.5.1 Quantization Error . . . . . . . . . . . . . . . . . . . . . 119

5.5.2 ADC Realizations . . . . . . . . . . . . . . . . . . . . . 1 21

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Chapter 1

Semiconductor Basics

Important elements of electric communications systems are devices capable of

amplifying the weak received electrical signals making further signal process-

ing 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 summa-

rized, 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 in-

creasing 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 , the permittivity and the permeability .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. Ruther-

ford, an English physicist showed with experiments that an atom consists of a very

small positive nucleus (diameter 1013 to 1012 cm) carrying positive charges

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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 positivenucleus 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 theblack-body radiation

W = h f (1.1)In equation 1.1 h = 6.6241034Ws2 stands for Plancks action quatum or Plancksconstant 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 Wx

W1

W2

W3

a) 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 Wj to the

lower energy level Wi by emitting an electromagnetic wave of frequency

f = 1h

(Wj Wi)

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

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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 structurewill 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 isoccupied by an electron is given by the so called Fermi-Dirac distribution [].

p(W) =1

exp(WWF

kBT) + 1

(1.2)

with kB = 1,381023 J/K (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, enhanc-

ing 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 oc-cupied. 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 we speak of anisolator. 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 e

and the elementary charge e

is up to an arbritray constant equal to the band energy. Thus the following relation

holds true:

e = 1e

WL +C1 = 1e

WV +C2 (1.3)

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

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a) metal b) semiconductor c) isolator

W W W

WV

WL

0

WV

WL

0

WV

WL

0

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 aretrapped 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

Si-atom covalent bonding

W

WC

WV

a) b)

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

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Si-atom

W

WC

WV

a) b)

E

electron 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 di-

rection. 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 creationof 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 pro-

cesses 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 inthe 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 = NCexp(WCWFkBT

) mit NC = 2(2mekBT

h2)3/2 (1.4)

p = NV exp(WFWVkBT

) mit NV = 2(2mpkBT

h2)3/2 (1.5)

Where me denotes the effective electron mass in the conduction band and mpthe 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

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periodic potential inside a cyrstal. The values NC and NV are called effective

density of states in the conduction band respectively valence band. Table 1.1gives some examples for the effective masses of electrons and holes for different

semiconductors. As already discussed earlier, the electron and hole density are

Semiconductor me/me mp/meSi 0,33 0,56

Ge 0,22 0,33

GaAs 0,067 0,48

InP 0,078 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 =

n p (1.6)

With the help of the equations 1.4 and 1.5 and W = WCWV we find:

ni = NLNV exp(W

2kBT) (1.7)

Example: Intrinsic charge density

Germanium: W = 0,63 eV, Silicon: W = 1,14 eV, T = 300K

ni Ge 1,8 1013 1cm3

ni Si 2,6 109 1cm3

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 mobilitye and the hole mobility 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

[].

= e(nn + pp) (1.8)

Example: Intrinsic conductivity of germanium and silicon at T = 300 K

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Crystal Electron Holes

Si 1300 500Ge 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 []

i Ge 2.3 102S/cmi Si 7.5 107S/cm

For example copper at the same temperature has a conductivity ofCu 5.9 105S/cmwhich 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 introducingimpurities into their crystal lattice. The process of adding controlled impurities to

W

WC

WV

a) b)

donor atom free electron

donorlevel WF

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

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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 group3 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 NDHence 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

WF WL kBTln(NCND

) (1.9)

Since the product np = n2i only depends on the band gap of the semiconductor wehave,

p =n2in

n2i

NDwhile the conductivity of the n-conductor is essentially given by

enND (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 useNA 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

WF WV + kBTln(NVNA

) (1.11)

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:

epNA (1.12)

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W

WC

a) b)

acceptor atom free hole

WVacceptorlevelWF

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 semi-

conductors, 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 crys-

rrr

rrr

rrrr rr

r rr rrrr rr rr rrr rr r

rr

rrr r rrr rrr rrr

rr

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 andthe 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 JDp (x)

the one dimensional diffusion current density of the holes and with JDn (x) the dif-fusion current density of the electrons, the diffusion current is described by the

following equations:

JDp (x) = eDpd p

dxJDn (x) = eDn

dn

dx(1.13)

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The positive sign in the equation of the electron current density reflects the defin-

tion of the positive technical current direction, which is contrary to the movementof the electrons. The constants Dp and Dn are called diffusion coefficients and

they are related to the mobility of the carriers by Einsteins relation [ ?, ]

Dp = pkBT

eDn = n

kBT

e(1.14)

Thus the total current density of the holes Jp and of the electrons in a semicon-

ductor is composed of the drift current due to an electric field and the diffusion

current.

Jp = eppE

eDpd p

dx

(1.15)

Jn = ennE + eDndn

dx(1.16)

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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 whilethe 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 arbi-

trarily 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 as-

sume 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 p-

and 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,

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a)

b)

E

free holefree electron fixed acceptorfixed donatorFigure 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 = e()

e(

) (2.1)

To calculate the hole distribution p(x) we use equation 1.15 and consider the factthat the diffusion process has stopped (JP = 0) and that one can deduce the electricfield by the gradient of the potential function, which is related to the valence band

energy WV(x) via equation 1.3:

kBTd p(x)

dx= p(x)

dWV(x)

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:

p(x) = NA exp

WV()WV(x)

kBT

(2.2)

In an analog manner we get for the distribution of the electrons using the boundary

condition n(x ) = ND:

n(x) = ND exp

WC(x)WL()

kBT

(2.3)

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To get a unique relation between the potential function e(x) and the band ener-

gies one uses the condition e(x ) = 0. As a result we get the followingrelation between the potential function and the band energies of the valence and

the conduction band:

e(x) = 1e

[WV(x) WV()] = 1e

[WC(x) WC()] (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:

p(x) = NA expee(x)

kBT

n(x) = ND exp

eUD e(x)

kBT

(2.5)

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

e(x):

UD =kBT

eln(

NAND

n2i) (2.6)

Example: Diffusion voltage of a pn-junction in silicon

NA =

ND = 10

15

cm3

, T = 300K

UD 25.9 mV ln

(1015)2

(2.6 109)2

= 660 mV

According to equation 2.2 and 2.3 the decline of the electron and hole distribu-

xwn

wp

(x)

eNA

eND

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

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according Fig. 2.2. This approximation is called abrupt pn-junction [2] and con-

siders a constant negative charge distribution NA in the region wp < x < 0 anda constant positive charge distribution ND in the region 0 < x < wn. Since thepn-junction is electrically neutral, the following equation must hold true:

NDwn = NAwp (2.7)

To calculate the internal electric field und potential function of an abrupt pn-

junction we use the one-dimensional divergence theorem of the electrical field,

which results in the following differential equations for the electric field:

dE

dx = e

NA for wp < x < 0 (2.8)dE

dx=

e

ND for 0 < x < wn (2.9)

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

wp < x < wn

E(x) = eNA

(x + wp) for wp < x < 0 (2.10)

E(x) = eND

(wn x) for 0 < x < wn (2.11)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 zeroagain at x = wn. The negative sign of the electric field reflects the fact that it isdirected 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 e(wp) = 0 ande(x = 0

) = e(x = 0+).

e(x) = eNA2 (x + wp)2 for wp x 0 (2.12)

e(x) =eND

2(wn(wp + 2x) x2) for 0 x wn (2.13)

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 wp and wn of the depletion zone may be evaluated.

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wp wn

E(x)

x

Emax

x

e(x)

UD

Figure 2.3: Electric field and potential function of an abrupt pn-junction without

bias

wp =

2e UD

NDNA(NA +ND)

wn =

2e UD

NAND(NA +ND)

As a result we get for the total width of the depletion zone:

w =

2

e

NA +NDNAND

UD (2.14)

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 casethe integral of the electric field, the easiest way to calculate its value is to evaluate

the area under the graph of the function:

UD =1

2

(wp + wn)Emax

Using equation 2.14 we find for the maximum of the electric field:

Emax =2UD

w=

2eUD

NAND

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 n-

conductor called diffusion voltage or build-in voltage UD and in a band bending.

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xwp wn

W

WCWF

WV

eUD

p-conductor n-conductor

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


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a)

WC

WF

WV

W

xwp wn

b)

wp wn x

WCWF

WV

W

s s

U

e(UD U)

e(UD U)

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 neglectedand 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 diffusioncurrent.

In the depletion zone no recombination takes place. As a result the totalcurrent thru the diode can be described as the diffusion current of the mi-

nority 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 n-conductor. The concentration of the electrons n(x) in the p-conductor can be

18


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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)

kBT) = pno exp(

eU

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 risewith U> 0 and decay for U> 0. To calculate the hole distribution in the n-regionwe use the rate equation ??, extended by the divergence term of the currents [2]:

p

t= 1

e

Jp

x p pno

In the stationary case ( t

= 0) this expression reduces to:

dJp

dx= ep pno

(2.17)

According to our assumption the current Jp is solely a diffusion current due to the

minority carrier and we get for the region wn < x < the following differentialequation:

d2p

dx2=

1

Dp(p pno) = 1

L2p(p pno) (2.18)

In the last term the constant Lp =

Dp was introduced, it posseses the dimensionof a length and hence denotes the mean length along which a minority carrier can

diffuse in its lifetime before it will recombinate. The solution of the abovedifferential equation has to reflect that for x = wn the hole density is given byequation 2.16 and hence we find as solution:

p(x) = (p(wn) pno) exp(x wnLp

) + pno (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:

Jp(wn) = eDp

Lppno

exp(

eU

kBT) 1

(2.20)

In an equivalent way one can also deduce an expression for the diffusion current

Jn(wp) and since we assume that there will be no recombination in the depletionzone we get for the total current thru a pn-junction:

J = Js

exp(

eU

kBT) 1

with Js = e n

2i

Dp

LpND+

Dn

LnNA

(2.21)

19


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According to equation 2.21 the current density thru the pn-junction will rise ex-

ponentially for positive voltages U, while it will decay for negative values. In thelimit it will reach a value of Js, hence this value is called reverse biased satura-

tion current density. If we multiply the current density with the area Apn of the

pn-junction we get the static I-U characteristic of an ideal pn-junction.

I = Is

exp(

U

UT) 1

with Is = ApnJs and UT =

kBT

e(2.22)

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

UV

ImA

ideal

RS = 1

0

20

40

60

100

0 0.2 0.4 0.6 1

Figure 2.6: Static I-U-characteristic of an ideal pn-junction with Is = 10nA

controlled by the reduced voltage URsI. To demonstrate the influence of thisresistor in Fig. 2.6 the static I-U-characteristic of an ideal pn-junction with Is =10nA and of the same diode with Rs = 1 is shown.

20


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2.1.3 Small-signal model of a pn-junction

The equations deduced in the preceding section describe the behaviour of a pn-

junction only for almost static time functions. To get an idea of its dynamic be-

haviour 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) isused to realize the small signal exitation.

D

r

r

r

r

i(t)

r

r

I

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 op-eration point (I, U). Due to the sinusoidal current source with amplitude I there

will also exist a sinusoidal voltage across the pn-junction with amplitude U. One

speaks of small signal exitation as long as the following relations hold true:

I I and U 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:

U dUdI

I =1

dI

dU

I = rD I

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 =UT

I(2.23)

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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

10 mA 2,6

Diffusion capacitance cD

As already discussed in section 2.1.2 a forward biased pn-junction will store mi-

nority 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:

Q(U) = eApn

wn

(p(wn) pno) exp(x wn

Lp)

dx

For a differential change of the applied voltage u we can write:

q dQ(U)dU

u =eADLppno

UTexp(

U

UT) u (2.24)

Equation 2.24 can be used to define the diffusion capacitance cD of a pn-junction.

cD =q

u=

eADLppno

UTexp(

U

UT) =

UTI =

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 2.5 103s

cD =

2,5 ms

25,9 mV 1mA 97 FAccording 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


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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 pn-

junction. 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 upn(t) is givenby the sum of a DC voltage Uo and a time varying voltage u(t). Hence we getfor the electric field in the pn-junction

E(t,x = 0) = 2e

NAND

NA +ND(UD Uo) (1 1

2(UD

Uo)u(t)) (2.26)

and for the displacement current density

Jv = dE(t,x = 0)

dt=

e

NAND

NA +ND

1

2(UD Uo)du(t)

dt(2.27)

Since we assume a homogenous distribution across the cross-section of the pn-

junction, the total displacement current can be calculated by multiplication with

the area Apn 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.

cJ = Apn

NAND

NA +ND

e

2(UD Uo) (2.28)

To describe the small signal frequency response of a real semiconductor diode in

s

s

rD cJ

LSs s

ss

CP

RS

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


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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 pn-

junction 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

s s

sss

iD

uDCD CJ

RS

ID

Figure 2.9: Spice model of a semiconductor diode

thru the diode is calculated using the following equation:

iD = ID + CDduD

dt+ CJ

duD

dt(2.29)

Static diode current ID

Forward biased, the static diode current ID is equal to the current of an ideal pn-

junction already given in equation 2.22, but with a further parameter N included,

called emission coefficient.

IDi = IS(T)

exp

uD

NUT

1

(2.30)

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Here the temperature dependence of the saturation current IS(T) is given by the

following expression:

IS(T) = IS

T

T0

(X T I/N) exp

EG(1 T0/T)

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 = ISRexp uDNR UT 1 1

uD

V J

2

+ 0,005M/2

(2.32)

If the reverse voltage of the diode is further enhanced reverse breakdown occurs

which is modeled by an exponential function:

ID = IBVexpuD BV

NBV UT

(2.33)

Dynamic diode current

To account for the dynamic behaviour of a real diode expressions for the junc-

tion 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

CJ = CJO

1 uDV J

M(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.

CD = T TdiD

duD=

T T

rD(2.35)

In the following table the essential SPICE parameters used to specify a real diode

are summarized

IS saturation current

N emission coefficient

ISR saturation current of correction diode

NR emission coefficient ofISR

BV reverse breakdown voltage

IBV current at break-down voltage

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NBV coefficient ofIBV

RS series resistanceT T minority carrier life time

CJ0 zero-bias junction capacitance

V J junction potential

M grading coefficient

FC coefficient for forward-bias depletion capacitance formula

EG activation energy

X T I temperature exponent ofIS

KF flicker noise coefficient

AF 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


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Zener diodes

The ordinary junction diode will be destroyed, if a reverse voltage is applied, ex-

tending their maximum reverse voltage and breakdown occurs. Zener diodes in

this sense are special diodes that will not be destroyed when the breakdown oc-

curs. 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 volt-

UD

IDUZ0

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.

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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 processis 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 Iph, is proportional to

the incident optical power Popt and the proportional constant is called responsivity

Rsp of the photo detector.

Iph = 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.

=C0 h

Wgf =

Wg

h

(2.37)

But this process may only take place in certain semiconductors, called direct band-

gap 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 reducedand 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


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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 compli-

cated. 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-U-

UDUth

ideal diode diode with threshold voltage Uth

ID

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 applica-

tions 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 behaviourof 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 con-

verted 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.

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Half-wave rectifier

r rrRL

rrr

r

r

uD

uRuS

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 resis-tance 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 remem-

ber, that the amplitude U 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) acrossthe diode and the voltage uR(t) across the load resistance. According to KVL thefollowing equation holds true:

uS(t) + uD(t) + uR(t) = 0Since 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 becomespositive 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 dropacross the resistor, but the whole voltage of the source will drop across the diode.

As an example Fig. 2.2.1 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 500 and the diode BA592. Essentiallyit shows the half-wave of the exiting voltage source, but there is a remarkable

30


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units < 1mm,1mm>x f rom0.00to80.00,y f rom0.00to68.1511 < 1mm> 1010< 0pt>

REF : 0 1V/Div[lb]at068.74[lb]at072.590Sec[lb]at 3.8523.25950mSec[lb]at76.15 3.259

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 > U =

2UN and IFmax >U

RL(2.39)

Of course the voltage shown in Fig. 2.2.1 is not yet a DC voltage but still a

periodic time function, with a DC part given by the following equation.

UDC =U

=

2

UN (2.40)

To further smooth the ouput voltage a capacitor may be used as shown in Fig.

r

r

r

r

uD

uS

r

r

C

r

r

r

r

RL

uR

Figure 2.14: Half-wave rectifier with smoothing capacitor

2.14. Fig. 2.15 shows the simulation results of the same half-wave rectifier where

according to Fig. 2.14 a capacitor of 100F was included for smoothing, also

shown is the time function without a capacitor. [lb] at 45.00 41.00

Figure 2.15: Ouput voltage of a half-wave rectifier with smoothing capacitor

31


34/125


REF : 0 1V/Div[lb]at068.74[lb]at072.590Sec[lb]at 3.8523.25950mSec[lb]at76.15 3.259C= 100F

So even if there seemed to be only a little change in the circuit due to the ca-

pacitor 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 U of the AC voltage. So, according to equation 2.38

the maximum reverse voltage may reach a value of 2U, thus the following condi-

tion must be fulfilled, in case of a half-wave rectifier with smoothing capacitor.

URmax > 2U (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 capac-

itor. 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 lim-

ited 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 half-wave rectifier is very simple, it is also very inefficient for power transfer, since

only one half-cycle is used.

Full-wave rectifier

The circuit that allows us to use every half-wave of a cycle is called full-wave

rectifier. Fig. 2.16 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

of the circuit each work like a half-wave rectifier, but if the anode of diode one ispositive, 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:

UDC = 2U

= 2

2

UN (2.42)

while the maximum reverse voltage will reach a value of 2U and thus the follow-

ing condition must be fullfilled.

URmax > 2U (2.43)

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rr

r

r

r

rrRL

uS(t)

uS(t)

uR(t)

uD1(t)

uD2(t)

Figure 2.16: Circuit schematic of a full-wave rectifier

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.

2.17 may be used. With the help of this circuit the costly transformer is omitted

D3rD4 r

D2 r

rrrrr r

RL

r D1r

r

uS(t)

uR(t)

Figure 2.17: Circuit schematic of a bridge rectifier

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. [lb] at 45.00 38.00

Figure 2.18: Output voltage of a brigde rectifier without and with smoothing ca-

pacitor

Fig. 2.18 shows the simulation results, again using the diode BA592 and a

33


36/125


REF : 0 1V/Div[lb]at068.74[lb]at072.590Sec[lb]at 3.8523.25950mSec[lb]at76.15 3.259C= 100F

500 load resistance. Comparing the maximum amplitude, with the simulationgiven in Fig. 2.2.1 shows, that in the case of the bridge recticfier the peak volt-

age 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 100F parallel to the load resistance isalso shown in Fig. 2.18.

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. 2.19a, 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,

r r r rrRP r Rprr

r rr r r r r rr RsRs

r r r

a) b)

Figure 2.19: Combined diodes to enhance reverse voltage or forward current ca-

pability

two diodes may be connected in parallel, as shown in Fig.2.19b. 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. 2.20 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

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C

r

uS(t)

uC(t)uo(t)

Figure 2.20: Half-wave rectifier with interchanged capacitor and diode

ouput voltage uo(t) of the circuit

uo(t) = uS(t) + uC(t)

Fig. 2.2.2 shows the simulation result for the time function uo(t) according to



Figure 2.21: Output voltage uo(t)

the circuit of Fig. 2.20. As source voltage uS(t) a sinusoidal time function with a5 V amplitude was used. Roughly spoken the output voltage uo(t) shows a maxi-mum 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. 2.22a. The principle

rr

r

r

r r rr

r

r

rrr

r r

r rUo

uS(t)

rr

r r

r

r

r

r rr r

r r

r

r

r rr r

r r

r

r

r rr r

r r

r

rr rrU0

uS(t)

Figure 2.22: Voltage doubler and multiplier circuits

of the voltage doubler shown in Fig. 2.22a was extended by Greinacher to reach

even higher voltage levels by adding further stages, as shown in Fig 2.22b. In

principle the voltages of the capacitors in the lower line will add up to the final

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Figure 2.23: Output voltage of a two stage voltage multiplier

voltage level U0. The time function of a two stage voltage multiplier ist given

in Fig. 2.2.2. 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

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. 2.24 a zener diode is used to stabilize the output

voltage U0 to the Zener voltage of the diode. To describe the performance of a

r r

Rs

rr

r rr rr r

Ui UoUZ

Io

IZ

UZ0 UZ

IZ

Figure 2.24: Simple circuit to regulate the ouput voltage

Zener diode usually the current and voltage directions given in Fig. 2.24a are

used. These results in the I-U characteristic of a Zener diode given in Fig. 2.24b.

In contrast to the very sophisticated models that can be used with SPICE, we will

restrict our considerations to idealized Zener diodes. As shown in Fig 2.24b 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.

2.24 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 itsmaximum possible dissipation power Pmax, which will limit the maximum current

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IZmax thru the diode and we have:

Izmax PmaxUZ0

(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

of the circuit and solve it for the resistor:

Rs =Ui UoIo + 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 out-put 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.

Rs Uimax Uo

Iomin + IZmax(2.47)

Only if the two inequalities are both fulfilled, the circuit according to Fig. 2.24 is

realisable with the chosen Zener diode. To compare different circuits to stabilize

the output voltage we define the following stability factor S

S =Ui/UiUo/Uo

dUi/UidUo/Uo

(2.48)

In principle the last form of equation 2.48 allows us to deduce expressions for thestability factor using small signal approximations. Of course, if we would assume

an ideal Zener diode with rZ = 0, the stability factor Swould become infinte sincea variation of the input voltage would not result in a variation of the ouput voltage

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. 2.24a the following

equations are valid:

Ui = RsI + rZIZ + UZ0 and Uo = rZIZ + UZ0

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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 + rZdIZ

and there will also be a change in the output voltage

dUo = rZdIZ

So we find for the ratio dUi/dUo:

dUi

dUo=

RsdI + rZdIZ

rzdIZ Rs

rZ

dI

dIZfor 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

dUo=

Rs

rz

So we get as final result for the stability factor of the circuit according to Fig.

2.24a:

S

Rs

rz

Uo

Ui

(2.49)

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 ofRs and the stability factor.

According to the equations 2.46 and 2.47 we get for the series resistance the fol-

lowing relations,

Rs < 100 and Rs > 90

so the ratings of the Zener diode are sufficient and the series resistor may be

chosen to be RS = 95. From the data sheet of the Zener diode one finds themaximum dynamic resistance rZ to be 7, so we get for the stability factor

S =95

7

15

30 6.8

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r rRs

r

rr rr r

r r

Ui

Uo

Figure 2.25: Circuit to stabilize low voltages

Stabilization of low voltagesUsually 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. 2.25. Here the series connection of diodes

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.

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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 dis-

cussion with the BJT since its operation principles are based on the the behaviourof 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 SBC-

technique (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 cm3). With the help of the p-zones on both sides thesingle 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 cm3. In a further oxidationand etching process the window for the emitter is formed and finally with the help

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s ss

E C

B

N P N

s s

sE C

B

a)

b)

c)

s s s

p-Silizium

n++p+npn

+p+

EB

C

Figure 3.1: a) simplified physical layout, b) circuit schematic and c) cross sec-tional view of a npn-BJT

of a diffusion process a donator concentration of approximately NDE 1022 cm3is 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 isforward 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 con-

ditions 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 elec-

trons 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 essen-

tially 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 cur-rent. 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

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x

WC

WF

WV

W

s s

s

E

B

C

UBE > 0 UBC< 0

s

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 di-

rections 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 = 0 IE (3.1)

The parameter 0 of the last equation is called static current gain in a commonbase circuit, despite the fact that due to the recombination of electrons in the base

its value is always lower than one (o 0,9 0,999). Since the BJT is a node

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ss

s

E C

B

IE IC

IB

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:

IC =0

1 0IB = 0IB (3.2)

The parameter of equation 3.2 is denoted as static current gain in a common emit-

ter circuit. Depending on the transistor 0 can reach values between approxi-mately 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 basecollector diode, which represent the electron current from the emitter to the col-

lector 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 behav-

iuor. 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.

IE = ISE

exp(UEB

UT) 1

(3.3)

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s s s s s

ss s

IE IC

oIE

UEB UCB

Figure 3.4: Simplified equivalent circuit according to Ebers-Moll under typical

operation conditions

IC = oIEISC

exp(UCBUT

) 1

(3.4)

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

s

s s s

s

s s

ss s

ss

UCE

IE

IC

IB

UBE

UCEUBE

0IE

Figure 3.5: BJT in common emitter configuration at its Ebers-Moll-model

now deduce an expression for the input characteristic IB = f(UBE,UCE) and forthe output characteristic IC = f(UCE,UBE). Staring point is again KCL for theBJT as a whole:

IB = IEIC = (1 0)IE +ISC

exp(UCBUT

) 1

Accounting the different definitions of the voltages UEB = UBE and UCB =UCEUBE we get for the base current:

IB = (1 0)ISE

exp(UBE

UT) 1

+ISC

exp(UCEUBE

UT) 1

For typical conditions of operation we have UCE UBE and the last term may beneglected, resulting in the following expression for the base current:

IB = (1 0)ISE

exp(UBE

UT) 1

= ISB

exp(

UBE

UT) 1

(3.5)

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The form of the last equation is equivalent to that of a normal pn-junction, which

has a saturation current of ISB = (1 0)ISE. Thus the input characteristic of atransistor 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,

IC = 0ISE

exp(

UBE

UT) 1

ISC

exp(UCEUBE

UT) 1

(3.6)

or with reference to the saturation current ISB

IC =0

1

0

ISBexp(UBE

UT) 1ISCexp(

UCEUBEUT

) 1 (3.7)

Example: Theoretical output characteristic of a BJT

As example Fig. 3.6 shows the output characteristic of a BJT according to equa-

UCE

V

ICmA

0,64V

0,66V

0,68V

0

2

4

6

10

0 1 2 3 5

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, 0 = 0,999. Comparing the theoretical predicted output char-acteristic 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 0 of the com-mon emitter configuration:

0 =0

1 0 (3.8)

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while the second term describes nothing else but the dependence of the base cur-

rent IB on the voltage UBE.

IB(UBE) = ISB

exp(

UBE

UT) 1

(3.9)

With the help of the introduced parameters equation 3.7 can be rewritten as

IC = 0IB(UBE) ISC

exp(UCEUBEUT

) 1

(3.10)

Since under normal operation conditions the base collector diode is based in re-

verse direction, the last term of equation 3.10 can be neglected, which reduces thisequation to

IC 0IB(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

rrrr

r0IBr IB CB

E

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 modelOne 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

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voltage source uBE(t) is used, with an amplitude UBE fulfilling the small signal

condition UBE UBE. In the ouput circuit a load resistanceRL is included to allowan alternating voltage uCE(t) to exist, also fulfilling the small signal conditionUCE UCE. A mathematical exact analysis of the circuit shown in Fig. 3.8 is

rr

RL

r

UCEUBE

uBE(t)

Figure 3.8: Common emitter circuit of a BJT with small signal exitation

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 performa Taylor approximation at the opertaion point.

IB =ISB

UTexp(UBE

UT)UBE |IBE |

UTUBE (3.12)

In analogy to the pn-junction one can introduce the dynamic resistance rBE of the

base emitter diode.

rBE =UT

|IB | (3.13)

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, al-ready 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

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rrBE

rr r

rr r r

rr rrrCE

rr r

rr

RL

uBE(t)

iB

0

iB

uCE(t)

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 = hieiB + hreuCE

iC = hf eiB + hoeuCE(3.14)

From equation 3.14 it becomes clear that the single parameters are defined ac-

cording to the following equations:

hie =uBE

iB|uCE=0 short circuit input resistance

hre =uBEuCE

|iB=0 open circuit reverse voltage ratiohf e =

iCiB

|uCE=0 short circuit forward current gainhoe = iCuCE|iB=0 open circuit output conductance

(3.15)

Since the first equation of 3.14 defines the small signal voltage uBE it may be

interpreted to be the result of Kirchhoffs voltage law at the input of the transistor

while iC of the second equation is the result of Kirchhoffs 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

s

s s

s

hoe

iChieiB

uBE hreuCE uCE

hf eiB

s s s

s

Figure 3.10: Small signal equivalent circuit according to the h-parameters

reveals the following equivalences:

hie = rBE hf e = hoe = 1/rCE (3.16)

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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 decribesthe 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 ac

electronicsscript_2

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