Module 2

Energy Band Diagram :-

The range of energies that an electron may possess in an atom is known as the energy band.

Three Important energy bands are,

• Valence Band
• Conduction Band
• Forbidden Band

Valence Band :-

The Range of Energy possessed by valence electrons is known as valence Bands.

Conduction Band :-

The valence electrons are less tightly bound with the nucleus. So that even an application of small electric field some of the valence electrons detached from the nucleus and it becomes free electrons. These free electrons are responsible for the conduction of current in good conductors. These electrons are also called conduction electrons.

→ The Range of energy possessed by these electrons is known as conduction band.

Forbidden Band (or) Energy Gap :-

The energy band in between the conduction band and the valence band is called forbidden Band.

Classification of Materials (or) Solids According to Energy Bands :-

Solids are classified in to there types Insulators

Conductors Semi conductors

Insulator Energy Bands

Most solid substances are insulators, and in terms of the band theory of solids this implies that there is a large forbidden gap between the energies of the valence electrons and the energy at which the electrons can move freely through the material (the conduction band).

Conductor Energy Bands

In terms of the band theory of solids, metals are unique as good conductors of electricity. This can be seen to be a result of their valence electrons being essentially free. In the band theory, this is depicted as an overlap of the valence band and the conduction band so that at least a fraction of the valence electrons can move through the material.

Semiconductor Energy Bands

Semiconductors, on the other hand, have an energy gap which is in between that of conductors and insulators. This gap is typically more or less 1 eV, and thus, one electron requires energy more than conductors but less than insulators for shifting valence band to conduction band.

The materials, in which the conduction and valence bands are separated by a small energy gap (1eV) as shown in figure are called semiconductors.

→ Silicon and germanium are the commonly used semiconductors.

→ A small energy gap means that a small amount of energy is required to free the electrons by moving them from the valence band in to the conduction band.

→ The semiconductors behave like insulators at 0K, because no electrons are available in the conduction band.

→ If the temperature is further increased, more valence electrons will acquire energy to jump into the conduction band.

“Fermi level” is the term used to describe the top of the collection of electron energy levels at absolute zero temperature.

The materials that are neither conductor nor insulator with energy gap of about eV (electron volt) are called semiconductors.

Semiconductors:

Some materials have a filled valence band just like insulators but a small gap to the conduction band.

At zero Kelvin the material behave just like an insulator but at room temperature, it is possible for some electrons to acquire the energy to jump up to the conduction band. The electrons move

Empty conduction band

Valence bands, filled with electrons

Small energy gap

easily through this conduction band under the application of an electric field. This is an intrinsic semiconductor.

At zero Kelvin – no conduction

Conduction band, with some electrons

So where are all these materials to be found in the periodic table ?

Top valence band now missing some electrons

At room temperature – some conduction

Intrinsic Semiconductors

As per theory of semiconductor, semiconductor in its pure form is called as intrinsic semiconductor. In pure semiconductor number of electrons (n) is equal to number of holes (p) and thus conductivity is very low as valence electrons are covalent bonded.

Silicon Energy Bands

Germanium Energy Bands

Electrons in intrinsic (pure) Silicon

• covalently bonded to atoms
• “juggled” between neighbors
• thermally activated: density  e^T
• move around the lattice, if free
• leave a positively charged `hole’ behind

Both electrons and holes contribute to current flow in an intrinsic semiconductor.

Extrinsic Semiconductors

As per theory of semiconductor, impure semiconductors are call extrinsic semiconductors. Extrinsic semiconductor is formed by adding small amount of impurity. Depending on the type of impurity added have two types of semiconductors: N – type and P-type semiconducto In 100 million parts of semiconductor one part of impurity is added.

Doping

When we add a small quantity of impurity in a semiconductor, then the impurity contributes either free electrons or holes to the semiconductor. As a result, the conducting property of semiconductor changes. The process of changing the conductive property of semiconductor by adding impurities is known as doping.

N type Semiconductor

In this type of semiconductor majority carriers are electrons and minority carriers are holes. N – type semiconductor is formed by adding pentavalent (five valence electrons) impurity in pure semiconductor crystal, e.g. P, As, Sb.

Four of the five valence electron of pentavalent impurity forms covalent bond with Si atom and the remaining electron is free to move anywhere within the crystal. Pentavalent impurity donates electron to Si that’s why N- type impurity atoms are known as donor atoms. This enhances the conductivity of pure Si.

The addition of pentavalent impurities such as antimony, arsenic or phosphorous contributes free electrons, greatly increasing the conductivity of the intrinsic semiconductor.

N type Semiconductor

N type Semiconductor

N type Semiconductor

P-Type Semiconductor

The addition of trivalent impurities such as boron, aluminum or gallium to an intrinsic semiconductor creates deficiencies of valence electrons, called “holes”

If instead of pentavalent impurity, a trivalent impurity is added to the intrinsic semiconductor, then instead of excess electrons there will be excess holes created in the crystal. Because when a trivalent impurity is added to the semiconductor crystal, the trivalent atoms will replace some of the tetravalent semiconductor atoms. The three (3) valance electrons of trivalent impurity atom will make the bond with three neighborhood semiconductor atoms. Hence, there will be the lack of an electron in one bond of the fourth neighboring semiconductor atom which contributes a whole to the crystal. Since trivalent impurities contribute excess holes to semiconductor crystal, and these holes can accept electrons, these impurities are referred as acceptor impurities. As the holes virtually carry positive charge, the said impurities are referred as positive – type or p – type impurities and the semiconductor with p type impurities is called p type semiconductor.

P-Type Semiconductor

P-Type Semiconductor

p-type doping in silicon

Column III elements accept an electron from the valence band

Holes and Intrinsic Semiconductors

When electrons move into the conduction band, they leave behind vacancies in the valence band. These vacancies are called holes.

Because holes represent the absence of negative charges, it is useful to think of them as positive charges.

Whereas the electrons move in a direction opposite to the applied electric field, the holes move in the direction of the electric field.

A semiconductor in which there is a balance between the number of electrons in the conduction band and the number of holes in the valence band is called an intrinsic semiconductor.

Examples of intrinsic semiconductors include pure silicon and germanium.

What are P-type and N-type ?

Semiconductors are classified in to P-type and N-type semiconductor

P-type: A P-type material is one in which holes are majority carriers i.e. they are positively charged materials (++++)

N-type: A N-type material is one in which electrons are majority charge carriers i.e. they are negatively charged materials ( )

extrinsic semiconductor: doped semiconductor

Terminology

donor: impurity atom that increases n

acceptor: impurity atom that increases p

N-type material: contains more electrons than holes P-type material: contains more holes than electrons

majority carrier: the most abundant carrier minority carrier: the least abundant carrier

intrinsic semiconductor: n = p = ni

pn Junction

The interface separating the n and p regions is referred to as the metallurgical junction.

Diffusion

Let us assume that we have two boxes- one contains red air molecules while another one contains blue molecules. This could be due to appropriate types of pollution.

Let us join these 2 boxes together and remove the wall between them.

Each type of molecules starts to move to the region of their low concentration due to the concentration gradient in the middle.

Eventually there would be a homogeneous mixture of two types of molecules.

pn Junction

As electrons diffuse from the n region, positively charged donor atoms are left behind. Similarly, as holes diffuse from the p region, they uncover negatively charged acceptor atoms. These are minority carriers.

The net positive and negative charges in the n and p regions induce an electric field in the region near the metallurgical junction, in the direction from the positive to the negative charge, or from the n to the p region.

The net positively and negatively charged regions are shown in Figure. These two regions are referred to as the space charge region (SCR). Essentially all electrons and holes are swept out of the space charge region by the electric field. Since the space charge region is depleted of any mobile charge, this region is also referred to as the depletion region

Diodes

Electronic devices created by bringing together a p-type and n-type region within the same semiconductor lattice. Used for rectifiers, LED etc

Diodes

It is represented by the following symbol, where the arrow indicates the direction of positive current flow.

Other electrons from the N region cannot migrate because they are repelled by the neg ions in the P region and attracted by the positive ions in the N region. The significanc this built-in potential across the junction, is that it opposes both the flow of holes electrons across the junction and is why it is called the potential barrier.

There are two operating regions and three possible “biasing” conditions for the standard Junction Diode and these are:

1. Zero Bias – No external voltage potential is applied to the PN junction diode.
2. Reverse Bias – The voltage potential is connected negative, (-ve) to the P-type material and positive, (+ve) to the N-type material across the diode which has the effect of Increasing the PN junction diode’s width.
3. Forward Bias – The voltage potential is connected positive, (+ve) to the P-type material and negative, (-ve) to the N-type material across the diode which has the effect of Decreasing the PN junction diodes width.

Forward Bias and Reverse Bias

Forward Bias : Connect positive of the Diode to positive of supply…negative of Diode to negative of supply

Reverse Bias: Connect positive of the Diode to negative of supply…negative of diode to positive of supply.

If the p-n junction diode is forward biased with approximately 0.7 volts for silicon diode or 0.3 volts for germanium diode, the p-n junction diode starts allowing the electric current. Under this condition, the negative terminal of the battery supplies large number of free electrons to the n-type semiconductor and attracts or accepts large number of holes from the p-type semiconductor. In other words, the large number of free electrons begins their journey at the negative terminal whereas the large number of holes finishes their journey at the negative terminal.

If the external voltage applied on the silicon diode is less than 0.7 volts, the silicon diode allows only a small electric current. However, this small electric current is considered as negligible.

When the external voltage applied on the silicon diode reaches 0.7 volts, the p-n junction diode starts allowing large electric current through it. At this point, a small increase in voltage increases the electric current rapidly. The forward voltage at which the silicon diode starts allowing large electric current is called cut-in voltage. The cut-in voltage for silicon diode is approximately 0.7 volts.

If the external voltage applied on the germanium diode is less than 0.3 volts, the germanium diode allows only a small electric current. However, this small electric current is considered as negligible.

When the external voltage applied on the germanium diode reaches 0.3 volts, the germanium diode starts allowing large electric current through it. At this point, a small increase in voltage increases the electric current rapidly. The forward voltage at which the germanium diode starts allowing large electric current is called cut-in voltage. The cut-in voltage for germanium diode is approximately 0.3 volts.

The process by which, a p-n junction diode blocks the electric current in the presence of applied voltage is called reverse biased p-n junction diode.

In n-type and p-type semiconductors, very small number of minority charg carriers is present. Hence, a small voltage applied on the diode pushes all th minority carriers towards the junction. Thus, further increase in the externa voltage does not increase the electric current. This electric current is calle reverse saturation current. In other words, the voltage or point at which th electric current reaches its maximum level and further increase in voltage doe not increase the electric current is called reverse saturation current.

However, if the voltage applied on the diode is increased continuously, the p- junction diode reaches to a state where junction breakdown occurs and revers current increases rapidly.

I-V characteristics of Ideal diode

Characteristics of Diode

Diode always conducts in one direction.

Diodes always conduct current when “Forward Biased” ( Zero resistance) Diodes do not conduct when Reverse Biased (Infinite resistance)

• Semiconductors contain two types of mobile charge carriers, Holes and Electrons.
• The holes are positively charged while the electrons negatively charged.
• A semiconductor may be doped with donor impurities such as Antimony (N-type doping), so that it contains mobile charges which are primarily electrons.
• A semiconductor may be doped with acceptor impurities such as Boron (P-type doping), so that it contains mobile charges which are mainly holes.
• The junction region itself has no charge carriers and is known as the depletion region.
• The junction (depletion) region has a physical thickness that varies with the applied voltage.
• When a diode is Zero Biased no external energy source is applied and a natural Potential Barrier is developed across a depletion layer which is approximately 0.5 to 0.7v for silicon diodes and approximately 0.3 of a volt for germanium diodes.
• When a junction diode is Forward Biased the thickness of the depletion region reduces and the diode acts like a short circuit allowing full current to flow.

• When a junction diode is Reverse Biased the thickness of the depletion region increases and the diode acts like an open circuit blocking any current flow, (only a very small

Diode Parameters

Maximum Forward Current

The Maximum Forward Current ( IF(max) ) is as its name implies the maximum forward current allowed to flow through the device.

Peak Inverse Voltage

The Peak Inverse Voltage (PIV) or Maximum Reverse Voltage ( VR(max) ), is the maximum allowable Reverse operating voltage that can be applied across the diode without reverse breakdown and damage occurring to the device.

Total Power Dissipation

This rating is the maximum possible power dissipation of the diode when it is forward biased (conducting).

Zener Diodes

The Zener diode is made to operate under reverse bias once a sufficiently high voltage has been reached. The I-V curve of a Zener diode is shown in Figure 11.15.

Notice that under reverse bias and low voltage the current assumes a low negative

value, just as in a normal pn-junction diode. But when a sufficiently large reverse bias voltage is reached, the current increases at a very high rate.

Figure 11.15: A typical I-V curve for a Zener diode.

Figure 11.16: A Zener diode reference

circuit.

Avalanche breakdown:

Avalanche breakdown occurs when the applied voltage is so large that electrons that are pulled from their covalent bonds are accelerated to great velocities. These electrons collide with the silicon atoms and knock off more electrons. These electrons are then also accelerated and subsequently collide with other atoms. Each collision produces more electrons which leads to more collisions etc. The current in the semiconductor rapidly increases.

Zener breakdown:

1. heavily doped therefore have narrow depletion layer
2. strong electric field is developed across this narrow layer.
3. covalent bonds break due to very strong electric field so even a small amount of reverse voltage is capable of producing large number of current carriers.

Light Emitting Diodes

LEDs are a particular type of diode that convert electrical energy into light. The LED is a specialised form of PN junction that uses a compound junction. The semiconductor material used for the junction must be a compound semiconductor. The commonly used semiconductor materials including silicon and germanium are simple elements and junction made from these materials do not emit light. Instead compound semiconductors including gallium arsenide, gallium phosphide and indium phosphide are compound semiconductors and junctions made from these materials do emit light.

Light Emitting Diodes

Another important kind of diode is the light-emitting diode (LED). Whenever an electron makes a transition from the conduction band to the valence band (effectively recombining the electron and hole) there is a release of energy in the form of a photon (Figure 11.17). In some materials the energy levels are spaced so that the photon is in the visible part of the spectrum. In that case, the continuous flow of current through the LED results in a continuous stream of nearly monochromatic light.

Figure 11.17: Schematic of an LED. A photon is released as an electron falls from the conduction band to the valence band. The band gap may be large enough that the photon will be in the visible portion of the spectrum.

Photodiode

A photodiode is a semi-conductor device, with a p-n junction and an intrinsic layer between p and n layers. It produces photocurrent by generating electron-hole pairs, due to the absorption of light in the intrinsic or depletion region. The photocurrent thus generated is proportional to the absorbed light intensity.

Working Principle

When photons of energy greater than 1.1 eV hit the diode, electron-hole pairs are created. If the absorption occurs in the depletion region of the p-n junction, these hole pairs are swept from the junction – due to the built-in electric field of the depletion region. As a result, the holes move toward the anode and the electrons move toward the cathode, thereby producing photocurrent.

A photodiode is a p-n junction or pin semiconductor device that consumes light energy to generate electric current. It is also sometimes referred as photo-detector, photo-sensor, or light detector.

Photodiodes are specially designed to operate in reverse bias condition. Reverse bias means that the p-side of the photodiode is connected to the negative terminal of the battery and n-side is connected to the positive terminal of the battery.

When external light energy is supplied to the p-n junction photodiode, the valence electrons in the depletion region gains energy.

If the light energy applied to the photodiode is greater the band-gap of semiconductor material, the valence electrons gain enough energy and break bonding with the parent atom. The valence electron which breaks bonding with the parent atom will become free electron. Free electrons moves freely from one place to another place by carrying the electric current.

The different materials used to construct photodiodes are Silicon (Si), Germanium, (Ge), Gallium Phosphide (GaP), Indium Gallium Arsenide (InGaAs), Indium Arsenide Antimonide (InAsSb), Extended Range Indium Gallium Arsenide (InGaAs), Mercury Cadmium Telluride (MCT, HgCdTe).

Solar Cell

Solar cell is also known as large area photodiode because it converts solar energy or light energy into electric energy. However, solar cell works only at bright light.

Photovoltaic Cells

An exciting application closely related to the LED is the solar cell, also known as the photovoltaic cell. Simply put, a solar cell takes incoming light energy and turns it into electrical energy. A good way to think of the solar cell is to consider the LED in reverse (Figure 11.18). A pn-junction diode can absorb a photon of solar radiation

by having an electron make a transition from the valence band to the conduction band. In doing so, both a conducting electron and a hole have been created. If a circuit is connected to the pn junction, the holes and

electrons will move so as to create an electric current, with positive current flowing from the p side to the n side. Even though the efficiency of most solar cells is low, their widespread use could potentially generate significant amounts of electricity. Remember that the “solar constant” (the energy per unit area of solar radiation reaching the Earth) is over 1400 W/m², and more than half of this makes it through the atmosphere to the Earth’s surface. There has been tremendous progress in recent years toward making solar cells more efficient.

Figure 11.18: (a) Schematic of a photovoltaic cell. Note the similarity to Figure 11.17. (b) A schematic showing more of the working parts of a real photovoltaic cell. From H. M. Hubbard, Science 244, 297- 303 (21 April 1989).

Bipolar Junction Transistors

Bipolar transistors are so named because the controlled current must go through two types of semiconductor material: P and N. The current consists of both electron and hole flow, in different parts of the transistor.

Bipolar transistors consist of either a P-N-P or an N-P-N semiconductor

“sandwich” structure.

The three leads of a bipolar transistor are called the Emitter, Base, and Collector.

A bipolar junction transistor is a three terminal semiconductor device consisting of two p-n junctions which is able to amplify or magnify a signal. It is a current controlled device. The three terminals of the BJT are the base, the collector and the emitter. A signal of small amplitude if applied to the base is available in the amplified form at the collector of the transistor. This is the amplification provided by the BJT.

The current-controlled devices wherein a small change in the base current, IB results in a large variation in the collector current, IC.

Common Emitter Characteristics

Input characteristics IB (Base Current) is the input current, VBE (Base – Emitter Voltage) is the input voltage for CE (Common Emitter) mode. So, the input characteristics for CE mode will be the relation between IB and VBE with VCE as parameter.

Procedure:

Connect the circuit as shown in the circuit diagram. Keep output voltage VCE = 0V by varying VCC.

Varying VBB gradually, note down base current IB and base-emitter voltage VBE.

Step size is not fixed because of non linear curve. Initially vary VBB in steps of 0.1V. Once the current starts increasing vary VBB in steps of 1V up to 12V.

Repeat above procedure (step 3) for VCE = 5V.

Output Characteristics Output characteristics for CE mode is the curve or graph between collector current (IC) and collector – emitter voltage (VCE) when the base current IB is the parameter.

Procedure

Connect the circuit as shown in the circuit diagram. Keep base current IB = 20µA by varying VBB.

Varying VCC gradually in steps of 1V up to 12V and note down collector current IC and Collector-Emitter Voltage(VCE).

Repeat above procedure (step 3) for IB = 60µA, 0µA.

Common Emitter Configuration

Current Relations in CE Configuration

In the above circuit connections observed that the supply voltage VB applied to the base terminal throu the load RB. The collector termin connected to the voltage VCC throu the load RL. Here both the loads RB a RL can limit the current flow throu the corresponding terminals. Here t base terminal and collector termin always contain positive voltages wi respect to emitter terminal.

The equation for base current of a bipolar NPN transistor is given by,

I_B = (V_B-V_BE)/R_B

Where,

I_B = Base current

V_B = Base bias voltage

V_BE = Input Base-emitter voltage = 0.7V

R_B = Base resistance

The output collector current in common emitter NPN transistor can be calculated by applying Kirchhoff’s Voltage Law (KVL).

The equation for collector supply voltage is given as V_CC = I_CR_L + V_CE (1)

From the above equation the collector current for common

emitter NPN transistor is given as

I_C = (V_CC-V_CE)/R_L

In a common emitter NPN transistor the relation between collector current and emitter current is given as

I_C = β I_B

In active region the NPN transistor acts as a good amplifier. In common emitter NPN transistor total current flow through the transistor is defined as the ratio of collector current to the base current IC/IB. This ratio is also called as “DC current gain” and it doesn’t have any units. This ratio is generally represented with β and the maximum value of β is about 200.

The emitter current is the sum of small base current and large collector current.

I_E = I_C + I_B

The equation for the collector current is given by, I_C= (V_CC-V_CE)/ R_L

The straight line indicates the ‘Dynamic load line’ which is connecting the points A (where V_CE= 0) and B (where I_C = 0).

The common emitter configuration characteristics curves are used to calculate the collector current when the collector voltage and base current is given. The load line is used to determine the Q-point in the graph. The slope of the load line is equal to the reciprocal of the load resistance. i.e. -1/R_L.

Transistor Biasing

All types of transistor amplifiers operate using AC signal inputs which alternate between a positive value and a negative value so some way of “presetting” the amplifier circuit to operate between these two maximum or peak values is required. This is achieved using a process known as Biasing. Biasing is very important in amplifier design as it establishes the correct operating point of the transistor amplifier ready to receive signals, thereby reducing any distortion to the output signal.

Amplifier

Common Emitter Amplifier

The aim of any small signal amplifier is to amplify all of the input signal with the minimum amount of distortion possible to the output signal, in other words, the output signal must be an exact reproduction of the input signal but only bigger (amplified).

R1 and R2 are the biasing resistors. This network provides the transistor Q1’s base with the necessary bias voltage to drive it into the active region.

Capacitor Cin is the input DC decoupling capacitor which blocks any DC component if present in the input signal from reaching the Q1 base. If any external DC voltage reaches the base of Q1, it will alter the biasing conditions and affects the performance of the amplifier.

Cout is the output DC decoupling capacitor. It prevents any DC voltage from entering into the succeeding stage from the present stage.

Ce is the emitter by-pass capacitor. The job of Ce is to bypass this alternating component of the emitter current. If Ce is not there , the entire emitter current will flow through Re and that causes a large voltage drop across it.

Characteristics of Common Emitter Amplifier

The voltage gain of common emitter amplifier is medium The power gain is high in the common emitter amplifier

There is a phase relationship of 180 degrees in input and output

Transistor as a Switch

The areas of operation for a transistor switch are known as the Saturation Region and the Cut-off Region.

We can use the transistor as a switch by driving it back and forth between its “fully-OFF” (cut-off) and “fully-ON” (saturation) regions.

Cut-off Region

Here the operating conditions of the transistor are zero input base current ( IB ), zero output collector current ( IC ) and maximum collector voltage ( VCE ) which results in a large depletion layer and no current flowing through the device. Therefore the transistor is switched “Fully- OFF”.

The input and Base are grounded ( 0v )

• Base-Emitter voltage VBE < 0.7v
• Base-Emitter junction is reverse biased
• Base-Collector junction is reverse biased
• Transistor is “fully-OFF” ( Cut-off region )
• No Collector current flows ( IC = 0 )
• VOUT = VCE = VCC = ”1″
• Transistor operates as an “open switch”

Saturation Region

Here the transistor will be biased so that the maximum amount of base current is applied, resulting in maximum collector current resulting in the minimum collector emitter voltage drop which results in the depletion layer being as small as possible and maximum current flowing through the transistor. Therefore the transistor is switched “Fully-ON”.

The input and Base are connected to VCC

• Base-Emitter voltage VBE > 0.7v
• Base-Emitter junction is forward biased
• Base-Collector junction is forward biased
• Transistor is “fully-ON” ( saturation region )
• Max Collector current flows ( IC = Vcc/RL )
• VCE = 0 ( ideal saturation )
• VOUT = VCE = ”0″
• Transistor operates as a “closed switch”

Oscillators

Any circuit that generates an alternating voltage is called an oscillator. To generate ac voltage, the circuit is supplied energy from a dc source. An oscillator generates ac output signal without any ac input signal. A part of the output is fed back to the input; and this feedback signal is the only input to the internal amplifier.

Types of feedback

Feedback is the process of taking a part of output signal and feeding it back to input circuit

Negative feedback:

Here the feedback voltage is out of phase with the input voltage. The net input voltage becomes the difference between input voltage and feedback voltage. So the net input is reduces, which in turn reduces the gain of the circuit

Positive feedback:

The feed back voltage can be in the same phase as the external input voltage. In such case effective input is increased. Obviously the gain increases

Barkhausen criterion

Barkhausen criterion are a set of two mathematical conditions which a linear electronic circuit must follow to act as an electronic oscillator.

According to Barkhausen criterion for sustained oscillation:

1. The magnitude of the product of open loop gain of the amplifier and the magnitude of the feedback factor is unity, i.e., |βA|=1| where A is the gain of the amplifying element in the circuit and β is the transfer function of the feedback path.
2. The total phase shift around the loop is 0 or integral multiples of 2π.

RC phase-shift oscillators

RC phase-shift oscillators use resistor-capacitor (RC) network (Figure 1) to provide the phase- shift required by the feedback signal. They have excellent frequency stability and can yield a pure sine wave for a wide range of loads.

Here the collector resistor RC limits the collector current of the transistor, resistors R₁ and R (nearest to the transistor) form the voltage divider network while the emitter resistor RE improves the stability. Next, the capacitors CE and Co are the emitter by-pass capacitor and the output DC decoupling capacitor, respectively. Further, the circuit also

shows three RC networks employed in the feedback path. This arrangement causes the output waveform to shift by 180° during its course of travel from output terminal to the base of the transistor. Next, this signal will be shifted again by 180° by the transistor in the circuit due to the fact that the phase-difference between the input and the output will be 180° in the case of common emitter configuration. This makes the net phase- difference to be 360°, satisfying the phase-difference condition. One more way of satisfying the phase-difference condition is to use four RC networks, each offering a phase-shift of 45°. Hence it can be concluded that the RC phase-shift oscillators can be designed in many ways as the number of RC networks in them is not fixed. However it is to be noted that, although an increase in the number of stages increases the frequency stability of the circuit, it also adversely affects the output frequency of the oscillator due to the loading effect. The generalized expression for the frequency of oscillations produced by a RC phase-shift oscillator is given by

Where, N is the number of RC stages formed by the resistors R and the capacitors C.