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Module-2

Contents

BJT as amplifier. Small signal equivalent circuits -h parameter model.Transistor Biasing circuits.

Stability factors. Thermal runaway. Small signal analysis of CE Configuration using hybrid π

model. Small signal analysis of CB Configuration using hybrid π model. Small signal analysis of

CC Configuration using hybrid π model.

BJT as an Amplifier:

Signal amplification is the simplest and most important signal procesing task and is

essential because of weak signal available from the transducer.Such signals are too small for

reliable processing signal magnitude should be large enough for easy signal processing without

error.The process of increasing the the amplitude of an input signal is called amplification and

the device used for this process is called amplifier.The magority of amplifiers are primarly

intended to operate on very small input signal.The objective is to make the signal magnitude

larger and therefore are thought of as voltage amplifier.These amplifiers are known as Small

signal voltage amplifier.

One of the basic feature of the amplifier is that the output wave form must be identical

in nature to those in the input waveform.The amplifier preserves the details of the signal

waveform and any deviation of the output waveform from the shape of the input waveform is

considered as distortion.

The main utility of transisitor lies in its ability of amplifying weak signals.The weak signal

is applied at input terminals and the amplified output is obtained across the output terminals as

illstrated in figure.

A transistor alone cannot prform the functions of amplification and some passive components

such as resistors and capacitors are needed.Common emitter (CE) configuration,because of its

high current,voltage and power gains,is much suited for most of amplifier circuits.

Biasing of an Amplifier

The purpose of biasing of an amplifier is to set dc operating point,ie to fix base current

IB,collector current IC and collector emitter voltage VCE such that transistor operate in the linear

region of the output characteristics even after the superposition of ac signal voltage at the base

.The process which set up the initial d.c values of IB ,IC and VCE in a transistor is called Biasing.

Necessity of Biasing

For most of applications,transistors are required to operate as linear amplifiers (i.e, to

amplify the output voltage as a linear function of the input voltage).To achive this, it is

necessary to operate the transistor over region of its characteristic curves which are

linear,parallel and equi-spaced for equel increments of the parameter.Such an operation can be

ensured by proper selection of zero signal operating point and limiting the operation of the

transistor over the linear operation of the characteristics.For proper selection of zero signal

operating point,proper biasing i.e,the application of d.c voltages at emitter to base junction and

collector to base junction is required.If transistor is not biased properly,it would work

inefficiently and produce distortion in the output signal.

The figure shows the basic amplifier in CE configuration with an NPN transistor.When an

input signal is applied in the emitter-base junction,the signal is superimposed in the d.c voltage

(VBB) at emitter-base junction.Therefore during the positive half-cycle of the input signal the

forward bias across the junction increases,because it is already positive with respect to

ground.This increase in forward bias increases the base current IB.Due to increase in base

current the collector curent also increases.In CE configuration the corresponding increase in

collector current will be ’β’ times the increase in IB.This increase in collector current produces

more voltage drop across the output terminal.During negative half-cycle of the input signal the

forward bias across emitter junction will be decreased and decreases base curent.This decrease

in base current results in corresponding (β-times) decrease in collector current.Consequently

the drop across output terminals will be decreased.It is clear that the collector current varies

according to the input signal applied and variation is β-times to that of input current

variation.Due to this action an amplified form of input signal can be obtained at the output

terminal.

D.C Load Line

In the amplifier circuit shown in figure, no signal is applied to its input. At this state transistor is

said to be quiescent condition. At quiescent state VCC sends a collector current IC through the

transistor. This current is called zero signal collector current. The magnitude of this current

depends on the supply voltage VCC,the load resistance Rc and collector emitter voltage VBE of

the transistor.As Ic flows through Rc and the transistor,it will make voltage drops across Rc and

transistor.

Applying krichoffs voltage law (KVL) to the collector circuit,We get

푉 = 퐼 푅 + 푉

퐼 푅 = 푉 − 푉

퐼 =푉 − 푉

푅

퐼 =푉푅 −

푉푅

퐼 = −푉푅 +

푉푅

퐼 = (− 1 푅⁄ )푉 +푉푅

As 푅 and 푉 are constants, the above equation is of the form

푦 = 푚푥 + 푏

which is standard form of the equation of a straight line AB.

If this equation is plotted on the output characteristics of the transistor, we get a straight line as

shown in figure. The slope of the line is (-1/Rc).This straight line is called d.c load line.

Here the co-ordinates of A are obtained by writing Ic=0 in equation of Vcc.

푉 = 0 × 푅 + 푉

푉 = 푉

Thus the co-ordinates of end ‘A’ are

푉 = 푉

퐼 = 0

The co-ordinates of end ‘B’ are obtained by writing 푉 = 0 in the equation 푉 = 퐼 푅 + 푉

푉 = 퐼 푅 + 0

퐼 = 푉푅

Thus the co-ordinates of end ‘B’ are

푉 = 0

퐼 = 푉푅

Knowing the values of Rc and Vcc the end points of the load line can be located. By joining the end points by a straight line, we get d.c load line.

Selection of Operating Point

In order to get faithful amplification,the operating point must be well within the active region of the transistor.For useful operation,the collector current should not exceed the maximum current rating. Also the collector to emitter voltage VCE should not exceed the value prescribed by the manufacturer.

Stabilisation

The collector current in a transistor changes rapidly when

(i) the temperature changes,

(ii) the transistor is replaced by another of the same type. This is due to the inherent variations

of transistor parameters.

When the temperature changes or the transistor is replaced, the operating point (i.e. zero

signal IC and VCE) also changes. However, for faithful amplification, it is essential that operating

point remains fixed. This necessitates to make the operating point independent of these

variations. This is known as stabilisation. The process of making operating point independent of

temperature changes or variations in transistor parameters is known as stabilisation.

Once stabilisation is done, the zero signal IC and VCE become independent of temperature

variations or replacement of transistor i.e. the operating point is fixed. A good biasing circuit

always ensures the stabilisation of operating point.

Need for stabilisation. Stabilisation of the operating point is necessary due to the following

reasons :

(i) Temperature dependence of IC

(ii) Individual variations

(iii) Thermal runaway

(i) Temperature dependence of IC. The collector current IC for CE circuit is given by:

IC = β IB + ICEO = β IB + (β + 1) ICBO

The collector leakage current ICBO is greatly influenced (especially in germanium transistor) by

temperature changes. A rise of 10°C doubles the collector leakage current which may be as high

as 0.2 mA for low powered germanium transistors. As biasing conditions in such transistors are

generally so set that zero signal IC = 1mA, therefore, the change in IC due to temperature

variations cannot be tolerated. This necessitates to stabilise the operating point i.e. to hold IC

constant in spite of temperature variations.

(ii) Individual variations. The value of β and VBE are not exactly the same for any two transistors

even of the same type. Further, VBE itself decreases when temperature increases. When a

transistor is replaced by another of the same type, these variations change the operating point.

This necessitates to stabilise the operating point i.e. to hold IC constant irrespective of

individual variations in transistor parameters.

(iii) Thermal runaway. The collector current for a CE configuration is given by : IC = β IB + (β + 1)

ICBO. The collector leakage current ICBO is strongly dependent on temperature. The flow of

collector current produces heat within the transistor. This raises the transistor temperature and

if no stabilisation is done, the collector leakage current ICBO also increases. It is clear from the

above equation that if ICBO increases, the collector current IC increases by (β + 1) ICBO. The

increased IC will raise the temperature of the transistor, which in turn will cause ICBO to increase.

This effect is cumulative and in a matter of seconds, the collector current may become very

large, causing the transistor to burn out. The self-destruction of an unstabilised transistor is

known as thermal runaway.

In order to avoid thermal runaway and consequent destruction of transistor, it is very

essential that operating point is stabilised i.e. IC is kept constant. In practice, this is done by

causing IB to decrease automatically with temperature increase by circuit modification. Then

decrease in β IB will compensate for the increase in (β + 1) ICBO, keeping IC nearly constant. In

fact, this is what is always aimed at while building and designing a biasing circuit.

The biasing network associated with the transistor should meet the following requirements:

(a) It should ensure proper zero signal collector current.

(b) It should ensure that VCE does not fall below 0.5 V for Ge transistors and 1 V for silicon

transistors at any instant.

(c) It should ensure the stabilisation of operating point.

Stability Factor (S)

To what extend the collector current has been stabilized in respect of variation in temperature

is expressed in terms of stability factor ‘S’ .The stability factor is defined as the rate of change of

the collector current with respect to the leakage current when other parameters (IB,β )are held

constant.

i.e ,푆 = ; I ,βconstant

Stability factor shows the change in collector current Ic because of change in collector leakage

current Ico. Here the smaller value of ‘S’ better will be the stability. Therefore stability factor

should be kept small as possible. The ideal value of stability factor is one, but it is not

practicable.

Expression for the Stability Factor

We Know

퐼 = 훽퐼 + (훽 + 퐼 )

Differentiating above expression w.r.t. IC, we get,

1 = 훽 + (훽 + 1))

1 = 훽 + ( ) 푆푖푛푐푒, 푆 =

푆 = ( )

( )

Stability Factor (S’)

The stability factor(S’) is defined as the rate of change of the collector current with respect to the base-emitter voltage VBE ,assuming Ico and are held constant.

i.e ,S’ = dIc/dVBE ; Ico, constant

Stability Factor (S’’)

The stability factor(S’’) is defined as the rate of change of the collector current with respect to the ,assuming Ico and VBE are held constant.

i.e ,S’’ = dIc/d ; Ico,VBE constant

Transistor Biasing Circuits:

The BJT can be biased to fix the quiescent operating point by different methods.

The following are the commonly used methods for transistor biasing;

Fixed bias (base resistor bias)

Collector to base bias (Feed back resistor bias)

Emitter resistor bias

Voltage divider bias

Fixed Bias (base resistor bias)

This is the simplest type of transistor biasing. In this method a single resistor RB is

connected between the base and positive terminal of Vcc as shown in figure. Here a single

battery Vcc is used to provide both Vcc and VBB. The value of RB has to be chosen suitably so as to

get the desired operating point.

Analysis of input section

Applying KVL to the input section,we get

푉 = 퐼 푅 + 푉

퐼 푅 = 푉 − 푉

퐼 = (푉 − 푉 )푅

퐼 = 푉푅 (푠푖푛푐푒푉 ≪푉 )

Then the value of the base resistor can be determined by

푅 = 푉퐼

Since the supply voltage Vcc is a fixed known value, once RB is chosen the base current IB will be fixed in value. For this reason this biasing circuit is called Fixed bias circuit.

Analysis of output section

Considering the output section, the collector current is given by,

퐼 = 훽퐼 + 퐼

The leakage current ICEO is small compared to the total collector current, there fore

퐼 = 훽퐼

Applying Kirchhoff’s voltage law to the output circuit, We get

푉 = 퐼 푅 + 푉

푉 = 푉 − 퐼 푅

This value of VCE and calculated value of Ic provide the co-ordinates of the operating point.

Advantages

Very simple in construction.

Operating point can be fixed at any desired part of the active region just by suitably

changing the value of RB.

Disadvantages

Temperature stabilization of the operating point is very poor; because rise in

temperature increases the collector current and thus the operating point is shifted.

High stability factor.

Collector to base bias (Feed back resistor bias)

Here the base resistor RB is connected to the collector instead of connecting to the Vcc as shown in figure.

Analysis of input section Applying KVL to the input section,we get

푉 = (퐼 + 퐼 )푅 + 퐼 푅 + 푉

푉 = 퐼 푅 + 퐼 (푅 + 푅 ) + 푉 푒푞(1)

퐼 =(푉 − 퐼 푅 ) − 푉

푅 + 푅 푒푞(2)

Then

푅 =(푉 − 퐼 푅 ) − 푉

퐼 − 푅 푒푞(3)

Analysis of output section Applying KVL to the output section,we get

푉 = 퐼 푅 + 푉

푉 = 푉 − 퐼 푅 푒푞(4)

Substituting this value of VCE in eqn(2) we get,

퐼 =푉 − 푉푅 + 푅

The collector current 퐼 is given by 퐼 = 훽퐼

퐼 = 훽푉 − 푉푅 + 푅

The co-ordinates of the operating point VCE and Ic are now obtained.

Emitter resistor bias Here the resistor RE is connected in between the emitter and ground as shown in figure.

Analysis of input section

Applying KVL to the input section,we get

푉 = 퐼 푅 + 푉 + 퐼 푅 푒푞(1)

퐼 = (푉 − 푉 − 퐼 푅 )푅

퐼 = (푉 − 퐼 푅 )푅 eq(2)

(푆푖푛푐푒푉 ≪ 푉 )

We know

퐼 = (훽 + 1)퐼

Substituting this in eq (1), we get

푉 = 퐼 푅 + 푉 + (훽 + 1)퐼 푅

퐼 = (푉 − 푉 )(푅 + (훽 + 1)푅 )

퐼 = 푉(푅 + 훽푅 )

(푆푖푛푐푒푉 ≪ 푉 푎푛푑1 ≪ 훽)

We know

퐼 = 훽퐼

퐼 = 훽푉(푅 + 훽푅 )

퐼 =푉

푅 + (푅 훽⁄ )

This gives the one co-ordinate 퐼 of operating point

Analysis of output section

Applying KVL to the output section, we get

푉 = 퐼 푅 + 푉 + 퐼 푅

푉 = 푉 − (푅 + 푅 )퐼

(푆푖푛푐푒퐼 ≅ 퐼 ) This gives the second co-ordinate (VCE) of the operating point.

Voltage Divider Bias

This is the most widely used method for providing biasing and stabilization for

transistor. The figure shows the voltage divider biasing arrangement for an NPN transistor. Here

two resistors R1 and R2 in series are connected across the supply voltage Vcc. A resistor RE is also

connected in the emitter circuit.

Analysis of input section

Here current through R1 is will almost completely flow through R2.ie, IB is very small.

퐼 = 퐼

Applying KVL to the input section, we get

푉 = 퐼 푅 + 퐼 푅

퐼 = 푉(푅 + 푅 )

Then the voltage at the base relative to the ground is given by

푉 = 퐼 푅

푉 =푉

(푅 + 푅 )푅

The drop across RE is the difference between VB and VBE.

퐼 푅 = 푉 − 푉

퐼 = (푉 − 푉 ) 푅⁄

퐼 = (푉 − 푉 ) 푅⁄ (푆푖푛푐푒퐼 = 퐼 )

퐼 = 푉 푅⁄ (푆푖푛푐푒퐵 ≪ 푉 )

Now one co-ordinate (Ic) of the operating point is obtained.

Analysis of output section

Applying KVL to the output section, we get

푉 = 퐼 푅 + 푉 + 퐼 푅

푉 = 푉 − (푅 + 푅 )퐼

(푆푖푛푐푒퐼 ≅ 퐼 ) Now the second co-ordinate of the operating point (VCE) is also obtained. Here the value of β

has never entered in to the calculation. Hence, change in β value has no effect on the operating

point.

Advantage

Super in stabilization hence it is most popular.

Design of Transistor Biasing Circuits

In practice, the following steps are taken to design transistor biasing and stabilisation circuits :

Step 1. It is a common practice to take RE = 500 − 1000Ω. Greater the value of RE, better is the

stabilisation. However, if RE is very large, higher voltage drop across it leaves reduced voltage

drop across the collector load. Consequently, the output is decreased. Therefore, a compromise

has to be made in the selection of the value of RE.

Step 2. The zero signal current IC is chosen according to the signal swing. However, in the initial

stages of most transistor amplifiers, zero signal IC = 1mA is sufficient. The major advantages of

selecting this value are :

(i) The output impedance of a transistor is very high at 1mA. This increases the voltage gain.

(ii) There is little danger of overheating as 1mA is quite a small collector current.

It may be noted here that working the transistor below zero signal IC = 1mA is not advisable

because of strongly non-linear transistor characteristics.

Step 3. The values of resistances R1 and R2 are so selected that current I1 flowing through R1 and

R2 is atleast 10 times IB i.e. I1 ≥ 10 IB. When this condition is satisfied, good stabilisation is

achieved.

Step 4. The zero signal IC should be a little more (say 20%) than the maximum collector current

swing due to signal. For example, if collector current change is expected to be 3mA due to

signal, then select zero signal IC ≅ 3.5 mA. It is important to note this point. Selecting zero signal

IC below this value may cut off a part of negative half-cycle of a signal. On the other hand,

selecting a value much above this value (say 15mA) may unnecessarily overheat the transistor,

resulting in wastage of battery power. Moreover, a higher zero signal IC will reduce the value of

RC (for same VCC), resulting in reduced voltage gain.

Two port devices & Network Parameters:-

A transistor can be treated as a two port network. The terminal behaviour of any two port network can be specified by the terminal voltages V1 & V2 at ports 1 & 2 respectively and current i1 and i2, entering parts 1 & 2, respectively, as shown in figure.

Two port network

Of these four variables V1, V2, i1 and i2, two can be selected as independent variables and the remaining two can be expressed in terms of these independent variables. This leads to various two port parameters out of which the following three are more important.

1. Z – Parameters (or) Impedance parameters 2. Y – Parameters (or) Admittance parameters 3. H – Parameters (or) Hybrid parameters.

Hybrid parameters (or) h – parameters:- If the input current i1 and output Voltage V2 are takes as independent variables, the input voltage V1 and output current i2 can be written as

V1 = h11 i1 + h12 V2

i2 = h21 i1 + h22 V2

The four hybrid parameters h11, h12, h21 and h22 are defined as follows.

h11 = [V1 / i1] with V2 = 0

>Input Impedance with output port short circuited.

h22 = [i2 / V2] with i1 = 0

>Output admittance with input port open circuited.

h12 = [V1 / V2] with i1 = 0

>reverse voltage transfer ratio with input port open circuited.

h21 = [i2 / i1] with V2 = 0

>Forward current gain with output port short circuited.

The dimensions of h – parameters are as follows:

h11 - Ω

h22 – mhos

h12, h21 – dimension less.

As the dimensions are not alike, (ie) they are hybrid in nature, and these parameters are called as hybrid parameters.

I = 11 = input ; 0 = 22 = output ;

F = 21 = forward transfer ; r = 12 = Reverse transfer.

Notations used in transistor circuits:-

h ie = h11e = Short circuit input impedance

h 0e = h22e = Open circuit output admittance

h re = h12e = Open circuit reverse voltage transfer ratio

h fe = h21e = Short circuit forward current Gain.

The Hybrid Model for Two-port Network:-

V1 = h11 i1 + h12 V2

I2 = h21 i1 + h22 V2

↓

V1 = h1 i1 + hr V2

I2 = hf i1 + h0 V2

Transistor Hybrid model:-

Use of h – parameters to describe a transistor have the following advantages.

1. h – parameters are real numbers up to radio frequencies . 2. They are easy to measure 3. They can be determined from the transistor static characteristics curves. 4. They are convenient to use in circuit analysis and design. 5. Easily convert able from one configuration to other. 6. Readily supplied by manufactories.

CE Transistor Circuit

To Derive the Hybrid model for transistor consider the CE circuit shown in figure.The variables are iB, ic, vB(=vBE) and vc(=vCE). iB and vc are considered as independent variables.

Then , vB= f1(iB, vc ) ----------------------(1)

iC= f2(iB, vc ) ----------------------(2)

Making a Taylor’s series expansion around the quiescent point IB, VC and neglecting higher order terms, the following two equations are obtained.

ΔvB = (∂f1/∂iB) Vc . Δ iB + (∂f1/∂vc) IB . ΔvC ---------------(3)

Δ iC = (∂f2/∂iB)Vc . Δ iB + (∂f2/∂vc)IB . ΔvC ----------------(4)

The partial derivatives are taken keeping the collector voltage or base current constant as indicated by the subscript attached to the derivative.

ΔvB , ΔvC , Δ iC , Δ iB represent the small signal(increment) base and collector voltages and currents,they are represented by symbols vb , vc , ib and ic respectively.

Eqs (3) and (4) may be written as

Vb = hie ib + hre Vc

ic = hfe ib + hoe Vc

Where hie =(∂f1/∂iB)Vc = (∂vB/∂iB)Vc = (ΔvB /ΔiB)Vc = (vb / ib)Vc

hre =(∂f1/∂vc)IB = (∂vB/∂vc) IB = (ΔvB /Δvc) IB = (vb /vc) IB

hfe =(∂f2/∂iB)Vc = (∂ic /∂iB)Vc = (Δ ic /ΔiB)Vc = (ic / ib)Vc

hoe= (∂f2/∂vc)IB = (∂ic /∂vc) IB = (Δ ic /Δvc) IB = (ic /vc) IB

The above equations define the h-parameters of the transistor in CE configuration.The same theory can be extended to transistors in other configurations.

The Hybrid- Model:

The hybrid model of a BJT is a small signal model, named after the “”-like equivalent circuit for a bipolar junction transistor. The Hybrid- Model is shown in Figure. It consists of an input impedance, r, an output impedance r0, and a voltage controlled current source described by the transconductance, gm. The hybrid- model is a linearized two-port network approximation to the BJT using the small-signal base-emitter voltage vbe and collector-emitter voltage vce as independent variables, and the small-signal base current ib and collector current ic as dependent variables.

Fig:Simplified, low-frequency hybrid- BJT model.

The various parameters are as follows.

is the transconductance in siemens.

where:

is the quiescent collector current (also called the collector bias or DC collector current)

is the thermal voltage, calculated from Boltzmann's constant k, the charge of an electron q, and the transistor temperature in kelvins, T.

in ohms

ris the small signal input resistance of bipolar transistor depends on current gain.

where:

is the current gain at low frequencies (commonly called hFE). Here IB is the Q-point base current. This is a parameter specific to each transistor, and can be found on a datasheet; β is a function of the choice of collector current.

is the output resistance due to the Early effect (VA is the Early voltage).

Related terms

The reciprocal of the output resistance is named the output conductance

.

The reciprocal of gm is called the intrinsic resistance

.

Hybrid- Model of CE Configuration

The small signal equivalent circuit of common emitter configuration is shown below.

The input resistance is the Thevenin equivalent resistance seen looking into the input.

The output resistance is the Thevenin-equivalent resistance seen looking in to the output with the input shorted, or

The open-circuit, or unloaded, voltage gain is

If the collector load resistor Rc is made very large then av become,

Where Ic is the collector current at the operating point,VT is the thermal voltage,VA is the Early voltage and is given by

The short circuit current gain is the ratio of output current to input current when the output is short circuited.For CE configuration,

Hybrid- Model of CB Configuration

Short circiut transconductance is

The input resistance is the resistance re:

The output resistance is given by

The open-circuit voltage gain is

The short-circuit current gain is

Hybrid- Model of CC Configuration

In common collector configuration,the signal is applied to the base and output is taken from the emitter.When the input voltage increases Vs increases ,the base emitter voltage of the transistor increases,which increases the output current io.How ever increasing io increases the output voltage Vo,which decreases the base emitter voltage by a negative feedback.Apply KCL at output node,we get

When rRs, >>1 and ro >> RL then the above equation can be approximated as

Calculation of input resistance We calculate the input resistance Ri by removing the input source, driving the input with a test current source it and calculating the resulting voltage vt across the input terminals. The circuit used to do this calculation is shown in the figure.

From KCL at the output node,

Then the voltage vt is

And thus

Calculation of output resistance

The output resistance Ro is calculated by removing the load resistance RL and finding the Thevenin equivalent resistance looking in to the output terminals. By doing this we insert a test current and calculating the resulting voltage or applying a test voltage and calculating the current.In this case calculation is simpler if a test voltage vt is applied as shown in figure.

The voltage V1 is given by

The total output current it is thus

Therefore,

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