am generation

AM GENERATION

1

ANALOG COMMUNICATION LABREPORT - SESSION 1

Submitted by:Anil Vishnu G K

Allen JobAsish Oommen Abraham

Ajmal V K

19th August

Contents

Introduction 2

Principles 3

Design 8

Experimentation 13

Observations and Calculations 16

Inference 18

Conclusion 19

Bibliography 20

1

Introduction

The evolution and development of human beings as a race can be quitebeautifully understood by the different modes of communication that we haveemployed over the centuries. From the invention of the telegraph machine inthe mid 1800s to the modern day multi-utility mobile handsets, communi-cation systems have come a long way. All communication systems basicallydeal with sending a given data from one point to another. For long distancecommunication (say across continents) this would essentially have to be donewirelessly, that is, the data would have to be transmitted. For the transmit-ting devices to be of adequate dimensions the signals would have to be of aminimum high frequency. Here comes the need for modulation, wherein thedata or message signals have to be carried across the transmitting mediumusing some high frequency signal called the carrier, which will carry the re-quired information through proportional changes in its parameters. Thereare three types of modulation schemes, namely amplitude, frequency andphase. In amplitude modulation the amplitude of the carrier signal is variedin proportion to the message signal. It is the scheme used in our normalradio systems. Such an operation would imperatively introduce extra fre-quencies other than those in the carrier and message signals. So to map fromthe message signal to the amplitude modulated wave we would have to usesome non-linear devices to introduce the extra frequencies. This is the ba-sic idea with which we should go about designing an amplitude modulationgeneration circuit on which this report is based.

2

Principles

The transfer of an analog waveform containing information from onepoint to another, or between users, is the basic mechanism involved in Ana-log Communication. This analog waveform that carries the information isusually called the message signal.The first step in understanding analog com-munication is to charecterise the message signal which we can denote as m(t).m(t) is a real valued energy signal with a fourier transform of M(f), say.The bandwidth of this message signal may be baseband or bandpass. So fortransmitting such a signal the equipments need not be practical as we knowthat the dimensions of the transmitting antenna depends inversely on thefrequency of the signal that is to be transmitted and it has to be changed fordifferent signals. If we could embed the information of the message signal ina particular high frequency signal it would solve the issue of transmission.Moreover we can use the same high frequency signal for sending a variety ofmessages. This is the basic idea of modulation.

The high frequency signal referred to above is called the carrier signal.Modulation basically involves the systematic alteration of some of the charecter-istics of this carrier signal according to the changes in the message signal. Incontinuous modulation schemes the carrier is usually a sinusoid. The threeparameters of a sinusoid that can be varied are its Amplitude, Frequency, and Phase. Any modulation scheme involves alteration of one or moreof these signal parameters in accordance with the message. In AmplitudeModulation the amplitude of the carrier signal is varied according to themessage signal, the phase and frequency remaining constant. Our purpose isto finally design a circuit that give an amplitude modulated signal as the out-put. The first step is to evolve a mathematical model of the overall processof modulation.

3

Let us denote the carrier frequency (a sinusoid) as ωc and the carrier am-plitude as Ac. Then the carrier signal c(t) is given by

c(t) = Accos(ωct) (1)

now we are modulating the amplitude of the carrier according to themessage. Say the amplitude of the modulated signal is A(t). Then

A(t) = Ac +m(t) (2)

then the modulated wave is given by

x(t) = A(t).cos(ωct) (3)

orx(t) = (Ac +m(t)).cos(ωct) = Accos(ωct) +m(t).cos(ωct) (4)

Where k is called the modulation index which is a measure of the extentof modulation. It is a quantity which gives a measure of how much themodulated parameter (the amplitude in this case) of the carrier signal variesaround its unmodulated level. Now the above equation can also be writtenin the foolowing form

x(t) = Ac.(1 + k.m(t))cos(ωct) (5)

In this case m(t) represents a normalised message signal of unit amplitude.Wemay denote it as mn(t)The figure given below gives a graphic description of a message signal andthe corresponding amplitude modulated wave. Notice the way in which themessage signal appears in the modulated wave and how it varies the ampli-tude of the carrier signal.From the equation above it is evident that the modulation index k is given

k =max|m(t)|

Ac

(6)

where Am is the maximum amplitude of the message signal.In case of a single tone message signal the above equations can be simplifiedfurther. The modulation index for example will then be given by

k =Am

Ac

(7)

4

now as we have seen earlier A(t) = Ac(1 + k.m(t)) , therefore

A(t)max = 1 + k (8)

A(t)min = 1− k (9)

As the minimum and maximum values of the normalised message signalm(t) or mn(t) to be precise is −1 and +1 respectively.Applying componendo and dividendo we get

k =A(t)max − A(t)min

A(t)max + A(t)min

(10)

If the single frequency of the message signal is given by ωm then themodulated signal can be further simplified and written as

x(t) = Ac.cos(ωct) +Ac.k

2cos((ωc + ωm)t) +

Ac.k

2cos((ωc − ωm)t) (11)

From equation5 we can see that when the value of k is greater than 1 thetotal amplitude function will undergo a phase reversal as A(t) becomes neg-ative. This is called overmodulation.And this leads to the loss of informationcontained in m(t)

Once we have the time domain representation of the system the nextlogical step is to determine the frequency components present in the same.

5

For this we have the fourier transform tool. On taking the fourier transformof the time domain expression we get X(f) as

X(f) =Ac

2.(δ(f − fc) + δ(f + fc)) +

k.Ac

2.(M(f − fc) +M(f + fc)) (12)

where M(f) is the fourier transform of the message signal.In the case of a single tone modulation the frequency spectrum essentiallyconsists of impulses at three distinct frequencies, namely ωc, ωc +ωm (calledthe upper sideband as it is of higher frequency), ωc − ωm (called the lowersideband as it is below the carrier frequency), each being weighted by theabovementioned amounts.Now for the single tone modulation case we now know the time domain andfrequency domain representations. The next information that we need is theamount of energy contained in each frequencies. From the frequency domainexpression we see that useful information .i.e. information of the messagesignal is contained only in the two side frequencies and not in the centralcarrier frequency.So the power used in transmitting this carrier frequency isactually a wasteful energy.So as such general AM does not make an efficientuse of the transmitted power.So,The efficiency η of the system is given by

η =total sideband power

total power(13)

The carrier frequency term is weighted by the amplitude of the carrier, Ac.Hence the carrier power is given by Ac

2

2.

Each side frequency is weighted by the amount Ac.k2

. Hence each sidebandpower is given by root mean square of that value. thus

Power in USB/LSB =(Ac.k

2)2

2=

(Ac)2.k2

8(14)

The total sideband power = Power in LSB + Power in USB = (Ac)2.k2

4

Total power is sideband power + carrier power

Total power =Ac

2

2+

(Ac)2.k2

4=

(Ac)2.(2 + k2)

4(15)

6

Hence, efficiency η is

η =(Ac)2.k2

4(Ac)2.(2+k2)

4

=k2

(2 + k2)(16)

Now we have a sound mathematical model of the amplitude modulationsystem. The next step is to map from the mathematics to the actual physicalrealisation of a circuit that generates the same.

7

Design

From the mathematical model of the system that we have developed ,wefind that the resultant signal consists of three distinct frequencies (assumingthat both carrier and message are pure sinusoids ).Hence we have a sys-tem that takes in two frequencies and gives out three frequencies.No systemthat behaves linearly can produce new frequency components from existingones.So the physical implementation of amplitude modulation has to be doneusing nonlinear systems.A square law device is a nonlinear device followingthe input-output relation output = K.(input)2 for some constant K.If theinput is the sum of two signals with distinct frequencies , the square lawdevice produces five frequencies at the output, which will contain the fre-quencies that correspond to an amplitude modulated wave.So square lawdevices are possible candidates for amplitude modulation systems. A JFETis a device whose output current (drain current) depends on the square of theinput voltage (gate to source voltage). Amplitude modulation can also beachieved by using devices which have nonlinear input-output characteristics,like exponential relationships.A diode is another nonlinear device, having anexponential relation between input and output. Apart from all of these, thenonlinear behaviour can be achieved in a more controlled manner by varyingthe biasing of a BJT, preferably in an amplifier circuit.

When taking BJT as a possible solution to the design problem any classof amplifiers can be used. But there comes the issue of power consumptionby the amplifier circuit.Taking this aspect into account a class C operationwould give the necessary non-linearity while at the same time significantlyreducing the power consumed. A class C amplifier is one whose output con-ducts load current during less than one half cycle of an input sine wave. Thetotal angle during which the current flows is less than 180 degrees and isknown as the conduction angle, θc. Usually the output of a class C amplifier

8

is a highly distorted version of the input.It is possible to make the outputcurrent of a Class C amplifier proportional to the modulating voltage byapplying this voltage in series with any of the dc supply voltages for thisamplifier.This property can be utilized to produce an amplitude modulatedsignal.To obtain the required amplitude modulation, the Class C BJT am-plifier is modified.The carrier is given as the input signal by applying to thebase of the BJT using a capacitive coupling.Since the frequency of the inputis high, a BJT capable of high frequency operation is to be used. A resis-tance of high value is connected across the base-emitter junction so as tokeep the time period of the capacitor discharging very high.As a result, thecapacitor will charge quickly,via the capacitor-base-emitter loop (which willessentially be a very low resistance path),but discharges very slowly via thecapacitor-resistor loop as its time constant is designed to be very high.Thecapacitor now acts almost like a constant voltage source,shifting the basevoltage level so as to drive the transistor to Class C operation. The BJTis initially biased in cutoff and reaches active region only when input signalreaches appropriate levels owing to the charged capacitor connected in seriesto it. Under Class C operation, the BJT will be in the active region onlyfor a time period less than a half cycle for one full cycle of the input signal.The conduction angle is given by θc =2arccos(V c/V p) where θc- Conductionangle; Vc - Conducting voltage ; Vp - Peak Voltage. From this,we see thatthe conduction angle of the amplifier is very less and is related to the inputat the base as well as the bias applied.

Now consider the amplitude of the output signal.The amplitude of theoutput signal must vary with the message signal.The output of the ampli-fier depends upon the Vcc, the bias voltage, as well as the input given.Sothe modulating signal is applied at the collector via transformer couplingi.e.,using an Audio Frequency Transformer (since the message signal has afrequency which is in the AF range).A tank circuit is also connected in seriesbetween the collector terminal and the message signal. It can be regardedas a highly selective i.e. high quality filter that suppresses the harmonics inclass C waveform and passes its fundamental frequency. The tank circuit isincorporated in the circuit by using an Intermediate Frequency Transformer(IFT). The tank circuit together with the amplifier constitute a tuned ampli-fier.The resonant circuit at the collector responds to an impulse by ’ringing’

9

at its resonant frequency. The impedance,and hence the gain of the tank ishigh for the resonant frequency and is very low for the harmonics, desensitis-ing the output from the harmonic distortion accompanying class C operation.The resonant frequency of the tank circuit is set to be the frequency of thecarrier signal. The output at the collector terminal of the BJT is a series ofcurrent pulses with their amplitudes proportional to the modulating signal.The current pulses initiates damped oscillations in the tuned circuit. Eachoscillation produced in this manner would have an initial amplitude propor-tional to the size of the current pulse and a decay rate dependent on thetime constant of the circuit. The train of pulses fed to the tank circuit wouldgenerate a series of complete sine waves proportional in amplitude to the sizeof the pulses. Thus the output from the tank would be an amplitude mod-ulated signal. Also, high power is required for amplitude modulated signalgeneration. The class C amplifier has very high efficiency and is used in highfrequency, high power applications. Since the time of operation is very brief,the damage to device due to joule loss is negligible. As discussed above, werequire the following components:

1. A high frequency BJT : BF195

2. A high value resistance: 1 MegΩ

3. A capacitor : 0.1µF

4. An audio frequency transformer (AFT)

5. An intermediate frequency transformer (IFT, preferably with tank res-onant frequency = IF 455kHz)

Finally, the circuit to be assembled would be:

10

To obtain a higher modulation depth, one way would be to decrease thetime of damping of the oscillations in the tank circuit. To do this we couldtrigger the tank more frequently. This can be done by increasing the fre-quency of the current pulses at the collector. This increases the numberof consequent damped oscillations in the tuned circuit. Also this will in-crease the accuracy of the modulated output. This enhanced rate of currentpulse generation can be achieved by connecting two identical collector mod-ulated circuits back to back-each for one half cycle. In this case we use anintermediate frequency transformer with a center tap to input the carriersignal. As designed earlier, the message signal is to be input using an au-dio frequency transformer and the output is taken out using an intermediatefrequency transformer again, with all other component values remaining thesame. The circuit would essentially look like the one below:

11

Experimentation

As discussed in the previous section, the design of the circuit for generatingan amplitude modulated wave was based on class C operation of a transistor.The circuit as provided was finally assembled and AM waves were observed.But many troubleshooting steps had to be taken to get the final workingcircuit. A negative biased (Vbb) RF Choke was planned to be kept at thebase terminal of BJT so that the base emitter region of the amplifier doesnot conduct until the input voltage is greater than 0.7+Vbb. Thus a nonlinearoperation could be obtained with an additional Vbb voltage. But then ratherthan using RF Choke the use of an RC circuit was suggested.

On using the RC Circuit the base terminal of the transistor effectivelybecame a parallel combination of a diode and resistor, the combination beingin series with the capacitor.The base emitter junction starts conducting afterthe input voltage crosses VBEon, which being generally 0.7 V.Then the diodestarts conducting and the capacitor starts charging through a low resistancepath in the base- emitter loop. Now during the negative half of the cyclethe capacitor discharges through the resistor. But since the resistor,R iskept very high (in the order of M) the time constant,RC is very high andso the capacitor discharges very slowly. Hence the capacitor retains most ofits charge. In the next positive half cycle, since the capacitor has retainedmost of its charge, it will act like a voltage source and keep the base emitterjunction reverse biased for an additional Vc (voltage across the capacitor inthe steady state) voltage.So the base-emitter junction will not conduct untilthe input voltage is greater than a particular voltage, the point of conductionbeing determined by the voltage across the capacitor.

In amplitude modulation,the output is tuned to the carrier frequency. Soan LC tank circuit which could selectively bring out the voltage at that fre-

13

quency was needed.An Intermediate frequency transformer (IFT) which hasan LC tank circuit in its secondary was used for this purpose. Now in orderto tune the circuit a carrier was applied at the input terminal and the outputwas taken across the LC tank circuit. The frequency of carrier was changedand the output observed. At a particular frequency the amplitude of the out-put was maximum. This frequency was maintained as the carrier frequency.This tuning was done before applying the message signal. The Vcc was variedslightly and it was observed that the amplitude of the output varied quitesatisfactorily with Vcc. While applying the message signal the voltage acrossthe tank circuit varies so by changing the Vcc it was just as if a slowly vary-ing message signal was applied. So the change in output voltage amplitudeshowed that the circuit was functioning properly upto that point.The mes-sage signal was fed to the collector through an Audio Frequency Transformer(AFT).On applying the message signal the desired output was not observed.While debugging the circuit we found that primary and secondary of the Au-dio Frequency Transformer (AFT) had been interchanged. Having noticed it,proper checking of the connections was done, and an amplitude modulatedsignal was observed. In order to make sure that the modulation was properover a range of message amplitude values,the amplitude of the message wasvaried. The modulation depth was observed to be very small, so the functiongenerator was replaced and then the amplitude of the output was observed tochange in accordance with the message signal over a wide range of messageamplitudes and frequencies. Once such a proper output was observed theamplitudes were varied to understand and observe normal modulation andovermodulation.

After obtaining the output from a single Class C amplifier, another designwas made in which two identical Class C amplifier circuits (each similar tothe ones used above) were connected back to back through a center tappedIFT.In this circuit,the input(carrier signal) was coupled to these amplifiersin such a way that a transistor was on in both half cycles.

After carefully assembling the circuit the carrier was applied and the out-put was observed to change with changes in Vcc. Having seen the change, themessage signal was applied. But the output did not show any change. So themessage signal was taken and connected directly to the CRO .A distortedmessage signal was observed and hence the signal generator was changed.

14

Again the message signal was directly observed on the CRO,which showed aproper sine wave.Now the message was given to the assembled circuit. Butstill the output only showed the carrier wave as such,and no influence ofthe message signal. The circuit design was checked and compared with theassembled circuit.Since this did not give any insight into the problem thecircuit was debugged segment to segment and the voltages at each part ofthe circuit was verified.

Then an anomaly was found that though the transistors were connectedto the same carrier,the voltages seen at the base of both the transistors weredifferent.These voltage were phase shifted by 1800 (as required) but their am-plitudes were different. This was found to be due to the imbalance in the ironcore used in the IFT. So the frequency of the carrier was changed to make thevoltage amplitudes the same, thus tuning the circuit again.But since we hadonly changed the characteristic of the carrier,as expected the output didnothave the desired result.Thus the debugging process was continued. After theentire debugging process was completed, it was seen that the voltages at allthe nodes of the circuit was proper except for the output node.So it wasintuitively decided to change the IFT at the output.Then it was seen thatthe IFT which was being used had improper soldering which had caused allthe trouble. As expected when the IFT was replaced,the desired amplitudemodulated output was observed.

Now the modulation index was calculated from the waveforms observed inthe CRO using the formulae discussed earlier.

15

Observations and Calculations

The following observations were made from the waveforms seen on the CRO:

For the single transistor circuit:

1. Carrier Frequency = 370 kHz

2. Message Frequency = 2.5 kHz

3. A(t)max = 3.2V

4. A(t)min = 1V

5. β = A(t)max−A(t)min

A(t)max+A(t)min= 0.524

6. Ac = A(t)max+A(t)min

2= 2.1 V

7. Am = A(t)max−A(t)min

2= 1.1 V

8. Carrier Power = (2.1)2

2= 2.205 W

9. Total Sideband Power = (1.1)2

4= 0.3025 W

10. Total Power = CarrierPower + TotalSidebandPower = 2.5075 W

11. Efficiency, η = TotalSidebandPowerTotalPower

= 12.064 %

16

For the improved transistor circuit:(back to back configuration)

1. Carrier Frequency = 434 kHz

2. Message Frequency = 1.3 kHz

3. A(t)max = 0.9V

4. A(t)min = 0.1V

5. β = A(t)max−A(t)min

A(t)max+A(t)min= 0.8

6. Ac = A(t)max+A(t)min

2= 0.5 V

7. Am = A(t)max−A(t)min

2= 0.4 V

8. Carrier Power = (2.1)2

2= 0.125 W

9. Total Sideband Power = (1.1)2

4= 0.04 W

10. Total Power = CarrierPower + TotalSidebandPower = 0.165 W

11. Efficiency, η = TotalSidebandPowerTotalPower

= 24.24 %

17

Inference

From the observations and calculations done using the waveforms seenat the output,the modulation depth and power efficiency were calculated.Itis quite evident from the results obtained that with the basic circuit notmuch depth can be obtained and even with an improved circuit (a back toback configuration) the power efficiency was quite low.this should serve as animpectus for better modulation schemes that give higher depth and powerefficiency.

18

Conclusion

Amplitude modulation is one of the basic analog modulation schemes.Itis beautiful in its simplicity.Though it has many short comings like a sus-ceptibility to noise interferences it has survived the technological surge overthe years as a rather modest modulation scheme for low cost systems.One ofthe best surviving examples is the Radio systems that still exist quite promi-nently.Thus the study of communication engineering would have to start withthe historically and theoritically basic system. Doing the experiment withthis view point and finally observing the output was an inspiring experience.

19

Bibliography

[1] A Bruce Carlson,Paul B. Crilly, Janet C. Rutledge, 2002. CommunicationSystems-An Introduction to signals and Noise in Electrical Communica-tion, McGraw Hill,4th Edition.

[2] John G Proakis and Masoud Salehi,1994. Communication Systems Engi-neering,Prentice Hall.

[3] Michael P. Fitz. Fundamentals of Communication Systems,McGrawHill,2007.

[4] Leon W Couch. Digital and Analog Communication Systems,Prentice HallIndia.

20

AM DETECTION

1




Ajmal V K

2nd September

Contents

Introduction 2

Principles 3

Design 9

Experimentation 12

Observations and Inferences 13

Conclusion 16

Bibliography 17

1

Introduction

The two basic components of any communication system are the transmit-ter and receiver blocks. The previous experiment dealt with the generationof a normal Amplitude Modulated wave. In practise once this wave is gener-ated and transmitted, it needs to be received and demodulated to retrieve theencoded message information. Hence, this experiment, as a logical continu-ation deals with the designing of the detection block of an AM system andassociated improvements that could be brought about in the demodulation.

In an Amplitude Modulated signal, the message is encoded in the am-plitude of the modulated signal. To detect the envelope of the modulatedsignal, the detector needs to interpolate the successive peaks (positive or neg-ative) of the modulated signal. It is with this basic algorithm of detectionin mind that one should go about designing the circuit. It is only when oneencounters practical issues in the design process like varying incoming signalpowers that one thinks about improvising and incorporating new blocks intothe existing system. An example of such an improvisation is the concept ofAutomatic Gain Control which efficiently utilizes the otherwise waste-ful dc offset component in the demodulated signal to dynamically controlthe gain of an amplifier stage preceding the detection block.This experimentaims to incorporate all the basics of detection and the above mentioned im-provisations that have driven the wheel of development of communicationengineering to bring it to where it stands now.

2

Principles

Once a method to efficiently generate and transmit amplitude modulatedwaves is devised and designed, the next logical step would be to look fora system that faithfully reconstructs the encoded message information(viz.the simple low frequency sinusoidal signal in this case) from the amplitudemodulated wave. As was observed in the previous experiment, in an AMwave the envelope of the transmitted signal carries the information of themessage signal, the envelope being generated by the sinusoidal variations ofthe high frequency carrier. The equation of the AM wave is given by,

xAM(t) = Ac(1 + km(t))cosωct (1)

Now to get the behaviour of the envelope of the wave we need its inphase-quadrature representation. Say, the received AM wave has a small phaseshift also associated with it. Then the expression for it is

xAM(t) = Ac(1 + km(t))cos(ωct+ φ) (2)

xAM(t) = Ac(1 + km(t))cos(φ)cos(ωct)− Ac(1 + km(t))sin(φ)sin(ωct) (3)

xAM(t) = mI(t).cos(ωct)−mQ(t).sin(ωct) (4)

Then the envelope of the wave is given by

E =√mI(t)2 +mQ(t)2 (5)

E =√A2c(1 + km(t)2)(cosφ2 + sinφ2) (6)

E = Ac(1 + km(t)) (7)

E = Ac + Ackm(t) (8)

The envelope function E is proportional to the variations in the messagesignal.

3

Detecting AM wave means detecting the envelope of the wave.The peaks ofthe AM waves appear at the carrier frequency. A continuous mapping of thepeaks would give a rough estimate of the envelope and hence the message.

Now, as discussed above we are interested only in the locus of the peaks,either the positive or the negative part. So to get one of these parts wecan use a diode to clip off the positive or negative parts. Then we require amethod to hold on to the current peak value until the next peak comes. Sucha process would give a rough approximation of the envelope. So this wouldmean a charging element that charges up very quickly but discharges at avery slow rate so that between peaks the voltage is fairly constant. We canuse a capacitor resistance combination with a high discharge time constantfor this so that the capacitor charges up during the positive rise of the AMand discharges until the next positive rise comes.

This is a fairly simple logical solution to the problem at hand. But thereare many nuances that needs to be taken care off. One is that the circuitor system designed must be capable of detecting each positive/negative peakthat arrives quite efficiently or in other words it should be well responsive tothe variations in the message signal. Another point is that while doing thesame it must still maintain a fairly constant value between consecutive peaks.These are two opposing conditions. So we must arrive at an expression thatrelates the time constant of the capacitance-resistance block to the messagefrequency and the modulation depth.Now, as we use sinusoidal signals forthe message and the carrier we shall derive the expressions using the same.So,

xAM(t) = Ac(1 + kcos(2πfmt))cos(2πfct) (9)

where xAM(t) is the AM wave, fm - message frequency, fc - carrier frequency

We assume the case of normal modulation (i.e. no over modulation) thenk¡1 always So the (1+kcos(2πfmt) term is positive for all t hence the envelopefunction can be written as

A(t) = 1 + kcos(2πfmt) (10)

Lets take a time t1 will be

A(t1) = 1 + kcos(2πfmt1) (11)

4

This voltage across the capacitor will discharge until the next positivepeak. Since the time between consecutive peaks = 1

fc, the discharge time

td = 1fc

The capacitor voltage at the next peak is,

Vc(t1 +1

fc) = Vc(t1).e

−1fcRLCL (12)

Where CL and RL are the values of the capacitance and resistance used. Butas per the requirement the discharge time has to be very high.

∴ fcRLCL 1 (13)

now, e−1x for x 1 can be approximated to the first two terms of the

expansion:

e−1x = 1− 1

x( for x 1) (14)

Here the equation becomes,

Vc(t1 +1

fc) = Vc(t1).(1−

1

RLCLfc)(∵ x = RLCLfc) (15)

A(t1 +1

fc) = 1 + kcos(ωm(t1 +

1

fc)) (16)

Vc(t1 + 1fc

) has to be less than or equal to A(t1 + 1fc

) to faithfully follow themessage signal.

A(t1 +1

fc) = 1 + k[cos(ωmt1)cos(

ωmfc

)− sin(ωmt1)sin(ωmfc

)] (17)

but fm fc

∴ cos(wm

fc) u 1 ; sin(wm

fc) u wm

fc= 2πfm

fc

Vc(t1 +1

fc) ≤ A(t1 +

1

fc) (18)

(1 + kcos(ωmt1)(1−1

RLCLfc) ≤ 1 + kcos(ωmt1)− k

wmfcsin(ωmt1) (19)

1− 1

RLCLfc≤

1 + kcos(ωmt1)− kwm

fcsin(ωmt1)

1 + kcos(ωmt1)(20)

5

1− 1

RLCLfc≤

1− kwm

fcsin(ωmt1)

1 + kcos(ωmt1)(21)

RLCL ≤1 + kcos(ωmt1)

kwmsin(ωmt1)(22)

or1

RLCL+

k

RLCLcos(ωmt1) ≥ kwmsin(ωmt1) (23)

1

RLCL≥ kwmsin(ωmt1)−

k

RLCLcos(ωmt1) (24)

If we consider a right angled triangle with

tanθ =1

RLCLωm(25)

∴1

RLCL≥ kccos(θ).sin(ωmt1)− kcsin(θ).cos(ωmt1) (26)

where

c =

√ω2m +

1

R2LC

2L

(27)

1

RLCL≥ kc[cos(θ).sin(ωmt1)− sin(θ).cos(ωmt1)] (28)

1

RLCL≥ kc.sin(ωmt1 − θ) (29)

orkcRLCL.sin(ωmt1 − θ) ≤ 1 (30)

this must be true even when sin(ωmt1 − θ) = 1

∴ kcRLCL ≤ 1 (31)

=⇒ k ≤ 1

cRLCL=

1√(ω2

m + 1R2

LC2L)

).RLCL(32)

6

k ≤ 1√1 + 1

R2LC

2Lω

2m

(33)

1

k2− 1 ≥ R2

LC2Lω

2m (34)

=⇒ RLCL ≤1

ωm

√1− k2k2

(35)

The above given inequality is the condition that must be satisfied by thecircuit parameters to efficiently detect the incoming AM wave. Once thisenvelope has been detected we have to get the message signal as a propersinusoid(provided it was transmitted as a sinusoid). Now the detected enve-lope waveform will contain distinct ripples arising from the discharging of thecapacitor. The frequency of these ripples will be same as that of the carrier.So if this envelope wave is passed through a block that detects these ripplesand smoothens them, the work is done. The block can be designed as aresistance capacitance combination with the output taken across the capac-itor. The time constants of this combination must be much higher than thetime period of the carrier. So that, between ripples the voltages is smoothedout. Once the signal is out of this block we have a fairly smooth sinusoid(that corresponds to the message) but with a dc offset. The dc offset can beremoved by taking the output through a blocking capacitor of some distinctvalue. In practical circuits, before this detection block there will be a seriesof amplifier stages and a receiver that receives the incoming transmitted AMwave. In usual transmissions the signals will have variable strength levelswith low and very high strength components. If normal amplifiers are usedboth the low strength as well as the high strength signals will be amplifiedto the same amount leading to a suppression of the low strength part whichmight contain valuable information.

To resolve this issue we use a system called Automatic Gain Control(AGC).In AGC, we take the dc offset that is present in the detected envelopeand feed it back to the amplifier stage so as to control its gain in such a wayso that for the stronger signal components the gain is reduced. To realizethis we take the ripple free envelope output but with the dc offset and passit through a low pass filter. The same resistor capacitor combination asdescribed above can be used but with the time constant now much greater

7

than the time period of the message signal. So that we get a nearly dc voltageat the output of the AGC terminal. This is the concept of AGC.Anotherimprovement that can be done is to keep a minimum value of the input wave,signals below which will not be given the advantage of AGC by giving adelay to the working of the AGC block.This can be done by using a diodewhose cathode is biased at the required voltage level so that only when theincoming envelope waves voltage value is greater than this set value,will theAGC block work. This is the working of the system referred to as a delayedAGC.

These are the various principles required to construct a basic AM detectionblock with added features like the normal and delayed AGC.

8

Design

From the concepts evolved in the previous section an exact circuit for AMdetection can be designed. There are many basic blocks that are requiredwhose theoretical sides have been discussed. The parts are:

1. A high frequency diode: To clip off the negative/positive half cycle,whichever is required. A high frequency type must be used as we areoperating in the range of 0.5 MHz

2. A parallel combination of a capacitor and resistor immediately followingthe diode. This is put in such a way that while charging there is a lowresistance path from the diode through the diode to the ground. Whiledischarging it discharges through the resistance connected parallel tothe capacitor. The time constant of this section is designed based onthe relation derived:

RLCL ≤1

ωm

√1− k2k2

(36)

or

RLCL =1

ωm

√1− k2k2

(37)

with k = 0.7 and fm = 2 kHz

we get RL =33kΩ ,CL =2nF

The primary part of the detector is designed.

9

3. After the diode and capacitor resistor sections, we need the ripple re-mover circuit. This is a series combination of a capacitor and resistorwith output taken across the capacitor. The time constant of this sec-tion must be much greater than the ripple time period

R2C2 = 100T (38)

R2C2 = 100.1

fc(39)

R2 =2.2kΩ ,C2 =0.1µF

After all these sections we get the message signal with an offset dc voltage.So at the terminal section a blocking capacitor Cb = 4.7µF

The circuit thus obtained is given below

At the lab the single capacitor CL and resistor RL where split into a πfilter type section. The effective circuit is given below:

For getting the AGC voltage we need the dc offset message signal and itneeds to be passed through a capacitor-resistor section whose time period ismuch greater than that of message. So effectively the dc -offset voltage isobtained.

∴ R3C3 = 100T ′ (40)

10

= 100.1

fm(41)

R3 = 3kΩ ,C3 = 10µFThe AGC circuit is given below

For getting the delayed AGC as discussed earlier we can use a diode whosecathode is biased at the required positive level. The delayed AGC circuit isgiven below

The above given were the various circuits used at the lab.

11

Experimentation

First the circuit for the detection was assembled. As mentioned abovethe envelope detector circuit and the RC filter (ripple remover circuit) wascombined and a filter circuit was assembled. Output at each point of thecircuit was observed. Thus the waveforms with ripples and without rippleswere observed. The output without ripples was the exact envelope of themodulated signal but with a dc offset. Then a blocking capacitor was keptand the output observed after it. The message signal without the dc offset wasobserved. Now to assemble the AGC circuit, the resistance capacitor pathwas kept and the dc voltage was observed across the capacitor. Observationswere made from the output of AGC. Then it was observed that the AMgenerator was not put in the modulation mode. After correcting this errorthe observations were again made. These observations were tabulated and agraph was plotted between the carrier amplitude and the AGC output.

Now after having made the observations the output of the AGC circuit, thedelayed AGC circuit was assembled in which an external voltage was appliedto the negative terminal of a diode through a potentiometer. But initiallythe change in the dc voltage couldnt be observed and so the circuit waschecked for any errors. It was seen that a minor loose connection had causedthe problem so after connecting the components properly the output wasagain observed. Then it was seen that as we increased the carrier amplitude,the output of the delayed AGC was constant for some range and then itvaried linearly with carrier amplitude. The changes in the Delayed AGCwere tabulated and a graph similar to the previous one was plotted.

12

Observations and Inferences

The designed circuits were assembled and the message signal was observedfor various frequencies and carrier and message amplitudes. The outputswere observed on the CRO with and without dc offsets and also at differentpoints of the circuit.Observations for the AGC output are:

13

Observations for the delayed AGC output are:

14

It was observed that the output of the AGC was linearly varying withthe carrier amplitude.It was observed that the output of the Delayed AGC was linearly varyingwith the carrier amplitude.

15

Conclusion

The AM detector circuit was designed and the output observed. The de-tection circuit is basically an envelope detector which gives a rippled output.Then it was passed through a resistance capacitor combination which re-moves the ripples and gives a signal with a dc offset. This signal is thenpassed through a blocking capacitor which removes the dc component.

Now since this circuit doesnt take advantage of the dc component presentin the output signal, the AGC circuit was designed. In this modified circuitthe dc component of the output signal is obtained by passing the signal (withdc offset) through a resistance capacitor combination which only passes thedc component. The AGC output has been observed and tabulated.

Now to improve it into a more practically applicable circuit the delayedAGC circuit was designed wherein the output of the delayed AGC is low fora particular range and then varies linearly with the carrier amplitude. Thiscircuit will not give an output for small carrier amplitudes.

The logical manner in which the experiment was done - introducing newblocks into the primitive circuit to improve the characteristics of the output-gave a basic experience of system design .Such a methodology could serve asa benchmark for designing better and more complex systems.

16

Bibliography




[4] Leon W Couch. Digital and Analog Communication Systems,Prentice HallIndia,2006.

[5] Dennis Roddy, John Coolen.Electronic Communication,Tata McGrawHill, India,2004,4th Edition.

[6] Herbert Taub, Schilling.Principles of Communication Systems,Tata Mc-Graw Hill,2004.

17

DSBSC GENERATION

1




Ajmal V K

16th September

Contents

Introduction 2

Principles 3

Design 6

Experimentation 10


Conclusion 13

Bibliography 14

1

Introduction

The previous experiments dealt with the generation and detection of ageneral amplitude modulated wave. These served as a means to understandthe use and need for modulation as a technique for transmitting the messageinformation.The basic aim was to lay a foundation for linear continuous wavemodulation schemes without as much focusing on the practical intricacies ofthe transmission and reception process. Once the basic idea of modulationis acquired the next step is to better the design as well as the theories soas to improve the practical feasibility.Two basic parameters that have to belooked into while designing any communication system are:

1. Bandwidth Requirement

2. Power requirement

As it can be seen, in the basic AM case the carrier does not carry any usefulinformation but at the same time immense power is wasted in transmitting it.This is the drive behind the modulation method called as Double SidebandSuppressed Carrier where only the sidebands are transmitted and notthe unwanted carrier signal.This can be done by just taking the product ofthe message and carrier signals, mathematically. One logical way to producesuch a wave would be to generate two AM waves modulated with the messagesignal and its inverse and then take the difference of the two generated AMwaves. This would mathematically eliminate the carrier. Practical realizationwould require non-linear devices for producing higher order terms and abandpass filter for filtering out the sidebands alone.

With this idea in mind the modelling of a DSBSC system can be done.

2

Principles

The equation of the basic amplitude modulated wave is given by

xAM(t) = Ac(1 + km(t))cosωct (1)

where Ac is the amplitude of carrier signal, m(t) is the message signal, k isthe modulation index and ωc is the frequency of the carrier.

On evaluating the frequency domain expression of the above equation weobtain

xAM(t) =Ac

2(δ(f − fc) + δ(f + fc)) +

kAc

2(M(f − fc) +M(f + fc)) (2)

The power of the above wave is given by

PAM(total) =A2

c

2+k2A2

c

4(3)

The first term is due to the carrier signal and the second is due to themodulated signal.

From the above expression we see that a sufficient amount of power iscontributed by the carrier to the total power. But at the same time it does notcarry any information(message signal, m(t)) that we wish to transmit.Thuswe waste a lot of energy in transmitting the unwanted carrier.

From this it is clear that we only need to transmit the sidebands (M(f−fc)and M(f + fc)) for conveying the information. So we need to eliminatethe carrier. Mathematically, this can be done by taking the product of themessage signal and carrier.

x(t) = m(t).c(t) (4)

3

Thus it is the simplest in terms of the mathematical description of modulationand demodulation operations.

Thus a DSBSC modulator will be basically a multiplier. If k’ is the am-plitude sensitivity of the modulator then

x(t) = k′m(t).c(t) = k′m(t).cosωct (5)

The frequency domain expression will then be given by evaluating the fouriertransform of the above expression.

X(f) =Ac

2[M(f − fc) +M(f + fc)] (6)

We observe that the modulation process only shifts the message spectrum by±fc. If the message is also a sinusoidal signal then we will have the frequencydomain expression as two weighted impulses at ωc + ωm and ωc - ωm whereωm is the angular frequency of the modulating sinusoidal message signal.

From the time domain expression, we can see that wherever the messagesignal value reaches zero there will be a corresponding null in the modulatedwave followed by a 180o phase reversal in the carrier wave in the modulatedsignal. This can be explained as follows. Till the first zero crossing themessage signal decreases from its most positive value to zero.So there will bea linear decrease in carrier amplitude in the modulated wave.After the zerocrossing the message increases from zero to its maximum value.Thus therewill be a phase reversal at the zero crossing owing to the inversion of themessage (if it is sinusoidal) signal in its second half cycle.

Now, when we look at the modulation process as such of the DSBSC sys-tem we see that the percentage of modulation is infinite because there is nocarrier component. Furthermore,the modulation efficiency of the DSBSC sig-nal will be 100 % since no power is wasted in transmitting a discrete carrier.Moreover, although the message and the carrier are periodic waveforms, theDSBSC itself need not necessarily be periodic. The phase reversal at thezero crossing may not be properly observable unless the zero crossing of themessage is accompanied by the zero crossing of the carrier. This observationthough small is significant in a laboratory implementation of the DSBSCmodulator generated using two independent signal generators for the carrier

4

and the message where the phase reversal can be observed only when mes-sage frequency is an integral factor of the carrier frequency. This is becausephase coherence of two independent sinusoidal sources has an extremely lowprobability of occurring, and hence the generated DSBSC waveform usuallydoes not reveal the phase reversal. This can be brought about by makingthe carrier signal an integral multiple of the message. This is a very delicateidea that is useful only for a laboratory implementation and not for evolvingthe theory as such.

Overall it can be seen that the DSBSC can be modelled as a product of themessage and carrier signal.As no device can produce only the product termsalone, there will be components in the higher degrees as well as standalonecarrier and message components. These have to be properly filtered and therequired sidebands obtained.

5

Design

As seen earlier the mathematical expression for a DSBSC signal is a prod-uct expression of the message and the carrier.

x(t) = Acm(t).c(t) = Acm(t).cosωct (7)

So we basically need a multiplier. This can be done using any non-lineardevice. But the output of the non-linear device will contain several higherorder terms. As discussed earlier DSBSC can be generated by a process calledBalanced Modulation wherein two modulators are connected back to backand the modulating signal is injected with reversed phase in one modulatorcompared to the other. The carrier is injected into both the modulators inthe same phase. Thus there will be two non-linear devices connected backto back which will act as the modulators mentioned. The inputs to each ofthem will be

x1(t) = c(t) +m(t) (8)

&x2(t) = c(t)−m(t) (9)

The phase reversed message can be given by applying the message at theprimary of an Audio Frequency Transformer and taking two outputs from thetwo ends of the secondary with the centre tap as a common terminal. Thenthe upper half of the secondary will couple message signal to the circuit andthe lower half will couple message signal to the circuit. If x(t) is the input tothe non-linear device say a JFET then its output will be given by the generalexpression

s(t) = a+ b(x(t)) + c(x(t))2 (10)

We know that x(t) has two values for the 2 non-linear devices used, namelyx1(t) and x2(t) So denoting the two corresponding outputs as s1(t) and s2(t)

∴ s1(t) = a+ b(c(t) +m(t)) + c(c(t) +m(t))2 (11)

6

s1(t) = a+ b(c(t) +m(t)) + c(c(t)2 +m(t)2 + 2c(t).m(t)) (12)

∴ s2(t) = a+ b(c(t)−m(t)) + c(c(t)−m(t))2 (13)

s2(t) = a+ b(c(t)−m(t)) + c(c(t)2 +m(t)2 − 2c(t).m(t)) (14)

Now if these outputs are given to the two ends of the secondary of atransformer the net voltage applied at the secondary will be

s(t) = s1(t)− s2(t) (15)

=⇒ s(t) = 2bm(t) + 4c(t).m(t) (16)

Now the message frequencies used in DSBSC are usually in the audio fre-quency range and are usually very small as compared to the carrier frequencywhich is in the intermediate frequency range. So if we use an IntermediateFrequency Transformer(IFT) as the transformer mentioned above will be fil-tered out and we get the required DSBSC signal at the output. For thediodes also similar analysis ensues. The only difference being that there willbe even more higher order terms as its output characteristic is exponential.But the basic idea and block structure is the same.

If we take single tone sinusoidal modulation and ignoring higher orderterms and the fundamental input signal(by assuming that the necessary filtersare available at the output) the output from each of the modulators will begiven by:

V1 = Accosωct+Am

2cos(ωc + ωm) +

Am

2cos(ωc − ωm) (17)

V2 = Accosωct−Am

2cos(ωc + ωm)− Am

2cos(ωc − ωm) (18)

Vo = V1 − V2 = Amcos(ωc + ωm) + Amcos(ωc − ωm) (19)

The carrier signal has to be obviously applied at the centre tap of the sec-ondary of the input transformer so that it is commonly coupled to the 2non-linear blocks. For the diode circuit the components required are:

1. 2 high frequency diodes :OA79

2. 1 IFT for coupling the carrier

7

3. 1 AFT for providing the message

4. 1 IFT for taking the output

The circuit thus obtained is as given below:

If they are perfectly balanced i.e. have equal carrier outputs, the carrierwill be completely suppressed and only the modulation sideband remainsabout all multiples of the carrier frequency. However, in most cases thesecomponents and modulating signal coupled through from the input are veryeasily removed by filtering. The circuit operation is as follows. The inputtransformer introduces the audio signals to the balanced diodes, which areturned off and on by the carrier voltage introduced in an In phase relationship. If the carrier amplitude is large with respect to the modulating inputvoltage, the only current flowing in the output transformer is due to theaction of the modulating voltage added to the carrier voltage. Because ofthe push-pull arrangement, the carrier components will be balanced out,and the output will consist of sideband components around all harmonicsof the carrier frequency as well as the original modulating signal. If thecarrier frequency is greater than twice the highest modulating frequency,these components can be removed by filtering to leave only the desired DSBsignals.

The shortcoming of the diode circuit is that it does not provide any scopefor fine tuning of the output and observation of the phase reversal at thezero crossing. For this a transistor can be used in place of the diode eithera FET or BJT. A JFET can provide a very high gain so it can be used.

8

While using a JFET special care has to be taken to ensure that the Gate toSource junction is always reverse biased. For this either a negative voltagecan be given across the gate and source or more preferably a resistor can beinserted in the source to keep it at the required higher potential as comparedto the gate. The main issue in using transistors is the difficulty in matchingthe two components used. If they are not matched then we cannot observethe DSBSC output as these will always be a stray carrier components thatprevents the output from having a zero crossing. For this a variable resistoris inserted between the two source resistors of the JFETs so that tuningthe pot will bring about matching of the 2 devices used. The componentsrequired for the JFET circuit are:

1. 2 high frequency JFET : BFW10

2. 1 IFT for coupling the carrier

3. 1 AFT for providing the message

4. 1 IFT for taking the output

5. DC supply to bias the JFETs

6. 10k resistors : 2 nos

7. A variable resistor(of 10k or 470k)


9

The capacitors are inserted across the source resistors to increase the gainof the response. The analysis and behaviour of the circuit is similar to thediode case except that in this case the output of the JFET will have onlysecond order terms.

10

Experimentation

As per the design evolved in the previous section the diode based circuitwas assembled on a breadboard. A message signal of 2kHz was coupledthrough the AFT to the circuit and a carrier wave of 455kHz was coupledthough as IFT. The output was taken from the 2nd IFT. At first only basicAM was observed but on adjusting the message frequency as an integralfactor of the carrier wave a rough version of the DSBSC was observed. Butthe problem with the output was that the phase reversal at the zero crossingcould not be clearly observed as the output seen was not properly resolvable.Since the circuit did not present any further scope for finetuning (owing tothe use of a diode) the circuit using JFET was next assembled.

After assembling the JFET circuit on the breadboard special care wastaken to ensure that the gate-source junction was never forward biased. Forthis the dc supply was first applied without giving any signal. The gate-source voltage was checked and verified to be negative. Now the carrierwas applied without giving the message. A sinusoidal output at the carrierfrequency was obtained. Ideally as per the design no output should be seenas the carrier has to be suppressed. But owing to the mismatches in theFETs used some carrier components were still present in the output. Thepot connected across the source resistances was adjusted so as to minimizethe output amplitude of the carrier. It was kept in the minimum possiblevalue

Now a message signal of 2kHz frequency was applied. There was initiallyonly a feeble effect of the message on the output. On slightly reducing thebias voltage of the JFET, a proper output was able to be observed. But stillthe output observed was only basic AM and not DSBSC. On increasing themessage frequency the output was found to be varying and at frequencies

11

that were integral factors of the carrier frequency, the nearest output toDSBSC was observed. But still the phase reversal at the zero crossing wasnot observed. The reason was finally attributed to the presence of the slightcarrier component at the output. Since the JFETs were already matched thereason for the problem was the error in the output IFT. Owing to improperinsertion of the core of the transformer into the windings, the two halves ofthe secondary were not symmetric. This was the explanation evolved for thecarrier components presence. The frequency of the carrier was now slightlyadjusted to reduce the carriers presence and at 464 kHz it was nearly zero.Now the message was again applied and its frequency was varied. Only ata single frequency of 11.06 kHz was proper DSBSC observed and the phasereversal at the zero crossing was clearly seen. It was ensured that the outputobtained was not a case of over modulation.

In the circuit many trouble shooting steps had to be performed before aproper output was obtained. The final tuning to observe the phase reversalwas a very delicate and critical process and it was a great experience inpractical assembly.

12


=⇒ For the diode circuit a DSBSC output was observed but the output couldnot be properly resolved and the phase reversal could not be clearly noted.=⇒ The output obtained for the JFET circuit is as given below.

The message frequency used was 11.06 kHzThe carrier frequency used was 464 kHzThe phase reversal in the zero crossing was clearly observed.

13

Conclusion

Observing a proper output for the DSBSC circuit demanded several finetunings of the primarily designed circuit.Once such a system is properlyevolved it is much better than a basic AM system as the power requirementsare drastically reduced owing to the suppression of the carrier componentsat the output by the use of a balanced modulation scheme.But the circuitdesigned does not allow for any compromises in matching.Hence the compo-nents used have to be perfectly matched as we use two symmetric sectionsfor the process and a slight imbalance can hamper the entire output.

Further transforming reveals that the information contained in both thesidebands are the same.This intuitively suggests that we need to transmitonly one of the sidebands as the information in the other is the same.Thiswould further reduce the power of transmission.Hence we arrive at singlesideband modulation which is a further improvement of the DSBSC system.

14

Bibliography







15

COLPITT’S OSCILLATOR

1




Ajmal V K

23rd September

Contents

Introduction 2

Principles 3

Design 6

Experimentation 14


Conclusion 16

Bibliography 17

1

Introduction

In all of the experiments that are done in any electronics lab we fre-quently encounter the necessity to use sinusoidal, square or other types ofperiodic waveform as inputs to different stages of an assembled circuit . Thisexperiment deals with that class of circuits that can produce as output suchperiodic waveforms, namely oscillators. They form the basic block of anyfunction generator that we use in our labs. So it is imperative that the de-sign of a common oscillator be done so that a better understanding of theoverall picture is obtained.

To oscillate means to fluctuate between two states or conditions. An os-cillator would then imply a device that produces oscillations. In electrical orelectronics domain an oscillator would be a device that produces repetitivecontinuous changes in voltage or current. There will be a periodic rate as-sociated with the output waveform obtained. Now if the oscillations (otherthan possibly a dc supply) they are called free-running oscillators. Thusessentially it converts a dc input voltage to an ac output voltage.

They can be designed in a variety of ways. But for a free-running oscillatora logical method would be to use a feedback loop to sustain the oscillation. Itshould also have frequency determining components like resistors, capacitors,inductors etc. One effective circuit would be a tuned circuit that is composedof inductors and capacitors. A Colpitts oscillator is one such circuit thatemploys a tuning block as its frequency determining circuit block.

2

Principles

An oscillator basically represents a circuit that generates a periodicsignal that fluctuates between voltage levels or current levels. As discussedabove, one way to implement this would be by using a feedback amplifierwith some part of output being fed back to the input and then again getsamplified. Imperatively, there can be two types of feedbacks , namely positiveand negative. While a positive feedback is regenerative, in the sense that itcumulatively increases the amplitude of the output. A negative feedback isdegenerative, that is it reduces the input voltage and keeps the amplitudeof the output in check. A negative feedback is normally used in all stablesystem designs as in a positive feedback network there is possibility for devicesaturation. So, let us first look at a negative feedback system as a blockstructure and try to arrive at the conditions for oscillations. It is quite clearthat for self sustained oscillations to occur the feedback has to be positive asthen only the input voltage would be replenished each time with a portion ofthe output. β stands for the fraction of the output that is actually fed back

to the input. Now, from the block

Vo = AoV (1)

3

V = Vi − βVo (2)

Vo = Ao(Vi − βVo) (3)

Vo = AoVi − AoβVo (4)

orVoVi

=Ao

1 + Aoβ= Af (5)

Af is the net gain with feedback.β > 0 for all practical circuits as it denotes just a fraction.∴ Af < AoThis shows that applying a negative feedback reduces the gain of the amplifier. Af is also called the closed loop voltage gain. Now if Ao and β are functionsof frequency then we can write

Af =Ao(s)

1 + Ao(s)β(s)(6)

In this case,if at any frequency Ao(s)β(s) becomes equal to -1 then the de-nominator of the equation above becomes zero.Or Af (s) =∞ =⇒ Vo

Vi= ∞

This mathematically means that there is an output for a zero value of theinput voltage. Or that the circuit is able to generate an output. The termAo(s)β(s) is called loop gain and is denoted as T(s). As T(s) approaches-1 the actual circuit becomes nonlinear, which means that the gain does notgo to infinity. If T(s) ≈ -1 so that positive feedback exists over a particularfrequency range. If a spontaneous signal (due to noise or mismatches) is cre-ated at the input in this frequency range the resulting feedback signal will bein phase with the input signal, say Vs. Therefore, the input to the amplifierstage is reinforced and increased. This cumulative process of reinforcementcontinues for only those frequencies for which the total phase shift aroundthe feedback loop is zero(so that the input and output are in phase).All theother feedbacks will diminish with time owing to phase differences. Thus thecondition for oscillation would be that at a specific frequency we have

T (jωo) = A(jωo)B(jωo) = −1. (7)

4

This is often referred to as Barkhausen criterion.From the above expressions it is evident that two conditions are necessaryto sustain oscillations.

1. The total phase shift around the feedback loop should be an integralmultiple of 2π.

2. The magnitude of the loop gain must be unity.

Once this basic block model of the oscillator is decided upon, the next focusmust be on the feedback network used. We have seen that the feedbacknetwork has to be a frequency selective one. One common frequency selectivenetwork is the LC tank circuit. The basic tank circuit operation involves theexchange of energy between kinetic and potential (like in a simple pendulum).Once a current is injected into the LC network the energy the derived from itis exchanged between the inductor and capacitor producing a correspondingac output voltage.When net impedance = 0 , the output will be maximum. This conditionis called resonance and the frequency at which this happens is called theresonating frequency.

Ls+1

Cs= 0 (8)

=⇒ s2 = − 1

LC= 0 (9)

or

ω =1√LC

(10)

=⇒ f =1

2π√LC

(11)

is the resonating frequency.

Thus we can use an LC block as the frequency selective feedback circuitryand a BJT circuit as an amplifier block.

One common oscillator configuration that employs the LC tank circuit iscalled the Colpitts oscillator which uses a capacitive divider arrangementto give the necessary feedback.

5

Design

In a Colpitt’s oscillator the capacitor of LC circuit is split and the feedbackis taken between the two split capacitors.As seen in the previous sectionthe feedback oscillator can be designed using an LC circuit as a frequencygenerating circuit.This circuit is the one given as the feedback. In a Colpitt’soscillator the capacitor of the LC section is split into two and there is avoltage feeding from the centre of the two split capacitances.

To start designing the circuit we have to first analyse the basic configura-tion that we plan to use and find out its poles and zeroes.Thereby generatethe conditions for the circuit to operate with the desired parameters.As seenearlier lets look at the feedback block but this time taking the special caseof the oscillator:

Theoretically, for the oscillator the input voltage is zero and there is onlythe feedback voltage.We have to look at both the open loop and closed loopcases(Open loop used to determine the loop gain)Open Loop Case

From open loop analysis, we know that the loop gain is given by

T =Return V oltage

Test V oltage(12)

hence the return voltage is Vf and test voltage is Vi

∴ T =VfVi

=VoVi

= A(s) (13)

lets generalize

A(s) =k.N(s)

D(s)(14)

6

Closed Loop Case

Vo = A(s)(Vi + Vf ) (15)

Vo = A(s)(Vi + Vo) (16)

Vo(1− A(s)) = A(s)Vi (17)

or

Af (s) =A(s)

1− A(s)(18)

7

Now from these basic blocks if we consider an oscillator circuit with an am-plifier and feedback stage(with the feedback given as the closed-loop analysisabove then the circuit will be

R represents total resistance in collector circuit. That is,Rc||ro = Rc.=⇒ In the closed loop Cπ will be part of C2 =⇒ Cµ negligible (compared toC1 and C2) =⇒ rπ large (as compared to 1

C2ω) If we consider the open loop

case the circuit will be

The above circuit can be used for the loop gain analysis.The equivalent cir-cuit will be as given in the next page.

VPi = Vπ (19)

Then loop gain is

A(s) =VfVπ

(20)

8

Node equation at V1,

(gm − sCµ)Vπ +V1R

+ sC1V1 +V1 − VfsL

= 0 (21)

Node Equation at Vf :Vf − V1sL

+ sC2Vf = 0 (22)

Rearranging the two equations:

(sC1 +1

sL+

1

R)V1 −

1

sLVf = (sCµ − gm)Vπ (23)

− 1

sLV1 + (sC2 +

1

sL)Vf = 0 (24)

∴ (s2C1L+ 1

sL+

1

R)V1 −

1

sLVf = (sCµ − gm) (25)

∴ − 1

sLV1 + (

s2C2L+ 1

sL))Vf = 0 (26)

Multiplying equation 25 with 1sL

and equation 26 with ( s2LC1+1sL

+ 1R

) andadding we get

(−(1

sL)2 + (

s2LC1 + 1

sL+

1

R)(s2LC+1

sL)Vf = (

sCµ − gmsL

)Vπ (27)

Multiplying throughout by sL2, expanding and arranging in terms of powersof sn,we get

(−1 + s4C1C2L2 + s2(LC1 +LC2) +

s3L2C2

R+sL

R= 1)Vf = (sCµ− gm).sL.Vπ

(28)

9

Simplifying and multiplying by RsL

,we get

(s3RC1C2L+ sR(C1 + C2) + s2LC2 + 1)Vf = (sCµ − gm)RVπ (29)

making the coefficients of the highest power of s = 1,we get

(s3 +s(C1 + C2)

C1C2L+

s2

RC1

+1

RC1C2L)Vf =

(s− gmCµ

)RCµ

RC1C2LVπ (30)

Thus

VfVπ

= A(s) = T (s) =

RCµRC1C2L

(s− gmCµ

)

s3 + s2

RC1+ s(C1+C2)

C1C2L+ 1

RC1C2L

=k.N(s)

D(s)(31)

Next we do the closed loop analysis,from results obtained earlier

Af (s) =A(s)

1− A(s)=

k.N(s)

D(s)− k.N(s)(32)

D(s) = s3 +s2

RC1

+s(C1 + C2)

C1C2L+

1

RC1C2L(33)

k =RCµ

RC1C2L,N(s) = s− gm

Cµ(34)

∴ Denominator of Af (s)

D(s)− k.N(s) = s3 +s2

RC1

+s(C1 + C2)

C1C2L+

1 + gmR

RC1C2L. (35)

since the open loop system is a passive RLC circuit,it’s poles are in theleft half of s-plane.So the open loop A(s) is stable with three poles-one isnegative real and tthe other two can be negative real or complex conjugates.

∴ D(s) = (s+ a)(s2 + ω2x) (36)

∴ D(s) = s3 + as2 + aω2x + sω2

x (37)

D(jω) = j(ωω2x − ω3) + (aω2

x − aω2) (38)

10

For oscillation and real parts are individually zero.Now for the closed loopcase the denominator is D(s)+k.N(s).For this to be stable we can compare itwith the open-loop case and arrive at the conditions.

∴ D(s) + k.N(s) = s3 +s2

RC1

+s(C1 + C2)

C1C2L+

1 + gmR

RC1C2L(39)

comparing with open - loop case we get

a =1

RC1

, ω2x =

1

( C1C2

C1+C2)L, aω2

x =1 + gmR

RC1C2L(40)

∴ From these equations we get

ωx =1√CTL

(41)

where

CT =C1C2

C1 + C2

(42)

and

gmR =C2

C1

(43)

gmR is the gain of the BJT amplifier circuit.∴ ωx is the oscillating frequency and to ensure startup of oscillation gmR ¿C2

C1.

Now the circuit with dc biasing for the transistor and practical feasibility isgiven below:

Now, as we have all the formulations required for the different parameterswe can now design the circuit for specific design values.

We need an oscillating frequency of 1.5 Mhz.Therefore ,

∴ f =1

2π√LCT

(44)

∴ putting L= 1mH we get CT = 11.25pFNow to design the biasing of the transistor amplifier, we first fix the gainfix the gain, Av = 185Gain from the equivalent circuit is gmRc = 185

11

let ICQ = 4mA ( for BF195) For the transistor BF195,typical β = 60.

∴ gm =ICQVT

= 0.1538 (45)

∴ Rc = 1.164kΩ ≈ 1.2kΩ (46)

Now if we take thevenin equivalent of R1 and R2 at the base of thetransistor we get the base emitter equations as :

VTH −ICβ.RTH − VBE(ON)− β + 1

βICRE = 0 (47)

VCC = 12V and VCE = 6V (the fixed Q point)Ve =0.1VCC = 1.2V∴ RE = 300Ω∴ RE ≈ 330ΩNow

VCCR2

R1 +R2

− ICβ

R1R2

R1 +R2

− β + 1

βRCRE = 0 (48)

For stability, RTH = 0.1(β + 1)RE

From these conditions, we get R1 ≈ 47KΩ, R2 ≈ 10KΩNow AV > C2

C1.

12

∴ let us fix C2

C1= 100 (as AV = 185)

That is, CT = C1C2

C1+C2.

Therefore C1 = 11pF, C2 = 1.1nF.Standard values C1 = 12pF, C2 = 1nF.All other external capacitances = 0.22 µFOnce the calculation of the values of components is done, the design is com-plete.The values of the components used in the lab areR1 = 47KΩ, R2 = 10KΩ in series with 1KΩ pot.C1 = 4.7pF, C2 = 1nF.L = 60 µHRC = 1.2K Ω, RE = 330 ΩCoupling and Bypass capacitances = 0.22 µF

13

Experimentation

The circuit with the designed values as given in the previous section wasfirst assembled on the breadboard. The variable inductance box was con-nected as the inductance element. Once the assembly was verified, the DCvoltage was applied. Now the DC conditions were verified with the designedvalues. But at the output only very feeble oscillations were observed. Afterchecking the circuit again and adjusting the inductances in the inductancebox a rough sinusoid was observed but with a frequency of nearly 8 Mhz. Sothe capacitance values were changed and the inductance was varied.

Finally a proper sinusoidal output was obtained but it had some harmonics.It was then suggested to put a variable resistor as the emitter resistor. Ondoing so a distorted output was obtained. This was the problem with thebiasing. So a pot was inserted in place of R2 and varying it reduced thedistortions. This was done till a proper sinusoid of desired frequency wasobtained.

14


The frequency of the sinusoid observed was 1.42 Mhz. The design was for1.5 Mhz.The waveform observed was

The distortions in the output were rectified using a variable resistor to adjustthe biasing .

15

Conclusion

Using a feedback amplifier’s oscillating conditions the required circuit wasdesigned and the output sinusoid was observed. Many frequency selectivecircuits could be employed . Here an LC tuning circuit was used. It isseen that delicate troubleshooting steps have to be taken to produce a pure,sustained sinusoid at the output. With this experiment we get the idea of theinternal circuitry of the devices used in the labs to generate various functions.This circuit is not used as a standalone one but rather its output is givento other circuits or subsequent stages. At that point loading effects, namelythe output impedance of the oscillator circuit and the input impedance ofthe subsequent stages are very critical for sustaining a clear, undistortedsinusoid. At that time loading compensations would have to be done.

16

Bibliography

[1] A Bruce Carlson,Paul B. Crilly, Janet C. Rutledge, CommunicationSystems-An Introduction to signals and Noise in Electrical Communi-cation, 2002.

[2] John G Proakis and Masoud Salehi, Communication Systems Engineer-ing,Prentice Hall,1994.

[3] Donald A Neamen Electronic Circuit Analysis and Design,McGrawHill,2nd Edition,2005.

[4] A S Sedra, K C Smith Microelectronic Circuits,Oxford UniversityPress,5th Edition,2005.

[5] Dennis Roddy, John Coolen.Electronic Communication,Tata McGrawHill, India,2004

[6] Lecture Notes from ESE319 Introduction to Microelectronics,Universityof Pennsylvania

17

RF MIXER

1




Ajmal V K

14th October

Contents

Introduction 2

Principles 3

Design 8

Experimentation 10


Conclusion 13

Bibliography 14

1

Introduction

Many a times while designing circuits, situations are encountered whereinit is necessary to generate or change the existing frequency components in thecircuit so as to feed this altered frequency value to subsequent stages of thecircuit. This process can be accomplished if we have two or more manipulatedto get the required frequency. Such a process is termed as mixing . Anintuitive way of implementation would be to utilize the non-linear natureof any three-terminal device(such as BJT) to generate the harmonics of thedifference frequencies which could be suitably filtered to get the requiredfrequencies. This experiment deals with such an implementation.

2

Principles

To begin with, a mixer circuit, when considered as a black box, willhave two input terminals and one output terminal. One input would corre-spond to a signal of some frequency that is existing in the circuit whereas theother input would correspond to a local oscillator that is designed to generatea definite frequency value tailor made for the required output frequency. Sothe designer will have the liberty of deciding at which frequency the oscilla-tor should work so as to get the required output provided the mathematicalmodel of the mixer block is known.

The mixer is then a non-linear resistance having two sets of input and oneset of output terminals. All mixer circuits make use of the fact that whentwo sinusoidal signals are multiplied together the resultant consists of sumand difference frequency components. This can be demonstrated as below:If we have two frequency signal say,

v1 = V1sinω1t (1)

v2 = V2sinω1t (2)

Then the signal obtained on multiplying these two signals is

v1.v2 =V1V2

2(cos(ω1 − ω2)t− cos(ω1 + ω2)t) (3)

3

We are usually interested in the term containing the frequency ω1 - ω2

and that is usually the frequency which is filtered out. This frequency isalso usually desired to be located in the Intermediate Frequency (IF) as it isrequired for various applications. For an ideal mixing operations neither ofthe two initial input frequencies are present in the output, only the sum anddifference frequencies will be present. Now, to produce such a multiplicationeffect of sinusoidal signals as described above,we can use the non-linearity inthe output characteristics of devices such as the BJT. The voltage/currentrelationship for the transistor is

IC = ISeVBEVT (4)

where IC is the collector current (usually taken as the output current),IS is the reverse saturation current, VBE is the base emitter voltage and VTthe thermal voltage. The collector current equation written above can befactored into a dc bias term and a periodic term

∴ IC = ISeVAVT .e

ViVT cosωt (5)

or generalizing,IC = ISe

a.ebcosωt (6)

where

a =VAVT

(7)

b =ViVT

(8)

Now, since is a periodic function it can be expanded into a fourier seriesas below

ebcosωt = I0(b) + 2I1(b)cosωt+ 2I2(b)cos2ωt+ ... (9)

Where the fourier coefficients In(b) are also modified Bessel functions.Here I0(b) can be regarded as the bias current of the amplifier circuit. IfI0(b) is stabilised then

IC = ISeaI0(b)(1 +

2I1(b)

I0(b)cosωt+

2I2(b)

I1(b)cos2ωt+ ...) (10)

letIQ = ISe

aI0(b) (11)

4

NowIC = ISe

aebcosωt (12)

IC =IQI0(b)

ebcosωt (13)

The ICIQ

characteristic is given below. With increasing input drive, the

current waveform becomes more and more peaky and the peak value can beexceed the dc bias by a large factor. Now if we can plot the rates of harmonicfrequencies to the dc coefficient current against b, ie. Vi

VT, We get the curve

below:

All the above, shows that the BJT output spectrum is rich in harmon-ics. Again, the exponential can be approximated to the first three terms

5

of its expansion. That is, we model the BJT as having a second order I-Vcharacteristic

∴ iC = a+ bvBE + cvBE2 (14)

as a general case.

Now, if we make the vBE voltage as the sum or difference of the two signalsof different frequencies which are given as inputs, the vBE voltage becomes

vBE = v1 − v2 = V1sinω1t− V2sinω2t (15)

Substituting this vBE in the output/input equation

iC = a+ b(V1sinω1t− V2sinω2t) + c(V1sinω1t− V2sinω2t)2 (16)

iC = a+ bV1sinω1t− bV2sinω2t+c

2V1

2 +c

2V2

2 +c

2V2

2sin2ω1t (17)

+c

2V2

2sin2ω2t− c′sin(ω1 + ω2)t− c′sin(ω1 − ω2)t

Since the RF input applied will be small, the circuits response to it will belinear (or weakly non-linear). But the oscillator input is substantial and itvaries the operating point of the circuit periodically. So the overall responseto the RF input is a linear-time varying one.

i0(t) = g(t).vin (18)

where g(t) is the transconductance. The transconductance varies periodicallyand can be expanded in a fourier series.

g(t) = g0 + g1cosωOt+ g2cos2ωOt+ ... (19)

ωO −→ frequency of local oscillator. when

vin = V1cosωSt (20)

is appliediO(t) = g(t)× V1cosωSt. (21)

Hence we get,

iO(t) = g0V1cosωSt+g12V1cos(ωO ± ωS)t+

g22V1cos(2ωO ± ωS)t+ ... (22)

6

This again shows that the output of a mixer has various harmonics of thesum and difference frequencies.

From the above equation we see that a lot of other frequencies are presentat the output which needs to be filtered out by providing a suitable tunedcircuit at the output. The output frequency is decided by our needs whetherwe need a higher frequency than the reference frequency input frequency ora smaller frequency than the reference. Accordingly we select the sum ordifference term and filter it out. Fine tuning of the local oscillator overwhich we have some control. While connecting the oscillator stage and theRF mixer stage the loading issues have to be taken care of. Otherwise thereis a possibility of the local oscillators output not being fed to the input ofthe mixer. This is fine tuning done during circuit assembly. This is the basicmathematical foundation for designing a mixer circuit.

7

Design

As discussed above a BJT can used to design the circuit. At the out-put(collector) side in order to filter out the required output frequency anIntermediate Frequency Transformer(IFT) can be kept. Now in order totake care of the problem of loading we have to give a substantial amount ofresistance at the emitter of the transistor, where the oscillator output will befed. The input frequency will be given to the base of the transistor so thatthe base-emitter voltage will be the difference of these two voltages. Therehas to be a resistor from collector to base for giving the necessary bias.The dc biasing is designed as below.BF195 high frequency transistor is used.ICQ = 4mA, VCC = 9V, VCE = 5V , β = 60

∴ IB =ICβ

= 66.6µA. (23)

The circuit will be as below:

Writing KVL,VCC −RE(β + 1)IB − VCE = 0. (24)

8

∴ RE ≈ 1KΩ. (25)

VCC − IBRB − 0.95 = IERE (26)

=⇒ RB = 64.5KΩ ≈ 62KΩ. (27)

The capacitors used for coupling are 0.1uF each.This is the design for the circuit.

The difference frequency if it falls in the IF range will be filtered out bythe IFT and observed as a sinusoid at VO. While doing the experiment avariable resistor needs to be kept at the emitter to properly give the outputimpedance for the oscillator and to ensure that the oscillations are properlycoupled to the emitter. The oscillator circuit used is given below:

The components required for the mixer are

1. BF195 -1 no

2. 62k resistor -1 no

3. 1k resistor -1 no

4. 10k potentiometer -1 no

5. IFT -1 no

6. 0.1uF capacitor -2 nos

With these components the circuit is assembled and output is tested/observed.

9

Experimentation

The circuits designed in the previous section were assembled step by step.At first the oscillator circuit was setup and fine tuned to get an output fre-quency of 1.5 Mhz. Fine tuning was done in the sense that the inductancevalues needed to be varied from the designed values to get the required fre-quency. This was possible as a variable inductance was used through aninductance box. The smaller value of capacitance in the colpitts oscillatorcircuit used (in the picofarad range) was also slightly changed for the same.Finally it was ensured that the output of the oscillator circuit was a pure,distortion free sinusoid of 1.5 Mhz.

After this the mixer circuit given in the previous section was assembled onthe breadboard. At first only a constant value of emitter resistance was used.The IFT used was tested seperately for its tuned frequency and the clarity ofthe output. It was found to be tuned to 476 Khz. Once the circuit was setupand DC conditions verified, the output from the oscillator stage was fed to theemitter of the mixer transistor while at the same time observing the output ofthe oscillator on the oscilloscope. It was seen that as soon as the connectionbetween the oscillator and mixer stages were made the output of the oscillatorgot highly distorted. The problem was attributed to the reduced impedenceoffered by the mixer stage to the oscillator stage. To resolve this issue an 1K pot was connected in series with the fixed emitter resistance and varied toimprove the output of the oscillator stage. Adjustments were also made inthe pots in the oscillator circuit.

Once the output of the oscillator stage was made satisfactorily distortion-less the signal of the input frequency was applied at the base of the mixertransistor. In the beginning no proper output was observed. This was be-cause the difference frequency did not fall in the tuned range of the output

10

IFT. To do this the input frequency was varied starting from 900 KHz. Allthe time it was ensured that the ouptut of the oscillator was proper. Onlywhen the input frequency was 1.023 MHz was a sinusoidal signal observedat the ouptut. The frequency of the ouptut observed was 477 KHz whichwas exactly the difference frequency. Thus the mixing action was observed.But it was a very delicate arrangement as even a slight change in the inputfrequency distorted the output observed.

11


The circuits were assembled and after various troubleshooting steps the re-quired outputs were observed.

The frequency of signal produced by the local oscillator was 1.5 MHz.

The frequency of the input signal given was 1.023 MHz.

Output frequency observed was 477 KHz.

The emitter resistance used was a series combination of 2.2 KΩ and an 1KΩ variable resistor.

A sinusoidal signal of 477 KHz was observed at the output of the IFT con-nected across the collector and supply voltage.

12

Conclusion

The mixer is a very important circuit in the communication sphere inthe sense that it is capable of altering or transforming one frequency to theother. This is a necessity of paramount importance as it is desirable tooperate many circuits in the IF or some other range and when such circuitscome as a part of some bigger circuits it is imperative to alter frequencyfrom previous stages and transform them into the IF range for proper circuitoperation. Using BJT’s non linear nature to implement the same is just oneway of doing it. Even a diode can be used for the same but fine tuningand circuit improvisation would be difficult. This experiment is also a verysignificant example of how loading can adversely affect circuit operations ina big way. Troubleshooting the issue was an enlightening learning process.

13

Bibliography







14

IF AMPLIFIER

1




Ajmal V K

5th November

Contents

Introduction 2

Principle 3

Design 14

Experimentation 16


Conclusion 21

Bibliography 22

1

Introduction

One aspect of design that is integral and imperative in the design of anycommunication system is the frequency selectivity of the system proposed tobe designed. Oftentimes than not the specifications demand the emphasison certain frequency components at the output as compared to others. thatis the ouptput of certain stages of the circuit has to be frequency selective.Implementation of such a selectivity is oftentimes referred to as tuning. Thatis we tune the circuit to operate at a particular frequency. Such a block(Tuning) finds its application in almost all kinds of communication systems.One practical example would be the radio systems that have been in existencefor some time now. The type of receiver that is employed in such systems isreferred to as the superheterodyne receiver. They usually employ an tunedamplifier to amplify the Intermediate frequency range (455 KHz). Even ifwe take any general circuit design situation we would always require to giveimportance and emphasis to certain frequencies. Hence the necessity fordesigning such blocks. An intuitive exploration would immediately bring oneto the conclusion that there are only two blocks required to design a simpleyet working single stage tuned amplifier. One would need an amplifier totake the input obtained to the necessary signal levels and a tuning circuit(like an RLC circuit ) to zero in on the required frequency. It is with thisbasic idea that the design of the circuit be approached.

2

Principles

When we go about designing such a tuned amplifier circuit the first logicalstep to which we reach is selection of circuit blocks that will be required.There is the necessity for an amplifier circuit to enhance the signal frequen-cies received and then of course the tuning block for tunig the output to thedesired frequency plus or minus some allowed side frequencies. The couplingof the amplifier block and the tuning block is another point to be taken careof-in what part of the amplifier circuit should the tuning block be kept. Be-fore we go into those complexities what is required first is a detailed analysisof the tuning block itself. The section that follows gives an analysis of thecommon parallel R-L-C tuning circuit.

R-L-C Analysis

Consider a parallel R-L-C circuit combination as given below. We have tosee both the zero state response and the zero input response of the circuitto a standard input signal say an impulse.An analysis based on impulseresponse is what is particularly important as it is the only input that canexcite all the poles and zeroes of a given circuit. There is the need to seeboth the transient as well as the stable state responses.We may distinguishthe transient behavior of an electrical circuit from its steady-state, in thatduring the transients all the quantities, such as currents, voltages, power andenergy, are changed in time, while in steady-state they remain invariant, i.e.constant (in d.c. operation) or periodical (in a.c. operation) having constantamplitudes and phase angles.

The cause of transients is any kind of changing in circuit parametersand/or in circuit configuration, which usually occur as a result of switch-ing (commuta- tion), short, and/or open circuiting, change in the operation

3

of sources etc. The changes of currents, voltages etc. during the transientsare not instantaneous and take some time, even though they are extremelyfast with a duration of milliseconds or even microseconds. These very fastchanges, however, cannot be instantaneous (or abrupt) since the transientprocesses are attained by the interchange of energy, which is usually storedin the magnetic field of inductances or/and the electrical field of capaci-tances. Any change in energy cannot be abrupt otherwise it will result ininfinite power (as the power is a derivative of energy, p = dw

dt), which is in

contrast to physical reality. All transient changes, which are also called tran-sient responses (or just responses), vanish and, after their disappearance, anew steady-state operation is established. In this respect, we may say thatthe transient describes the circuit behavior between two steady-states: anold one, which was prior to changes, and a new one, which arises after thechanges.

There are basically two methods of transient analysis the classical differen-tial equation based and the transform based.Comparing the classical methodand the transformation method it should be noted that the latter requiresmore knowledge of mathematics and is less related to the physical matter oftransient behavior of electric circuits than the former.The basic RLC analyishere will be done using the classical method whereas in later sessions theanalysis of the entire tuning circuit including the amplifier stage will be doneusing the transform method so as to introduce both.

The parameters L and C are characterized by their ability to store en-ergy:magnetic energy wl = ψ.i

2= L.i2

2(since ψ = L.i), in the magnetic field

and electric energy wc = q.v2

= C.v2

2(since q = C.v ), in the electric field of

the circuit. The voltage and current sources are the elements through whichthe energy is supplied to the circuit. Thus, it may be said that an electricalcircuit, as a physical system, is characterized by certain energy conditions inits steady-state behavior. Under steady-state conditions the energy storedin the various inductances and capacitances, and supplied by the sources ina d.c. circuit, are constant; whereas in an a.c. circuit the energy is beingchanged (transferred between the magnetic and electric fields and suppliedby sources) periodically.When any sudden change occurs in a circuit, thereis usually a redistribution of energy between L− s and C − s, and a changein the energy status of the sources, which is required by the new conditions.

4

These energy redistributions cannot take place instantaneously, but duringsome period of time, which brings about the transient-state.

The main reason for this statement is that an instantaneous change of en-ergy would require infinite power, which is associated with inductors/capacitors.As previously mentioned, power is a derivative of energy and any abruptchange in energy will result in an infinite power. Since infinite power is notrealizable in physical systems, the energy cannot change abruptly, but onlywithin some period of time in which transients occur. Thus, from a physicalpoint of view it may be said that the transient-state exists in physical sys-tems while the energy conditions of one steady-state are being changed tothose of another. Our next conclusion is about the current and voltage. Tochange magnetic energy requires a change of current through inductances.Therefore, currents in inductive circuits, or inductive branches of the circuit,cannot change abruptly. From another point of view, the change of current inan inductor brings about the induced voltage of magnitude L.di

dt. An instan-

taneous change of current would therefore require an infinite voltage, whichis also unrealizable in practice. Since the induced voltage is also given as dψ

dt,

where ψ is a magnetic flux, the magnetic flux of a circuit cannot suddenlychange.

Similarly, we may conclude that to change the electric energy requiresa change in voltage across a capacitor, which is given by v = q

C, where

q is the charge. Therefore, neither the voltage across a capacitor nor itscharge can be abruptly changed. In addition, the rate of voltage change isdvdt

= 1C.dqdt

= iC

, and the instantaneous change of voltage brings about infinitecurrent, which is also unrealizable in practice. Therefore, we may summarizethat any change in an electrical circuit, which brings about a change in energydistribution, will result in a transient-state.

In other words, by any switching, interrupting, short-circuiting as well asany rapid changes in the structure of an electric circuit, the transient phe-nomena will occur. Generally speaking, every change of state leads to a tem-porary deviation from one regular, steady-state performance of the circuit toanother one. The redistribution of energy, following the above changes, i.e.,the transient-state, theoretically takes infinite time. However, in reality thetransient behavior of an electrical circuit continues a relatively very short

5

period of time, after which the voltages and currents almost achieve theirnew steady-state values. The change in the energy distribution during thetransient behavior of electrical circuits is governed by the principle of energyconservation, i.e., the amount of supplied energy is equal to the amount ofstored energy plus the energy dissipation. The rate of energy dissipationaffects the time interval of the transients. The higher the energy dissipa-tion, the shorter is the transient-state. Energy dissipation occurs in circuitresistances and its storage takes place in inductances and capacitances. Incircuits, which consist of only resistances, and neither inductances nor ca-pacitances, the transient-state will not occur at all and the change from onesteady-state to another will take place instantaneously. However, since evenresistive circuits contain some inductances and capacitances the transientswill practically appear also in such circuits; but these transients are veryshort and not significant, so that they are usually neglected.

Now, with this basic knowledge of transient responses in mind we go aboutdoing the analyis of the above given R-L-C circuit. The circuit given abovehas no driving input given to it. So the analysis that results can be calleda zero input response.Let the currents through the inductor, capacitor andresistor be il, ic and ir and the respective voltages across them be vl, vc andvr. On observing we see thatvl = vc = vr and, from KCLic + ir + il = 0

vr = R.ir (1)

orir = G.vr (2)

vl = L.di

dt, il(0) = I0 (3)

6

Therefore,

il(t) = I0 +1

L.

t∫0

vl(t).dt (4)

ic = C.dvcdt, vc(0) = V0 +

1

C

t∫0

ic(t).dt (5)

From the above given equations we take one convenient variable and writethe most convenient equation in terms of that variable and solve for the othervariables using this. If we choose the inductor current il as the variable thefollowing two equations are obtained from which we get the third secondorder differential equation.

C.dvcdt

+Gvr + il = 0 (6)

since vl = vc = vr , we get

C.dvldt

+Gvl + il = 0 (7)

and hence we get following second order differential equation

LCd2ildt2

+GLdildt

+ il = 0 (8)

The above given second order equation may be considered as the homogenouspart of a general second order equation with some finite RHS value. The RHSvalue should then intuitively correspond to any excitation to the system (that is some arbitrary source). If we adopt a generalised representation ofthe above given homogenous diffrential equation as below with arbitraryparameters α and ω0 then the equation becomes

d2ildt2

+ 2αdildt

+ ω20il = 0 (9)

Then α is called the damping factor and ω0 as the resonating frequency givenby

α =G

2C(10)

7

and

ω0 =1√LC

(11)

The solution of the given differential equation will be of the form

ih = k1.es1t + k2e

s2t (12)

where ih is the homogenous solution for the current, k1 , k2 are constantsand s1 and s2 are given by

s1, s2 = −α±√α2 − ω2 (13)

Based on the values of above two variables the circuit will demonstrate over-damped, underdamped or critically oscillating conditions. To understandthis better lets take the impulse response of the circuit.

impulse response

As described earlier with an input applied tothe previous R-L-C circuitthe differential equation gets modified only in its RHS, with the RHS’s 0replaced by the excitation applied. So for an impulsive current δ(t) appliedto the R-L-C circuit the differential equation gets modified as

LCd2ildt2

+GLdildt

+ il = δ(t) (14)

and the inintial conditions would be

il(0−) = 0,

dildt

(0−) = 0 (15)

for t > 0+ The impulse input is charecterised by the conditions for t > 0+

δ(t) = 0 (16)

8

An impulse at t = 0 creates an initial condition at t = 0+ The impulse re-sponse for t > 0 is simply the zero input response due to the initial conditionscreated by that impulse response Integerate from t = 0− to t = 0+ to get theinitial conditions we get

LCdil

dt(0+)− LC dil

dt(0−)+GLil(0

+)−GLil(0−) +

0+∫0−

il.dt = 1 (17)

The inductor current cannot jump at time zero or that the inductor cur-rent is a continuous function therefore the integeral above is zero and

il(0+) = il(0

−) (18)

If it were not continuous the derivative of the current would contain animpulse and the second derivative would contain a doublet and the secondequation above wont be satisfied. Now applying all the initial conditions weget the solution for the current as

il(t) =ω20

ωd.e−αt.sinω − dt (19)

here ω0 and α have same values as described earlier and

ω2d = ω2

0 − α2 (20)

As is evident the mathematics of the analysis has been cut short for a qual-itative description of the situation. Suppose we approximate the impulseresponse as a pulse that extends for a very short duration ∆. As ∆ tends tozero the pulse approaches an impulse Thus at t = 0+ all current from sourcegoes to capacitor. Therefore

ic(0+) = is(0

+) =1

∆(21)

ir(0+) = il(0

−) = 0 (22)

Current in the capacitor forces a gradual rise of voltage across it at an initialrate given by

dvcdt

(0+) =icC

(0+) =1

C∆(23)

9

With the assumption that in a small interval slope of the voltage curve re-mains constant. That is at time ∆ voltage reaches 1

CThen at that point

current in the resistor, ir is proportional to vc. Thus it is linear in t. And,

il ∝∫vc (24)

Thus il will be parabolic in t Now as ∆→ 0 the input becomes an impulsivecurrentvc jumps from 0 to 1

C

ic becomes an impulseir undergoes sudden change to 1

RC

il(0+) = il(0

−),dildt

(0−) = 0,dildt

(0+) =1

LC(25)

This is the general behaviour of the parallel R-L-C Circuit to an impulseresponse leaving aside the mathematical intricacies and considering it intu-itively Now the next step is to couple this tuning circuit to a basic amplifier.Let us put the tuning block at the collector of a BJT based amplifier. Oncethis is done we need to locate where the poles aqnd zeroes of the entire circuitare and how the introduction of a tuning block modifies the already existingpoles of the amplifier. To analyse this circuit we make use of the Laplacetransform method. The analysis ensues

Tuned Amplifier Analysis

For the tuned amplifier analysis, we have to make use of the incrementalmodel of the BJT.This is required to take into account the poles and zeroescontributed by the active device.For simplified analysis, the Base-Collectorcapacitance Cµ is analyzed by considering Miller effect.Miller effect intro-duces a gain dependent capacitance into the circuit.

10

Vπ(s) =I(s)

1R1

+ 1R2

+ 1rπ

+ Cπs+ C1s(26)

Vπ(s) =I(s)

1R1||R2||rπ + Cπs+ C1s

(27)

VO(s) =−gmVπ(s)

1R

+ 1Ls

+ C2s(28)

VO(s) =−gmI(s)

( 1R

+ 1Ls

+ C2s)(1

R1‖R2‖rπ + Cπs+ C1s)(29)

Here C1 is the miller capacitance

C1 = Cµ(1 + gmXLoad)

where

XLoad =1

1R

+ 1Ls

+ C2s

TakeReq = R1 ‖ R2 ‖ rπ

=⇒ VO(s) =−gmI(s)

( 1R

+ 1Ls

+ C2s)(1Req

+ Cπs+ Cµs(1 + gmXLoad))(30)

=⇒ VO(s) =−gmI(s)

( 1R

+ 1Ls

+ C2s)(1Req

+ Cπs+ Cµs(1 + gm1

1R+ 1Ls

+C2(s)))

(31)

Now we know that

H(s) =VO(s)

I(s)(32)

∴ H(s) =VO(s)

I(s)=

−gmRLs(RLC2s2 + Ls+R)( 1

Req+ Cπs+ Cµs) + gmCµRLs2

(33)

H(s) =−gmRLs

RLC2

Reqs2 +RLC2(Cπ + Cµ)s3 + L

Reqs+ L(Cπ + Cµ)s2 + R

Req+R(Cπ + Cµ)s+ gmCµRLs2

11

H(s) =−gmRLs

RLC2(Cπ + Cµ)s3 + (RLC2

Req+ L(Cπ + Cµ) + gmCµRL)s2 + ( L

Req+R(Cπ + Cµ))s+ R

Req

From the expression for transfer function that has been obtained it is clearthat the system under consideraton has one zero and three poles.The zero isat s = 0.To find the nature of the poles consider the coefficients of sn in thefollowing expression:

RLC2

Req

s2+RLC2(Cπ+Cµ)s3+L

Req

s+L(Cπ+Cµ)s2+R

Req

+R(Cπ+Cµ)s+gmCµRLs2

(34)All the coefficients are real and positive for the present system.If there isa complex root for this equation, another root will be its complex conju-gate.The circuit contains poles from both the RLC circuit as well as theamplifier circuit.These poles of the circuit can be expressed as a + jb,a − jband c.The expression becomes

∴ (s− a− jb)(s− a+ jb)(s− c) (35)

=⇒ ((s− a)2 + b2)(s− c)6 (36)

=⇒ (s2 − 2as+ a2 + b2)(s− c) (37)

=⇒ s3 − 2as2 + a2s+ b2s− cs2 + 2acs− ca2 − cb2 (38)

=⇒ s3 − (2a+ c)s2 + (a2 + b2 + 2ac)s− c(a2 + b2) = 0 (39)

Since all the coefficients are positive for the present system, we have

−(2a+ c) > 0 (40)

a2 + b2 + 2ac > 0 (41)

−c(a2 + b2) > 0 (42)

From equations (15) and (17)

=⇒ 2a+ c < 0, c < 0 (43)

=⇒ a < − c2

(44)

Also=⇒ a2 + b2 > −2ac (45)

12

The pole ‘c’ lies on the left hand side of the jω axis.The value ‘a’ can beany value less than − c

2.That it can either be positive,zero or negative.If ‘a’

is positive then the poles a+ jb and a− jb lie on the right side of the jω axisand the system becomes unstable.If ‘a’ = 0, the poles lie on the jω axis.Thesytem will now oscillate.For the system to be stable, we have to restrict thepoles to the left hand side of the jω axis.Hence ‘a’ has to be > 0 for stableoperation.

In the tuned amplifier incremental model we have analyzed, it was foundout that the real pole ‘c’ depended on the gain of the amplifier.So it ispossible for the pole to migrate to the right hand side of the jω axis andhence cause unstability.The root-locus plot of the transfer function givesthe overall picture of the system behaviour.To examine this behaviour - orthe impulse response - of the system ,the output of the tuned amplifier fordifferent gains is to be observed. The variation in gain is bound to changethe position of the poles and hence the nature of the output response.

13

Design

From the earlier discussion, it is clear that the circuit we need is a amplifiercircuit with an R-L-C circuit at the load, designed such that its gain can beadjusted.One of the methods used for controlling the gain is feedback.Thefeedback from the emitter stabilizes the amplifier and varying this feedbackcan help vary the gain.Partial feedback method is employed in this circuit.Thedesign of the amplifier is as follows:Take VCC = 9V,IE = 2mAFrom the dc analysis of the circuit,we have a short between the collector andthe power supply due to the inductor.Taking VCE = 0.5 VCC ,we get

VCC − VCE − IERE = 0 (46)

=⇒ RE = 2.2KΩ (47)

Now R1 and R2 are designed so as to provide the necessary bias.

R1

R2

=VCBVBE

(48)

∴R1

R2

=3.8

5.2(49)

=⇒ R1 = 1.8kΩ, R2 = 2.2kΩ (50)

The emitter resistor along with the potentiometer is used to provide thepartial feedback which modifies the gain.Making use of a variable resistanceat the emitter will allow varying of the gain to some extent. The bypasscapacitor value is 0.1µF .An Intermediate Frequency Transformer is used inthe place of the load.The IFT comprises of a transformer and a capacitortuned to an intermediate frequency,preferably 455KHz.This internal L-C

14

circuit also has some inherent resistance which acts as the resistive componentfor realizing the required R-L-C circuit load.


For the tuning amplifier circuit the components required are:

1. 1 high frequency transistor :BF195

2. 1 IFT for the R-L-C load

3. 2 Resistors of 2.2 KΩ

4. 1 potentiometer of 1 K Ω

5. 1 capacitor of 0.1 µF for providing partial feedback

15

Experimentation

As per the design described above the circuit was planned to be assembled.At the lab the circuit was assembled in stages. Since the amplifier was totune to the Intermediate Frequency (455 Khz) an IFT was planned to bekept as the tuning block described earlier. First the IFT obtained was testedseperately to ensure that it was tuning and not giving just some responseto an input that increased with frequency. The tuning frequency of the IFTwas found out to be 420 KHz. Next the amplifier circuit using BF195 wasassembled with the designed values of resistors and capacitors. Betweenthe collector of the transistor and the supply voltage the secondary of theIntermediate Frequency Transformer was connected. After this the supplyvoltage was given and the DC conditions of the transistor was checked. Nowa very small input of 100 mV peak to peak was given at the base of thetransistor. The input frequency was gradually increased from a few Hertz.The output voltage was very small (almost negligible ) for almost all thefrequencies. A very high gain (160) was obtained at 408 KHz. this was closeto the tuning frequency of the IFT ( 420 KHz ). A Very small yet significantpeak was observed at 205 KHz. But this was found out to be due to the sidelobes of the frequency response function. With this we got an idea about thelocation of the poles and zeroes.

Once the basic working of the IF amplifier was verified and observed itwas time for proper testing. What was required was to observe the dampingeffect in the output or the ringing. As discussed earlier there are three polesin the circuit. One pole located on the real axis and two on the left half of thes plane. Now from the expressions of each poles it is clear that the real axispole has a dependancy on the miller capacitance which inturn depends on thetransconductance gm of the transistor and hence it’s gain. So theoriticallyadjusting the gain of the circuit would move this pole along the real axis

16

which would inturn move the other two poles. The other two poles have aweak and opposite dependancy on the gain of the transistor arising from theother miller capacitance put across the collector and emitter. The effect ofthis capacitance is usually ignored in all analysis. So the horizontal motionof these two poles dominates the vertical motion. This is graphically shownbelow.

Now when the gain of the amplifier circuit is increased, owing to the inversedependance of the real axis pole on gain, the pole moves towards the origin orwe can say its magnitude decreases. As a result the other two poles move inthe oppositte direction that is away from the imaginary axis. The distance ofthe two conjugate poles from the imaginary axis is a measure of the dampingintroduced in the circuit. As these poles determine the tuning frequencyof the circuit as well their location from the imaginary axis determine thedecay rate of damped oscillations when they occur. Continuing, as the polesmove away from the imaginary axis on increasing the gain the damping factorincreases, the decay rate increases and hence the time constant of dampedoscillations decreases. This was observed in the lab by giving a pulse inputat comparitively lower frequencies (10-20 KHz) to observe the ringing causeddue to damped oscillations. The time constant of the decay was observed.The opposite phenomenon was also observed. That is the gain was decreasedand as a result the decay rate decreased and the time constant increasedowing to the movement of the conjugate poles towards the imaginary axis.Asa critical condition when the gain was continuously decreased the systembecame unstable and sustained oscillation was observed at one point. This

17

was due to the crossing over of the conjugate poles to the right half plane.Measurements with the increased time constant and reduced gain were alsotaken tabulated. Thus an exhaustive experimental study of the assembledcircuit was done and the correlation with derived theoritical points wereverified.

18


The designed circuit was assembled in the lab and the dc conditions wereverified. To this a small input signal of 100 mV peak to peak amplitude wasgiven from a very low frequency. The frequency was then gradually increasedand output observed.

Proper sinusoidal output of gain 160 was observed at 408 KHz. The normaltuning frequency of the IFT was 420 KHz. Only one very small side lobewas observed at 205 KHz

Pulse input at a lower frequency of 45 KHz was given to observe the dampingand ringing effects. The outputs observed were like those given below

19

The gain of the amplifier was varied to study the pole movements. Themovements were verified by measuring the decay rate and time constantsof the damped oscillations observed on the CRO. The tabulated results aregiven below.

For all the damped oscillations observed the frequency was nearly equalto 406 KHz.

20

Conclusion

The design and implementation of the intermediate frequency amplifierwas performed by analyzing the basic behaviour of a R-L-C circuit first aloneand with an amplifier.The process progressed by observing the behaviour ofthe poles and zeroes of the system.The different possible responses of thesystem for different configurations were examined.The pulse testing of thecircuit gave a picture about the present state of the system.The ringing inthe output was observed and analyzed qualitatively and quantitavely.Thegradual changes in decay rates and damping constants with variations ingain were observed.

21

Bibliography

[1] John G Proakis, Masoud Salehi. Communication Systems Engineer-ing,Prentice Hall,1994.

[2] Charles A. Desoer, Ernest S. Kuh.Basic circuit theory,Tata McGraw-Hill,1984

[3] Donald L Schilling,Charles Belove.Electronic Circuits, Discreet andIntegerated,McGraw-Hill, 1979



[6] Paul E Gray , Campbell L Searly Electronic Principles: Physics, Modelsand Circuits

22

FM GENERATION

1




Ajmal V K

21st October

Contents

Introduction 2

Principles 3

Design 6

Experimentation 9


Conclusion 12

Bibliography 13

1

Introduction

The concept of modulation is the backbone of analog communicationprinciples.The need for modulation is obvious from the power, SNR andbandwidth considerations in signal theory.Fourier analysis enables any infor-mation signal to be broken down to a series of sinusoidal variations, addedup after multiplying with corresponding weights. Each distinct component ischaracterized by two parameters - the amplitude and the angle. The descrip-tion of the signal is obtained from the values of the amplitude and angle.Theparameter angle can be resolved into the function of two new parameters -frequency and phase.So for modulating a given signal, we can either vary theamplitude or angle of the carrier with the modulating signal.That is bothamplitude and angle modulation is possible.

The amplitude modulation scheme was discussed by considering the DS-BSC and DSBFC AM. To implement angle modulation, we can vary thefrequency of the carrier signal with the message signal. Such a modula-tion scheme is called frequency modulation.Frequency modulation is morecomplex than amplitude modulation. This arises from the higher number offrequency components generated in frequency modulation.

2

Principles

Consider a signal s(t) given by

s(t) = AssinΘ(t) (1)

here, As is the amplitude and Θ(t) is the angle of the signal.In amplitudemodulation schemes, we had varied the parameter As with the message sig-nal.Now by varying the value of the parameter Θ with the message signal,we can generate angle modulated waves.This variation can be done in twoways :

Θ(t) = k1.m(t) + ωst (2)

ordΘ(t)

dt= k2.m(t) + ωs (3)

where k1 and k2 are constants.The former gives rise to phase modulated signal whereas the latter gives riseto frequency modulated signal. Now, consider the scenario where the signals(t) is obtained by frequency modulating a carrier wave c(t) with a messagesignal m(t).The message signal is given by

m(t) = Amsinωmt = Amsin(2πfmt) (4)

and the carrier signal

c(t) = Acsinωct = Acsin(2πfct) (5)

Then the angle modulated signal is given by

s(t) = Acsin(ωct+ φ(t)) (6)

3

where φ(t) is a function of the modulating signal m(t).More specifically, fora frequency modulated wave,from equation (3), we get the phase φ(t) variesas

dφ(t)

dt∝ m(t) (7)

∴ r(t) = Acsin(ωct+ k3

t∫−∞

m(τ)dτ)) (8)

where k3 is a constant.That is, the frequency deviation is proportional to m(t).Now for realizing aFM generator, a circuit which can vary the frequency with the message signalis required.Consider the circuit given below: The circuit is a LC tank circuit with an

inductor of value Lo , a fixed capacitor of value Co and a variable capacitanceof value C.The frequency of oscillation of this tank circuit is given by

f =1

2π√Lo(Co + C)

(9)

Now assume that the value of the capacitance varies with the message signalm(t).That is,

C = k.m(t) (10)

=⇒ f =1

2π√LoCo

.1√

1 + kCom(t)

(11)

Now, the frequency of the tank varies with the message.Hence we get afrequency modulated output from the LC oscillator.

For realizing a capcitive impedance exhibiting this behaviour, we can makeuse of an active device such as a BJT.Consider the circuit given below: Fromthe circuit,

Vπ =Vo × Z2 ‖ rπZ2 ‖ rπ + Z1

(12)

4

AlsoVo = −IoRc (13)

Vo = Vi.Z2 ‖ rπ + Z1

Z2 ‖ rπ(14)

Now, Io = gmVi and Vi=Iogm

Vo =Iogm.Z2 ‖ rπ + Z1

Z2 ‖ rπ(15)

=⇒ Zo =1

gm.Z2 ‖ rπ + Z1

Z2 ‖ rπ(16)

Let Z1 be capacitive. =⇒Z1=−jXC and Z2‖rπ Z1

=⇒ Zo =−jXC

gmZ2 ‖ rπ(17)

Take Z2 as purely resistive.

=⇒ Xeq =XC

gmR(18)

This circuit gives the required capacitive impedance.So we can provide thisreactance in parallel to an oscillator to obtain the required FM circuit.

5

Design

The design is done by providing the varying capacitive reactance from amodified BJT amplifier to a colpitts oscillator,whose design had been doneearlier. Take gain,Av = 200.Also,

Xc R (19)

=⇒ 1

Cω= 10R. (20)

Take R =1KΩ

=⇒ 1

C × 2× 2π × 103= 10× 1× 103. (21)

=⇒ C = 7.9nF. (22)

The dc biasing is designed as in the case of the Mixer circuit.BF195 high frequency transistor is used.The circuit will be as below:

6

let ICQ= 4mA ( for BF195) For the transistor BF195,typical β = 60.

∴ gm =ICQ

VT= 0.1538 (23)

∴ Rc = 1.164kΩ ≈ 1.2kΩ (24)

Now if we take thevenin equivalent of R1 and R2 at the base of thetransistor we get the base emitter equations as :

VTH −ICβ.RTH − VBE(ON)− β + 1

βICRE = 0 (25)

VCC = 12V and VCE = 6V (the fixed Q point)Ve =0.1VCC = 1.2V∴ RE = 300Ω∴ RE ≈ 330ΩNow

VCCR2

R1 +R2

− ICβ

R1R2

R1 +R2

− β + 1

βRCRE = 0 (26)

For stability, RTH = 0.1(β + 1)RE

From these conditions, we get R1 ≈ 47KΩ, R2 ≈ 10KΩNow AV > C2

C1.

∴ let us fix C2

C1= 100 (as AV = 185)

That is, CT = C1C2

C1+C2.

Therefore C1 = 11pF, C2 = 1.1nF.Standard values C1 = 12pF, C2 = 1nF.All other external capacitances = 0.22 µFOnce the calculation of the values of components is done, the design is com-plete.The values of the components used in the lab areR1 = 47KΩ, R2 = 10KΩ in series with 1KΩ pot.C1 = 4.7pF, C2 = 1nF.L = 60 µHRC = 1.2K Ω, RE = 330 ΩCoupling and Bypass capacitances = 0.22 µF

7

The oscillator circuit used is the same as in the mixer design and is givenbelow:

For the FM generator, the following components were used.

1. BF195 -1 no

2. 47 kΩ resistor -1 no

3. 10 kΩ resistor -1 no

4. 1.2 kΩ resistor -1 no

5. 330 Ω resistor -1 no

6. 10 kΩ potentiometer -1 no

7. 1 kΩ potentiometer -1 no

8. 0.22 µF capacitor -3 nos

9. 7.9 nF capacitor -1 nos

The above components were assembled and the circuit was tested.

8

Experimentation

The circuit as described and designed in the previous sections was as-sembled. First the oscillator circuit was assembled and the variable resistorsin the circuit was adjusted to obtain a distortion free sinusoidal signal offrequency 3.7 MHz. After this the reactance modulating circuit was assem-bled. First the dc conditions of this circuit were checked to be concordantwith the designed values. Next this circuit was tested for its basic ampli-fier action without connecting the external impedances that make its outputimpedance purely reactive. Next the external impedances were connectednamely a resistance from base to ground and a capacitance across the baseand collector. To the collector of this circuit the output of the oscillatorcircuit was coupled. The output of the oscillator suddenly got distorted onconnecting. As in the previous experiments the biasing of the amplifier wasadjusted to bring the output of the oscillator back to distortion- free state.Next the required message signal was given across the base biasing resistorR2 and ground. Now the output of the output was observed.

It was observed that at high frequencies not much difference from normaloscillator output was observed. Then the frequency was consistently reduced.On bringing it into Hertz range the output started wobbling indicating a fre-quency variation. Now the frequency was slightly increased. After sufficientlyincreasing the frequency the output was observed. Near the maximum am-plitude points of the message signal the output was shifting and changingfaster than in the other parts indicating an increased frequency and hencefrequency modulation. Now to confirm the above inference a square wavewas given as input. Since the square was has only two discreet voltage levels,frequency modulation of such a wave would effectively result in an outputwith only two frequencies. This was clearly observed on the oscilloscope.

9

Next a ramp signal was given as the input. For a ramp the amplitude goeson linearly increasing and after its maximum value it suddenly drops to. Soits frequency modulated wave would be a signal with continuously increasingfrequency upto a certain point and then sudden change in frequency to a lowvalue. A similar result was observed on the oscilloscope on applying an inputramp.

Thus by applying different kinds of input signals the characteristics offrequency modulation was clearly observed and studied.

10


The circuits were assembled and after various troubleshooting steps the re-quired outputs were observed.

The frequency of signal produced by the local oscillator was 3.7 MHz.

The frequency of the input signal was varied from 1 Hz to 5 KHz

Output waveform observed was frequency modulated, showing pronouncedwobbling at very low frequencies.

The output waveforms for various amplitude variations - square,ramp andsinusoidal waveforms - were observed.

11

Conclusion

Frequency modulation requires the mapping of voltage levels to frequencyvalues.Such a mapping can be obtained in different ways.This can be achievedby employing a circuit that performs reactive modulation.The frequencymodulation generator was designed using a colpitts oscillator and a modi-fied BJT amplifier.The idea is to introduce a variable reactance into a LCoscillating circuit in such a way that the variable reactance ∝ message.Thedesign of the varying reactance makes use of network analysis techniques tosynthesize a viable solution.

12

Bibliography









13

am generation

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