Angle Modulation

Introduction

There are three parameters of a carrier that may carry information:AmplitudeFrequencyPhase

Frequency and Phase modulation are closely related and grouped together as Angle modulation

Frequency Modulation

In this the instantaneous frequency of the carrier is caused to vary by an amount proportional to the amplitude of the modulating signal. The amplitude is kept constant.

Simple FM generation

Resistant to Some Noise

time

Some key Observations

More complex than AM this is because it involves minute changes in frequency

Power in an FM signal does not vary with modulation

FM signals do not have an envelope that reproduces the modulation

FM is more immune to effects of noise

Frequency Deviation

Frequency deviation of the carrier is proportional to the amplitude of the modulating signal as illustrated

e(t) = E sin (t +)

Emkf Maximum freq deviation

kf= frequency deviation/V = kf kHz/V

•This shift in frequency compared with the amplitude of the modulating voltage is called the deviation ratio.

•ExampleGiven that the deviation constant is 1kHz/10mV, what is the shift in frequency for a voltage level of 50 mV?

•Frequency deviation =

Deviation ratio Kf

kHz510150

m

mcCFM

ftSintSinEe

tSinmtSinEe mfcCFM

f

m

mf

Where modulation index is

Emkf Maximum freq deviation

Mathematical Expression for FM

e(t) = E sin (t +)

m

m

cCFM

ftSinEK

tSinEe mf

% modulation FM

% Modulation = Actual freq deviation/ allowed freq deviation

Example:An FM broadcast-band transmitter has a peak deviation of ±60 kHz for a particular input signal. Determine the percentage of modulation.

-Frequency Deviation is maximum departure of instantaneous freq. of FM wave from career frequencyMaximum Freq of FM is fmax= fc+ is independent of modulating freq. and proportional to only amplitude of information

KfEm

f

m

mf

Modulation index is proportional to deviation

and inversely proportional to modulating freq.

This decides the BW of the FM wave also decides the no of side bands In FM the modulation index can be greater than 1

Important Definitions

DR =Maximum deviation /maximum modulating freq in FM is 75KHz

is 15KHz

maxmax

mfDR

max

maxmf

Deviation Ratio

Examples

In an FM system when the audio frequency is 300 Hz and the audio voltage is 2.0V, the deviation is 5kHz. If the audio voltage is now increased to 6V what is the new deviation? If the voltage is now increased to 9V and the frequency dropped to 100Hz what is the deviation? Find the modulation index in each case.

VkHzV

V

m

m

/5.225

15kHz62.5 6vV

when

22.5kHz92.5 9vV

when

67.163.0

5

mf f

m 503.0

15

mf f

m 225

1.05.22

m

f fm

Find the carrier and modulating frequencies, the modulating index, and the max. deviation of an FM wave below. What power will the wave dissipate in a 10 ohm resistor?

ttv 1250sin5106sin12 8

Compare this with:

tf

tAv mm

c sinsin

MHz5.952

1062

8

ccf 9Hz19

21250

2

m

mf

Modulating index =5 as given.

Power,W

RVP rms 2.7

1072

102

122

Frequency spectrum of FM Wave –

Sin of sin function is solved by Bessel function tSinmtSinEe mfcCFM

.......44

3322

4

3

2

1

0

tSintSinmJtSintSinmJtSintSinmJ

tSintSinmJtSinmJEe

mcmcf

mcmcf

mcmcf

mcmcf

cfCFM

Fc+3fm

J0EcJ1Ec

J2EcJ3Ec

J1EcJ2Ec

J3Ec

fc Fc+fm

Fc+2fm

Fc-fm

Fc-2fm

Fc-3fm

Modulation index

Jn

Carson’s Rule

This is an approximate method used to predict the required bandwidth necessary for FM transmission

maxmax2 sfBW

About 98% of the total power is included in the approximation.

What bandwidth is required to transmit an FM signal with intelligence at 12KHz and max deviation 24 kHz

21224

m

f fm

Consult Bessel function table to note that for modulating index of 2, components which exist are J1,J2,J3,J4 apart from J0.

This means that apart from the carrier you get J1 at +/-10kHz, J2 at +/- 20kHz, J3 at +/- 30kHz and J4 at +/- 40 kHz.

Total bandwidth is therefore 2x40=80kHz.

ttv 38 102sin2104sin60

For an FM signal given by

the carrier frequency•the transmitted power•the modulating index•the intelligence frequency•the required bandwidth using Carson's rule and tables•the power in the largest and smallest sidebands

If this signal is input into a 30 ohm antenna, find

In both systems a carrier wave is modulated by an audio signal to produce a carrier and sidebands. The technique can be applied to various communication systems eg telephony and telegraphy

Special techniques applied to AM can also be applied to FM

Both systems use receivers based on the superheterodyne principle

AM Vs FM systems

•In AM, the carrier amplitude is varied whereas in FM the carrier frequency is varied

•AM produces two sets of sidebands and is said to be a narrowband system. FM produces a large set of sidebands and is a broad band system

•FM gives a better signal to noise ratio than AM under similar operating conditions

•FM systems are more sophisticated and expensive than AM systems

TransmittersIn an AM transmitter, provision must be made for varying the carrier amplitude whilst for FM the carrier frequency is varied.

AM and FM modulators are therefore essentially different in design. FM can be produced by direct frequency modulation or by indirectly phase modulation.

The FM carrier must be high usually in the VHF band as it requires large bandwidth which is not available in the lower bands.

ReceiversThe FM and AM receivers are basically the same, however the FM receiver uses a limiter and a discriminator to remove AM variations and to convert frequency changes to amplitude variations respectively. As a result they (FM) have higher gain than AM.

FM receivers give high fidelity reproduction due to their large audio bandwidth up to 15 kHz compared with about 8 kHz for AM receivers.

Frequency Modulation Index

Another term common to FM is the modulation index, as determined by the formula:

mf f

m

Phase Modulation

In phase modulation, the phase shift is proportional to the instantaneous amplitude of the modulating signal, according to the formula:

mp ek

Relationship Between FM and Phase Modulation

Frequency is the derivative of phase, or, in other words, frequency is the rate of change of phase

The modulation index is proportional to frequency deviation and inversely proportional to modulating frequency

Modulating Signal Frequency

Converting PM to FMAn integrator can be

used as a means of converting phase modulation to frequency modulation

]...}4sin4[sin

]3sin3[sin

]2sin2[sin

]sin[sin

sin{

4

3

2

1

ttmJ

ttmJ

ttmJ

ttmJ

tmJAv

mcmcf

mcmcf

mcmcf

mcmcf

cfo

This solution may be shown to be given by

To evaluate the individual values of J is quite tedious and so tables are used.

Observations

•Unlike AM where there are only three frequencies, FM has an infinite number of sidebands•The J coefficients decrease with n but not in any simple form and represent the amplitude of a particular sideband. The modulation index determines how many sideband components have significant amplitudes•The sidebands at equal distances from fc have equal amplitudes•In AM increase depth of modulation increases sideband power and hence total transmitted power. In FM total transmitted power remains constant, increase depth of modulation increases bandwidth•The theoretical bandwidth required for FM transmission is infinite.