CHAPTER 5 ANGLE MODULATION Prepared by Dr M

CHAPTER - 5 ANGLE MODULATION Prepared by Dr M. Murugappan

COURSE CONTENT Discuss and analyze Frequency Modulation (FM), Phase Modulation (PM), Bessel Function. Narrowband wideband Power Bandwidth and Distribution Explain the generation and detection circuit of FM and Discuss the application of FM.

TOPICS COVERED IN THIS CHAPTER Introduction to Angle Modulation in Frequency and Time domain Mathematical models of Phase and Frequency Modulation Analysis of FM signals Frequency spectrum of FM signals Narrow Band FM signal Power in FM signal Generation

INTRODUCTION Angle modulation is the process by which the angle (frequency or phase) of the carrier signal is changed in accordance with the instantaneous amplitude of modulating or message signal. Used for : Commercial radio broadcasting Television sound transmission Two way mobile radio Cellular radio Microwave and satellite communication system

ADVANTAGES OF ANGLE MODULATION OVER AM: Freedom from interference: all natural and external noise consist of amplitude variations, thus receiver usually cannot distinguish between amplitude of noise or desired signal. AM is noisy than FM. Ø Operate in very high frequency band (VHF): 88 MHz-108 MHz Ø Can transmit musical programs with higher degree of fidelity. Ø

Carrier Resting fc Increasing fc Decreasing fc Increasing fc Resting fc Modulating signal FM

ANGLE MODULATION In angle modulation, information is embedded in the angle of the carrier. Results whenever the phase angle θ of a sinusoidal wave is varied with respect to time Angle modulated wave expression: ---Eqn (1) m(t) = angle modulated wave Vc = peak carrier amplitude (volts) ωc = carrier radian frequency ф(t) = instantaneous phase deviation (rad) Instantaneous change in the phase of the carrier at a given instant of time and indicates how much the phasor of the carrier

ANGLE MODULATION θ(t) as a function of the modulating signal if vm(t) = modulating signal the angle modulation: Eqn (2) θ(t) = F[vm(t)] - - - - Where vm(t) = Vmsin(ωmt) ω mt = angular velocity (2πfm) fm = modulating signal frequency Vm = peak amplitude of the modulating signal

ANGLE MODULATION Angle Modulation Frequency Modulation (FM): Varying the frequency of a constant - amplitude carrier directly proportional to the amplitude of the modulating signal at a rate equal to the frequency of the modulating signal Phase Modulation (PM): Phase Modulation (PM) Varying the phase of a constant-amplitude carrier directly proportional to the amplitude of the modulating signal at a rate equal to the frequency of the modulating signal Difference between FM and PM: which property of the carrier (freq or phase) is directly varied by the modulating signal and which property is indirectly varied

ANGLE MODULATION IN FREQUENCY DOMAIN Angle modulated signal fc m[t] in frequency domain is changed when acted on by a modulating signal – vm[t] The relative displacement of the carrier freq. in Hz in respect to its un-modulated value – frequency deviation (Δf) The magnitude and direction of the freq. shift (Δf) is proportional to the amplitude and polarity of the

ANGLE MODULATION IN FREQUENCY DOMAIN The rate at which freq. changes are occurring is equal to the freq. of the modulating signal (fm) Example positive modulating signal produces an increase in frequency negative modulating signal produces a decrease in frequency

ANGLE MODULATION IN TIME DOMAIN Phase (θ) of the carrier is changing proportional to the amplitude of the modulating signal - vm[t] The relative angular displacement (shift) of the carrier phase in radian in respect to the reference phase – phase deviation (Δθ) The magnitude of the freq & phase deviation is proportional to the amplitude of the

ANGLE MODULATION FM – maximum frequency deviation (change in the carrier freq. ) occurs during the maximum positive and negative peaks (zero slope) of the modulating signal (freq. deviation is proportional to the amplitude of the modulating signal) PM – the maximum freq. deviation occurs during the zero crossings (maximum slope) of the modulating signal (the freq. deviation is proportional to the slope or first derivative of the modulating signal)

COMPARISON BETWEEN AM, PM & FM

ANGLE MODULATION Carrier Signal Modulating Signal Frequency Modulation Phase Modulation

Mathematical Representation of Angle Modulation It is shown that information signal, vm(t) can be transmitted with the amplitude of the carrier signal is held constant and the angle either the phase or frequency of the carrier is varied linearly with the information signal, vm(t). Let the unmodulated carrier signal: - - - - Eqn (3) - - Eqn (4) And the instantaneous angle value: c(t) i(t) Ec ct c(t)

ANGLE MODULATION Angular frequency ωc is the average rate of change of phase Instantaneous frequency or Instantaneous Angle Frequency is the derivative of phase with respect to time. Phase is the integral of instantaneous frequency. Therefore, the instantaneous angle frequency and instantaneous angle value are given by: - - Eqn (5) From Eqn (5): - - Eqn (6) Or - - Eqn (7)

PHASE MODULATION (PM) Ø PM implies that the phase deviation of the carrier, c is proportional to the modulating signal, vm(t): - - - - Eqn (8) Therefore, Eqn(4) is => - - - - Eqn (9) Ø where kp is the phase deviation constant in radians/sec/volt Ø And the instantaneous angle frequency (it’s a derivative of phase with respect to time) from Eqn (5): - - - - Eqn (10) Therefore: - - - Eqn (11)

FREQUENCY MODULATION (FM) Ø FM implies that the frequency deviation of the carrier, is proportional to the modulating signal, vm(t): ==> - - - Eqn (12) Ø where kf is the frequency deviation constant in radians /volt Integrate: Eqn(6) => Sub Eqn (12) in Eqn(6) => Therefore FM signal : - - - Eqn (13)

FREQUENCY MODULATION (FM) Ø Assuming that the modulating signal, vm(t): - - - Eqn (14) Ø Substitute in the equation Eqn (13): - - - Eqn (15)

Take: rad/s, as a maximum frequency deviation - - - Eqn (16) • Define the modulation index as a ratio of maximum frequency deviation to modulating signal frequency: - - - Eqn (17) Narrowband FM: << 1 Wideband FM: >> 1 • Hence equation FM yields: - - - Eqn (18)

Trigonometric identities: A= B= • Hence : - - - Eqn (19) Where cos[βsin(ωmt)] and sin[βsin(ωmt)] is a trigonometric series called as Bessel Function (Fungsi Bessel). Expand using Fourier series yields: n = even - - - Eqn (20) n = odd - - - Eqn (21)

Using Bessel identities. Sub : Eqn (20) and Eqn (21) in Eqn (19) ;

Hence FM equation also known as WBFM: - - - Eqn (22) Symmetry at x axis Symmetry at y axis Expand the equation yields : Carrier band Sideband 1 Sideband 2 Sideband 3 Sideband 4 Sideband n

FREQUENCY SPECTRUM OF FM SIGNAL The number of sidebands depend on value: β = 0. 25 BW β=2 BW=2 nfm=8 fm β=5 BW=2 nfm=16 fm

Bessel Function Table

Summary of FM spectrum: Frequency spectrum consists of carrier component at fc and also sideband at fc±nfm where n is an integer (n = 1, 2, 3, …) The number of sideband depends on modulation index value, β. Magnitude of carrier signal (represented by J 0 value) decreases as β increases. Amplitude of the frequency spectrum depends on value of Jn(β). The bandwidth of modulated signal increases when index modulation, β increases. BW > 2∆fm is expected.

RELATIONSHIP BETWEEN FM AND PM In frequency modulation, the phase angle varies linearly with the integral of modulating signal vm(t). In phase modulation, the phase angle varies linearly with modulating signal vm(t). This suggests that FM can be obtained from a PM Modulator and vice versa. vm(t) Integrator PM Modulator v. FM(t) Generation of FM Differentiator vm(t) FM Modulator v. PM(t) Generation of PM From the above block diagrams, it can be shown that the generation of FM and PM signals are mutually related. Pemodulatan Sudut

v. FM(t) PM Demodulator Differentiator vm(t) FM Demodulator v. PM(t) FM Demodulator Integrator vm(t) PM Demodulator Demodulation process is used to get back the information signal. For FM demodulator in order to get back information signal from FM signal : PM modulator is used and the signal is pass through differentiator. In contrast for PM demodulator : FM demodulator is used and the signal is pass through the integrator. This shows the close relationship between FM and PM. Hence we can discuss only either one technique in angle modulation. Pemodulatan Sudut

NARROW BAND FM (NBFM) For FM signal with the small index modulation i. e β < 0. 2, is called Narrow Band FM For FM signal that we have studied previously also known as WBFM and the equation is given by : (from slide ) Let : Hence, the equation yields: NBFM with β = small , therefore; We know that -1<=sin t <=1

; • Therefore : and • Hence NBFM equation yields : • Compared with am. DSB-FC signal: (from chapter AM) • It is shown from both equations for NBFM and AMDSB-FC consist of one carrier component and two sidebands components. But LSB component for NBFM the phase shift is varies for 90° (quadrature).

POWER IN FM SIGNAL Power signal depends on the amplitudes but not on the frequencies. The amplitude of the FM signal is constant and therefore the power transmitted depends only on the amplitudes of the signal. It does not depends on the modulation index. For AM signal the power transmitted depends on the modulation index. It can be seen from the Bessel equation: In other word, the total power of FM signal consists of the power in carrier component and all the power in the sidebands. Why there is “ 2”?

FM equation is given by: (from slide 10) And therefore the total power transmitted :

Ex. 1 : A carrier with a peak value of 2000 V is frequency modulated with a message signal of 5 k. Hz. The modulation index obtained is 2. Calculate the average power in: (i) Lowest sideband (ii) Highest sideband. Given R = 50 Ω. Solution : For β = 2 from Bessel table : The highest sideband is : The lowest sideband is : (i) => (ii)

Ex. 2 : (a) Determine the BW required to transmit FM signal when the modulating frequency, fm = 10 k. Hz and maximum frequency deviation is 25 k. Hz. Solution : From Bessel table the components obtained is J 0 , J 1 , J 2 , J 3 , J 4 and J 5 That means J 1 will be at 10 k. Hz, J 2 at 20 k. Hz, J 3 at 30 k. Hz etc. Therefore BW = BFM = 2 nfm = 2 x 5 x 10 = 100 k. Hz Amplitud J 1 J 0 J 5 fc-fm fc fc+fm fc+2 fm f (k. Hz)

(b) Repeat (a) with fm = 5 k. Hz , and maximum frequency deviation is 20 k. Hz. Solution : From Bessel table the highest component is J 7 Therefore BW = 2 x 7 x 5 = 70 k. Hz

Ex. 3 : A FM signal, 2000 cos (2π x 108 t + 2 Sin π x 104 t) is transmitted using an antenna with the resistance of 50 Ω. Determine (i) Carrier frequency (ii) Modulation index (iii) Information signal (iv) Power transmitted (v) Bandwidth (vi) Power in highest and lowest sidebands Solution : The basic Eqn of FM is By Comparing the given Eqn with basic FM Equation (i) fc = 2π108 / 2π Hz = 100 MHz P = (0. 58 x 2000/ 2)2 / 50 Ω (iii) fm = π 104 / 2 π = 5 k. Hz (iv) Ec = 2000 V => Ec (rms) = 2000 / 2 = 13. 27 k. W for only one sideband Two sidebands = 2 x 13. 27 k. W = 26. 54 k. W Power transmitted, PT = V 2 (rms) / R = (2000 / / 50 (v) β = 2 => total sidebands 4 For lowest sideband, Peak value for J 1 = 0. 58 x 2000 (ii) β = 2 2)2 (vi) = 40 k. W BW = BFM = 2 nfm = 2 x 4 x 5 = 40 k. Hz For highest sideband J 4 P = (0. 03 x 2000/ 2)2 / 50 Ω = 36 W Both side bands are at fc fm = 100 MHz 5 k. Hz n = No of carrier frequency subbands from Bessels Table

Exercise 4 Consider an FM signal where , , , Calculate (i) Frequency deviation (ii) Bandwidth (iii) Power transmitted on the fc Assume modulation index is very small, obtain signal equation for NBFM

Solution : n = No of carrier frequency subbands from Bessels Table From slide 15

Generation of FM signal • • • Oscillator circuits are more essential for generating the FM signals by varying either the capacitance or the inductance of an LC oscillator – Can vary the frequency. variation of capacitance or inductance is directly proportional to the modulating voltage. several methods by which we can vary the capacitance or inductance by varying a control voltage. If such a voltage variable reactance is placed across the tank we can achieve FM modulation. • Frequency of the output signal is varying in accordance to an information signal amplitude.

Generation of FM signal 2 techniques – direct and indirect methods Direct method: 1. Varactor diode 2. Reactance modulation 3. Voltage Controlled Oscillator

1. Varactor diode Ø Varactor diode’s capacitance depends on the voltage across it. Ø Audio signals placed across the diode cause its capacitance to change, which in turn, causes the frequency of the oscillator to vary. Varactor diode L Varactor diode characteristic when only carrier signal exist, no other sidebands When vm= 0 ; ; C = kvm where k is constant and vm is voltage for information signal

Using Binomial expansion : ; if is small From the equation it can be seen that the FM signal can be obtained because the output frequency is dependant on the information signal amplitude, vm. Remember what you have learned in AM, information is contained in sidebands, Wm < Vm

Varactor diode based FM Generation

2. Reactance modulator Ø A reactance modulator is a circuit in which a transistor is made to act like a variable reactance. Ø The reactance modulator is placed across the LC circuit of the oscillator and as the modulator’s reactance varies in response to an applied audio signal, the oscillator frequency varies as well. RL change, freq changes 3. VCO Ø The VCO’s output frequency is proportional to the voltage of the input signal (control voltage). Ø If audio is applied to the input of a VCO, the output is an FM signal. Limitations of Direct Method of FM Generation Ø Not suitable for WBFM generation Ø Since, no one method uses crystal oscillator for generating the carrier frequency Ø Where the stability is tightly controlled by FCC Ø To address this issue, the Crosby modulator was developed.

2. Reactance modulator based FM Generation

Home Work: Indirect Methods based FM Generation

COMPARISON BETWEEN FREQUENCY MODULATION AND AMPLITUDE MODULATION Advantages SNR is much better than AM can be obtained, if the BW is greater enough. SNR can be increased by increasing the transmitted power. Constant amplitudes resulted in efficient use of non linear preamplifier. Disadvantages BW is usually larger than AM. Circuitry is more complex than AM.

DEMODULATION (DETECTION) OF FM SIGNAL FM discriminator extracts the intelligent signal that has been modulated onto the carrier via frequency variations Amplitude is depend on instantaneous carrier frequency deviation Frequency depends on rate of carrier frequency deviation Demodulation process is done in order to recover/get back the information signal transmitted. Basic concepts of demodulation circuit is to detect the frequency variation. Two techniques can be used: FM Demodulation Direct • Discriminator Indirect Phase Lock Loop(PLL)

CONVERSION CIRCUIT - FM TO AM (DISCRIMINATOR) This technique is required to convert FM signal to AM signal and then by using AM demodulation circuit is to get back the information signal. This technique is called slope detection or discriminator. Block diagram of the detection circuit is as shown below: Envelope Detector v. FM(t) y(t) v. FM(t) t t t

Mathematical analysis : FM equation : The basic equation of FM Differentiate; yields : • From the above equation it can be seen that the amplitude of the signal contains the information signal. • The amplitude of the signal is an envelope of the signal and the equation is given by :

PHASE-LOCKED LOOP (PLL) – INDIRECT METHOD vin(t) X vvco(t) ve(t) LPF vo(t) Voltage-Controlled Oscillator (VCO) Above is a block diagram of FM detector using Phase-Locked Loop (PLL). The input is FM signal:

PHASE-LOCKED LOOP Use this trigo identity: VCO output: where Multiplier in the circuit will function as a phase variation detector: LPF will pass all the lower frequency components and filtered all the higher frequency components: Assume: If Then

FM APPLICATIONS: FM RECIEVER FM TRANSMITTER

Application: Radio FM Receiver

FM band covers 88 -108 MHz. RF amplifier selects and amplifies the desired station from the many. This is called TUNING. The selected frequency is applied to the mixer. The output of an oscillator is also applied to the mixer. The output from the mixer is the intermediate frequency (i. f. ) The i. f. is a fixed frequency of 10. 7 MHz. No matter what the frequency of the selected radio station is, the i. f. is always 10. 7 MHz. The i. f. signal is fed into the i. f. amplifier. The advantage of the i. f. amplifier is that its frequency and bandwidth are fixed, no matter what the frequency of the incoming signal is. The amplified i. f. signal is fed to the demodulator. This circuit recovers the audio signal and discards the r. f. carrier. (discard carrier, take only fm-info) Some of the audio is fed back to the oscillator as an AUTOMATIC FREQUENCY CONTROL voltage. This ensures that the oscillator frequency is stable in spite of temperature changes. The audio signal voltage is increased in amplitude by a voltage amplifier. The power level is increased sufficiently to drive the loudspeaker by the power amplifier.

APPLICATION: RADIO FM TRANSMITTER

The microphone (transducer) converts sound pressure wave to electrical signals. These audio voltages are amplified by the audio amplifier. The amplified audio is used to control the deviation of the frequency controlled oscillator. The oscillator frequency is at the carrier frequency, in the 88 -108 MHz FM band. The low power of the frequency modulated carrier is boosted by the Radio Frequency amplifier. The aerial is driven by the amplifier and produces an electromagnetic wave. Under normal conditions the transmitted signal will travel as far as the horizon.

THE END OF CHAPTER

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