Frequency Modulation

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PRINCIPLE OF COMMUNICATIONS

ANGLE MODULATION

INTRO TO FREQUENCY MODULATION A major problem in AM is its susceptibility to noise superimposed on the modulated carrier signal. To improve on this, the first frequency modulation (FM) radio communication system was developed in 1936, which is much more immune to noise than its AM counterpart. Unlike the AM, FM is difficult to treat mathematically due to the complexity of the sideband behavior resulting from the modulation process.

Angle Modulation

In AM, the amplitude of the carrier signal varies as a function of the amplitude of the modulating signal. But when the modulating signal can be conveyed by varying the frequency or phase of the carrier signal, we have angle modulation. Angle modulation can be subdivided

by

a. frequency modulation (FM) and b. phase modulation (PM).

A. Frequency Modulation- the carrier’s instantaneous frequency deviation from its unmodulated value varies in proportion to the instantaneous amplitude of the modulating signal.

B. Phase Modulation, the carrier’s instantaneous phase deviation from its unmodulated value varies as a function of the instantaneous amplitude of the modulating signal.

Below is the figures illustrates the FM and PM waveforms for sine wave modulation

FREQUENCY MODULATION

- Type of angle modulation wherein the frequency of the constant-amplitude carrier signal is varied or changed according to the instantaneous amplitude of the modulating signal.

- As the amplitude to the information varies, the carrier frequency varies above and below its normal center frequency.

Frequency Deviation,

δ

- the peak frequency shift that occurs in the carrier.

- - is the amount by which the carrier frequency is varied from its unmodulated value.

Carrier swing - is the peak to peak frequency deviation.

CS=2 δ

Modulation Index,

m

f

- The modulation index for an FM signal is defined as the ratio of the maximum frequency deviation to the modulating signal frequency.

m

_f

=

δ

f

_m Where:

m

_f

=

¿

_{modulation index of fm}

δ=¿ _{maximum frequency deviation of the} carrier by the amplitude of the modulating signal.

v

_m

(

t )=V

_m

sin 2 π f

_m

t

v

_C

(

t )=V

_c

sin 2 π f

_c

t

(2)

f

_m

=

¿

_{frequency of the modulating signal}

FM was developed to cope with the undesirable noise, which competed with the desired signal when it is amplitude modulated.

MATHEMATICAL DESCRIPTION OF AN FM SIGNAL

The instantaneous frequency of the FM signal is given by the equation,

f =f

_c

₍

1+k V

_m

sin 2 π f

_m

t

₎

Where:

f

_c

=

¿

_{unmodulated carrier frequency, HZ}

K=

¿

_{proportionality constant}

V

_m

sin 2 π f

_m

t=

¿

_{instantaneous modulating}

voltage, V

The instantaneous value of the FM signal is given by the equation,

2 π f

_m

t

2 π f

_c

t+

δ

f

_m

sin

¿

V

FM

(

t)=V

c

sin

¿

Where:

f

_c _{= unmodulated carrier frequency, Hz}

f

m _{= modulating signal frequency, Hz} δ _{= Frequency deviation}

V

_c _{= peak amplitude of the carrier voltage,} V

Recall that in AM, the frequency component consists of a fixed carrier frequency with upper and lower sidebands equally displayed above and below the carrier frequency. The frequency spectrum of the FM wave is much more complex, that it will produce an infinite number of sidebands

In expanded form,

FREQUENCY SPECTRUM OF A FM SIGNAL The frequency spectrum of the FM signal can be obtained using the Bessel function table. (Please refer to the book) Example:

For a FM signal with a modulation index of 0.5, draw the frequency spectrum of the FM signal. (Bessel Function Table)

BANDWIDTH REQUIREMENT FOR AN FM SIGNAL

The exact bandwidth obtained using the Bessel function is given by the equation,

BW =2 x n sideband pairs x f

_m

Approximate minimum bandwidth using Carson’s rule is

δ+f

_m₍_max_❑₎

BW =2

¿

Narrow Band FM with low modulation index values, the minimum bandwidth is given by the equation,

BW =2 f

_m

Wideband FM with high modulation iindex values, the minimum BW is given by the equation,

BW =2 δ

PERCENT MODULATION

- Ratio of the actual frequency deviation to the maximum allowable frequency deviation allowed by law.

- For FM broadcast band, the maximum allowable frequency deviation is 75KHz. - For sound portion of TV broadcast, the maximum allowable frequency deviation is 25KHz

%M=

δ

actual

δ

maximum

x 100

DEVIATION RATIO, DR

- - is the worst-case modulation index and is equal to the maximum peak frequency deviation divided by the

maximum modulating signal

frequency.

- - The worst case modulation index produces the widest output frequency spectrum.

DR= δMAX f_m(max❑)

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For FM broadcast band, the maximum allowable frequency deviation is 75KHz. For sound portion of TV broadcast, the maximum allowable frequency deviation is 25KHz.

- For FM broadcast band and sound portion of TV broadcast, the maximum modulating signal frequency is 15KHz.

POWER CONTENT OF AN FM SIGNAL

P

T

=

P

C

+

P

1

+

P

2

+

…+P

n

P

_T

=

V

c 2

2 R

+

V

12

2 R

+

V

22

2 R

+

…+

V

2n

2 R

ADVANTAGES OF FM OVER AM 1.FM is more immune to noise than AM. 2.Rejection of interfering signals because of “capture effect”.

3.Lower power output requirements.

4.Better transmitter efficiency since class C amplifier may be used.

5.Improved signal to noise ratio. DISADVANTAGES OF FM OVER AM 1. Wider channel is required for FM

2. FM transmitter and receiver circuits are more complex and expensive.

3. Since the reception is line of sight, the area of reception for FM is smaller than for AM.

4. Maximum deviation is limited. PHASE MODULATION

– modulation technique in which the amount of phase shift of a constant frequency carrier is varied in accordance with the modulating frequency.

-FM can be obtained from PM by the use of the Armstrong systems.

Phase shift – the separation between two sinewaves of the same frequency.

Principle of PM

In PM, when the modulating signal goes positive, the amount of phase lag increases with the amplitude of the modulating signal. The effect is the frequency being increased, when the modulating signal goes negative, the amount of phase decreases thus frequency is lowered.

Expression for PM signal:

Comparison between PM and FM:

1. Modulation Index is defined differently in each systems

mp = is proportional to the amplitude of the modulating signal.

Note: In PM, the max frequency deviation takes place at the crossing points. While in FM, maximum frequency deviation takes place t the peak amplitude of the modulating signal. The value of change in the carrier fm; mf - indirectly proportional to fm.

2. FM is a form of PM.

3. mf indicates the amount of frequency deviation, while mp indicates the phase change introduced by the modulating signal. mp is independent of the modulating frequency.

Example: A 25 MHz carrier is modulated by a 400 Hz audio sine wave. If the carrier voltage is 4V and the maximum deviation is 10 KHz, Write the equation of this modulated wave.

a. FM b. PM

c. if the modulating frequency is increased to

2 KHz,

What is now the resulting FM and PM equation?

Noise Triangle

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There is a non- uniform distribution of noise. Noise of the higher modulating frequencies is inherently greater in amplitude than noise at lower frequencies.

For information signal with a uniform signal level, a uniform S/N is produced. The higher modulating frequencies have lower S/N than the lower than the lower frequencies. To Compensate for this pre-emphasis is used.

Pre-emphasis – process of emphasizing or boosting in amplitude of the high frequency signals prior to performing modulation.

Pre-emphasis circuit:

De-emphasis – reciprocal of pre-emphasis; restores the original amplitude vs. frequency characteristics of the information signal.

De-emphasis circuit:

Pre-emphasis Network:

- A high pass filter that amplifies the high frequency components more than the low frequency components.

- A differentiator circuit.

De-emphasis Network

- A low pass filter or an integrator.

Advantage of Pre-emphasis and De-emphasis Network:

1. increased S/N ratio 2. increases fidelity

3. increased intelligibility of FM signals. 4. uniform S/N indicates stable flow of signals.

Fu = upper break frequency where the signal enhancement flatten.

The break frequency, (the frequency whrere pre emphasis and de emphasis begins) is determined by the RC and L/R time constant of the network. The break frequency occurs at the frequency Where Xc or XL equal R.

Mathematically:

FM DUE TO AN INTERFERING SINUSOID

The noise vector is super imposed on the carrier circulating about it with a relative angular Wn-Wc

Peak phase:

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The peak frequency deviation

The S/N ratio at the output of an FM demodulator due to unwanted frequency deviation from an interfering sinusoid is the ratio of the peak frequency due to the information signal to the peak frequency deviation due to the interfering signal.

S/N =

Example:

For an angle modulated carrier

Vc(t )=0.5 cos (2 π 200.01 MHzt )

Determine: a. the frequency of the demodulated interference signal.

b. Peak phase and frequency deviation due to the interfering signal

c. S/N at the output of the demodulator. Solution:

FM GENERATION

a. Direct Method – producing FM by directly varying the output frequency of a carrier oscillator.

-is angle modulation in which the frequency of the carrier is varied (deviated) directly by the modulating signal.

Three common methods: a. Varactor diode Modulators

b. FM Reactance Modulator

c. Linear Integrated-circuit direct FM modulators

b. Indirect method - producing FM by use of phase modulation

forward bias – current will flow depletion region collapses

reverse bias – no current will flow in the depletion region, acts as a capacitor

Note: Width of the depletion region depends upon the amount of reverse bias, the wider the depletion the lower the reverse bias, the thinner the depletion region.

FM modulator w/ Wc f0 =

1 2 π

√

LC

f0 =

LC

₁

+

C

¿

L

¿

2 π

√

¿

1 ¿

Note: Capacitance of the Wc ( D1) is

controlled by:

a.) Fixed dc voltage (set by R1 and R2)

b.) Modulating signal voltage(that adds to and subtracts from the fixed dc bias)

- If reverse bias increases, the

capacitance of D1 is lowered, thus the frequency of the oscillator is increased - If the reverse bias decreases, the

capacitance of D1 is increased and the

frequency is decreased. Reactance

- employs a reactance circuit that presents inductive or capacitive reactance to the tank circuit of an oscillator.

- the variation in reactance causes the frequency of the oscillator to shift in accordance with the modulating signal thereby producing FM.

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Basic FET reactance modulator

The impedance is entirely reactive. The value of the reactance is proportional to the

transconductance of the device which is dependent on the gate voltage and its variations.

to det. Z:

z =

v

_i

impedance looking at the terminals

Note: For z to be pure reactance

1. The bias network current in must be

negligible compared to the drained current, id. The impedance of the bias network must

be large to be ignored.

2. The drain-to-gate impedance (Xc) must be

greater than the gate to source impedance (R) By analysis: Vg = Rib where: ib =

V

R− jXc

Vg =

VR

R− jXc

id= gmVg id = gm

(

VR

R− jXc

)

then, z =

V

i

_d =

V

gm

(

VR

R− jXc

)

z =

R− jXc

_{gm R}

=

_gm

1 −

¿

R− jXc

_{R gm}

z =

_gm

1

(

−

jXc

_R

)

The impedance is a capacitive reactance

Xeq =

Xc

R gm

1 2 π f

_ceq

=

1 /2 π fc

R gm

Ceq= gmRc

Note: The equivalent capacitance depends in the device transconductance as given by Ceq2 gmRc and can therefore be varied, with bias

voltage since gm =

id

vg

, also the gate-to-drain impedance (Xc) is made 5 to 10 times greater than gate-to-source impedance ( R ), that is Xc = nR where n = multiplying factor. Therefore,

Ceq =

g

_m

2 πfn

TYPES OF REACTANCE MODULATOR Name Zgd Zgs Condition Reactance Formula RC Capacitive C R Xc>>R Ceq = gm RC RC Inductive R C R>>Xc Leq = Rc/gm RL Inductive L R XL>>L Leq = L/gm R RL Capacitive R L R>>XL Ceq= gmL/R Indirect FM

- The frequency of the carrier is deviated indirectly by the modulating signal and is accomplished by changing the phase of the carrier, which is a farm of direct FM.

- With the use of FM, carrier oscillator can be ophimized per frequency accuracy and of the stability.

Note:

Primary disadvantage of direct FM is that relatively unstable LC oscillators are used to produce carrier frequency.

Advantage of direct FM is the relatively high frequency deviation and modulation indices attainable because of the LC oscillators’ instability.

Primary Advantage of direct PM (indirect FM) is the use of high stable crystal oscillators. Disadvantage of direct PM the difficulty to achieve high phase deviation and modulation indices because of the inherent stability of the crystal oscillator.

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A VVC Phase Modulator Circuit

Direct FM transmitter

- uses AFC to meet FCC specifications on oscillator stability.

1. Crosby Direct FM Transmitter

- used in commercial broadcast band transmitter.

- includes an AFC Loop AFC – Automatic Frequency Control

- provides the necessary correction in case of unwanted frequency drifts

- uses discriminator circuit. AFC Operation:

- Discriminator will give positive dc voltage if input frequency is higher than what is tuned

and negative voltage if input frequency is lower.

- Voltage output of the discriminator is in series with the AF input of the reactance modulator and therefore the

transconductance will either increase or decrease. The output capacitance or inductance is increase or decrease, thus lowering or increasing the frequency of the master oscillator. Any unwanted shift in the frequency is connected.

Up-Conversion Method 1. Heterodyning or Mixing - change the fc only

2. Frequency Multiplication - changes fc, mf, δ, etc.