Angle modulation
• Process of varying the total phase angle of a carrier wave in accordance with
the instantaneous value of the modulating signal, keeping the amplitude of the
carrier constant.
• Consider an unmodulated carrier 𝜑(t)=Acos (𝜔𝑐 t + 𝜃0 )
or 𝜑(t)=A cos 𝜑 (1)
where ∅ 𝑡 = 𝜔𝑐 𝑡 + 𝜃0 is the total phase angle of the carrier wave.
• Eqn (1) can be considered as a real part of a rotating phasor A𝑒 𝑗𝜑 and can be
ෙ A𝑒 𝑗𝜑 or ∅ 𝑡 = Re[A𝑒 𝑗𝜑 ] = A Re[cos 𝜑 + j sin 𝜑]
represented as ∅= (2)
• The phasor ∅ෙ rotates at a constant angular frequency 𝜔𝑐 provided 𝜃0 is the
phase angle of the unmodulated carrier at t=0.
Instantaneous frequency
• The constant angular frequency 𝜔𝑐 of the phasor is related to its total phase angle.
𝜑= 𝜔𝑐 t + 𝜃0 (3)
• Differentiating (3) we get d 𝜑/dt = 𝜔𝑐 [𝜃0 is independent of time] (4)
• This derivative varies with time and hence the angular frequency of the phasor 𝜑ු
will also change with time
• The time dependent angular frequency is called as instantaneous angular frequency
and is denoted as d 𝜑/dt = 𝜔𝑖 (5)
• 𝜔𝑖 is time dependent
Fig: Waveform of a carrier wave with varying frequency
Types of Angle modulation
• Two types of angle modulation
• Frequency modulation
• Phase modulation
• Phase modulation – The phase angle ψ(t) is varied linearly with the modulating
signal f(t) about an unmodulated phase angle 𝜔𝑐 t
• Frequency modulation – The instantaneous frequency 𝜔𝑖 varies linearly with a
modulating signal f(t) about an unmodulated frequency 𝜔𝑐 .
Representation of Frequency modulated signal
• The instantaneous value of the angular frequency ωi is equal to the frequency
ωc of the unmodulated carrier plus a time varying component proportional to f(t)
.
• Mathematically ,𝜔𝑖 = 𝜔𝑐 + 𝐾𝑓 f(t) (6)
• Where 𝐾𝑓 is the frequency sensitivity (Hz/V)
• The total phase angle of the FM wave can be obtained by
integrating (5) [d 𝜑/dt = 𝜔𝑖 ]
𝜑𝑖 = 𝑖𝜔dt = 𝑐𝜔 [+ 𝐾𝑓 f(t)] = 𝜔𝑐 t + 𝐾𝑓 )𝑡(𝑓 dt (7)
• The corresponding FM wave can be given by ∅𝐹𝑀 (t) = Acos 𝜑𝑖 (8)
• Sub (7) in (8) we get ∅𝐹𝑀 (t) = Acos[𝜔𝑐 t + 𝐾𝑓 )𝑡(𝑓 dt] (9)
Representation of Frequency modulated signal- Contd
• We know f(t) =𝐸𝑚 cos 𝜔𝑚 t (10)
𝐸𝑚
• Sub (10) in (9) ∅𝐹𝑀 (t) = A cos[𝜔𝑐 t + 𝐾𝑓 sin 𝜔𝑚 t] (11)
𝜔𝑚
∆𝜔
• Let ∆𝜔= 𝐾𝑓 𝐸𝑚 . Hence ∅𝐹𝑀 (t) = Acos[𝜔𝑐 t + sin 𝜔𝑚 t] (12)
𝜔𝑚
Where ∆𝝎 is the frequency deviation
∆𝜔
• Let 𝑚𝑓 = Then ∅𝐹𝑀 (t) = A cos [𝜔𝑐 t + 𝑚𝑓 sin 𝜔𝑚 t] (13)
𝜔𝑚
Where 𝒎𝒇 is the modulation index - Ratio of frequency deviation to the
modulating frequency
Representation of Phase modulated signal
• The total phase angle of the carrier wave is given by 𝜑𝑖 (t) = 𝜔𝑐 t + 𝜃0
• For a phase modulated signal, the phase angle is varied linearly with the
modulating signal.
Hence 𝜃α f(t)
𝜃= 𝐾𝑝 f(t)
• The phase modulated signal ∅𝑃𝑀 (t) = A cos𝜑𝑖 (t)
∅𝑃𝑀 (t) = A cos[𝜔𝑐 t +𝐾𝑝 f(t) ]
The maximum change in total phase angle from the centre phase is known as
phase deviation (∆𝜃)
∆𝜔
∆𝜃= 𝑚𝑓 =
𝜔𝑚
Relationship between PM and FM
FM generation using Phase modulator PM generation using Frequency modulator
g(t)=)𝑡(𝑓 𝑑 𝑓(𝑡)
g(t)= 𝑑𝑡
Integrator Phase modulator Differentiator Frequency
f(t) FM f(t) modulator PM
Acos 𝜔𝑐 t Acos 𝜔𝑐 t
Carrier generator Carrier generator
Transmission bandwidth of FM signal
• Bandwidth=2n𝜔𝑚 Bandwidth of PM signal
where n is the number of sidebands
BW(PM) ≈ 2∆ω
n≈𝑚𝑓
= 2Kp Emωm
• BW=2𝑚𝑓 𝜔𝑚=2∆𝜔=2∆f
Bandwidth using Carson’s rule Modulation index of PM signal
BW=2(∆𝝎+ 𝝎𝒎) = 2(∆𝒇+ 𝒇𝒎) mp = Kp Em = θd
Depending upon the value of ∆𝜔, FM
is classified as narrowband FM
(NBFM) and wideband FM (WBFM)
Figure: Phasor diagram of FM
Comparison between NBFM and WBFM
NBFM WBFM
• Frequency deviation is very small • Frequency deviation is very large
• BW = 2𝜔𝑚 • BW = 2∆𝜔
• 𝐾𝑓 is very small • 𝐾𝑓 is very large
• BW is narrow • BW is wide
• 𝑚𝑓 is very small • 𝑚𝑓 is very large
• Only two sidebands • ‘n’ number of sidebands
∆𝜔
𝑚𝑓 = 𝜔 ∆𝜔= 𝐾𝑓 𝐸𝑚
𝑚
International regulation for FM signal
• The following values are prescribed by CCIR (Consultative Committee for
International Radio) for commercial FM broadcast stations.
❖ Maximum frequency deviation ±75𝐾𝐻𝑧.
❖ Frequency stability of the carrier ±2𝐾𝐻𝑧.
❖ Allowable bandwidth per channel = 200KHz.
Power content in FM signal
Generation of Narrowband FM
Carrier signal Acos 𝜔𝑐 t
Phase shifted carrier - 𝐴sin 𝜔𝑐 t
Message signal f(t)= 𝐸𝑚 cos 𝜔𝑚 t
g(t)= 𝑚𝐸 = )𝑡(𝑓 = )𝑡(𝑓 cos 𝜔𝑚 t
𝐸𝑚
= sin𝜔𝑚 t
𝜔𝑚
Output of balanced modulator is
𝐸
- 𝐴sin 𝜔𝑐 t * 𝑚 sin𝜔𝑚 t
𝜔𝑚
𝐸𝑚
∅𝑁𝐵𝐹𝑀 (t) = Acos 𝜔𝑐 t - KA𝜔 sin𝜔𝑚 t sin
𝑚
𝜔𝑐 t
𝐸𝑚
Let K𝜔 = 𝑚𝑓
𝑚
∅𝑁𝐵𝐹𝑀 (t) = Acos 𝜔𝑐 t - A𝑚𝑓 sin𝜔𝑚 t sin 𝜔𝑐 t
Varactor diode FM modulation
Principle of Operation:
• Modulating signal directly modulates the carrier
that is generated by an electronic circuit.
• The oscillator circuit involves a parallel circuit.
• Frequency of oscillation of the carrier generator is
1
𝜔𝑐 = 𝐿𝐶
• The Carrier frequency 𝜔𝑐 can be made to vary
according to the modulating signal f(t), if L or C is
varied according to f(t).
Varactor diode FM modulation Contd..
Operation:
• Varactor diode is a semiconductor diode whose junction capacitance changes
with the applied d.c bias voltage.
• The varactor diode is shunted with the oscillator tank circuit.
• 𝐶 < 𝐶𝑑 to keep the r.f voltage from the oscillator across the diode small as
compared to 𝑉𝑜 , the polarizing voltage.
• 𝑋𝑐 at highest modulating frequency is kept large as compared to R.
• 𝑉𝑜 is reverse bias voltage across the varactor diode.
Varactor diode FM modulation Contd..
• The capacitance 𝐶𝑑 of the diode is given by 𝐶𝑑 = K/ 𝑉𝐷 (1)
K the proportionality constant.
Where 𝑉𝐷 is the total instantaneous voltage across the diode
𝑉𝐷 =𝑉𝑜 + f(t) (2)
• The total capacitance of the oscillator tank circuit is (𝐶𝑜 + 𝐶𝑑 )
• The instantaneous frequency of oscillation
𝜔𝑖 = 1ൗ 𝐿 (𝐶 +𝐶 ) (3)
𝑜 𝑜 𝑑
• Sub (1) in (3), we get 𝜔𝑖 = 1 (4)
൘ 𝐿𝑜 (𝐶𝑜 +K/ 𝑉𝐷 )
• 𝜔𝑖 is dependent on 𝑉𝐷 which in turn depends on the modulating signal f(t).
Varactor diode FM modulation Contd..
• Distortion due to non-linearity:
• From (4) it is understood that 𝜔𝑖 does not change linearly with 𝑉𝐷 .
• This non-linearity produces distortion due to the frequency variations caused by the
higher harmonics of the modulating frequency.
• Assume that the oscillator tank circuit comprises only the diode capacitance 𝐶𝑑 and
𝐶𝑜 is absent.
(4)becomes = = (5)
The R.H.S of the above equation can be represented by a Taylor series about the
polarizing voltage 𝑉𝑜 as given below.
= = + - (6)
Varactor diode FM modulation Contd..
• The higher order terms can be neglected if (𝑉𝐷 − 𝑉𝑜 ) is small.
• Let (𝑉𝐷 − 𝑉𝑜 )=∆V = f(t) = 𝑉𝑚 sin𝜔𝑚 t (7)
= = (1- cos2 (8)
• Sub (7) and (8) in (6)
= = + - + (9)
Varactor diode FM modulation Contd..
• % second harmonic distortion is the ratio of amplitude of the 𝑐𝑜𝑠2𝜔𝑚 term
and the fundamental term
3𝑉𝑚
% second harmonic distortion = x 100
8𝑉𝑜
By adjusting proper ratio of 𝑉𝑚 and 𝑉𝑜 second harmonic distortion may be
reduced.
Ignoring the effect of second harmonic of f(t)
= + = + (∆ )
𝑉𝑚
Modulation index 𝑚𝑓 = 3 1
4𝜔𝑚 (𝐿𝑜 𝐾𝑉𝑜 2 )2
