Chapter 3.1
Lecture Outcomes
After this lecture, studying the examples, and solving the problems, you should be able to:
Write time-domain AM equation,
Define the modulation index, calculate it, and measure it,
Describe the effect of overmodulation,
Calculate the bandwidth of an AM signal,
Calculate power and voltage of an AM signal,
Analyze full carrier, suppressed carrier, and single-sideband suppressed carrier in both time and frequency domains.
Amplitude Modulation
Introduction
Full-Carrier AM (DSB-FC)
Suppressed-Carrier AM (DSB-SC)
Introduction
AM modulation is a process of changing amplitude of carrier in proportion with amplitude of information signal.
AM is cheap, less complexity, low quality
AM is used for commercial broadcasting of both audio and video signals.
Broadcast in medium and high frequency bands
Aircraft communication in VHF band
Principles of AM
AM modulators are nonlinear devices with two
inputs:
High-frequency sinusoidal carrier
DSBFC AM
Time Domain
If the baseband signal is sine wave and carrier is
then, the DSBFC AM modulated signal is given by
(
)
(
)
( ) ( ) sin sin sin = c m c c m m c v t E e t t E E t t ω ω ω = + +( )
( ) sin m m m e t = E ω t( )
( ) sin c c c e t = E ω tDSBFC AM
Time Domain
A carrier wave with rms voltage of 2 V and frequency of 1.5 MHz is modulated by a sine wave with frequency of 500 Hz and amplitude of 1 V rms. Write the equation for the resulting signal.
6 6 3 2 2 2.83 2 1 1.41 2 1.5 10 9.42 10 / 2 500 3.14 10 / c m c m E V E V rad s rad s ω π ω π = × = = × = = × × = × = × = × ( )
(
3) (
6)
( ) sin sin 2.83 1.41sin 3.14 10 sin 9.42 10 c m m c v t E E t t t t V ω ω = + ⎡ ⎤ = ⎣ + × ⎦ ×DSBFC AM
Modulation Index
Modulation index describes amount of amplitude change present in AM waveform.
Mathematically, it can be expressed as
Using m, AM formula can be expressed as
m c E m E =
(
)
(
)
( ) sin sin 1 sin sin = c m m c c m c v t E E t t E m t t ω ω ω ω = + +DSBFC AM
Modulation Index
( ) ( ) ( ) sin sin 1 sin sin = c m m c c m c v t E E t t E m t t ω ω ω ω = + +DSBFC AM
Modulation Index
When modulation index becomes greater than 1, overmodulation occurs.
DSBFC AM
DSBFC AM
Modulation Index
When there are two or more sine waves of different
frequencies modulating a single carrier, m is calculated
by
mT = total resultant modulation index
m1, m2, … = modulation indices due to individual modulating components
2 2
1 2
T
DSBFC AM
Modulation Index
Find the modulation index if a 10 V carrier is
modulated by three different frequencies with
amplitudes 1 V, 2 V, and 3 V, respectively.
1 2 3 2 2 2 1 2 3 2 2 2 1 2 3 0.1 0.2 0.3 10 10 10 0.1 0.2 0.3 0.374 T m m m m m m m = = = = = = = + + = + + =
DSBFC AM
Modulation Index
By inspecting the figure, we get
(
)
(
)
max min 1 1 c m c c m c E E E E m E E E E m = + = + = − = − max min max min E E m E E − = + B ADSBFC AM
Modulation Index
Calculate the modulation index for the waveform shown in figure. max min max min max min 75 , 35 , -75 35 0.364 36.4% 75 35 E mV E mV E E m E E or = = = + − = = +
DSBFC AM
Frequency Domain
AM mathematical expression for sine wave modulating signal is given by
By using the trigonometric identity:
We have
( ) ( )
1
sin sin cos cos 2
A B = ⎡⎣ A B− − A+ B ⎤⎦
(
)
( ) 1 sin sin
sin sin sin
= c m c c c c m c v t E m t t E t mE t t ω ω ω ω ω = + + ( ) ( ) ( ) ( )
( ) sin cos cos
2 2 c c c c c m c m o c c mE mE v t E t t t mE mE ω ω ω ω ω ω ω ω ω ω = + − − + ⎡ ⎤ + − + + +
DSBFC AM
Frequency Domain
(
)
(
)
(
)
(
)
(
)
( ) sin cos cos
2 2
sin 2 cos 2 cos 2
2 2 = c c c c c m c m c c c c c m c m mE mE v t E t t t mE mE E f t f f t f f t ω ω ω ω ω π π π = + − − + + ⎡⎣ − ⎤⎦ − ⎡⎣ + ⎤⎦
DSBFC AM
Bandwidth (BW)
Glance at DSBFC spectrum will show that
For complex modulating signal, BW is twice the highest modulating frequency (fm(max))
(
)
2usb lsb c m c m m
BW = f − f = f + f − f − f = f
( ) 2 ( )
DSBFC AM
Frequency Domain
A 1 MHz carrier with amplitude 1 V peak is modulated by a 1 kHz signal with m = 0.5. Sketch the voltage spectrum.
An additional 2 kHz signal modulates the carrier with
DSBFC AM
Frequency Domain
For a DSBFC AM modulator with Carrier frequency
f
c=100 kHz and maximum modulating signal
frequency f
m(max)= 5 kHz, determine:
Frequency limits of upper and lower sidebands
Bandwidth
Upper and lower sidebands produced by modulating signal of single-frequency 3 kHz.
DSBFC AM
DSBFC AM
Bandwidth (BW)
103 97 6 2 m 2 3 6 BW kHz f kHz = − = = = × =DSBFC AM
DSBFC AM
Power Relationships
The power that appears across resistor R is
(
)
(
)
( ) sin cos cos
2 2 c c c c c m c m mE mE v t = E ω t + ω ω− t − ω ω+ t
(
) (
2) (
2)
2 2 2 2 2 2 2 2 2 2 / 2 / 2 2 / 2 2 2 4 2 4 2 4 4 1 t c lsb usb c c c c c c c c c P P P P E mE mE R R R E m E m E R R R m m P P P m m P P P = + + = + + = + × + × = + + ⎛ ⎞ = + = ⎜ + ⎟DSBFC AM
Power Relationships
t lsb c usb
DSBFC AM
Power Relationships
For a Full-Carrier AM wave with carrier voltage Ec=10V, a load resistance RL=10Ω, and modulation coefficient
m = 1, determine:
Powers of carrier and upper and lower sidebands
Total sideband power
Total power of modulated wave
Draw power spectrum
DSBFC AM
DSBFC AM
Generation and Detection
Square-Law Modulation: Let’s consider nonlinear
device which has input-output characteristics as shown in Figure
If the input to the nonlinear device is
x y 2 2 1x x y =α +α x y 2 2 1x x y =α +α ( ) ( ) sin 2 x t = e t + E
π
f tDSBFC AM
Generation and Detection
Square-law AM Modulator ( ) m( ) c sin 2 c x t =e t + E π f t y(t ) =α1x(t )+α2x2(t ) 2 2 2 1 2 2 1 1 2 ( ) m( ) m( ) sin 2 c c 1 m( ) sin 2 c y t α e t α e t α π f t Eα α e t π f t α ⎡ ⎤ = + + + ⎢ + ⎥ ⎣ ⎦ 2 1 1 2 ( ) c 1 m( ) sin 2 c u t Eα α e t π f t α ⎡ ⎤ = ⎢ + ⎥ ⎣ ⎦
DSBFC AM
Generation and Detection
Envelope Detector: consists of a diode and an RC circuit ( Lowpass filter)
DSBFC AM
Limitations
From frequency domain, we find that DSB-FC
suffers from two limitations:
It is wasteful of power because of the transmission of the carrier.
It is wasteful of bandwidth, as it requires twice the message bandwidth.
To overcome these limitations, we use:
Double sideband suppressed carrier (DSB-SC)