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The Voltage Controlled Oscillator
Soumyadip Mondal
210108048
In the last class we were talking about wideband frequency modulation.We saw that spectral
analysis of the wide-band FM wave is
s(t) = Ac cos[2πfc t + βsin(2πfm t)]
(1)
s(t) = Re[Ac ej(2πfc t+βsin(2πfm t) ]
(2)
This can be written as
= Re[s̃(t)ej2πfc t ]
After doing some calculation we have got an expression of s̃(t) as
s̃(t) =
∞
X
Ac Jk (β)ej2πkfn t
(3)
Ac Jk (β)cos(2π(fc + kfm )t)
(4)
k=−∞
And s(t) can be written as
∞
X
s(t) =
k=−∞
where
Jk (β) =
∞
X
(−1)l (β/2)k+2l
l!(k + l)!
n=1
Let’s discuss some properties of bessel function:
1) It is an odd function in k. i.e
Jk (β) = (−1)k j−k (β)
2) The power contained within all sideband of F.M signal is fixed. i.e
∞
X
|jk (β)|2 = 1
k=−∞
3) For small values of β, J0 (β) = 1 and Jk (β) is generally zero for β<k .
J0 (β) is the componenet corresponding to k = 0 and It is called carrier component.
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Fig. 1: Plot of bessel functions
In general,J0 (β) reperesent strength of kth sideband and Jk2 (β) represent the power contained within the sideband.
4) Jk (β) has peaks around β ≈ k+1
I.
CARSON ’ S RULE
The theoretical bandwidth of an FM transmission is unlimited, but the practical bandwidth
may be approximated using the number of significant sidebands. In general for a modulation
signal with a maximum frequency fm , the bandwidth is approximated as
B = 2(1 + β)fm
(5)
II. G ENERATEION OF FM SIGNAL
We have analyzed FM signal.Now the question is how to generate and demodualte a FM
signal.We can generate FM signal in two ways.
A. using voltage controlled oscillator with varactor diode
Incase of FM signal, the frequency fi of the output is function of voltage of message signal.Hence we need an oscillator whose frequency of oscillation can be controlled via an external
applied voltage.This type of setup is called Voltage controlled oscillator.
let’s first discuss about basic LC oscillator.It is an inductor connected in parallel with a capacitor.The resonant frequency of the output for the LC oscillator is
fc =
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1
√
2π LC
(6)
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3
Next is Varactor Diode.Consider an pn junction in reverse bias(figure attached above).As the
reverse bias voltage is increased, current flowing through the diode is increased.As a result the
width of the depletion layer increases.We can consider it as a capacitor and the capacitance is
C=
ϵA
w
(7)
where w is the width of the depletion region.
Since capacitance is dependent on length, and here we are able to change this length(length
of the depletion region increases with increasing current) by varying the current, thus what we
have is a variable capacitor in the diode ,or Varactor Diode.
But after a certain reverse bias voltage, capacitance of the diode becomes almost linear .Now
if we add the varactor diode in parallel with the LC oscillator in reverse bias,the capacitance
becomes
cnet = c0 + kVin
(8)
where Vin is the input voltage and k is the sensitivity factor.This capacitance is called parasitic
capacitance.
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Fig. 2: Voltage Controlled Oscillator
We can easily calculate the frequency of our output using equation 8.
1
2π LCnet
1
= p
2π L(C0 + km(t)
1
1
q
= √
2π LC0 1 + km(t)
fi (t) =
√
C0
km(t) −1
1
√
(1 +
)2
C0
2π LC0
1
km(t)
km(t)
= √
(1 −
) = fc (1 −
)
2C0
2c0
2π LC0
=
This is approximately a VCO which generates F.M waves with negative slope.
B. Using 555 Timer
[See the figure attached below]
When an input voltage (say V) is given to Control Voltage pin(pin 5), the upper and lower
comparator reference changes to voltages V and V/2. So when the capacitor voltage becomes
less than V/2, output becomes high and the capacitor starts charging to Vcc through resistor R1
and diode D. When the the capacitor voltage becomes greater than V, output becomes low and
the capacitor starts discharging through resistor R2 and 7th pin of the IC. So the time period is
proportional to the input voltage V. So as V increases, time period of the output wave increases
and when V decreases time period of the output wave decreases.
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Fig. 3: FM generation using 555 timer
III. D EMODULATING F.M
SIGNAL
We will discuss mainly two ways to demodulate FM signal.One way is to convert FM signal
to AM signal using a low pass filter to reduce the signal to its low pass form and then use
envelope detector to demodulate the AM signal.
We know, transfer function of a series RLC circuit is dirac delta.And frequency response of FM
signal is similar to that of a LCR resonance circuit (figure 3) where,
fc + δf =
1
√
2π LC
(9)
Fig. 4: Frequency Response
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This means that a LCR circuit can be used to produce the response that is required.It can be
shown that FM signal can be converted into a AM signal by the LCR lowpass filter circuit. This
AM signal can now be passed through an envelope detector to recover the message signal.
The second and main way to demodulate FM signal is using Phase locked loop(PLL). We will
discuss about it in next class.
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