DEFINITION OF PLL ::
The Phase-Locked Loop (PLL) is a feedback system that may be used to extract a
base band signal from a FM carrier, especially under low SNR conditions. Thus
PLL tracks the phase and the frequency of the carrier component of an incoming
signal.
A PLL has three basic components: -
- A voltage-controlled oscillator (VCO)
- A multiplier, serving as a phase detector or a phase comparator
- A loop filter having response H(s)
The operation of PLL is similar to that of a feedback system except that the
quantity feedback and compared is phase, but not amplitude.
OPERATION OF VCO ::
An oscillator whose frequency can be controlled by an external voltage is a
Voltage Controlled Oscillator (VCO). In a VCO, the oscillation frequency varies
linearly with the input voltage. If a VCO input voltage Eo(t), its output is a
sinusoid of frequency given by,
ωVCO = ωc + Ceo(t)
Where C is a constant of the VCO and ω
c is the free-running frequency of the VCO.
The multiplier output is further low pass filtered by the loop filter and then
applied to the input of the VCO. This voltage changes the frequency of the
oscillator and keeps the loop locked, i.e. the frequency and phase of the input
and output sinusoidal signals becomes identical.
OPERATION OF PHASE COMPARATOR ::
A Phase Comparator is a device with two input ports and a single output port.
If periodic signals of identical frequency but with a timing difference are
applied to the inputs, the output is a voltage, which depends on the timing
difference. After phase comparator the signal is low pass filtered to get the
error voltage.
PLL ACTING AS A DEMODULATOR ::
In PLL the output Eo(t) of the loop filter H(s) acts as an input to the VCO. The
free-running frequency of the VCO is set at the carrier frequency ωc. The
instantaneous frequency of the VCO is given by,
ωvco =ωc + Ceo(t) ---------(1)
If the VCO output is,
Bcos [ωct +
θo(t)],
then its instantaneous frequency is
[ωct + d(θo(t))].
Therefore,
d(θo(t)) = Ce
o(t) ----------(2), where C and B are constants of the PLL.
Let the incoming signal be,
Asin [ωct +ωi (t)]. At the multiplier this incoming
signal and the VCO output are fed so that the output X(t) is given by,
X(t) =
A B sin(ω
ct +
θi)cos(ω
ct +
θ0)
=[½AB {sin (θi -θ0) + sin(2ωct +θi
+θ0)}]
---------(3)
The sum frequency term is suppressed by the loop filter, Hence the effective
input to the loop filter is
[½AB {sin (θi(t) -θ0(t))]. If h(t) is the unit impulse
response of the loop filter,
e
o(t) = h(t) * [
½ABsin{
θi(t) -θ0(t)}] = [
½(AB)]
0∫
t h(t –
x)sin[
θi(t) -θ0(t)]dx
-(4)
Substituting eq.(2) in eq.(4) we get
d(θo(t)) = AK
-α∫
th(t –
x)sin[
θe(x)]dx
----------------(5)
where K =CB
and
θe
(t) is the phase error,
defined as
θe
(t) =
θi(t) –
θo(t).
When the incoming FM carrier is Asin[ω
ct +
θi(t)],
θi(t)= k
f-α∫
tm(α)dα -------------(6)
Hence,
θo(t)
= [k
f-α∫
tm(α)dα] – 0e(t)
and assuming a small error e(t) we get from eq.(2)
e
o(t) =1/c[
d(θo(t))]~
1/ck
f m(t) ---------------(7)
Thus, the PLL acts as an FM demodulator
