Phase Locked Loop (PLL)

 


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: -

1. A voltage-controlled oscillator (VCO)
2. A multiplier, serving as a phase detector or a phase comparator
3. 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 ω_c. 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) = θ_i(t) – θ_o(t).
When the incoming FM carrier is Asin[ω_ct +  θ_i(t)],

θ_i(t)= k_f-α∫^tm(α)dα -------------(6)
Hence,
θ_o(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