1. TELE3113 Analogue and Digital
Communications
Phase-Locked Loop
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales

2. Phase-Locked Loop (1)
Block diagram. (Demodulation of FM Signals.)
×
FM wave
s(t) e(t) Loop v(t)
filter
r(t)
Voltage
controlled
oscillator
Voltage controlled oscillator (VCO) performs FM as
r(t) = Av cos[2πfc t + φ2 (t)],
where Av is the amplitude and the angle
t
φ2 (t) = 2πkv v(τ )dτ.
0 TELE3113 - Phase-Locked Loop. August 26, 2009. – p.1/

3. Phase-Locked Loop (2)
Suppose the incoming FM wave is
s(t) = Ac sin[2πfc t + φ1 (t)]
and the angle
t
φ1 (t) = 2πkf m(τ )dτ. (1)
0
After the multiplication of s(t) by r(t), the low-frequency
component is
e(t) = km Ac Av sin[φe (t)],
where km is the multiplier gain and φe (t) = φ1 (t) − φ2 (t).
TELE3113 - Phase-Locked Loop. August 26, 2009. – p.2/

4. Phase-Locked Loop (3)
When the phase error φe (t) = 0, the PLL is in phase-lock.
When the phase error is small compared to 1,
sin[φe (t)] ≈ φe (t), the PLL is near-phase-lock.
Therefore, if φe is small compared to one,
K0
e(t) = km Ac Av φe (t) = φe (t),
kv
where K0 = km kv Ac Av is called the loop-gain parameter of the
PLL.
TELE3113 - Phase-Locked Loop. August 26, 2009. – p.3/

5. Phase-Locked Loop (4)
Linearized model of the PLL.
K 0 / kv
φ1 (t ) φe (t )
×
e(t) v(t)
∑
+
h(t)
__
φ2 (t )
× ∫0
t
dτ
2πkv
TELE3113 - Phase-Locked Loop. August 26, 2009. – p.4/

6. Closed-loop Gain
Consider a negative feedback amplifier, as follows.
µ
×
I(t) O(t)
∑
+
__
Network gain
β
The closed-loop gain of the amplifier is
O(t) µ
G = .
I(t) 1 + µβ
1
If µβ is large compared to one, G ≈ β .
TELE3113 - Phase-Locked Loop. August 26, 2009. – p.5/

7. Phase-Locked Loop (5)
If K0 is large compared to 1, then the closed-loop gain of the
PLL is
v(t) 1
≈ ,
φ1 (t) β
where β denotes the gain of the network generating FM waves.
Therefore,
1 dφ1 (t)
v(t) ≈ .
2πkv dt
Using (1), we continue to obtain
kf
v(t) ≈ m(t).
kv
Now the message signal m(t)is recovered from s(t).
TELE3113 - Phase-Locked Loop. August 26, 2009. – p.6/