Conventional Synchronisation Techniques - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 349

the output of the PD, is driven to zero eventually. As a result, the phase of the output signal is
locked with that of the input signal.
Figure 22.2(b) shows the control structure of a simple PLL, where the PD unit is a multiplier,
the LF is a low-pass filter (LPF) and the VCO consists of a PI controller, an integrator and
a sinusoidal function. For an input signal v = V m cos θ g with phase θ g = ω g t + φ g and an

output signal y = sin θ with phase θ = ω t + φ , the output of the PD unit is

v = v y = V m sin θ cos θ g

V m
2

V m
2

= sin( θ − θ g ) + sin( θ + θ g ) . (22.1)

V m
2

The two components in (22.1) can be rewritten as sin[( ω − ω g ) t + ( φ − φ g )] and

V m
2

φ + φ g )]. It is obvious that the first term is a low frequency component
that contains the phase difference between

sin[( ω + ω g ) t + (

and y and the second term is a high frequency
component, which is out of interest and can be filtered out with the loop filter. The output d of
the LF is

v

V m
2

d = sin[( ω − ω g ) t + ( φ − φ g )] ,

˙

ω = θ =

which is then fed into a PI controller to generate the estimated frequency until d 0.

= θ

The estimated frequency is integrated to form the phase of the output signal y , which

is sent back to the PD unit to complete the loop. In the steady state, d is driven to zero and
θ = θ g , i.e.

sin

ω = ω g and φ = φ g . The phase of the output signal y is said to be locked with

that of the input signal .

It is worth noting that the input signal and the output signal are actually 90 ◦ shifted in order
for the DC component d to be zero when

v

is a cosine
curve and y is a sine curve. This is not a problem because a constant can be added to

θ = θ g . Indeed, in the case shown here, v

θ to

obtain any phase angle needed.

22.4 PLL in the Synchronously Rotating Reference Frame (SRF-PLL)
A common technique in three-phase applications is a PLL in the synchronously rotating
reference frame (SRF-PLL) (Amuda et al . 2000; da Silva et al . 2010; Kaura and Blasko 1997),
which is shown in Figure 22.3. Similar operating concepts can also be found, e.g. in (Chung
2000).

αβ

θ

αβ

θ θ

αβ

Figure 22.3 Three-phase PLL in the synchronously rotating reference frame (SRF-PLL)