POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
Compa a i e Analysis o La ice-based All-Pass
Fil e and Second O de Gene alized In eg a o as
O hogonal Sys em Gene a o o a PLL
Luciano Emilio BELANDRIA1, Nancy Alejand a AGUDELO1,
Joan BERGAS-JANE 2
1Depa men o Elec onic Enginee ing, Na ional Expe imen al Uni e si y o Tachi a,
Uni e sidad, 5001 San C is obal, Venezuela
2Cen e o Technological Inno a ion in S a ic Con e e s and D i es, Depa men o Elec ical Enginee ing,
Poly echnic Uni e si y o Ca alonia, Diagonal 647, 08028 Ba celona, Spain
lb[email p o ec ed]e, [email p o ec ed], joan.gab iel.be [email protected]
DOI: 10.15598/aeee. 19i1.4002
A icle his o y: Recei ed No 13, 2020; Re ised Jan 31, 2021; Accep ed Feb 10, 2021; Published Ma 31, 2021.
This is an open access a icle unde he BY-CC license.
Abs ac . This pape p esen s a s eady-s a e compa i-
son o wo me hods ha gene a e an o hogonal ol age
sys em o a single-phase Phase-Locked Loop (PLL)
s uc u e: a widely accep ed one based on a Second
O de Gene alized In eg a o (SOGI) and a new
one based on a All-Pass Fil e (APF) wi h La ice
s uc u e. Bo h me hods a e e y a ac i e because
o hei simple digi al implemen a ion, low compu a-
ional load and good pe o mance unde ha monically
dis o ed g id condi ions and a iable equency, so
hey a e a good al e na i e o o he known me hods.
The pape de i es and analyzes he ull s a e space mod-
els o he wo me hods. I is shown ha hese wo
me hods a e equi alen in he mos common ope a ion
condi ions o dis ibu ed ene gy esou ces, al hough
he APF s uc u e is clea ly be e han he SOGI
one because i main ains i s o hogonal gene a ion
abili y o any highe no ch equencies and any lowe
sampling equencies. The compa a i e analysis we e
alida ed by simula ion using MATLAB/Simulink and
expe imen al esul s using a ixed-poin DSP.
Keywo ds
All-Pass Fil e , O hogonal Signal Gene a o ,
Phase-Locked Loop, Single-Phase PLL, No ch
Fil e , Second O de Gene alized In eg a o .
1. In oduc ion
The use, de elopmen and deploymen o Dis ibu ed
Ene gy Resou ces (DERs), especially enewable e-
sou ces, has inc eased d ama ically in he las decade.
This is changing he pa adigm o elec ic gene a ion
[1], [2], [3], and [4]. Single-phase g id-connec ed in-
e e s a e ound in many DERs, such as pho o ol aic
in e e s and ene gy s o age de ices [5] and [6].
In e e sys em con ol mus ensu e ha he powe
gene a ion sys em is synch onized wi h he g id, and
ha phase angle jumps a e de ec ed o eliable powe
deli e y [7] and [8]. Mo eo e , i mus ensu e ha g id-
connec ed sys em pe o mance complies wi h ope a ion
equi emen s unde he mos common dis o ions, such
as line ha monics, no ches, ol age dips, ises and alls,
and equency a ia ions.
Phase, equency and ampli ude cha ac e ize he
single-phase g id ol age signal and knowledge o
hese pa ame e s is undamen al in he design o g id-
connec ed in e e sys ems [9], like he one in Fig. 1.
In o de o mee he new equi emen s o ne wo k codes
and op imize in e e pe o mance, a PLL is used o
apid synch oniza ion wi h he g id.
The main ask o he PLL is o accu a ely de ec
he ac ual ol age phase angle a he Poin o Com-
mon Coupling (PCC), e en in he p esence o ol age
ha monics and unbalance [10] and [11]. This s uc u e,
o example, should be used o p o ide any uni y powe
ac o ope a ion which in ol es synch oniza ion o he
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 1
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
DC
Powe
Sou ce
PLL
In e e Fil e T ans o me G id
PWM
PCC
ac iac
q
, , V
RMS
Con olle
Fig. 1: G id-connec ed powe con e sion sys ems.
in e e ou pu cu en wi h he g id ol age and o
gi e a clean sinusoidal cu en e e ence, among o he
unc ions.
Fo h ee-phase applica ions, some o he ecen and
mos popula g id synch oniza ion echniques o de-
ec ing he phase and equency o he mains ol age
signal using PLL a e p esen ed in [12], whe e a com-
ple e s udy was ca ied ou on he con ol s a egies o
Dis ibu ed Gene a ion Powe Sys ems (DGPSs) unde
ideal and non-ideal ne wo k condi ions. These ech-
niques a e: basic s uc u e adi ional Synch onous
Re e ence F ame PLL (SRF-PLL), Enhanced PLL
(EPLL), Dual Second O de Gene alized In eg a o
PLL (DSOGI-PLL), Mo ing A e age Fil e (MAF) and
Decoupled Double Synch onous Re e ence (DDSRF).
Indeed, he single-phase s uc u e o PLLs limi s he
use o some well-known h ee-phase con ol s a egies
[13] and [14]. In single-phase sys ems, less is known
abou he ne wo k ope a ing condi ions han in h ee-
phase sys ems. Tha is why he mos ad anced me h-
ods used o o e come his limi a ion mus c ea e an
o hogonal ol age sys em [15] and hen exploi he
exis ing h ee-phase con ol me hods. In his line o
esea ch, se e al ad anced PLL echniques, ha e been
p oposed o single-phase applica ions [16], [17], [18],
and [19].
dq
Vol age
Moni o ing
F equency
and Phase
Es ima o
O hogonal
Sys em
Gene a o
a b
V
V
VV
V
V
a
b
RMS
q
d
q
Fig. 2: PLL using he pa k ans o ma ion.
Gene al s uc u e o a single-phase PLL algo i hm
based on O hogonal Signal Gene a o (OSG), also
called Quad a u e Signal Gene a o (QSG), o g id
synch oniza ion is p esen ed in [15], [16], [20], [21], [22],
[23], and [24]. This s uc u e can use he Pa k T ans-
o ma ion, as shown in Fig. 2 o he a c- angen unc-
ion depic ed in Fig. 3. The main di e ence be ween
OSG-based single-phase PLLs lies in he way o hogo-
nal ol age sys ems a e gene a ed.
Vol age
Moni o ing
F equency
and Phase
Es ima o
O hogonal
Sys em
Gene a o
V
V
V
V
a
b
RMS
q









b
a

an 1_
+
q
Fig. 3: PLL using he a c angen unc ion.
The OSG-based single-phase PLL s uc u es, Fig. 2
and Fig. 3, ound in he s a e-o - he-a me hods a e
basically o med by wo blocks. In he i s , an o hog-
onal sys em in phase wi h he abo e signal is gene a ed
om a single e e ence sinusoidal signal. The second
block uses ei he a eedback loop h ough he αβ o dq
ans o ma ion o he a c- angen unc ion o de e -
mine he phase angle o he e e ence signal.
The me hods o OSG o a single-phase PLL mus
be easy o apply in p ac ice. Mo eo e , he OSG mus
deli e he il e ed ou pu wi hou any delay, because
o i s esonance a he undamen al equency, no o
be a ec ed by equency changes. The mos common
me hods used in he li e a u e o gene a e he o hog-
onal ol age a e p esen ed in [15], [19], [20], [21], [25],
[26], [27], [28], [29], [30], [31], and [33]. One using
a block o a T anspo Delay unc ion is shown in Fig. 4.
I in oduces a 90 deg ee phase shi wi h espec o
he inpu signal [21] and [28]. Ano he me hod uses he
Hilbe T ans o ma ion [29], depic ed in Fig. 5, and he
In e se Pa k T ans o ma ion [19], [30], and [31], shown
in Fig. 6. Howe e , hese me hods ha e one o mo e o
he ollowing de iciencies: equency dependence, high
complexi y, non-linea i y, and poo o no il a ion.
V
V
V
a
b
-1
T/4
delay
Fig. 4: T anspo delay unc ion.
To imp o e he de iciencies men ioned in he OSG
me hods, an imp o ed A e age Fil e (AF) is p oposed
in [32], wi h a simple s uc u e and implemen a ion,
which, in eg a ed in o OSG, a enua es he nega i e e -
ec o ol age ha monics and unbalances on o hogonal
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 2
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
V
V
V
a
b
-1
Hilbe
T ans o m
Fig. 5: Hilbe ans o ma ion.
V
V
b
a b
dq
LPFLPF
V
q
Vd
q
V
a
Fig. 6: In e se pa k ans o ma ion.
d-q signals o PLL. P o iding as phase de ec ion and
as dynamic esponse wi hou using he second o de
il e s, which educe he dynamic esponse ime and
he de ec ion o he synch oniza ion uni .
In [34], a F equency Lock Loop (FLL) wi h a Gen-
e alized In eg a o (GI) was p oposed. In he GI-FLL,
he GI pa is a OSG, while he FLL pa uses he sig-
nals gene a ed o unknown equency es ima ion pu -
poses. The GI wo ks as an adap i e bandpass il e
using coo dina e ans o ma ion, which allows o im-
p o e he dynamic uning ange, wi h an excellen bal-
ance be ween he con e gence speed and he maximum
accep able peak es ima ion e o , bu wi h some com-
pu a ional cos addi ional o ha needed o accu acy
and he FLL has o add he no maliza ion o he gain
ha he GI does no ha e.
The echnique using he g id ol age demodula ion
o ob ain a OSG is p oposed in [35], in a Demodu-
la ion Type PLL (DT-PLL) wi h imp o ed DC o se
ejec ion capaci y, o he adap i e es ima ion o he
phase angle and he equency o a single-phase sys-
em, which a oids he use o any low-pass il e . The
demodula ion has good dynamic pe o mance and dis-
u bance ejec ion abili y. Howe e , due o he p es-
ence o igonome ic quan i ies in he es ima o dy-
namics, small-signal modelling-based pa ame e uning
can be complica ed o DT-PLL. Mo eo e , eal- ime
implemen a ion o igonome ic unc ions is compu a-
ionally expensi e.
A single-phase PLL s uc u e based on SOGI which
o e comes he abo e p oblems and a oidance o il e -
ing delays due o i s esonance a he undamen al e-
quency was p esen ed in [15], [20], [21], [36], [37], [38],
[39], [40], and [41]. Thus, he way in which he wo sig-
nals a e gene a ed is imp o ed. In [42], [43], and [44]
a h ee-phase PLL s uc u e based on a Double Second
O de Gene alized In eg a o (DSOGI) is p esen ed.
The SOGI s uc u e has also been applied o o he
aspec s o powe elec onic con ol, especially in he
cu en con ol loop [37] and [38], de ec ion o ha mon-
ics [39] and ac i e an i-islanding me hods [22].
Howe e , as al eady indica ed in [15], [37], and
[38], he SOGI was designed in he con inuous ime
and quad a u e phase delay and ampli ude ipples a e
p esen in i s disc e e applica ion. Mo eo e , i is di -
icul o apply his s uc u e on a ixed-poin DSP o
FPGA due o hei limi ed p ecision and sensi i i y o
coe icien ounding.
A second-o de APF wi h La ice s uc u e was p o-
posed in [45], [46], [47], [48], [49], and [50]. This il e
gene a es o hogonal signals necessa y o he PLL,
wi h good noise il e ing capabili y bu ampli udes di -
e en om ha o he single-phase inpu signal. He e
a APF wi h La ice s uc u e is p oposed as OSG which
mee s all equi emen s and o e comes all he d aw-
backs o he me hods used in single-phase PLL while
main aining any uni y gain wi h espec o he inpu .
This pape pe o ms a compa a i e s eady-
s a e analysis o he s uc u es based on APF
and SOGI as pa o a PLL. These s uc u es
a e widely used o il e ing he powe supply
signal o ensu e he bes possible synch oniza-
ion sys em e e ences o use in single-phase
con e e s ope a ing in highly dis u bed en i onmen s.
Solu ions o he disc e e applica ion o he wo s uc-
u es a e also p o ided. Simula ions and a ixed-poin
DSP implemen a ion alida e he e ec i eness o one
o ano he s uc u e.
The es o he pape is o ganized as ol-
lows. Sec ion 2. desc ibes he la ice APF
as OSG wi h i s main diag ams and equa ions.
In Sec. 3. , he SOGI algo i hm is p esen ed. Sec-
ion 4. p o ides disc e e ime simula ion esul s o
bo h OSGs using as inpu a 50 Hz no malized sinu-
soidal signal in bo h il e s. In Sec. 5. , he expe i-
men al esul s o a ixed-poin DSP implemen a ion o
bo h OSGs unde he same simula ion condi ions a e
discussed. Finally, Sec. 6. d aws he conclusions.
2. La ice-Based APF
In he app oach p oposed in [49], no uni y gain has
been conside ed while, in his pape , a new s uc u e o
OSG based on APF, as illus a ed in Fig. 7, is p oposed
cha ac e ized by uni y gain [50].
In he sys em, he ou pu signals x1(n)and x2(n)
ha e a −90 deg ee and 0 phase shi , espec i ely,
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 3
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
-1
-1 -cos 1
-sin 1
sin 1
u(n)
x (n)
1
sin -1
2
-sin 2
-cos 1
sin 2
x (n)
2
-sin -1
2
y(n)
Fig. 7: APF as OSG wi h uni y gain.
wi h espec o he inpu signal wi h uni y gain,
a e de ined as:
x1(n) = cos θ1(1−sin θ2)z−1
1 + sin θ1(1 + sin θ2)z−1+ sin θ2z−2u(n),
(1)
x2(n) = sin θ1(sin θ2−1)z−1+ (sin θ2−1)z−2
1 + sin θ1(1 + sin θ2)z−1+ sin θ2z−2u(n).
(2)
The space-s a e equa ion o he APF wi h La ice
s uc u e can be ob ained om he sys em in Fig. 7, is
de ined as:


x1(n+ 1)
x2(n+ 1)
y(n)

=
=
−sin θ1cos θ1sin θ2cos θ1(1 −sin θ2)
−cos θ1−sin θ1sin θ2sin θ1(sin θ2−1)
0−(1 + sin θ2) sin θ2

·

x1(n)
x2(n)
u(n)

,
(3)
whe e θ1is associa ed wi h he no ch equency ω0, a
which he APF o e s a phase shi o π adians, and
θ2is associa ed wi h he 3 dB a enua ion Bandwid h
BW o he il e . They a e de ined as:
θ1=ω0
s−π
2,(4)
θ2= a csin 



1− an BW
2
1 + an BW
2




,(5)
BW =2πB
s
,(6)
whe e sand Bco espond o he sampling equency
and bandwid h in Hz, espec i ely. Independen
adjus men o no ch equency and bandwid h is a de-
si able a ibu e. Adjus ing he bandwid h o any
sampling equency om Eq. (6), ei he di ec ly o
adap i ely, allows he APF o ejec e y low equen-
cies, e en ejec ing he DC o se , wi hou adding an-
o he ype o il e and wi hou in e e ing wi h he
uning equency. This ea u e o he APF can comply
wi h hose o he OSG wi h DC o se ejec ion ca-
pabili y p oposed in [33], wi h only he adjus men o
a single pa ame e .
The s uc u e in Fig. 7, is heo e ically
s able and nume ically well beha ed in ime- a ying
en i onmen s [49]. Each o a ion angle ωk(k = 1, 2) is
di ec ly con olled, so ha θ1and θ2a e con e ed in o
he adjus able pa ame e s o adap i e pe o mance.
Elemen s Aand Bo he s uc u e a e ex ac ed
om Eq. (3):
A=−sin θ1cos θ1sin θ2
−cos θ1−sin θ1sin θ2,(7)
B=cos θ1(1−sin θ2)
−sin θ1(1−sin θ2).(8)
Figu e 8, shows he Bode diag am o he APF as
no malized OSG o a uning equency o 50 Hz.
Magni ude (dB)
Phase (deg)
F equency (Hz)
-80
-60
-40
-20
0
100101102103
-180
-90
0
90
180
x (n)
x (n)
1
2
Fig. 8: Bode diag am o la ice-based APF as no malized OSG.
3. The SOGI
A SOGI is equi alen o wo P opo ional-In eg al (PI)
con olle s in synch onous e e ence ames compensa -
ing he sequences o posi i e and nega i e [15], [20],
[37], [38], [39], and [40].
SOGI is p oposed o ob ain a ze o s eady s a e e o
using sinusoidal e e ences wo king on he αβ s a ion-
a y e e ence ame. This me hod has been included
in ha monic elimina ion algo i hms (because ha mon-
ics ac in a e y na ow band a ound hei esonance
equency); g id sequence de ec ion and quad a u e
signal gene a ion algo i hms; algo i hms o con e e
synch oniza ion o he powe supply; and o mul i-
equency de ec ion.
The ans e unc ion o a SOGI o a single sinu-
soidal signal is [37] and [38]:
G(s) = 2s
s2+ω2
0
,(9)
whe e ω0is he esonance equency and s he
Laplace ope a o . The in eg a o ou pu con ains no
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 4
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
only he in eg a ed inpu bu also an insigni ican ad-
di ional componen . In o de o use hem o gene a e
quad a u e signals, he o iginal opology is modi ied,
as shown in Fig. 9. The esul ing ma hema ical ex-
p ession is:
G(s) = x(s)
y(s) =ω0s
s2+ω2
0
.(10)
(s)
y(s)
_
+
w
1
s
1
s
x(s)
0
w
0
Fig. 9: Con inuous- ime SOGI.
The new ans e unc ion has wo poles a ±jω0and
a ze o a he o igin, jus as Eq. (9). The only di e ence
be ween he wo exp essions is in he gain, which is no
signi ican as a as he inal pe o mance is conce ned.
Howe e , he unc ion leads o a mo e gene al s uc u e
ha can be used o bo h powe con e e s con ol and
synch oniza ion asks.
The SOGI s uc u e o gene a ion o o hogonal sig-
nals, also known as OSG-SOGI, is ou lined in Fig. 10.
As can be seen, he basic elemen is a SOGI [15], [20],
[37], [38], [39], and [40]. The con inuous- ime ans e
unc ions a e:
’( s)
( s)=Ksω0s
s2+Ksω0s+ω2
0
,(11)
q ’( s)
( s)=Ksω2
0
s2+Ksω0s+ω2
0
.(12)
’(s)
_
+
w
1
s
1
s
x(s) 0
w
0
_
+
K
(s)
s
q ’(s)
SOGI
Fig. 10: Con inuous- ime OSG-SOGI.
Disc e e- ime implemen a ion o he SOGI can be ac-
complished by disc e izing he con inuous- ime ans e
unc ions o by using disc e e in eg a o s [15], [37], and
[38].
Figu e 11 shows he disc e e- ime SOGI s uc u e.
I s ou pu beha es like ha o he con inuous- ime
SOGI in Fig. 9 upon applica ion o a s ep signal o am-
pli ude 1 a inpu x(n), ha is a sinusoidal signal o
pulsa ion ω0and ampli ude 1. The choice o his SOGI
is based on he use o disc e e Eule Backwa d In eg a-
o wi h compu a ional delay added in se ies wi h he
eedback gain, modeling he inhe en delay caused by
he p og amming p ocess. Mo eo e , i s s uc u e is
mo e simila o ha o he classical PI con olle .
++
++
x(n) y(n)
_
+
Z-1
Z-1
(n) Z-1
w T
0 S
w T
0 S
Eule Backwa d In eg a o
Eule Backwa d In eg a o
Fig. 11: SOGI based on Eule Backwa d In eg a o and com-
pu a ional delay.
The ans e unc ion o he disc e e- ime SOGI is:
G(z) = ω0Ts−ω0Tsz−1
1+(ω2
0T2
sz−1−2) + z−2.(13)
Figu e 12 displays he SOGI based on disc e e Eu-
le Backwa d In eg a o wi hin an OSG-SOGI, which
ensu es ha signals ’ and q ’ a e quad a u e signals
a all ope a ing equencies. The OSG-SOGI allows
independen adjus men o he no ch equency ω0and
o he 3 dB a enua ion bandwid h BW, conside ing
he sampling pe iod Ts, acco ding o:
K =ω0Ts,(14)
Ks=BW
ω0
√0.98.(15)
++
++
u(n) ’(n)
_
+
Z-1
Z-1
q ’(n) Z-1
Eule
Backwa d
In eg a o
_
+
K
s
Z-1
Eule
Backwa d
In eg a o
SOGI
K
K
Fig. 12: Disc e e- ime OSG-SOGI s uc u e.
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 5

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
The OSG-SOGI can also be exp essed by s a e equa-
ion o m based on he inal ci cui o Fig. 13, which is
equi alen o ha o Fig. 12, whe e x1(n)and x2(n)a e
he signals co esponding o he o hogonal sys em and
a e deno ed as in he APF, wi h a −90 and 0 deg ee
phase shi , espec i ely, and uni y gain wi h espec
o he inpu signal o powe u(n)a ins an n.
+
+
u(n) ’(n)
_
+
Z-1
q ’(n)
_
+
K
s
Z-1
+
+
x (n)
1
x (n)
2
K
K
Fig. 13: De ini i e OSG-SOGI s uc u e.
This o hogonal sys em is gi en by:
x1(n) =
=KsK2
z−1
1+(KsK −2 + K2
)z−1+ (1−KsK )z−2u(n),
(16)
x2(n) =
=KsK z−1−KsK z−2
1+(KsK −2 + K2
)z−1+ (1−KsK )z−2u(n).
(17)
Figu e 14 o e s he Bode diag am o he OSG-SOGI,
o a uning equency o 50 Hz. Conside ing ha ou -
pu s x1(n+ 1) and x2(n+ 1) a e he same answe s
q ’(n) and ’(n), he s a e equa ion is:


x1(n+ 1)
x2(n+ 1)
y(n)

=
=

1−K2
K (1 −KsK )KsK2
−K 1−KsK KsK
0 1 0

·

x1(n)
x2(n)
u(n)

.
(18)
Elemen s Aand Bo he s uc u e can be ex ac ed
om Eq. (18):
A=1−K2
K (1 −KsK )
−K 1−KsK ,(19)
B=KsK2
KsK .(20)
Magni ude (dB)Phase (deg)
F equency (Hz)
-80
-60
-40
-20
0
100101102103
-180
-135
-90
-45
0
45
90
135
180
q (n)
(n)
Fig. 14: Bode diag am o he OSG-SOGI.
4. Beha io and Compa ison
o OSGs
The s uc u es o he la ice-based APF in Fig. 7 and
he OSG-SOGI in Fig. 13 can be easily implemen ed
as ma ix sys em consis ing o elemen s Aand Bonly,
hus we will ha e he equa ion o s a e o he o m:
x1(n+ 1)
x2(n+ 1)=A B·x(n)
u(n).(21)
Bo h s uc u es ha e he abili y o gene a e, in phase
wi h he inpu signal, an o hogonal sys em. Figu e 15
shows he implemen a ion o he la ice-based APF and
OSG-SOGI s uc u es in s a e equa ion. Two pa ame-
e s iden i ied as x1(n)and x2(n), which cons i u e he
o hogonal sys em, a e obse ed.
+
A
B
u(n) x (n)
1
x (n)
2
+
Z-1 x (n)
Fig. 15: OSG in s a e equa ion.
Bo h s uc u es a e simula ed unde he same con-
di ions o compa e hem as Band-Pass Fil e s (BPF)
which gene a e a dis u bance- ee o hogonal sys em
wi h one signal image o he undamen al inpu signal
and ano he one delayed 90 deg ees wi h espec o he
same inpu signal. The wo il e s a e e alua ed using
any uni y ampli ude sinusoidal signal a he equency
o 50 Hz as he undamen al inpu signal. Is se s a sam-
pling equency o 20 kHz, 50 Hz uning equency and
a low bandwid h o 4 Hz, na ow enough o many ap-
plica ions whe e a e y selec i e magni ude esponse is
equi ed, wi h a high quali y ac o . The simula ion is
implemen ed in MATLAB/Simulink. The s a e equa-
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 6
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
ion om Eq. (3) o he APF wi h he alues es ima ed
unde hese condi ions is:
x1(n+ 1)
x2(n+ 1)=
=0.9998766 0.0156876 0.0000197
−0.0157073 0.9986209 0.0012557·

x1(n)
x2(n)
u(n)

.
(22)
And he s a e equa ion om Eq. (18) o he OSG-
SOGI wi h he alues es ima ed:
x1(n+ 1)
x2(n+ 1)=
=0.9997532 0.0156884 0.0000195
−0.0157080 0.9987560 0.0012440·

x1(n)
x2(n)
u(n)

.
(23)
Figu e 16 gi es he equency esponse o ou pu s
x2(n)/u(n), o Eq. (2) and Eq. (17), co esponds o he
APF and he OSG-SOGI. Fo equencies o e 10 Hz
bo h s uc u es main ain a s ong simila i y in bo h
magni ude and phase. Bu o equencies below 10 Hz
hey al eady begin o di e en ia e, he APF begins o
inc ease i s phase o alues g ea e han 90 deg ees.
Fo equencies below 1 Hz he di e ence in magni ude
begins o be mo e no iceable.
Magni ude (dB)Phase (deg)
F equency (Hz)
-80
-60
-40
-20
0
100101102103104
-180
-90
0
90
180
OSG-SOGI
APF La ice
Fig. 16: Bode diag am o x2(n)/u(n)in La ice-based APF and
OSG-SOGI.
As shown in Eq. (22) and Eq. (23), he alues o
he coe icien s o bo h il e s a e nea ly iden ical un-
de hese ope a ing condi ions, he di e ences be ween
he coe icien s a e less han 0.0001233. Hence, hei
equency esponses should ha e a e y simila beha -
io .
The equency esponses a e ob ained o
x2(n)/u(n), in he APF and in he OSG-SOGI
o a ange o undamen al equencies, om 100 Hz
o 10 kHz, wi h a bandwid h o 4 Hz, a sampling
equency o 20 kHz, o bo h il e s. In Fig. 17,
one can obse e ha bo h OSG beha e like a BPF,
bu o undamen al equencies g ea e han 400 Hz,
he magni ude esponses begin o ha e di e ences,
al hough hey main ain hei uning. F om equencies
g ea e han 3 kHz, he SOGI begins o lose i s uning,
bo h in i s magni ude and phase esponse.
-150
-100
-50
0
Magni ude (dB)
101102103104
-180
-90
0
90
180
Phase (deg)
F equency (Hz)
OSG-SOGI
APF La ice
Fig. 17: Bode diag am o x2(n)/u(n)in La ice-based APF and
OSG-SOGI o a ious uning equencies.
102103104
-80
-60
-40
-20
0
Magni ud(dB)
APF La ice
OSG-SOGI
(a) Fo a ious uning equencies.
Magni ud(dB)
102103104105
-25
-20
-15
-10
-5
0
5
APF La ice
OSG-SOGI
(b) Fo a ious sampling equencies.
101102103104
-35
-30
-25
-20
-15
-10
-5
0
5
F ecuency(Hz)
Magni ud(dB)
APF La ice
OSG-SOGI
500Hz 14 kHz
17 kHz
20 kHz
23 kHz
29 kHz
11 kHz
8 kHz
5 kHz
2 kHz
26 kHz
(c) Fo a ious sampling equencies and a ia ion o he
uning equency.
Fig. 18: Magni ude o x2(n)/u(n)in he La ice-based APF
and OSG-SOGI.
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 7
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
-3
-2
-1
0
a21
-1
0
1
a22
102103104
-2
0
2x 10 -3
F ecuency(Hz)
a23
-6
-4
-2
0
2
a11
0
1
2
3
a12
0
2
4x 10-3
a13
10210 10
F ecuency(Hz)
3 4
APF La ice
OSG-SOGI
(a) Fo a ious uning equencies.
-10
-5
0
5
a11
0
1
2
3
a12
-0.5
0
0.5
1
a13
10210310410
5
F ecuency(Hz)
-3
-2
-1
0
a21
-1
0
1
a22
-0.4
-0.2
0
0.2
a23
102103104105
F ecuency(Hz)
APF La ice
OSG-SOGI
(b) Fo a ious sampling equencies.
Fig. 19: Beha io o he ma ix pa ame e s o he s a e equa-
ion o APF and OSG-SOGI.
Figu e 18 displays he esul o he magni ude o
x2(n)/u(n)o he la ice APF and he OSG-SOGI
wi h a bandwid h o 4 Hz and a sampling equency o
20 kHz a a ious equencies, o a uning equency o
50 Hz wi h sampling equencies anging om 100 Hz
o 100 kHz and o a ious sampling equencies and
a ia ions o he equency o uning. As can be no-
iced, om a uning equency o 600 Hz and a sam-
pling equency below 1 kHz, he uning o he SOGI
is no accu a e, and so a e i s cha ac e is ics as OSG,
which does no happen wi h he APF.
The magni ude in dB o x2(n)/u(n)is ob ained by
e alua ing he APF and he OSG-SOGI wi h a band-
wid h o 4 Hz, depic ed in Fig. 18(a) o a sampling
equency o 20 kHz and he a ia ion o he uning
equency. In Fig. 18(b) he uning equency is kep
a 50 Hz wi h a ia ion o he sampling equency om
100 Hz o 100 kHz and in Fig. 18(c) he esul o he
magni ude is shown o se e al sampling equencies
(500 Hz, 2 kHz, 5 kHz, 8 kHz, 11 kHz, 14 kHz, 17 kHz,
20 kHz, 23 kHz, 26 kHz, and 29 kHz) and a ia ion o
he uning equency. As can be seen, he APF main-
ains a cons an magni ude o 0 dB o all he a ia-
ions in all he e alua ed anges. On he o he hand,
he SOGI loses he uning s a ing om a uning e-
quency o 500 Hz and a sampling equency lowe han
1 kHz, and he e o e hence i s cha ac e is ics as OSG.
Figu e 19 shows he beha io o he ma ix pa ame-
e s o he s a e equa ion o he APF Eq. (3) and he
OSG-SOGI Eq. (18) o a ious uning and sampling
equencies. The pa ame e s begin o di e mo e sig-
ni ican ly om uning equencies g ea e han 500 Hz
and sampling equencies below 1 kHz in mos cases.
This is demons a ed by he esul s in Fig. 16, Fig. 17,
and Fig. 18.
Figu e 20, has he magni ude esponse su ace o
x2(n)/u(n), as a unc ion o he uning equency and
he sampling equency. Obse ing ha o he APF,
he esul is a comple ely la su ace wi h a magni-
ude o 0 dB cons an o any a ia ion o he equen-
cies. This does no occu o he OSG-SOGI, whe e i
is obse ed ha he magni ude esponse a ies as he
equencies a y, mo ing away om he equi ed pass-
band. Being he wo s case when he uning equency
inc eases and he sampling equency dec eases.
0
1
2
x 10
4
0
1
2
3
x 10
4
-300
-200
-100
0
F ecuency(Hz)
Sampling F ecuency(Hz)
Magni ud(dB)
-350
-300
-250
-200
-150
-100
-50
0
(a) APF.
0
1
2
x 10
4
0
1
2
3
x 10
4
-300
-200
-100
0
F ecuency(Hz)
Sampling F ecuency(Hz)
Magni ud(dB)
-350
-300
-250
-200
-150
-100
-50
0
(b) OSG-SOGI.
Fig. 20: Su ace o he magni ude o x2(n)/u(n) o a ious
sampling equencies and a ia ion o he uning e-
quency.
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 8
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH
5. Simula ion and
Expe imen al Resul s
In o de o alida e he analysis, he APF
and he OSG-SOGI a e simula ed by using
MATLAB/Simulink p og am and an expe imen al
se up was implemen ed. A CDM2480 es pla o m,
wi h a TMS320F2812 ixed-poin DSP om Texas
Ins umen s as he cen al elemen , sui able o mo ion
con ol and powe elec onics applica ions, was used
o implemen and es bo h s uc u es as OSG. The
ixed poin DSP, wi h a clock equency o 150 MHz,
was used o gene a e he inpu signal, he OSG
algo i hms and he ou pu signals wi h a 12 bi D/A
con e e .
All simula ed and expe imen al esul s, we e ob-
ained using he s uc u es in Fig. 7 and Fig. 13. The
ou pu s x1( )and x2( )o he APF and he OSG-SOGI
a e ob ained. The OSG inpu signal and pa ame e s
a e he same o he simula ion and he expe imen al
pa , whe e he inpu signal is a sinusoid wi h uni y
ampli ude and equency equal o he uning equency
o he il e . A bandwid h o 4 Hz was se o he de-
sign o he OSGs. In he DSP, a ixed-poin Q15 base
was used o global calcula ions whe eas a Q30 base
was used o calcula ions o he il e s.
Vol age(V)
0.5 0.505 0.51 0.515 0.52 0.525 0.53 0.535 0.54
-1
Time(sec)
-1
0
1
-1
0
1
-1
0
1
-1
0
1V e
V e 90º
V e
V e 90º
Vol age(V)Vol age(V) Vol age(V)
2APF
1APF
2SOGI
1SOGI
(a) Simula ion.
V e 2APF
V e 90º
1APF
V e 2SOGI
V e 90º
1SOGI
(b) DSP Implemen a ion.
Fig. 21: x1( )and x2( )o he APF and he OSG-SOGI o
a sampling equency o 20 kHz and uning equency
o 50 Hz.
In he expe imen al esul s ob ained o he OSGs,
channels A and B in blue a e x2( )and x1( )o he
APF wi h [1 V/di ]; channels C and D in g een a e
x2( )and x1( )o he SOGI wi h [1 V/di ]. Co e-
sponding e e ence signals (in ed), an o he unda-
men al inpu signal on channels A and C and delayed
signals 90 deg ees wi h espec o his inpu signal on
channels B and D, we e added.
The esul s o a equency o 50 Hz o he inpu sig-
nal and uning, wi h a sampling equency o 20 kHz,
a e p esen ed in Fig. 21, whe e i is e idenced ha
bo h OSGs beha e in acco dance wi h he es ablished,
bo h in he simula ion and in he DSP implemen a ion.
The ou pu s x1( )and x2( ) ollow hei e e ences and
e ain hei wa e o m, and main ain he gain o 1, nec-
essa y condi ion o bo h OSGs ope a ing as PLLs.
-1
0
1
-1
0
1
-1
0
1
0.5 0.5005 0.501 0.5015 0.502 0.5025 0.503 0.5035 0.504
-1
0
1
Time(sec)
Vol age(V)
V e
V e 90º
V e
V e 90º
Vol age(V)
Vol age(V) Vol age(V)
2APF
1APF
2SOGI
1SOGI
(a) Simula ion.
V e
2APF
V e 90º 1APF
V e
2SOGI
V e 90º
1SOGI
(b) DSP Implemen a ion.
Fig. 22: x1( )and x2( )o he APF and he OSG-SOGI o
a sampling equency o 20 kHz and uning equency
o 500 Hz.
The sampling equency is kep a 20 kHz and he
equency o he inpu and uning signal was inc eased
o 500 Hz in Fig. 22 and 1000 Hz in Fig. 23. In he
simula ion o bo h uning equencies, he signals o
he o hogonal sys em co esponding o he APF ha e
a good ollow-up o he e e ence signals wi h a gain o
1. The OSG-SOGI, loses i s cha ac e is ics like OSG,
wi h an ad ance o 0.0002 seconds, ela i e o he e -
e ence signals, which in oduces a phase shi , as well
as a dec ease in he gain, which is no eaching 1.
©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 9