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The integral method to calculate the power states in electrical circuits
Conference Paper in PRZEGLĄD ELEKTROTECHNICZNY · June 2009
DOI: 10.1109/CPE.2009.5156032 · Source: IEEE Xplore
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Marek T. HARTMAN
Gdynia Maritime University/ Department of of Marine Electrical Power Engineering, Gdynia, Poland
The integral method to calculate the power states in electrical
circuits
Streszczenie. W artykule przedstawiono uwagi i propozycje dotyczące opisów stanów energetycznych w obwodach elektrycznych. Na podstawie
obserwacji Emanuela, Erlickiego i Czarneckiego opisanych we wcześniejszych publikacjach, autor przeprowadza analizę matematyczną otrzymując
zmodyfikowaną formułę mocy biernej QI zaproponowaną przez Iliovici. Formuła ta ma związek z energią pola elektrycznego i magnetycznego
zgromadzonego lub występującego w otoczeniu elementów reaktacyjnych. Autor proponuje rozróżnienie przyczyn powstawania mocy biernej QI
oraz mocy nierównokształtności K. Obie te moce sa składnikami mocy nieaktywnej N. Sugeruje się przeprowadzenie procesów redukcji lub
eliminacji składników mocy nieaktywnej w dwóch etapach. (Metoda całkowa obliczenia stanów energetycznych w obwodach elektrycznych)
Abstract. Some remarks on the power states calculation in the electrical circuits have been described. The new equation for Iliovici’s reactive power
has been proposed. Based on Iliovici’s concept of the reactive power, Emanuel’s, Erlicki’s and Czarnecki’s observation concerning the reactive
power properties, the new term of power based on equiformity of voltage and current waveforms has been introduced. The two steps of non-active
power reduction or elimination has been also proposed.
Słowa kluczowe: moc bierna, moc nieaktywna, analiza obwodów, teoria mocy
Keywords: reactive power, non-active power, circuit analyze, power theory
Introduction
The problem of determining or defining the reactive
power and the non-active power has existed for many
years. An attempt to define these powers can be found in
the American standard IEEE Std 1459-2000 [1], which gives
the following:
 the definition of the reactive power Q according to
Iliovici’ conception [2] for single-phase sinusoidal voltage
u (t ) and current i (t ) waveforms
(1)
Q  QI 
1
2
 udi  
1
2
 idu 

kT
  kT
 i[ udt ]dt
 the definition of the reactive power QB according to
Budeanu’s conception for single-phase non-sinusoidal
voltage and current waveforms
(2)
QB   U h I h sin  h
h
where:
 h is a shift-phase angle between the voltage and
the current of h -th harmonic, h is the
 the definition of the non-active power N
(3)
harmonic order
N  S 2  P2
The standard [1] contains also the information that
Czarnecki [3] and Lyon [4] questioned the usefulness of the
power QB .
In the author’s opinion there are still ambiguities
concerning the notion of “reactive power” and “non-active
power”, their mutual relations or interpretations. The lack of
clear, unambiguous physical interpretation of both the
powers has resulted in scientific polemics [5].
Reactive power and nonactive power
Fryze [6][7] introduced the notion of ”multiplicity of the
function” – he wrote that: “the power factor  is equal to the
unity, which means it reaches the maximum, when at every
single moment the instantaneous current in the load is
proportional to the instantaneous voltage of the load. That
194
is: when the function i (t ) is the multiple of the function
u (t ) so that u (t )  Ri (t ) .
Czarnecki [8] introduced the notion of “mutually
proportional”
characteristics:
which
means
such
y (t )  cx(t )
characteristics, for which we have
and y  c x .
On the basis of Fryze’s and Czarnecki’s proposals it is
possible to come up with the following definition of the
“proportional characteristics of the voltage u (t ) and the
current i (t ) ”:
Def: The necessary and sufficient condition for the
periodical current and voltage waveforms to be proportional
is that the linear equality u (t )  R  i (t ) occurs, where
proportionality factor R is constant in the whole time
interval [0  t  T] and R is the natural number.
Proportional waveforms are characterised by two
properties:
- there is no time-shift between the voltage characteristics
u (t ) and the current characteristics i (t ) , therefore;
u (t   )  R  i (t   ) or u (t   )  Ri (t   )
- the voltage u (t ) and the current i (t ) waveforms have the
same shape so they have the equiform waveforms.
On the basis of the above, it is possible to consider a few
special cases:
a. Only the constancy of the factor R in the whole time
interval [0  t  T] causes the equality S  P  U RMS I RMS
occurs and the power factor   P / S  1 .
b. If there appears a time-shift between the voltage and
the
current
satisfying
the
following
equality
u (t )  A  i (t   ) or u (t )  A  i (t   ) , then the reactive
power QI appears in the circuit. For the sinusoidal voltage
and current characteristics it is QI  U RMS I RMS sin  and
S 2  P 2  QI2 .
c. If the load is resistive but non-linear or non-stationary
and if its resistance is the time function
Rt  f (t ) the situation is more complicated. Author take
into account Steinmetz’s remark included in [9]. For such a
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010
load characteristic the voltage and current are in phase but
they have not the equiform waveforms. This means, in the
author’s opinion that, the reactive power QI (1) is equal to
zero, but S  P .
d. For any single-phase circuit with the load of the nonresistive character and for the nonsinusoidal voltage and
current characteristics, except the reactive power, the nonactive power N characterizing also the shape of the
voltage and the current waveforms.
The conception of reactive power by Iliovici
According to Ohm’s law, on the terminals of any twoterminal network representing any kind of load (linear,
nonlinear, time - variant and so on), the fraction of voltage
u (t ) and current i (t ) describes the load character. Let
mark this fraction as z (t ) so that
(4)
(u  0  i  0)
z (t ) 
If
 functions u (t ) and i (t ) are of the class C1 (have
derivatives and are continuous in the whole range of the
function domain)
 functions u (t ) and
i (t ) are not of the class C1, but
have finite derivatives in the points, where the function is
not smooth (e.g. the characteristics of voltages and
currents describing the circuits with converters, in which
fast changes of the function of switches states occur),
then it is possible to apply Green’s theorem to (10) as
follows
(11) Y  
1
1


[  (udi  idu )]  { [ (i )  (u )]didu}
u
2 M
2 F i
Transforming the equality (11), we obtain :
Y 
u (t )
i (t )

(12)
during the time interval (0 < t < T).
Derivative the both side of (4) as Majewskij’s proposed [10]
we obtain
du
di
u
dz
 dt 2 dt
dt
i
1
1
du (t )di (t )  F
 

where: F is the area of the surface inside the contour M .
The equation (12) can be written in the following form :
The equation (5) can be rewritten as
(6)
1
{ [(1)  (1)]di (t )du (t )} 
2 
F
F
i
(5)

1
[  (udi  idu )] 
2 
M
Y
(13)
dz 2
du
di
di du
i i
 u  (u  i )
dt
dt
dt
dt
dt
1
F 1
[  (udi  idu )]  
2 


M
T
Y 
(14)
T
dz 2
di du
0 dt i dt  0 (u dt  i dt )dt
(7)

Divide the both side of (7) by 2 we get
(8)
1
2
T
dz
1
T
di
0
du
T


T
(0 < t < T) marks with its end a line l . If the line l is
continues and constitutes a closed curve, i.e. a closed
contour M , then it is possible to substitute the integration
in time [in the equation (9)] with the integration over the
curve l , that is: over the contour M . Treating i (t ) and
u (t ) as the parametric equations of the curve l (the
contour M ), the equation (9) can be written down as
follow:
T
1
di
du
1
{ [u (t )  i (t ) ]dt} 
dt
dt
2 0
2
 [udi  idu ] 
M
1
 udi     idu  2Q
I
M
M
T
1
di
du
1
di
du
(u  i )dt  
[(u (t )  i (t ) ]dt


dt
dt
dt
2 0 dt
2 0
Y 

1
QI 
we can ask: what the Y stand for ?
It can be concluded from the equality (9) that the vector Y
on the plane [ u (t ), i (t ) ] while moving in the time interval
(10)
F
1
di
du
1
{ [u (t )  i (t ) ]dt}  
2 0
dt
dt
2
0
Marking the right hand side of (8) by letter "Y "
(9) Y  
M
In that way we obtained the answer on given question: the
formula (9) marked by Y is equal to half of the reactive
power QI
 dt i dt   2  (u dt  i dt )dt
2
M
Based on (1)(9) and (13) we can write
Integrating (6) during the period of time [0, T], we obtain :
T
1
 udi     idu
 [udi  idu]
M
(15)

Y
1
di
du
  { [u (t )  i (t ) ]dt} 
2
4 0
dt
dt
1
4
 [udi  idu ] 
M
F
1


2 2
1
 udi   2  idu
M
M
The equation (15), similar to proposed by Majewskij [10],
includes the new formula of Illovici’s reactive power
(16)
QI 
1
4
T
T
dz 2
1
di du
0 dt i dt   4 0 (u dt  i dt )dt
In the author’s opinion, limiting the formula (1) only to the
sinusoidal voltage and current waveforms is unjustified. The
equality (15) has the universal character and does not
depend on the shape of the characteristics. So, the value of
the reactive power Q  QI can be calculated on the basis
of (15) or (16).
Emanuel and Erlicki [11][12] pointed out a possibility of
practical use of (15) forty years ago. In their opinion
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010
195
“nonlinear resistance behaves like a reactive-power
generator while having no energy-storing elements”.
Emanuel and Erlicki’s observations were confirmed among
others by Czarnecki, who wrote [8] that: ”the reactive power
in single-phase circuits with sinusoidal current and voltage
characteristics can be associated with the energy
accumulation in the reactance elements but it is not the
general property of the reactive power Q and we that can
interpret the reactive power Q as the measure of the
influence of the phase-shift of the current relative to the
voltage on the apparent power of the supplying source. This
shift can be caused by the presence of reactance elements
or simply by the periodical switch …….” In the author’s
opinion, the quoted Authors are right with regard to the
“reactive power”. It seems, however, that two causes of
presence of the “non-active power” in the circuits should be
clearly emphasized and interpreted.
Nonequiformity power K
In Budeanu’s conception it is possible to notice an
attempt to identify the causes of non-active power
generation. Not getting involved in the controversy over the
rightfulness of Budeanu’s conception, it is necessary to
notice that Budeanu suggested distinguishing ”the reactive
power QB ” as the measure of the influence of the phaseshift of the current with regard to the voltage (2) and “the
distortion power D ” as the measure of the mutual distortion
of the nonsinusoidal voltage and current waveforms.
Making use of the reactive power QI (15), it is possible to
ask the question: in what conditions is the reactive power
QI equal to zero, which means: when does the equality
T

T
di (t )
0 u (t )di(t )  0 [u (t ) dt ]dt   [U m sin t

U m2
R
 sin t cos td t

T
 i(t )du (t )  [i(t )
0

0
R
Um
sin t )
R
]d t 
(18)
dt

T
U m2
d(

U
d (U m sin t )
du (t )
]dt   [ m sin  t
]d t 
dt
R
dt

(19)

 sin t cos tdt

This results from the comparison of the results of
calculations of the integrals from (18) and (19) that the
equality (17) is satisfied and the reactive power QI in the
circuit is equal to zero ( QI  0 ).
The circuit from figure 1 has only a resistive character.
The resistive character of the circuit was reported earlier in
the paper [5]. Based on (16) one can draw some specific
conclusion:

if the load is a linear, time-invariant resistor so
z (t )  R , hence
dz (t ) dR

 0 and QI  0 ,
dt
dt
QI  0 appear ? On the basis of (15) and (9), it was found
out that the reactive power QI  0 , only when the equality:
(17)
T
T
0
0
 u (t )di(t )   i(t )du (t )  0
(a)
is satisfied. Yet the condition of zeroing the reactive power
QI (QI  0) is not the sufficient condition for the voltage
and current characteristics to be the proportional
characteristics, for which the equality S  P occurs. The
condition of the equiformity of the voltage and current
characteristics is explained by the following example. Let
the load in the circuit consist of a series connection of a
triac and a resistor as in
fig. 1
(b)
Fig. 2 Electrical circuit with the nonlinear resistance load: a)
characteristics of the voltage u (t ) and the current i (t ) , b)
characteristics

196
if the load is a linear but time-variant resistor so
z (t )  Rk in k – time interval, hence
dz (t ) dRk

 0 and QI  0
dt
dt
Fig. 1. Circuit with resistive load and triac
The voltage u (t ) has the sinusoidal waveforms and the
switching angle is    (deg). The time characteristics of
the voltage and current in addition to the characteristics on
the plane [ u (t ), i (t ) ] are presented in figure 2b.
For the given voltage and current waveforms, the values
of the integrals from the equality (16) are :
u (i )

if the load is a nonlinear resistor and voltage has
antysimmetric waveform there is
T
2
T
dz 2
dz 2
0 dt i dt   T dt i dt and QI  0 .
2
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010
As the current i (t ) waveform from figure 2 does not have
the same shape as the voltage u (t ) waveform, the nonequiformity power K will occur in the circuit.
Therefore the question arises how to calculate the nonequiformity power K . The author advocates the use of the
conception of current decomposition proposed by Fryze
[6][7] and Czarnecki [8][14]. The current active component
iaF (t ) according to the conception by Fryze, is the
characteristics proportional to the voltage characteristics
u (t ) . It determines the possible minimal effective value of
the current, at which for a given voltage u (t ) the same
active power P will dissipate in the circuit :
iaF (t ) 
(20)
P
u
2
u (t )
T
1
where: P   u (t )i (t ) dt is the active power, u
T 0
( K  0 ) to S 2  P 2  QI2

the phase-shift does not occur between the voltage
and the current ( QI  0 ), but the waveforms are not
equiform, then K  0 and S 2  P 2  K 2
the phase-shift does not occur between the voltage
and the current ( QI  0 ) and moreover, the waveforms

are equiform, then ( K  0 ) to S  P .
The author proposes to use :

the word “compensation” to make reduction of reactive
current component ir (t ) in order to decrease reactive

2
U
2
RMS
where ir (t ) is the current reactive component and ik (t ) is
the current nonequiformity component. When the reactive
power in the circuit is equal to zero
QI  0 so ir (t )  0 ,

the current in the circuit i (t ) , apart from the active
iaF (t ) , will have only the nonequiformity
*
k
component i (t ) :
(22)
In special cases when :
the phase-shift occurs between the voltage and the
current ( QI  0 ), but the waveforms are equiform, then
power QI ,
Based on Czarnecki’s CPC concept of current
decomposition, the supply current in a circuit with any load
(linear, nonlinear, time-variant etc.) according to the
author’s different interpretation, can be expressed as:
(21)
i (t )  iaF (t )  ir (t )  ik (t )
component

the word “equiformisation” to make reductions of
nonequiform current component ik (t ) so that to
decrease nonequiformity power K .
Traditionally passive shunt compensator (e. g capacitors)
can be used to carry out full compensation ( QI  0 ) but
only to decrease equiformity power K . This can be the first
step to achieve total non-active power N elimination. In the
second step of this procedure (when QI  0 ), the
nonequiformity power K should be reduced. This can be
done by the active filter or in some cases by the special
kind of passive filter. Proposed methodology can be explain
by using the power factor PF definition [15]
where:
i (t )  iaF (t )  i (t )
*
*
k
PF 
where * means under condition QI  0 .
Non-active power N
On the basis of the analyses presented in chapter
Nonequiformity power K, it is possible to formulate
unambiguously the components of the non-active power
with regard to the causes of its occurrence. Hence :

the reactive power QI is the measure of the energy
accumulation in the reactance elements (the measure
of the influence of the phase-shift of the current with
regard to the voltage).

the non-equiformity power K is the measure of
nonlinearity of the load without the energy
accumulation (the measure of the current non-equiform
with regard to the voltage).
Both the powers QI and K can be compensated and/or
corrected in the same way. This phenomenon has been
firstly reported by Czarnecki [13]. In a given circuit,
situations occur during which the powers QI and K are
not simultaneously equal to zero [ QI  0 , K  0 ]. On the
basis of the IEEE Standard, which introduced the notion of
the non-active power N (3), it is possible to write :
(23)
N  f (QI , K )
and thus, we obtain the square equation of the power in the
form :
(24)
PF  DPF  DF
(25)
P
P

is so called the true power factor,
S U RMS I RMS
DPF 
U1RMS I1RMS cos 
 cos 
U1RMS I1RMS
is commonly known as
the displacement power factor (for the purely sinusoidal
voltage/current waveforms),  is the power factor angle,
DF 
1
1  THD
2
u
1  THDi2
is so called the distortion
power factor.
If the voltage has the sinusoidal waveform, the distortion
power factor DF is simplified to the relation
DF 
1
1  THDi2
The displacement power factor DPF is related to the
reactive power QI , the distortion power factor DF is
related to the equiformity power K .
The practical example of the specific passive filter is
Lineator TM. “Lineator” is a wide spectrum filters, tri-limbed
reactor fitted with a small capacitor bank as illustrated in Fig
3. [16].
The Lineator filter is connected to the load as per a
standard AC line reactor [i.e. between the mains supply and
the rectifier(s)]. This makes the current equiformisation to
the voltage shape as is shown at Figures 4 and 5. It means
that Lineator reduces mainly the the distortion power factor
DF .
S 2  P 2  N 2  P 2  f (QI , K ) 2
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010
197
Conclusions
The presented conception of the description of the
power states in electrical circuits with any voltage and
current waveforms does not contain any limitations
connected with the analysis method (the convergence of a
Fourier series, the orthogonalization of the function) or the
kind of load (linear, nonlinear or time-dependent) and it can
be used in any single-phase circuit.
Particular power state values are calculated with the
help of integrals, so

the active power P
T
P
Fig. 3 Lineator wide spectrum filter schematic [16]
T
1
1
p(t )dt   u (t )i (t )dt
T 0
T 0

the reactive power QI

dz 2
1
di du
0 dt i dt   4 0 (u dt  i dt )dt
the apparent power S
T
1
4
S
1 2
1 2
u (t )dt
i (t )dt
T o
T o
T

T
QI 
T
the non-active power N
N  S 2  P2
Fig. 4 The power supply voltage and current waveforms with
150kW AC PWM drive load (with 3% DC bus reactor) [16]
The figure 5 illustrates that Lineator not reduce the
reactive current component significantly. The phase-shift
between power supply voltage and current still exists.
Actually Lineator does increase the displacement power
factor DPF slightly (nothing like active filter).
Fig. 5 The power supply voltage and current waveforms after
TM
application and with 150kW AC PWM drive load (with
Lineator
3% DC bus reactor )[16,17]
Due to that fact, the reactive power compensation
should be repeated and detuned. It is worth emphasizing
that the displacement power factor DPF can be
improvement only by e. g adding shunt capacitors or active
reactive power compensator via active filter or not static
VAR compensator.
If the active filter is applied in the second step of the
non-active power elimination, the reference current as ik (t )
can be calculate from equation (22).
198
For these reasons the author suggests naming this
proposal the integral method to calculate power states
values in the single-phase circuit.
The proposed decomposition of the instantaneous
current is similar to the way Czarnecki described in His
CPC power theory [8,14]. However, there are some
differences between CPC and the author’s concept and the
different interpretation are proposed:
1. It has been proposed to calculate the reactive power QI
from (16).
2. The condition whether the reactive power QI is not equal
to zero is based on (16)
3. The reactive power QI is the measure of the energy
accumulation in the reactance elements (the measure of the
influence of the time-shift of the current with regard to the
voltage).
4. The non-active power N consists of two components:
the reactive power QI and the nonequiformity power K .
5. The nonequiformity power K is the measure of
nonlinearity of the load without the energy accumulation
(the measure of the current non-equiform with regard to the
voltage shape).
6. The compensation is the action required to reduce or
eliminate the reactive power QI .
7. The equiformisation is the action required to reduce or to
eliminate the nonequiformity power K .
8. It has been proposed to carry out the non-active power
N reduction or elimination in two steps. In the first one, the
reactive power QI can be compensated by the passive
shunt filter e.g. capacitor. In the second step the
nonequiformity power K can be eliminated by the active
filter or in some cases by the special kind of passive filter.
Acknowledgment
The author thanks to Czesław Krawczyk, Ph. D, from
Department of Mathematics, and to Tadeusz Piotrowski Ph.
D. form Marine Power Engineering Department, both from
Gdynia Maritime University, Gdynia, Poland for their
assistance and fruitful discussion and as well as to Ian C.
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010
Evens from the Harmonic Solutions Co. UK for his
assistance and information about the Lineator industrial
application.
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Author: prof. dr hab. inż. Marek T. Hartman,
Gdynia Maritime University, 81-87 Morska Str.,
80-225 Gdynia, Poland, E-mail: mhartman@am.gdynia.pl
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010
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