ee/educ

1. Basic Power Factor Consideration in Simple Series RLC Impedance Load
Roa, F. J. P.
Ref’s
[1] Hugh Young, Roger Freedman, Francis Sears, Mark Zemansky, University Physics,
12th edition, 1949, Pearson Education
[2]AC Power Circuit Analysis, http://webstaff.kmutt.ac.th/~dejwoot.kha/
[3]http://www.slideshare.net/AcacioDantas1/power-system-analysisstevenson
[4]http://www.allaboutcircuits.com/textbook/alternating-current/chpt-11/power-resistive-
reactive-ac-circuits/
Suppose we are to maintain or to keep constant the (absolute) magnitude of load active
power βcos
mDOLP being supplied to an impedance load DOLz that consists for
example, of a series qqL CLR −−′ circuit. This circuit load we are to set up then as
connected to a perfect ac voltage source gv via a line resistance that we call iR .
For a series qqL CLR −−′ circuit, the power factor is equal to the ratio of the load
resistance LR′ to the absolute magnitude DOLz of the load impedance of this series
circuit. That is,
(1)
DOLmg
Lmg
DOL
L
zi
Ri
z
R
2
2
cos
′
=
′
=β
βcos=DOLPF

2. Here, by power factor, we simply refer to the cosine of the phase angle β between the
load current gi and load voltage DOLv . Note here that this load voltage is not equal to the
source voltage since there is a resistance iR that exists between them as we have set up.
For our present purposes, say that this power factor DOLPF has a linear dependence on
the load resistance with the absolute magnitude of the load impedance held constant. So
in this way, we can say that to lower values of load resistance there correspond lower
values of DOLPF in the case for a series circuit, while keeping the absolute magnitude of
load impedance fixed.
Because of the presence of line resistance not all of the (total) active power supplied by
the source voltage goes to the load impedance. Only a fraction of this total active power
goes to the said load.
(2)
gmg
Li
L
mDOL P
RR
R
P ψβ coscos
′+
′
=
(Note: This is expressed as absolute magnitude.)
In (2), the ratio of the magnitude of load active power to the magnitude of the total active
power gmgP ψcos is equal to the factor
(3)
Li
L
RR
R
′+
′
The other part of gmgP ψcos is dissipated at the line resistance and we can actually
express this in terms of the magnitude of load active power
(4)
βcos
2
mDOL
L
i
mgimiR P
R
R
iRP
′
==
We may call the ratio of line resistance iR to the load resistance LR′ as feedback factor
in (4) as we have expressed it (4) using the (absolute) magnitude of load active power.
This amounts to saying that the power being dissipated at the line resistance is a fraction
of the magnitude of the load active power fed back to the said line resistance and this

3. power is inversely proportional to the load resistance. Meaning, line resistance dissipates
more power at lower values of load resistance, while using the same amount of load
active power and keeping the value of line resistance fixed. Note here that in the case for
series impedance, lower values of load resistance imply lower DOLPF , while holding
constant the magnitude of load impedance (reflect on (1)). So, line power dissipation is
greater at lower values of DOLPF , while requiring the same amount of load active power
and keeping the value of line resistance fixed.
We have to note that with ever decreasing values of load resistance (with corresponding
decrease of DOLPF for a fixed series load impedance), while keeping the same
magnitude of load active power, the total active power requirements increase with that
decreasing load resistance, and in the limits where Li RR >> , this total active power
required from source becomes approximately equal to the line power dissipation
(5)
miRgmg PP ≈ψcos
This increases further with ever decreasing load resistance (or DOLPF ) for a given fixed
value of line resistance with the same magnitude of load active power.
In addition, let us note that if we are holding the absolute magnitude of this series load
impedance constant, then decreasing the power factor of this load means increasing the
reactive loading effects as to be noticed with
(6)
βsinDOLqe ZX =
This is because as 0cos →β , consequently, 1sin →β so that DOLqe ZX = at 0=′LR .
Although in this limiting case where the load impedance is acting more of a reactive load
with low DOLPF , nonetheless the same load active power is being maintained at this load
and the result since this load is more reactive than resistive some of this load active
power that is being maintained is given back to the line resistance iR as dissipative
power (4).
For a series qqL CLR −−′ impedance load DOLz , we have relation (6) as an additional to
(1) in which we assumed DOLz as constant, and through these relations we can define
the phase angle β by

4. (7)
L
qe
R
X
′
=βtan
For the moment say, we also keep fixed the absolute magnitude
mDOLv of load voltage
as well, while we are maintaining the same load active power. Consequently, this well
make the magnitude of load current
mgi vary in inverse proportion as the DOLPF and
since tconsv
mDOL tan= , the load power factor ( DOLPF ) can also be varied directly as
the magnitude of load impedance. However, this no longer holds for a series
qqL CLR −−′ load because as we recall in series, DOLPF varies inversely with DOLz .
In a more general case where DOLz could either be a series or paralled impedance,
(To insert a schematic)
the absolute magnitude gz of total impedance gz can be expressed as
(8)
)(coscos DOLgDOLgig zRz ψψψ −+=
where gψ is the phase angle between gi and gv and DOLψ is the phase angle between
gi and DOLv . It is more useful to regroup the terms in this by re-writing this expression
in the following form
(9)
gDOLDOLgDOLDOLig zzRz ψψψψ sinsincos)cos( ++=
and identify the active and reactive parts respectively as
(10.1)
DOLDOLigg zRz ψψ coscos +=
and
(10.2)
DOLDOLgg zz ψψ sinsin =

5. Consequently, from this identifications we can define the phase angle gψ by
(10.3)
DOLDOLi
DOLDOL
g
zR
z
ψ
ψ
ψ
cos
sin
tan
+
=
The magnitude
mg actP )( of total active power )(actPg supplied by the source is given
by the following definition
(10.4)
ggmggmgmg ziPactP ψψ coscos)(
2
==
and this can be expressed in terms of (10.1) as
(10.5)
DOLmDOLmgigmg PiRP ψψ coscos
2
+=
where we identify as the magnitude of load active power
(10.6)
DOLDOLmgDOLmDOLmDOL ziPactP ψψ coscos)(
2
==
and the magnitude of power dissipated at iR as
(10.7)
2
mgimiR iRP =
In these, the magnitude of total apparent power is given by
(10.8)
gmgmg ziP
2
=
while the magnitude of load apparent power as

6. (10.9)
DOLmgmDOL ziP
2
=