Download additional maximum voltage and maximum power transfer conditions

ADDITIONAL MAXIMUM VOLTAGE AND MAXIMUM
POWER TRANSFER CONDITIONS
[Technical letter]
Fayez Mohammed EL-Sadik
http://fayezmohammed.googlepages.com/
Department of Electrical & Electronic Engineering,
University of Khartoum, Sudan
ABSTRACT
Results of a generalized load impedance statement that will include as special
cases the conditions for maximum power transfer with conjugate impedance
matching and the maximum load voltage condition are presented.
These are based on an analytic solution to the problem of stability limits of radial
transmission circuits with inter-nodal constraints as shown in Figure 1 where
maximum received power is defined by the resulting magnitudes of sending and
receiving end voltages as the angular separation between the two variables
approaches the transmission angle.
Figure 1
In this Letter, a presentation in terms of graphical demonstrations of the results
of the generalized load impedance function is given. This leads to proofs of the
well-known maximum power and maximum voltage conditions where the
terminal and all other circuit voltages will come out as a result.
1. RESULTS OF A GENERALIZED LOAD
IMPEDANCE STATEMENT
In a previous paper [1], the impedance
conditions for Brainerd’s hypothesis of
maximum load voltage and Jacobi’s
theorem for maximum power transfer
with complex conjugate impedance
matching were verified as two special
cases of a continuous P-V stability
boundary function that has been based
on a transducer circuit model with an
internal node voltage constraint. This
has, in a fulcrum-cantilever type of
action, imposed restrictions on the
magnitude of the source voltage of any
given transducer circuit as the
conditions are identified through two
different equilibrium states.
•
In this Letter, the load impedances
for both conditions will be derived as
special states of a generalized
power-maximizing
impedance
function where transducer circuit
source voltage is independently
Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48
65
retained. As a result, a given
transmission circuit can be made to
deliver maximum powers less than
that which can be obtained from the
upper limit of conjugate matching;
with the extreme lower limit of power
maximums evolving in Brainerd’s
hypothesis for the maximum load
voltage condition.
•
Received power
The results to be presented will
show that generality of the powermaximizing
load
impedance
statement is derived from an
inherent property of concurrent load
voltage maximization. This will in
addition introduce a new cut-off
impedance
condition
for
the
termination of voltage and hence
power transfer of any given
transducer circuit.
2. CONDITION FOR CONCURRENT
MAXIMUM VOLTAGE AND MAXIMUM
POWER
•
Terminal voltage
ZL
Figure 3
•
Consider the case of a total series
impedance ZS = 1.0 + j√3.0 p.u at a
supply voltage ES = 1.0 /_0o p.u as
shown in Figure 2.
Figure 2
•
•
When conjugate matched, the
available transfer capability of the
circuit gives Pmax = 0.25 p.u and
this occurs at a receiving end
voltage VR = 1.0 p.u and a load
impedance of magnitude ZL = 2.0
p.u
Now consider the terminal voltage
and power curves shown in Figure 3
where it can be seen that:
66
3.
a.
Maximum voltage of 1.0 p.u is
synonymous with the maximum
power of 0.25 p.u ; both
occurring at the load impedance
magnitude 2.0 p.u
b.
The two curves will terminate at
a specific impedance value of
4.0 p.u.
The
curves
are
plotted
as
demonstration of the existence of a
capacitive load impedance function
ZL = R (ZL) - j X (ZL) that will extract
powers with a maximum of circuitmatched value which is concurrent
with the corresponding maximum
value of terminal voltage. In addition,
the simultaneous discontinuities of
voltages and powers shown are
indicative of the existence of a cutoff load impedance value above
which no transfer of voltage and
hence power will take place.
CONDITIONS FOR REDUCED
POWER MAXIMUMS
• In its generalized form, the
power-maximizing load impedance
statement is in fact both voltage and
power-dependent i.e., ZL = R (ZL ,
pv) - j X (ZL , pv), where pv is the
defining
parameter
for
those
terminal conditions.
Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48
•
The sample set of power curves in
Figure (4.a) and the associated load
voltage curves in Figure (4.b)
showing the influence of this
parameter for the present case study
reveals the natural tendency of
terminal voltage increase with
decreased load powers as well as
the associated differences in the
transfer cut-off impedance limits.
Notice that no such limit is attached
to the lower curve-c powers within
the range of ZL considered.
0.25
a
0.2
0.15
b
0.1
0.05
may however be conducted. This
entails finding the load impedance
elements
that
will
generate
associated power and voltage
curves
while
observing
the
occurrence of the maximums of
these
curves
at
the
circuit
transmission angle.
4. PROOF OF BRAINERD’S MAXIMUM
LOAD VOLTAGE CONDITION
•
In addition to its verification of the
upper power point of conjugate
matching, a significant feature in
support of the continuous powerand
voltage--maximizing
load
impedance function is provided by
the lower limiting zero-state of the
voltage-producing power curve (a)
in Figure 5; as this continuous state
evolves in Brainerd’s impedance
condition for maximum voltage.
c
2
0
0
5
10
ZL
ZL
15
1.8
1.6
Figure 4a: Load powers
1.4
1.2
2
1
1.8
0.8
c
1.6
0.6
1.4
b
1.2
0.2
1
0
0
c
0.8
0.6
0.2
0
5
10
ZLZL
15
Figure 4b: Corresponding load voltages
•
0.5
1
1.5
2
2.5
3
3.5
ZL
4
Figure 5: Impedance for maximum
voltage
0.4
0
(a) lower limiting
zero-power states
0.4
While the complete derivation
procedure of the analytic load
impedance function is left for a full
publication, a numerical solution for
the behaviors of the voltage- and
power- producing elements of ZL
As stated in his paper published in 1933
[2], while the conditions for current and
power maximums are commonplace,
specific
problems
of
voltage
maximization
are
solved
by
differentiation and, according to the
author’s knowledge, the condition for
maximum voltage has evolved in a class
discussion involving logical reasoning as
to the nature of physical loads followed
by differentiation.
Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48
67
•
•
Bernard Miller, a senior student in
the Moore School, was in particular
credited for the discussion leading
to the hypotheses which states that
maximum voltage at the terminals
of a passive linear transducer of
constant source voltage is obtained
when the load impedance is purely
capacitive and has the magnitude
ZS2/XS.
This can be seen as concurring with
the impedance result obtained from
Figure 5 giving a corresponding
maximum voltage of 2.0 p.u for the
case considered.
68
References
1.
EL-Sadik,
F.M.:
“Voltage
Constraints for the Maximum
Power Transfer Theorem &
Brainerd’s Resonance Voltage
Impedance Condition ” ; Sudan
Engineering
Society
Journal,
Volume 52, Number 45, January
2006.
2.
Brainerd, J.G.: “Some General
Resonance Relations and A
Discussion
of
Thevenin’s
Theorem”; Proceedings of the
Institute of Radio Engineers,
Volume 21, Number 7, July 1933.
Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48