Propagation of New Power Waves On a Complex
Impedance Transmission Line
Ronald H Johnston
Dept. of Elect. & Comp. Engg.
University of Calgary
Calgary, AB, Canada
rhjohnst@ucalgary.ca
Michal Okoniewski
Dept. of Elect. & Comp. Engg.
University of Calgary
Calgary, AB, Canada
okoniews@ucalgary.ca
Abstract— Travelling waves for real and complex reference
impedance are reviewed. New power waves are defined and their
behavior on a lossy transmission line are examined.
(7)
ࡽ = ݆(aܾ ‫ כ‬െܽ ‫)ܾ כ‬.
For a real Zo Eqns (6) and (7) are exact. If Zo has a imaginary
component both of these equations are incorrect.
Keywords— Travelling power waves, real and complex Zo, new
power waves.
B. PowerWaves,ComplexZo
To deal with the cases where Zo is complex, Youla-Kurokawa
developed other definitions [3].
I. INTRODUCTION
The traditional power waves are briefly reviewed and then
“New Power Waves” are developed and their behaviors on a
transmission line are examined.
II. POWERWAVE REVIEW
The concept of the existence of travelling voltage waves on a
uniform transmission line (constant Zo) is simple and intuitive.
In passive transmission lines forward and reverse travelling
voltage waves are attenuated and phase delayed as they travel
in their respective directions. This leads to simple and compact
equations. The two travelling waves may be summed at any
given location to give voltage (V(x)).
A. PowerWaves,RealZo
The forward and reverse voltage waves may be divided byඥܼ଴
to convert them into “Power Waves” [1,2]. They are defined
as:
ܽ = (ܸ + ‫ܼܫ‬ை )/(2ඥܼ଴ )
(1)
ܾ = (ܸ െ ‫ܼܫ‬଴ )/(2ඥܼ଴ )
(2)
ܽ௬௞ = (ܸ + ‫ܼܫ‬ை )/(2ඥܴ݁(ܼ୓ ))
ܾ௬௞ = (ܸ െ ‫ܼܫ‬ை‫) כ‬/(2ඥܴ݁(ܼ୓ ))
It is interesting to note that if we substitute ܽ௬௞ and ܾ௬௞ into (5)
we will obtain the correct result for the real power but the wrong
value for the reactive power.
III. NEW POWER WAVE DEFINITIONS
It is the principle intent of this paper to introduce new power
wave definitions that will remove problems noted with the above
two definitions of power waves. It is informative to rewrite (5)
using an impedance (ܼ) and an admittance (ܻ).
ࡿ = ‫ ܫ| = כ ܫ · ܼܫ‬ଶ |ܼ
ࡿ = V · (VY)‫ ܸ| = כ‬ଶ |ܻ ‫כ‬
ܽே௉ = (ܸ ඥܻ௢‫ כ‬+ ‫ ܫ‬ඥܼ଴ )/2
ܾே௉ = (ܸ ඥܻ௢‫ כ‬െ ‫ ܫ‬ඥܼ଴ )/2
(3)
(4)
One can now determine the complex power (Watts and VARs)
at specific location in a transmission line or a circuit.
ࡿ = ࡼ + ࢐ࡽ = ܸ‫כ ܫ‬
(5)
By noting that Zo is real only, one can substitute (3) and (4) into
(5) and obtain a simple and intuitive result. The real and
imaginary powers are:
(6)
ࡼ = |ܽଶ | െ |ܾ ଶ |,
978-1-5090-2886-3/16/$31.00 ©2016 IEEE
(10)
(11)
From these expressions we can note that the magnitude of
current squared times the impedance and that magnitude of
voltage squared times the conjugate of admittance both relate
to power. Therefore we can try a new definition for forward and
reverse power waves [4].
and this calculation may be reversed.
ܸ = (ܽ + ܾ)ඥܼ଴
‫ ܽ( = ܫ‬െ ܾ)ඥܼ଴
(8)
(9)
(12)
(13)
If values from (12) and (13) are substituted into (6) and (7)
correct values are obtained for the real and imaginary power.
The above expression shows the symmetry of the two terms in
each equation. These two equations differ from each other only
by the choice of positive current direction as is the case for the
power wave Eqns. (3) and (4). The reflection coefficient of a
load can be shown to be:
1607
߁ே௉ = (ܼ௅ െ |ܼ௢ |)/(ܼ௅ + |ܼ௢ |)
(14)
AP-S 2016
Note that a complex Zo TL has a self-reflection ߁ௌோ coefficient.
Calculated reflection coefficients based on this equation can be
plotted on a Smith chart without giving a reflection coefficient
larger than one as long as the load does not have a negative real
part. The magnitude of the complex ܼ௢ should be the
normalizing impedance of the Smith chart. A short has a
reflection coefficient of exactly -1.0 and an open has a
reflection coefficient of exactly 1.0.
We can calculate the node voltage and branch current in a
similar manner to (3) and (4).
V = (ܽே௉ + ܾே௉ )ඥܼ଴‫כ‬
I = (ܽே௉ െ ܾே௉ )ඥܼ଴
(15)
(16)
IV. NEW POWER WAVES ON A TRANSMISSION LINE
The determination of the forward and reverse New Power
waves on a uniform transmission line is mathematically simple
but lengthy. The most general expressions for voltage and
current on a TL use hyperbolic functions.
Fig.1. ܽே௉ (solid lines) and ܾே௉ (dashed lines) on a TL with a
Zo EODFN DQGíȍ UHG WHUPLQDWLRQV+RUL]RQWDOVFDOHVDUH
in radians (a) Vertical scale is magnitude in dB, (b) vertical
scale is in degrees. The phases of the waves of the íȍ UHG termination only are shown.
terminations, the first termination is ܼ௢ (essentially making the
line to appear to very long) and the second termination is a
negative resistance with a value of -1.25 ɏ. The magnitudes of
(17)
ܸ(‫ܸ = )ݔ‬௅ (cosh(ߛ‫ )ݔ‬+ ܼ଴ sinh(ߛ‫ )ݔ‬/ܼ௅ )
ܽே௉ and ܾே௉ for both terminations are plotted in Fig. 1a and
(18)
‫ܫ = )ݔ(ܫ‬௅ (cosh(ߛ‫ )ݔ‬+ ܼ௅ sinh(ߛ‫ )ݔ‬/ܼ଴ )
the phases of ܽே௉ and ܾே௉ are plotted in Fig. 1b. These plots
show that ܽே௉ is exponentially attenuated and delayed as it
Equations 15 & 16 should be substituted in (17) and (18) and
travels left to right but the self reflection of the TL produces a
the results should be expanded into exponential terms and then
ܾே௉ (right to left) that is equal to ܽே௉ times ߁ௌோ . Near the
substituted into (12) and (13). Simplification of those results
highly reflective termination ܾே௉ is much larger than the ܽே௉
give (19) and (20) which also defines a Transfer (T) matrix.
and it is travelling right to left. The ܾே௉ is large enough that it
produces a component ܽே௉ that interacts with the component of
(19)
ܽே௉ (‫( = )ݔ‬ห݇௣ଶ ห݁ ఊ௫ +|݇௡ଶ |݁ ିఊ௫ )ܽே௉ +
ܽே௉ coming from the source. In the middle of the TL, the ܾே௉
‫כ‬
‫ି כ‬ఊ௫
ఊ௫
(݇௣ ݇௡ ݁ + ݇௣ ݇௡ ݁ ) ܾே௉
has two sources which are not in phase alignment thereby
(20) causing constructive and destructive interferences.
ܾே௉ (‫݇( = )ݔ‬௣ ݇௡‫ ݁ כ‬ఊ௫ + ݇௣‫݇ כ‬௡ ݁ ିఊ௫ ) ܽே௉ +
ଶ |݁ ఊ௫
ଶ
ିఊ௫
+|݇௣ |݁ )ܾே௉
(|݇௡
V. CONCLUSIONS
Where
Definitions for “New Power Waves” are presented. These
݇௣ = (ܼ௢‫ כ‬/|ܼ௢ | +1)/2 and ݇௡ = (ܼ௢‫ כ‬/|ܼ௢ | -1)/2
The best way provide an understanding of the behavior of the
New Power Waves is to plot some examples of the waves on a
lossy and complex impedance TL. This transmission line has
values of ߛ = ͳ‫ס‬ͺͲ.91° nepers and radians per unit length and
ܼ௢ = 1‫ס‬-9.09° Ohms. The TL is given two different
waves produce correct real and imaginary power calculations
for complex reference impedances. On a TL it is found that the
forward power wave generates a continuously reflected reverse
power wave due to the self-reflection property of the
complex ܼ௢ transmission line.
ACKNOWLEDGMENT
NSERC discovery grant funding is gratefully acknowledged.
REFERENCES
C.G. Montgomery, R.H. Dicke, and E.M. Purcell (eds),
PrinciplesofMicrowaveCircuits, McGraw-Hill Book Company,
New York, 1948.
[2] R.E. Collin, Foundations for Microwave Engineering, New
York: McGraw Hill, 1966.
[3] D. C. Youla, “On Scattering Matrices Normalized to Complex
Port Numbers”, Proc. IRE, vol. 49, no. 7, Jul. 1961, p. 1221.
[4] R.H. Johnston.”New Power Waves Simplify Complex
Impedance S-Parameter Network Analysis”, Paper submitted to
IEEE Trans. Circuits Syst. II, Exp. Briefs, Dec., 2015.
[1]
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