1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
The use of Geometrical Inversion Method
to Calculate the Non-Dissipative Impedance Adaptor
Eugeniu Man and Lucian Man
Abstract — It is discussed the possibility to adapt known load impedance with a non-dissipative two-port, using the geometrical
inversion in the complex semi plan of the impedances and admittances.
The problem has minimum 2 and maximum 4 solutions; the two-port has only two components, in the ‫ ך‬and Γ scheme.
Keywords: impedance adaptor, non-dissipative two-port, geometrical inversion.
1. INTRODUCTION
The adaptation of the known load impedance
Z S = RS + jX S to a needed value Z i = Ri + jX i , also
knowing the angular frequency ω , can be done by
intercalation of a two-port non-dissipative, linear and
passive, which has the fundamental parameters
A , B ,C , D (figure 1):
Zi =
Figure 3. Γ equivalent scheme
AZ S + B
;
CZ S + D
A D − BC = 1 .
2. ADAPTING WITH ‫ך‬
(1)
TWO-PORT
The unknowns X and B are real solutions of the complex
equation:
(Z
)
−1
−1
S
− jB + jX = Z i .
(2)
The geometrical approach uses the inversion in the
complex impedance and admittance semi plan (figure 4),
the solutions being possible if the route from A (Z S ) to
G (Z i ) is included, in the specified conditions.
Taking into consideration the unitary power inversion, it
is defined the beginning of the route:
*
−1
B ZS ,C ZS .
(the symbol * represents the complex conjugate of the
complex value)
Figure 1. The two-port non-dissipative,
linear and passive
( ) ( )
The reference [3] has established algebraically the
existence of an infinite number of solutions for the three
components of the two-port, T or π equivalent. Between
those solutions are also the two-ports realized with only
two components, equivalent scheme ‫( ך‬figure 2) and Γ
(figure 3).
(
)
−1
−1
The point F [ Z S − jB ] is searched for on the
straight line (Di ) , which passes through G and is
parallel with the imaginary axis, followed by the addition
with jX .
(
)
−1
(
−1
)
*
The points D Z S − jB and E [ Z S − jB ] are on the
curve symmetrical to the straight line (Di ) , which is the
circle (Ci ) with the diameter between the origin (0 ,0 )
(
)
and the point Ri−1 ,0 .
On the other hand, to arrive from C to D it is added
− jB , so C and D are on the straight line (D0 ) , parallel
with the Oy axis.
Figure 2. ‫ ך‬equivalent scheme
Graphical, the solution exists if the straight line (D0 )
intersects the circle (Ci ) :
Eugeniu Man is with the Technical University of Cluj-Napoca,
Romania, Electrical Engineering Faculty, phone: +40-264-401-462;
e-mail: Eugeniu.Man@yahoo.com.
Lucian Man is with the Technical University of Cluj-Napoca,
Romania, Electrical Engineering Faculty, phone: +40-264-401-462;
e-mail: Lucian.Man@et.utcluj.ro.
331
1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
Figure 4. Geometrical Inversion Method for 1 < Ri < R S
L=
−1
Re { Z S } ≤ Ri−1 ;
X S2
.
RS
; C=−
B
ω
.
(4’)
3. ADAPTING WITH Γ TWO-PORT
⎧
⎫
1
−1
Re ⎨
⎬ ≤ Ri ;
R
+
jX
S ⎭
⎩ S
Ri ≤ RS +
X
ω
The scheme from Figure 3 is characterized by the
complex equation:
[( Z
]
−1
+ jX )−1 − jB = Z i ,
(5)
and to this equation, the path from A(Z S ) to G (Z i )
cross in the complex plane from Figure 7 the points:
*
−1
B Z S + jX ; C ⎡ Z S + jX ⎤ ; D (Z S + jX ) ;
⎥⎦
⎢⎣
(3)
Even more, the straight line (D0 ) and the circle (Ci )
intersect in two points, giving the second solution
resulted from D ′ ≡ E and E ′ ≡ D , continuing with F ′ ,
point conjugated with F.
In the example from figure 4, X < 0 and B > 0 leads to
a capacitor respectively to a coil:
1
1
; L=
.
(4)
C=−
ωX
ωB
In Figure 4 the graphical construction is illustrated for the
case 1 < Ri < RS .
Nothing is modified in the reasoning, if Ri < 1 < RS
(Figure 5), or if Ri < RS < 1 (Figure 6). The straight line
(
E [(Z
S
)
(
]
{[
[
)
]
]
}
+ jX ) − jB ; F (Z S + jX ) − jB
.
The unknowns of the problem, X and B , have real
solutions (positive or negative) only and only if the
straight line (DS ) , parallel with Oy , cuts the circle
(C S ) :
−1
S
−1
[ { }]
−1
RS ≤ Re Z i
RS ≤ Ri +
(Di )
passes through the intersection points between the
circles of radius 1, respectively ( Ci ) .
Of course, X > 0 leads to the use of a coil, and B < 0
impose the use of a capacitor, with the values:
332
−1
−1
;
X i2
X2
; Ri ≥ RS − i .
Ri
Ri
(6)
1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
Figure 5. Geometrical Inversion Method for Ri < 1 < RS
Figure 6. Geometrical Inversion Method for Ri < R S < 1
333
1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
Figure 7. Geometrical Inversion Method – adapting with Γ two-port
4. CONCLUSIONS
Eugeniu Man was born on 01.26.1947, in Braisor, ClujNapoca, Romania. Graduated at the Technical
University of Cluj-Napoca, Faculty of
Electrotechnics (1969), PhD (1987), Professor
at the Technical University, Electrotechnics
Department (1992). Author at 62 scientific
papers and 13 books, in electrical circuits,
sequential commands, management of quality.
The number of the distinct solutions, the two-ports Γ and
‫ך‬, confirm the conclusions of the reference [3], in
accordance with the table:
Table 1. Number of solutions
Ri
Γ
‫ך‬
Total
Rs −
0
0
2
2
1
2
3
X i2
Ri
Rs +
2
2
4
2
1
3
X s2
Rs
∞
2
0
2
Lucian Man was born on 04.07.1980, in Cluj-Napoca,
Romania. Graduated at the Technical
University of Cluj-Napoca, Faculty of
Electrical Engineering (2004), post-graduated
at the CAD Department (2005), PhD Student
at the Electrotechnics Department (2004).
Author at 20 scientific papers and 3 books.
Major fields of study: electromagnetic
compatibility, electrical circuits.
Choosing any of the values Z S and Z i , at a given
pulsatance ω , there are always 2 - 4 solutions to realize
the adaptor using non-dissipative two-ports ‫ ך‬or Γ. The
option is related to costs, reliability, stability or other
criteria.
5. REFERENCES
[1]. Simion, E., Maghiar, T., Electrotehnica, Bucuresti, Editura
Didactica si Pedagogica, 1981.
[2]. Smith, K.C.A., Alley, R.E., Electrical Circuits, Cambridge
University Press, 1992.
[3]. Man, E., Viorel, A., Man, L., Adapting the Impedance using
non-dissipative Two-Ports, 1st International Conference on
Modern Power Systems, Cluj-Napoca, 2006.
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