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T

The invariant properties of two-port circuitsAlexandr A. Penin

Abstract —Application of projective geometry to the theory of two-ports and cascade circuits with a load change is considered.The equations linking the input and output of a two-portare interpreted as projective transformations which have theinvariant as a cross-ratio of four points. This invariant has placefor all regime parameters in all parts of a cascade circuit. Thisapproach allows justifying the denition of a regime and itschange, to calculate a circuit without explicitly nding the a -parameters, to transmit accurately an analogue signal throughthe unstable two-port.

Index Terms —circuit regime, geometric circuit theory, projec-tive geometry, two-port.

I. INTRODUCTION

T HE theory of two-port circuits is widely used for theanalysis of linear electric circuits. It is convenient to use

the methods of projective geometry in the case of a load’schange in a wide range. It allows visually establishing therelationship between the various parameters of regime forindividual parts of cascade connection of two-port circuits. Inparticular, this approach justies the denition of the regimein the relative view (relative to these characteristic regimes asan open circuit OC and a short circuit SC ). Also, it allowscomparing the regimes of the different circuits.

In projective geometry, corresponding expressions denemapping between the elements of one-dimensional manifolds,such as direct lines. In this case, the invariant properties suchmaps or projective transformations are become apparent. Thus,if there is a value equal to all parts of the circuit or network,it is presents of interest to the theory of two-port and maybe useful in practice. This paper presents the results obtainedearlier by the author.

I I . T HE PROJECTIVE TRANSFORMATIONS IN THE CASCADE

CIRCUIT

Let we consider a cascade connection of the two two-port

T P 1 and T P 2 with the changeable conductivity or resistanceof the load within a wide range on Fig. 1. Load characteristicsor load lines for all parts or sections of the circuit can becalibrated in corresponding values of conductivity, current orvoltage. The Fig. 1 shows there is the specied mapping aload characteristics of different sections to each other, whichit is conditionally shown dashed lines.

Let the conductivity of the load Y H 2 be changed from thevalue Y 1H 2 to the value Y 2

H 2 , i.e. a changing of the load orof the parameters of regime is dened by the length of thesegment Y 2H 2 Y 1H 2 .This change determines the correspondchanges of the conductivity Y H 1 and Y H 0 . It may be noted

A.A. Penin is with the Institute of electronic engineering and industrialtechnologies, Academy of Sciences of Moldova, e-mail: [email protected]

Manuscript received August 24, 2009; revised November 11, 2009.

Fig. 1. A cascade connection of the two two-port with the changeableconductivity of the load

that the length of segments for all the load lines is differentfor the usually used Euclidean geometry. However, if themapping is viewed as a projective transformation (in thesense of projective geometry), the invariant, a cross-ratio of four points, is performed, which denes the same length of segments.

The entire circuit regarding load Y H 2 represents an activetwo-pole or an equivalent generator. The demonstration of the projective geometry and the denition of regimes of

International Journal of Electrical and Computer Engineering 4:12 2009

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the equivalent generators are considered in [1], [2]. In thisregard, let we use and complement the denitions valid forthe cascade circuit:

a) Regime of a circuit is a point (the work point) with aspecied (projective) coordinate on the load characteristicsand coordinate axes;

b) Change of the regime is a movement of a point forall characteristics, which determines the segments withcorresponding length;

c) There is an independence or invariance of the regimeand its changes to the parts of a circuit.

III . P ROJECTIVE CONFORMITY OF THE INPUT -OUTPUT OF

A TWO -PORT

Let we consider the rst asymmetrical two-port T P 1 andgive the necessary expressions. We use the specic values

of elements for visibility and dimensions of values are notindicated for simplifying of the record. As it is known,equation T P 1 through the Y - parameters or parameters of conductivity, taking into account the directions of currentshas the form [3]:

I 1

−I 0 = −Y 11 Y 10

Y 01 Y 00. U 1

U 0

where:

Y 00 = Y 10 + Y 0 = 5 , Y 11 = Y 10 + Y 1 = 6 .4, Y 10 = Y 01 = 4

The determinant of the Y -matrix is:

∆ = Y 00 Y 11 −Y 01 Y 10 = 16 , √ ∆ = 4

The characteristic conductance of T P 1 to the input and outputis:

Y BX 1 .C = Y 00

Y 11∆ = 3 .535, Y H 1 .C = Y 11

Y 00∆ = 4 .525

It is also known that the equation of the T P 1 through a-parameters or transmission parameters is as follows:

U 0I 0 = a11 a12

a21 a22 · U 1I 1 =

= 1.6 0.254 1.25 ·

U 1I 1 (1)

where:

a 11 = Y 11

Y 10, a12 =

1Y 10

, a21 = ∆Y 10

, a22 = Y 00

Y 10

The inverse expression has the form:

U 1I 1 = a22 −a 12

−a 21 a11 · U 0I 0 =

= 1.25 −0.25

−4 1.6 · U 1I 1

The determinant of the a -matrix is:

∆ a = a11 a 22 −a 12 a 21 = 1

This feature of the a -parameters allows entering the hyper-bolic functions:

∆ a = cosh 2 γ −sinh 2 γ = 1

where: γ is an attenuation coefcient.

Then the equation of the T P 1 is given by:

U 0

I 0=

cosh γ sinh γ

sinh γ cosh γ ·U 1 Y H.C

I 1 (2)

where:

I 0 = I 0Y BX.C

, Y H.C = Y H.C √ ∆ , I 1 = I 1√ ∆In turn, the conductivities at the input and output are connectedalready by the linear- fractional expression according to (1):

Y BX 1 = I 0U 0

= a22 Y H 1 + a 21

a 12 Y H 1 + a 11=

= 1.25Y H 1 + 40.25Y H 1 + 1 .6

(3)

This expression is in a normalized form takes the formaccording to (2):

Y BX 1

Y BX 1 .C =

tanh γ + Y H 1 /Y H 1 .C

1 + (tanh γ )Y H 1 /Y H 1 .C (4)

where:tanh γ = 1 / √ 2 = 0 .707

Now, consider the second T P 2. Let it be symmetrical withthe specic parameters. Then, similar to expressions (1), (3)we get:

U 1I 1 = 1.25 0.25

2.25 1.25 · U 2I 2 (5)

The inverse expressions have the forms:

U 2I 2 = 1.25 −0.25

−2.25 1.25 · U 1I 1 ,

Y BX 2 = I 1U 1

= 1.25Y H 2 + 2 .250.25Y H 2 + 1 .25

(6)

The condition of the given cascade circuit is performed:

Y H 1 = Y BX 2 .

A concrete mapping of the load characteristics for U 0 = 10is drawn on Fig. 2. Let the conductivity Y H 2 be equalto the extreme values Y H 2 (0) = 0 , Y H 2 (∞) = ∞ . The


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corresponding values of conductivities Y H 1 , Y BX 1 according to(6), (3) are:

Y H 1 (0) = 1 .8, Y H 1 (∞) = 5 ,

Y BX 1 (0) = 3 .048, Y BX 1 (∞) = 3 .595.

Next, we nd the values of current I 0 :

I 0 (0) = Y BX 1 (0) ·U 0 , I 0 (∞) = Y BX 1 (∞) ·U 0 .

We nd the values of currents

I 1 (0) = 8 .78, I 1 (∞) = 17 .54,

I 2 (0) = 0 , I 2 (∞) = 14 .03

and voltages

U 1 (0) = 4 .88, U 1 (

∞) = 3 .51,

U 2 (0) = 3 .89, U 2 (∞) = 0

using the inverse expressions (1), (5). The projective

Fig. 2. A concrete mapping of the load characteristics

mapping or transformation is dened by three pairs of points[4]. Therefore, in addition to these two extreme points, it isconvenient as the third or the unit (or scale) to use the pointof maximum power P 2 M of the load Y H 2 . In this case:

current I 2 (P 2 M ) = I 2 (

∞)/ 2 = 7 .016,

voltage U 2 (P 2 M ) = U 2 (0) / 2 = 1 .95,

and Y H 2 (P 2 M ) = 3 .597.

Next, we nd the corresponding values of currents, voltages at

the remaining parts of the circuit. Let the initial regime of the circuit (point i on the all load characteristics) be set bythe voltage of the load U 12 = 1 . Let we calculate the values of all currents, voltages, conductivities at the remaining parts of the circuit. In terms of projective geometry the voltages andcurrents are homogeneous coordinates, and conductivities areheterogeneous. The idea is that the homogeneous coordinatesalways have a nite value. The projective coordinate of apoint i at all lines is dened by cross-ratio m 1 of four pointsand takes the same value.We have for the part or terminals of load Y H 2 :

m 1 (Y H 2 ) = ( Y H 2 (0) Y 1H 2 Y H 2 (P 2 M ) Y H 2 (∞)) =

= Y 1H 2 −Y H 2 (0)Y 1H 2 −Y H 2 (∞) ÷

Y H 2 (P 2 M ) −Y H 2 (0)Y H 2 (P 2 M ) −Y H 2 (∞)

=

= Y 1H 2

Y H 2 (P 2 M ) =

10.413.597

= 2 .89, (7)

m 1 (U 2 ) = ( U 2 (0) U 12 U 2 (P 2 M ) U 2 (∞)) =

= U 12 −U 2 (0)U 12 −U 2 (∞) ÷

U 2 (P 2 M ) −U 2 (0)U 2 (P 2 M ) −U 2 (∞)

=

= 1 −3.89

1 −0 ÷1.95 −3.89

1.95 −0 = 2 .89 ÷1 = 2 .89, (8)

m 1 (I 2 ) = ( I 2 (0) I 12 I 2 (P 2 M ) I 2 (∞)) =

= I 12 −I 2 (0)I 12 −I 2 (∞) ÷

I 2 (P 2 M ) −I 2 (0)I 2 (P 2 M ) −I 2 (∞)

=

= 10.41 −010.41 −14.03 ÷

7.016 −07.016 −14.03

= 2 .89 ÷1 (9)

We have for the part or terminals of the load Y H 1 :

m 1 (Y H 1 ) = ( Y H 1 (0) Y 1H 1 Y H 1 (P 2 M ) Y H 1 (∞)) =

= 3.96 −1.8

3.96 −5 ÷3.14 −1.83.14 −5

= 2 .89, (10)

m 1 (U 1 ) = ( U 1 (0) U 11 U 1 (P 2 M ) U 1 (∞)) =

= 3.86 −4.883.86 −3.51 ÷

4.2 −4.884.2 −3.51

= 2 .89 ÷1 (11)

m 1 (I 1 ) = ( I 1 (0) I 11 I 1 (P 2 M ) I 1 (∞)) =

= 15.28 −8.7815.28 −17.54 ÷

13.15 −8.7813.15 −17.54

= 2 .89 ÷1 (12)


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For the current I 0 is:

m 1 (I 0 ) = ( I 0 (0) I 10 I 0 (P 2 M ) I 0 (∞) ) == ( 30 .48 34.55 33.22 35.95 ) = 2 .89

÷1

Thus, the invariant property of cross-ratio is performed, be-cause the projective transformations are dened by linear-fractional expression like (3). These invariant relations inall parts of cascade circuit dene the original ”principle of conformity of regimes”.Therefore, the denition of regime by cross-ratio (7) - (12)removes the indeterminateness in the choice of possible rel-ative expressions across all parameters and parts of a circuit.The value of the cross-ratio is not changed if to divide or tonormalize all members by the same value. For example, thenormalized value to the load Y H 2 will be Y H 2 (P 2 M ) , and thenormalized value to the voltage U 2 will be U 2 (0) .In turn, a special case of a cross-ratio, an afne ratio of threepoints or usual proportion is for the currents and voltages. Thisfollows from the fact that for those already homogeneous coor-dinates, the transformation is a linear type (1), correspondingto afne transformation or afne geometry.The rst and the second fractions for the currents and voltagesfor all parts are equal among themselves due to the implemen-tation of proportion. Thus, the afne transformation is denedby two pairs of corresponding points. Therefore, if we useonly the currents and voltages one can use an invariant in theform of an afne ratio.

Conversion (2) is characterized also by another invariant. Thisexpression for the symmetric T P takes the form:

U 0

I 0 √ ∆=

cosh γ sinh γ

sinh γ cosh γ ·U 1

I 1 √ ∆This transformation can be seen as a rotation of the radius-vector 0Y H of constant length at the angle γ to the position0Y BX in the pseudo-Euclidean space I , U on Fig. 3. Then theinvariant is given by the expression:

U 21 −I 21 ∆ = U 20 −I 20 ∆

and ones is the length of the vector 0Y H .This approach corresponds to the Lorenz transformations inthe mechanics of the relative motion. In turn, (4) correspondsto the rule of addition of relativistic velocities [5].

Let the regime of the circuit be changed (the point f at all the load characteristics), so that the load voltage isU 22 = 3 . Let we calculate similarly all values and make thecross-ratio:

m 2 (Y H 2 ) = ( 0 Y 2H 2 Y H 2 (P 2 M ) ∞ ) = 1.0733.597

= 0 .298,

m 2 (U 2 ) = ( U 2 (0) U 22 U 2 (P 2 M ) U 2 (∞) ) =

= ( 3 .89 3 1.95 0 ) = 0.296 ÷1,

Fig. 3. A rotation of the radius-vector of constant length at the angle γ

m 2 (Y H 1 ) = ( Y H 1 (0) Y 2H 1 Y H 1 (P 2 M ) Y H 1 (∞) ) =

= ( 1 .8 2.37 3.14 5 ) = 0.216 ÷0.72 = 0 .299,

m 2 (U 1 ) = ( U 1 (0) U 21 U 1 (P 2 M ) U 1 (∞) ) =

= ( 4 .88 4.56 4.2 3.51 ) = 0 .299 ÷1,

Let we express the changes of the regime Y 1H 2 →Y 2H 2 :

m 21 (Y H 2 ) = ( 0 Y 2H 2 Y 1H 2 ∞) = Y 2H 2

Y 1H 2

=

= 1.07310.414

= 0 .103 = 1

m 12 (Y H 2 ) =

19.7

,

m 21 (U 2 ) = ( U 2 (0) U 22 U 12 U 2 (∞) ) =

= ( 3 .89 3 1 0 ) = 0.297 ÷2.89 = 0 .103,

m 21 (Y H 1 ) = ( Y H 1 (0) Y 2H 1 Y 1H 1 Y H 1 (∞) ) =

= ( 1 .8 2.37 3.96 5 ) = 0.215 ÷2.074 = 0 .103,

m 21 (U 1 ) = ( U 1 (0) U 21 U 11 U 1 (∞) ) =

= (4 .88 4.56 3.86 3.51) = 0 .299 ÷2.89 = 0 .103

It is seen that the group property is carried out:m 21 = m 2

÷m 1


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This follows from the structure of the cross-ratio m 1 , m 2 . Thesecond fractions for the scale value U 1 (P 2 M ) are reduced, i.e.such a denition of the regime change is matched with thedenition of regime itself.

IV. VARIANTS OF THE EXPRESSIONS OF THE CROSS -RATIO

There are different variants of the expressions of the cross-ratio depending on the choice of the extreme points and thescale point. Let us consider the rst T P 1. Expression (4)

Fig. 4. A conformity or mapping of the points of line Y H 1 on the points of line Y BX 1 , including the negative values

on Fig. 4 also sets conformity or mapping of the points of line Y H 1 on the points of line Y BX 1 , including the negativevalues. We assume that the normalized values Y H 1 /Y H 1 .C ,Y BX 1 /Y BX 1 .C are just equal to Y H 1 and Y BX 1 .The repre-sentative load value of two-port, which transmits energy, is acharacteristic value. In this case Y BX 1 = Y H 1 =

±1.

We make the cross-ratio m1

(Y H 1 ) , using the extreme valuesY H 1 (0) = 0 , Y H 1 (∞) = ∞ and characteristic value Y H 1 (1)as a scale point:

m 1 (Y H 1 ) = ( 0 Y 1H 1 1 ∞ ) = Y 1H 1

Then the cross-ratio takes the form:

m 1 (Y BX 1 ) = ( tanh γ Y 1BX 1 1 1

tanh γ ) =

= Y 1BX 1 − tanh γ 1 − Y 1BX 1 · tanh γ

We express the regime changes when Y 1H 1 −→ Y

2H 1 and

respectively Y 1BX 1 −→Y 2BX 1 :

m 21 = m 2

÷m 1 = Y 2BX 1 −tanh γ 1 −Y 2BX 1 · tanh γ ÷

÷Y 2BX 1 −tanh γ

1 −Y 2BX 1 · tanh γ = Y 2H 1 ÷Y 1H 1

It is seen that the group property is also carried out. It maybe noted that the expressions for the regime changes andthe expressions for the conductivity changes at the input andoutput are analytically expressed in different ways, but theirnumerical values are equal. In this connection, we can chooseextreme points and a scale point in such a way so that theexpressions for these changes are the same. To do this we use

the characteristic values −1, 1 as the extreme values.Then:

m 21 (Y H 1 ) = ( −1 Y 2H 1 Y 1H 1 1 ) =

Y 2H 1 + 1Y 2H 1 −1 ÷

Y 1H 1 + 1Y 1H 1 −1

=

= 1 + ( Y 2H 1 −Y 1H 1 )/ (1 −Y 2H 1 ·Y 1H 1 )1 −(Y 2H 1 −Y 1H 1 )/ (1 −Y 2H 1 ·Y 1H 1 )

The resulting expression shows that there is solid argumen- tation to enter the value of load change in the form :

Y 21H 1 = ( Y 2H 1 −Y 1H 1 )/ (1 −Y 2H 1 ·Y 1H 1 )

Then, regime change is:

m 21 (Y H 1 ) = 1 + Y 21

H 1

1 −Y 21

H 1

Similarly, the same regime change and the conductivity changeare resulted at the input:

m 21 (Y BX 1 ) = ( −1 Y 2BX 1 Y 1BX 1 1 ) = 1 + Y 21

BX 1

1 −Y 21BX 1

where:

Y 21BX 1 = ( Y 2BX 1 −Y 1BX 1 ) ÷ (1 −Y 2BX 1 ·Y 1BX 1 )

We are using as a scale or a unit point Y H 1 (1) = 0 forthe setting of the regime. This point corresponds to the pointY BX 1 (1) = tanh γ on Fig. 4. Then:

m 1 (Y H 1 ) = ( −1 Y 1H 1 0 1 ) = 1 + Y 1H 1

1 −Y 1H 1

Because the same extreme points are used in the cross-ratio

Fig. 5. A movement of the point from position Y H 1 to position Y BX 1 fordifferent initial values Y H 1

for the input and output, we can consider the point Y H 1 andY BX 1 on the superposed axis on Fig. 5. This gure representsa movement of the point from position Y H 1 to position Y BX 1

(or motion of segment Y H 1 Y BX 1 ) for different initial valuesY H 1 , as shown by arrows. Then, the points ±1 are xed. Youcan make a cross-ratio to the points Y H 1 , Y BX 1 relatively xed

points, which determines the ”length” of segment Y H 1 Y BX 1 :m(Y H 1 Y BX 1 ) = ( −1 Y H 1 Y BX 1 1 )


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It is known the properties of such cross-ratio in projectivegeometry, its value does not depend on values Y H 1 , Y BX 1

. Therefore, we set Y H 1 = 0 and Y BX 1 = tanh γ forsimplifying of the record:

m(Y H 1 Y BX 1 ) = 1 −tanh γ 1 + tanh γ

= K P M

It is seen that the ”length” of segment Y H 1 Y BX 1 equals to themaximum of the efciency K P M of a two-pole. Therefore, itturns out that the concept of maximum efciency is alreadyinto this geometric interpretation, and do not have to ”think of” his denition.Thus, the proposed geometric approach provides variants forthe justied denition of a regime and his changes for thedifferent parts of a circuit.

V. A PPLICATION OF THE INVARIANT PROPERTIES OFCIRCUITS

The invariance of a cross-ratio can be applied in practice of measurements, because many measuring means exercise linear-fractional transformation on the input value [6].The cross-ratio allows obtaining also the convenient calculatedexpression Y BX (Y H ) of the type (3), but without the explicitnding of a -parameters. It is known, for the nding of the a-parameters necessary to exercise three experiments measuringvalues Y BX , Y H .The most simple expressions for the calculation of the a -parameters are obtained from experiments, in which the regime

of SC and OC at the output (two experiences) and one of such experiments at the input when the circuit is suppliedfrom the output. Clearly, such manipulations are not alwaysconvenient to implement (the length of the network, thetechnical complexity of switching power supply from the inputto the output). It is therefore, possible the following algorithmof the calculation:

-Circuit is tested (three experience are exercised) for thethree arbitrary values of the loads: Y H (0) , Y H (∞) are theextreme values and Y H (1) is the scale value. If possible, it isbelieved Y H (0) = 0 , Y H (∞) = ∞, as a regime of the SC and OC . Then, the scale value Y H (1) is convenient to chooseso that the voltage U H (1) = U H (0)/ 2 .

- The respective values Y BX (0) , Y BX (∞), Y BX (1) aremeasured.

- For the current load value Y 1H the cross-ratio m1 is

calculated.- The current value Y 1

BX is found through the m 1 from asimilar, but a reciprocal expression.Because

m 1 (Y BX ) = Y 1BX −Y BX (0)Y 1BX −Y BX (∞) ÷

÷Y BX (1) −Y BX (0)Y BX (1) −Y BX (∞)

=

Y 1BX −Y BX (0)Y 1BX −Y BX (∞) ÷N BX (1) (13)

we get the nal calculation expression:

Y 1BX = Y BX (0) −m 1 N BX (1)Y BX (∞)

1

−m 1 N BX (1)

(14)

For the following values of the load Y 2H value of the m2 is

rstly found and value of the Y 2BX is further found.

Fig. 6. System of accurately transmission of the measuring signal

Cross-ratio permits accurately to transmit analog (measuringsignal) via the unstable two-pole [7]. The scheme of suchsystem is presented on Fig. 6. In a short time when theparameters of a circuit does not signicantly are changed,the four samples of load are transmitted by connecting therespective loads Y H (0), Y 1H , Y H (1), Y H (∞) .Three of them Y H (0) , Y H (1) , Y H (∞) are extremely and scalevalues, and the sample Y 1

H is a signal or an informationsample. Therefore, its value is calculated by the formula whichis obtained similar to (14) through the prepared cross-ratio m 1 :

Y 1H = Y H (0) −m 1 N H (1)Y H (∞)

1 −m 1 N H (1) = F (m 1 ) (15)

We accept that value of the m 1 = U C .At the input of a circuit the cross-ratio or the signal is calcu-lated similar to (13), using the measured values of samples of the input conductivity:

m 1 = Y 1BX −Y BX (0)Y 1BX −Y BX (∞) ÷N BX (1) = F − 1 (Y 1BX ) (16)


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We can also use only three samples where a circuit is loadedon the sources of voltages on Fig. 6.Because

m 1 (U 2 ) = ( U 2 (0) U 12 U 2 (P 2 M ) U 2 (

∞)) =

U 12 −U 2 (0)

U 12 −U 2 (∞)

hence it follows the value of a signal sample, similar to (15)

U 12 = U 2 (0) −m 1 U 2 (∞)

1 −m 1

At the input of a circuit the cross-ratio or the signal is calcu-lated similar to (16), using the measured values of samples:

m 1 = U 11 −U 1 (0)U 11 −U 1 (∞)

The structure of this expression shows that the errors of mea-surement of samples of voltages mutually are reduced. Morethan a simple expression for the signal sample is obtained,

using the invariant in the form of an afne ratio:

m 1 (U 2 ) = ( U 12 U 2 (0) U 2 (∞)) = U 2 (0) −U 12U 2 (0) −U 2 (∞)

Then the value of the signal sampling is:

U 12 = U 2 (0) + n 1 (U 2 (∞) −U 2 (0))

VI. C ONCLUSION

1. The invariant relations for all regime parameters in allparts of cascade circuit are shown.

2. The denition of a regime by the cross-ratio removesthe indeterminateness in the choice of the possible relativeexpressions across all parameters and parts of a circuit.

3. The expressions for calculating the cascade circuit with-out explicitly nding the a -parameters are obtained.

4. The cross-ratio is an effective way to transmit an analog(measurement signal) via the unstable two-port.

5. The results are applicable to AC circuits.

REFERENCES

[1] A. A. Penin,”Linear- fractional relation in the problems of anal-ysis of resistive circuits with variable parameters”, Electrich-estvo , 11, 1999, pp.32-44, (Russian).

[2] A. A. Penin,”Determination of Regimes of the Equivalent GeneratorBased on Projective Geometry.The Generalized Equivalent Genera-tor”, International Journal of Electrical Systems Science and Engi-neering , vol.1, n.3, 2008, pp.169-177.

[3] C. A. Desoer and E. S. Kuh, Basic Circuit Theory , McGraw - Hill,New York, 1969.[4] O. Veblen and J. W. A. Young, Projective Geometry , vol.1, Blaisdell,

New York, 1938- 1946.[5] C. K. Kittel, W. D. Knight and M. A. Ruderman, Mechanics:

Berkely Physics Course , vol.1, McGraw - Hill, New York, 1973.[6] A. I. Gerasimov and V. D. Masin, ”Aplication of the properties of

linear- fractional transformatoins in the measuring converters”, Pribory,sistemy upravlenia , n. 9, 1983, pp. 22- 23, (Russian).

[7] A. A. Penin , ”Projective- afne properties of the resistive two-port with the changeable load”, Electrodinamika , n. 2, 1991, pp.38-42, (Russian).

Alexandr A. Penin graduated from Radio Department Polytechnic Institute

in 1974 of Odessa city, Ukraine. The area of research relates to the theory of electrical circuits with variable elements or regimes. The area of interest inengineering practice is the power electronics.


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jurnal two port

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