About the definition of parameters and regimes of active two-port networks with variable loads on the basis of projective geometry

Transcription

1 About the definition of paraeters and regies of active two-port networks with variable loads on the basis of projective geoetry PENN ALEXANDR nstitute of Electronic Engineering and Nanotechnologies "D hitu " Acadey of Sciences of Moldova Acadeiei str / MD-8 Chisinau REPUBLCA MOLDOA aapenin@ailru Abstract: - Disadvantages of soe well- known ethods of analysis of electric circuits with variable loads are analyzed To interpret a utual influence of the loads soe ethods of the projective geoetry are used The application of the projective coordinates allows receiving the equation of the active two-port network in a noralized or relative for as well as defining the scales for the currents and conductivity of the loads Such approach akes it possible to estiate the qualitative characteristics of the current regies to copare the regie efficiency of the different circuits The forulas of the recalculation of the currents which possess the group properties at change of conductivity of the loads are obtained t allows expressing the final values of the currents through the interediate changes of the currents and conductivities The generalized equivalent generator of the active two-port network in the for of the passive two-port network and a set of the sources of a current and voltage is proposed The paraeters of these sources do not depend on certain conductivities of the passive two-port network Key-Words: - Thevenin s theore load characteristics projective geoetry two-port networks ntroduction n the theory of electric circuits an attention is given to the circuits with variable paraeters of eleents n practice it can be the power supply systes for exaple of the direct current are containing the power supply of finite capacity and quantity (let it be two) of resistive loads with variable resistances Therefore a utual influence of the loads takes place in the siilar active two-port network One of analysis probles is deterination of dependence of the changes of characteristics or paraeters of the circuit regie on the respective change of paraeters of loads Also it is iportant to present a circuit by the equivalent generator to define noralized or relative values of the regie paraeters with use for exaple of the characteristic or axiu valuestherefore such definition of the regies allows estiating the qualitative characteristics of the current regies and coparing regies of different systes Consequently the proble of the representation of two-port network paraeters in the noralized for results fro of the - paraeters for exaple A range of properties theores and ethods is wellknown and their use siplifies the decision of the probles of this kind However the known approaches do not copletely disclose the properties of such circuits which reduces the efficiency of the analysis For exaple in the theore of utual changes of currents and resistance (the variation theore) the changes of the load resistances are set in the for of increents [] Therefore at circuit recalculation in case of a nuber or group of changes of these resistances these increents should be counted concerning the initial circuit t shows that the group properties are not carried out and the possibilities of this theore are liited The group property defines the resultant change through interediate changes of the resistance and current (the group operation takes place in the for of addition or ultiplication) Thus the obvious definition of the changes looks like foral and does not reflect the substantial side of the observed utual influences: resistance current SSN: ssue 5 olue May

2 The ethod of the equivalent generator or Thevenin s theore represents an active two-pole as a source of the voltage with an internal resistance [] [] [] For analysis of the regie efficiency it is convenient to present the external or load characteristic of an active two-pole through the noralized or relative expression Scales for corresponding paraeters of the circuit regie are the open circuit voltage the internal resistance and the short circuit current f the noralized load voltage is equal to \ or noralized load resistance is equal to it shows iediately that the load regie corresponds to the axiu power However if any resistance is changed in an active two-pole the open circuit voltage is changed also Therefore it is not convenient to use this voltage as the paraeter of the equivalent generator Siilarly the ethod of the equivalent generator represents the active two-port network as the passive two-port network and the separated sources of the voltage or current at the terinals of two loads [] The paraeters of these sources correspond to the open circuit voltage or the short circuit current of both loads at the sae tie But these paraeters or quantities do not allow presenting directly the syste of the equations of the active two-port network in the noralized kind Moreover if any resistance of the passive two-port network is changed also the recalculation of the values of these sources of the voltage or the current is necessary The ethod of geoetrical places or circular charts allows investigating the influence of various paraeters only of the alternating current circuits on the regie characteristics [] But this ethod is not developed enough since such basic geoetrical concepts as the transforation or oveent of point are not used To a larger extent this ethod represents the graphical constructions siply n a nuber of articles of the author the approach is developed for interpretation of changes or "kineatics" of the circuit regies on the basis of projective geoetry [4] [5] [6] t allows disclosing the invariant properties of circuits ie such relationships which are identical to all paraeters of the regie and sub circuits Such invariant relationships are also the basis of well-founded definition of the relative regies Also the theore of the generalized equivalent generator of an active two-pole which develops the well-known Tevenin s and Norton s theores is forulated t appears that the external characteristic is transfored into a bunch of straight lines for various values of the internal resistance of this twopole Since the coordinates of the center of a bunch do not depend on this changeable eleent they can be accepted as the paraeters of the generalized equivalent generator Proceeding fro the obtained results it is possible to draw a conclusion that such atheatical apparatus is adequate to the observed interrelations n the present article the obtained results are applied to the analysis of active two-port networks Projective coordinates of an active two-port network Let us consider an active two-port network with changed conductivities of loads L L in Fig Taking into account the specified directions of the currents a circuit is described by the following syste of the equations: where - paraeters are: Σ Σ 58 6 Σ Σ 95 Σ 4 Σ Σ () SSN: ssue 5 olue May

3 alue diensions are not indicated for siplifying of the record A bunch center a point corresponds to the bunch of the straight lines with the paraeter L Physically the bunch center corresponds to such regie of the load L which does not depend on its values t is carried out for the current and voltage at the expense of a choice of : paraeters of the second load L 65 Fig An active two-port network with changed conductivities of loads Taking into account the voltages / L / L fro the syste of equations () the equation of two bunches of straight lines with paraeters L L are obtained: 5 () Negative value of conductivity of the load L which already gives energy corresponds to these paraeters: L ( ) 8 (4) / ( ) L The paraeters of the center of the bunch L are expressed siilary: ( ) () L The bunches of these straight lines are presented in Fig (5) 5 ( ) L 5 (6) Regies which are defined at qualitative level we nae as characteristic regies The regies which set the centers of bunches are such characteristic regies One ore for of such regies is the short circuit regie on both loads that is presented by a point in Fig n this case the currents are equal: (7) Fig The bunches of load straight lines The open circuit regie on both loads is also characteristic and corresponds to the beginning of coordinates a point SSN: ssue 5 olue May

4 Let the initial or current regie corresponds to a point M which is set by the following paraeters: L L Let us use a definition of projective geoetry coordinates [7] [8] [9] Currents are the Cartesian non-unifor coordinates in the oordinate syste n turn conductivities L L define non-unifor projective coordinates which are set by a coordinate triangle and a unit point A point is the beginning of coordinates and a straight line is the infinitely reote straight line The non-unifor projective coordinate is set by a cross ratio of four points three of these correspond to the points of the characteristic regies and the fourth corresponds to the point of the current regie: ( L L L L L L ) L L : L (8) ( L L / / L L L L ) L L L (9) The conductivities L L also are the scale values as allow obtaining the noralized values of the load conductivities Non-unifor coordinates of points of a straight line have not final values Therefore hoogeneous projective coordinates are entered which set nonunifor coordinates as follows: () where is a coefficient of proportionality Hoogeneous coordinates are defined by the ratio of the distances of points M to parties of a coordinate triangle in Fig 4: () The confority of the values L and is shown in Fig The points correspond L L L to extree or base values there The point is the unit point L Fig The confority of a conductivity L and non-unifor projective coordinate The cross ratio for the projective coordinate expressed siilarly: is Fig4 The distances of points M a coordinate triangle to parties of SSN: ssue 5 olue May

5 Taking into account a choice of the coordinate syste the distances are given: For a finding of distances to a straight line the equation of this straight line is used: () Then the distance equal to where and is a noralizing factor: ( ) ( ) 97 Siilarly: 889 () (4) Then by expression () the hoogeneous coordinates have a view: (5) n turn the non-unifor coordinates are equal to: that coincides with the values (8) (9) Let us express the projective coordinates through the Cartesian coordinates We present the syste of the equations (5) by a atrix for: where [ C] C] atrix [ [ C] [ ] (6) Then the values ( ) are hoogeneous Cartesian coordinates For our exaple the atrix of transforation (6) is: SSN: ssue 5 olue May

6 ] [C The reverse transforation to the transforation (6) is expressed as follows: (7) Fro here we pass to the non-unifor Cartesian coordinates: (8) For our exaple the transforation (8) leads to the sae known values: Thus for the preset values of conductivities L L we find the coordinates and the transforation (8) allows finding the currents t is possible to present the syste of the equations (8) by the noralized or relative for: ) ( ) ( ) ( ) ( (9) Fro this syste of the equations it is possible to obtain the equations of two bunches of the straight lines corresponding to expression (): / / / / () The syste of the equations (9) allows obtaining also in a relative for the equations of load characteristics ) ( ) ( which correspond to the two first equations of the syste () For this purpose we express the non-unifor SSN: ssue 5 olue May

7 coordinates through the currents and voltages: ( L L L / / ) / ( / / ) / () / ( / ) / Fro this it follows that - paraeters are expressed by the paraeters of the characteristic regies: Having substituted these values in syste (9) we receive the required equations ) ( ) accordingly having ( excluded the currents : () The obtained syste of the equations represents the purely relative expressions Therefore such values as represent the «noralized» N N N N N N N / N N / -paraeters: Let us pass to absolute values of the regie paraeters in the syste (): 95 () 5 The systes of the equations (9) () are iportant relationships as represent the purely relative expressions t allows defining at first the coordinates (as relative values) for different load conductivities and then the values of the noralized currents Also as the equation () these expressions allow to estiate at once a qualitative condition of such circuit how uch the current regie is close to the characteristic values Then the currents represent scales n this sense the initial systes of the equations () () containing the actual or absolute values of the - paraeters currents and voltages are a little inforative because do not give of a direct representation about the qualitative characteristics of a circuit n addition it is possible to notice that it is difficult to obtain the relative expressions of type (9 - ) fro the systes of the equations () () directly Therefore the application of the geoetrical interpretation solves such proble by the easy and foralized ethod SSN: ssue 5 olue May

8 ariation of load conductivities Let a initial regie corresponds to a point M and a subsequent regie corresponds to a point M with the concrete paraeters of loads L L An arrangeent of points of the initial and subsequent regies is shown in Fig 5 Non-unifor coordinates are defined siilarly to (8) (9): ( L L L ) 4 L L ( L L L ) 6486 L L then: Let us check up the values of the non-unifor coordinates: Fig5 An arrangeent of points of the initial and subsequent regies t is possible to define a valid change of a regie using a group property of a cross ratio: : 4 : 5 6 : 6486 : then: (4) Therefore the change of the regie is expressed naturally through the cross ratio: ( L L L L L L L LH : ) L : ( ) (5) L L L Let us define the hoogeneous coordinates of a point M using () (4) (5): (6) 94 Using the transforations (8) we receive the current Taking into account () we present the nonunifor coordinates and in a for: Therefore the current ( ) / SSN: ssue 5 olue May

9 Then taking into account the expression (7) we receive the transforation: (7) Using (6) we receive the resultant transforation as a product of the two atrixes: or in a definitive for: (8) For our exaple the transforation (8) has a view: Let us check up the values of the currents of the subsequent regie: Let us notice that a group property of the transforation (8) is carried out owing to group properties of the expressions (4) Let a regie once again change ie we receive the values of the changes Then the final value of the regie is expressed as: (9) Using (8) we receive the resultant transforation as a product of the atrixes: SSN: ssue 5 olue May

10 or in a definitive for: () Special case of the transforation (8) Let us take interest only by a change of the current t is necessary to receive a transforation which defines the subsequent current only through the initial current For the explanatory we consider Fig 6 Let the initial regie corresponds to a point B with the concrete paraeters of loads and current L L B Then the regie passes into a point B at the expence of a change L L We nae such case as own change of a regie Further the regie changes at the expence of a change and passes into a pointc L L Such case is utual change of a regie The specified points B C correspond to above considered points M M Further we use the results [5] Let us receive expressions of a change of the regie B B For this purpose we ap the points B B B of a straight line with the paraeter L on the axis and we introduce the corresponding values of currents Using () we ake the cross ratio for this own change of the regie: ( B B B ) ( ) 6 () L L L Fig6 Changes of the current Also this cross ratio is expressed through the introduced currents: B B ( B B ) : B B : Let us ake the cross ratio for the utual change of the regie For this purpose it is necessary to ap the points C B D of a straight line with paraeter L on the axis also Then we receive the expression: ( ( L L C L B D ) ( D L ) C B ) () The choice of points D as the base points leads to the sae base points as well as for () The given circuit concerning of the first load is a generator of the current at the expence of the SSN: ssue 5 olue May

11 second load conductivity equal to D L The D physical sense of the value consists in this fact L D Let us define the value of conductivity L For this purpose we present expression () in such for: ) ( / ) ( L Let the current D L L 966 Then the conductivity Also the sae value is obtained fro the second expression () For this purpose we find the corresponding value of the cross ratio: / / / D 55 t is apparently the interediate value of the current B is excluded Thus knowing changes of load conductivities we find the values of changes and resultant change (5) by () C () A final value of a current is obtained fro (6) Coparative analysis with a traditional recalculation of a circuit Let us lead the basic stateents of a traditional approach to recalculation of currents of an active two-port network Expressions for the currents are obtained fro the equation (): Then the value of the conductivity fro expression (9) is found D L (7) Now we calculate C B : C B using the expression (): : L L L L : D D L L L L : () (4) As coon base points are used for the own and utual change the group properties are carried out Therefore the resultant change is expressed by their product: ( C B) (5) ( C B) ( C B C C ) : B B (6) The short circuit currents are considered as initial values of the load currents Now we introduce the load resistors as increents R L RL concerning initial zero values in Fig 7a Therefore the relationships take place for the subsequent values of the currents and voltages: RL RL We transfor the expression (7) as follows: or: ( R R R L L L R L ) ( R R R L L L R L ) SSN: ssue 5 olue May

12 R L RL Thus the structure of a denoinator of expression (8) shows that group properties are not carried out for the interediate changes of the load resistances Therefore the subsequent changes ust be counted concerning initial zero values However if we count off the changes R R L L concerning values R L RL the schee of other two-port network with conductivity paraeters g is obtained in Fig 7b Thus ij recalculation of paraeters of new two-port network is required The offered approach on the basis of the projective geoetry does not require recalculation of paraeters and naturally possesses greater flexibility and convenience Fig7 A traditional approach to recalculation of currents: a)- load resistors R L RL are as increents concerning initial zero values; b)- the changes R R are concerning values L L R L RL The decision of this syste of the equations gives calculated relationships: R R L L ( ( ) ) (8) 4 eneralized equivalent generator of an active two-port network The expression (7) shows that an active two-port network represents a passive part which is set by paraeters of conductivity ij and two generators of a current that is shown in Fig8 n turn the passive two-port network can be defined by various equivalent schees We notice that the currents are set by paraeters which depend practically on all eleents of the twoport network except where ( RL )( RL ) ( ) R L R L Let us carry out the analysis of these relationshirs We introduce interediate changes R L RL thus: R R R L L L R R R L L L Therefore the denoinator will contain both changes R R and initial values of resistance L L Fig8 A traditional equivalent generator of an active two-port network Therefore at possible changes for exaple of a conductivity it is necessary to ake recalculation of values of these generators of a current that is inconvenient The conductivity can SSN: ssue 5 olue May

13 be a part of a possible third load n this sense the paraeters of generalized equivalent generator offered for an active two-pole do not depend on such eleent of a circuit [4] [5] Let us introduce siilarly the generalized equivalent generator for the active two-port network on the basis of coordinates of the centers of bunches of straight lines and the relationships (-6) We consider at first the generalized equivalent generator of this two-port network concerning load L which is the active two-pole with a changeable eleent L in Fig 9a The faily of external or load characteristics ) with a changeable ( paraeter L is presented in Fig 9b This faily represents the bunch of the straight lines with failiar center and center s coordinates correspond to expressions (5) (6) Accordingly at change of the load L the bunch of the straight lines is fored with the center coinciding with the beginning of the coordinates The equation of this bunch is obvious: L The equation of the bunch with the paraeter L is obtained fro the expressions () () and accepts a view: ( ) ( ) (9) i where a internal conductivity of the active twopole is: i ( ) (4) L Let us transfor the expression (9) to a relative for: i L (4) Thus the value L is a scale for the internal conductivity The expressions (9) (4) allow aking directly the schee of the generalized equivalent generator in Fig 9c Fig9 An active two-pole with a changeable eleent L : a)- schee of this active two-pole; b)- faily of load characteristics; c)- generalized equivalent generator The value 5 is the characteristic i L value for the conductivity i n this case a voltage SSN: ssue 5 olue May

14 of the conductivity i is equal and opposite in a direction to the voltage of the source Therefore for all values of conductivity L Siilarly relationships of type of the expressions (9) (4) are obtained for a second load L Therefore the whole schee of the equivalent generator of the active two-port network can be constituted in Figa This schee gives a evident representation about utual influence of loads and allows to carry out calculations Also it is possible to obtain one ore schee of the equivalent generator in Fig b For this purpose we consider the expression () for paraeter : J U This expression defines input conductivity of the passive two-port network at short circuit of the first load and short circuit of the second pair terinals of the two-port network in Figa The siilar relationship is obtained for paraeter in Fig b The association of these two schees leads to the schee in Figb Taking into account the superposition theore the values of all entering sources of a current and voltage are decreased twice in the schee in Fig b Calculations prove such schee of the equivalent generator We will obtain a syste of equations which describes the given schee of the equivalent generator Using the specified designations in Fig b it is possible to write down: U U J J (4) Fig The proposed equivalent generator of the active two-port network: a)- is as an association of two active two-poles: b)-is as generalization of the equivalent generator of active two-port network Following relationships take place for the passive two-port network: J U U J U U Then taking into account (4) after a nuber of transforations the expression is obtained: (4) SSN: ssue 5 olue May

15 The offered schee of the equivalent generator is convenient at odeling of an active two-port network Paraeters of voltage and current sources of the generalized equivalent generator of the active two-port network in Fig do not depend on the conductivity t represents a practical interest This conductivity can be a part of third changeable load Therefore only - paraeters of the passive two-port network are recalculated 5 Conclusions The projective geoetry adequately interprets "kineatics" of a circuit with changeable paraeters of eleents allows to carry out deeper analysis and to receive relationships useful in practice The given approach is applicable to the analysis of «flow» kind processes of the various physical natures Fig The definition of input conductivity of the passive two-port network: a)- at short circuit of the first load and short circuit of the second pair terinals; b)- at short circuit of the second load and short circuit of the first pair terinals Let us copare this for to the expression (7) Then the expression in brackets should be equal to the short circuit current We can check up its value: 5 5 n addition the structure of the short circuit current expression shows coponents of this current A coponent / corresponds to own current of the two-port network and depends on its paraeters A coponent / is defined by a current source A coponent / corresponds to a utual current on account of a voltage source on the second pair of terinals An expression for a current is obtained siilarly to the equation (4) The application of the projective coordinates allows receiving the equation of the active two-port network in a noralized or relative for to define scales for the currents and conductivities of loads Such approach allows estiating qualitative characteristics of current regies to copare the regie efficiency of different circuits The forulas of the recalculation of the currents which possess the group properties at change of conductivity of the loads are obtained t allows expressing the final values of the currents through interediate changes of the currents and conductivities 4 The generalized equivalent generator of the active two-port network in the for of the passive two-port network and a set of the sources of a current and voltage is proposed The paraeters of these sources do not depend on conductivity which represents a possible third load References: [] Bessonov L A Basic electrical engineering theory: Electric circuits Ninth edition High school: Moscow 996 (russian) [] Johnson D E Basic electric circuit analysis Third edition Prentice Hall Enlewood cliffs: New Jersey 986 SSN: ssue 5 olue May

16 [] Bakshi U A Bakshi A Electrical network analysis and synthesis Technical publications pune p [4] Penin A A Linear- fractional relations in the probles of analysis of resistive circuits with variable paraeters Electrichestvo No 999 pp - 44 (russian) [5] Penin A A Deterination of Regies of the Equivalent enerator Based on Projective eoetry: The eneralized Equivalent enerator nternational Journal of Electrical and Coputer Engineering ol No5 8 pp (date of access: 4) [6] Penin A A The invariant properties of twoport circuits nternational Journal of Electrical and Coputer Engineering ol 4 No 9 pp pdf (date of access: 4) [7] eblen O oung J W A Projective geoetry ol Blaisdell New ork [8] Busean H Kelly P J Projective geoetry and projectiveetrics Acadeic press inc: New ork 95 [9] Casse R Projective geoetry: An introduction Oxford university press: New ork 6 SSN: ssue 5 olue May