Feusner, Friedrich Wilhelm

Friedrich Wilhelm Feusner
Friedrich Wilhelm Feusner
	him. Friedrich wilhelm feussner
Date of Birth	February 25, 1843
Place of Birth	Hanau
Date of death	September 5, 1928 (85 years old)
Place of death	Marburg
A country	Germany
Place of work	;

Feusner, Friedrich Wilhelm German. Friedrich Wilhelm Feussner (1843-1928) - German scientist and natural scientist. In his works, Ueber Stromverzweigung in netzformigen Leitern and Zur Berechnung der Stromstarke in netzformigen Leitern, published in the journal Annalen der Physik , laid the foundation for a circuit approach to the analysis of electrical circuits.

A Special Link in Symbol Analysis

Friedrich Wilhelm Feusner was the first to point out the shortcomings of the topological formulas of Gustav Robert Kirchhoff ^[1] and James Clerk Maxwell ^[2] , explaining in 1902 why they are not used by physicists and are absent in physics reference books. The main, in his opinion, reason was the difficulty in choosing the accepted combinations of resistances (conductivities) from a very large number of possible combinations. Therefore, Feusner developed a number of methods for the stepwise decomposition of the numerator and denominator of the circuit function. I noticed that the concept of “circuit function” leads to a study of the work of Maxwell ( 1873 ), who submitted an emf along one conductor and found the resulting current in another conductor.

V. Feusner’s interest in electrical engineering was far from accidental, because Kirchhoff himself was his teacher, and the title of his dissertation, the first serious scientific work, Über die Messung der Wärme durch die Veränderung des elektrischen Widerstandes mit der Temperatur (On measuring the amount of heat by taking into account the dependence of electrical resistance on temperature ”) speaks for itself. Meanwhile, in the history of science among the students of the founder of electrical engineering, the name of Feusner does not appear. Perhaps this is due to the fact that after obtaining a Ph.D. degree, V. Feusner dramatically changes the direction of research, and returns to the theory of electrical circuits only after 35 years.

In his works ^[3] , published in 1902-1904 in the authoritative journal Annalen der Physik und Chemie, Feusner developed the results of Kirchhoff and Maxwell practically to their present state with respect to passive electric circuits without mutual inductances. However, in contrast to the works of Kirchhoff and Maxwell , which set forth a topological approach to the analysis of electric circuits, the results of Feusner are still essentially unknown to specialists.

Milestones of scientific activity

The German scientist and naturalist Friedrich Wilhelm Feusner was born on February 25, 1843 in the city of Ganau , the birthplace of the famous Brothers Grimm . He was lucky to get an academic education under the leadership of two great compatriots at once - the world famous G. R. Kirchhoff in Heidelberg and Christian Ludwig Gerling in Marburg ^[4] ^[5] .

In 1867, after successfully defending his dissertation "Über die Messung der Wärme durch die Veränderung des elektrischen Widerstandes mit der Temperatur" ("On measuring the amount of heat by taking into account the dependence of electrical resistance on temperature") in Heidelberg, V. Feusner received a Ph.D. lifelong right to teach physics at the university (the so-called “venia docendi” - translated from Latin as “the right to teach”).

“In this work, we are talking about the expedient execution and design of the device (as von O. Svanberg, a Swedish mathematician and astronomer, had briefly pointed out before), which is now called a bolometer. Feusner’s dissertation contained (at least at the time of the obituary’s publication, according to F. A. Schulz) some of the data and provisions worthy of attention today. ”

The bolometer is a very thin blackened metal wire or strip inserted into one of the branches of the S. Wheatstone bridge ^[6] and placed on the path of the radiant energy flow. Due to its small thickness, the plate heats up quickly under the action of radiation and its resistance increases. The bolometer is sensitive to the entire spectrum of radiation. But they use it mainly in astronomy to record radiation with a submillimeter wavelength (intermediate between microwave and infrared): for this range, the bolometer is the most sensitive sensor . The source of thermal radiation can be the light of stars or the Sun, transmitted through a spectrometer and decomposed into thousands of spectral lines, the energy in each of which is very small.

For reasons unknown to us, V. Feusner soon abruptly changed the topic of his research and moved closer to his father’s house in the city of Marburg (cradle of the federal state of Hesse ), and already on January 14, 1869 made a report “Über der Bumerang” (“On Boomerang”) ^{[7 ]} at a meeting of the Marburg Society for the Promotion of Natural Sciences . At the same time, he first became a freelance, and then, since 1881 , - and a full member of this society.

In 1878-1881, S.P. Langley, who entered the history of science as a formal inventor of this device, was engaged in the improvement of the bolometer.

It is worth noting that the formation of physics as a scientific and educational discipline at the University of Marburg began with the appointment of Gerling in 1817 as professor of mathematics, physics and astronomy. Gerling was a close friend of K.F. Gauss , who at that time headed the department in Göttingen . Gerling is known for his research in the field of geodesy, in which he used the Gaussian least square method ^[8] .

Since 1871, Feusner has been working as a private assistant professor of physics and mathematics at the University of Marburg . During these years, V. Feusner published a number of works in the journal Annalen der Physik und Chemie (On Two New Methods for Measuring the Height of Clouds) ( 1871 ), Ueber die von Hrn. Sekulic beschriebene Interferenzerscheinung ” (“ On the description of the phenomenon of interference ”) ( 1873 ) ^[9] , “ Neuer Beweis der Unrichtigkeit der Emissionstheorie des Lichts ” (“ New evidence of the incorrectness of the emission theory of light ”) ( 1877 ) ^[10] , “ Über die Interferenzerscheung dünner Blättchen mit besonderer Reucksicht auf die Theorie der Newtonschen Ringe ” (“ On Interference in Thin Films with the Newton Ring Theory ”) ( 1881 ) ^[11] .

As can be seen from the titles of the publications of Feusner of those years, the German scientist worked fruitfully in various branches of physics, however, the studies of optics, in which he achieved considerable success, were of most interest. He was considered a recognized specialist, and his interpretations of the phenomena of interference and polarization were included in the manual on the physics of A. Winkelmann ^[12] . Feusner was the compiler of the chapter on interference in the second edition of this manual. Later, after the dismissal of Feusner, the material on interference, after considerable processing in collaboration with L. Yanicki and supplemented with new research results, was included in the textbook on optical physics “Dem Handbuch der Physikalischen Optik” edited by E. Gehrkke ^[13] .

Since 1880, V. Feusner has been teaching theoretical physics at the University of Marburg, first as a freelance professor, and since 1908 already as a full-time professor. Peter Thomas , professor of the Department of Theoretical Physics of Semiconductors, Dean of Physics, University of Marburg, a specialist in the history of this university, notes that in Marburg, up to the last decades of the nineteenth century, theoretical physics has not yet been formed ^[14] . Feusner was essentially the first theoretical physicist in Marburg, and in 1910 founded a regular scientific seminar on this subject. If at the time of Gerling physicists were content with a room of six small rooms, then by 1915 his successor Feusner, along with his colleagues, had at their disposal a large mansion equipped with the most modern equipment, built under the guidance of Professor Richartz .

The interests of V. Feusner in the second half of his creative life were very versatile. Along with the completion of his work in the field of theoretical physics ^[15] ^[16], he developed the basis for the formation and development of topological analysis of electric circuits ^[17] . Surprisingly, these articles, published in the prestigious journal Annalen der Physik und Chemie , remained almost unnoticed by Feusner's contemporaries! The first references to them in the literature date back to the fifties of the twentieth century ^[18] ^[19] , and F. A. Schulz , who wrote an obituary in Feisner's memory in 1930 , does not even mention these works among the achievements of a German scientist.

After fifty years dedicated to the University of Marburg, in 1918, Feusner resigned. In 1927, he had a unique opportunity to celebrate both the 400th anniversary of the University and his own anniversary - 60 years from the date of defense of the dissertation (Dozenenjubilaeum). The life path of Feusner was surprisingly smooth and smooth for the alarming and turbulent time of social revolutions and world wars. “Quiet work and reliable performance of duty were the happiness of his life” ^[8] . The remaining years he spent on well-deserved rest surrounded by his family. Friedrich Wilhelm Feusner died on September 5, 1928 in Marburg at the age of 85.

Parameter allocation method

The essence of the computational advantages of the topological methods for the decomposition of the Feusner determinants consists, firstly, in eliminating the exhaustion of unnecessary combinations of circuit branches and, secondly, in the formation of a bracket expression for the determinant, that is, an expression with common factors taken out of brackets. The latter repeatedly reduces the number of required computational operations. By the determinant of the Z-circuit (Y-circuit), like Feusner, we mean the determinant of the corresponding matrix of circuit resistances (nodal conductivities). This emphasizes the fact that topological methods are designed to obtain a circuit function, bypassing the formation of the matrix of the circuit.

Feusner proposed formulas for extracting parameters ^[3] ^[17] , which allow one to reduce the decomposition of the determinant of a passive circuit to the decomposition of determinants of simpler derived schemes in which there is no distinguished branch z or y:

\Delta =z\Delta ^{z}+\Delta _{z}\qquad (1)

{\ displaystyle \ Delta = z \ Delta ^ {z} + \ Delta _ {z} \ qquad (1)}

\Delta =y\Delta _{y}+\Delta ^{y}\qquad (2)

{\ displaystyle \ Delta = y \ Delta _ {y} + \ Delta ^ {y} \ qquad (2)}

Where $\Delta$ ${\ displaystyle \ Delta}$ - determinant of the passive circuit. Subscript or superscript character $\Delta$ ${\ displaystyle \ Delta}$ indicate contraction or removal of the selected branch, respectively. Pulling a branch is equivalent to replacing it with an ideal conductor. As a result of contracting and removing branches, degenerate schemes can be formed whose determinant is identically equal to zero, which simplifies the decomposition of the determinants. The figure illustrates the application of formulas (1) and (2).

The recursive application of formulas (1) and (2) reduces the initial formulas to the simplest, the determinants of which are derived from Ohm's law.

Enumeration of graph trees

In the mid-60s, it was found that the simplest algorithm for enumerating graph trees is based on formula (2) ^[20] . In symbolic form, the set S (G) of all trees of the graph G must satisfy the condition ^[21] :

S(G)=eS(G/e)\cup S(G\ e)\qquad (3)

{\ displaystyle S (G) = eS (G / e) \ cup S (G \ e) \ qquad (3)}

Where $e$ ${\ displaystyle e}$ is an edge of the graph $G$ ${\ displaystyle G}$ , $G/e$ ${\ displaystyle G / e}$ and $G\backslash e$ ${\ displaystyle G \ backslash e}$ - graphs obtained from the source as a result of contraction and removal of the edge $e$ ${\ displaystyle e}$ respectively.

The prominent software theoretician Donald Knuth in the fourth volume of his monumental work “The Art of Programming ” cites Feusner as the founder of the efficient generation of graph trees through allocation formulas (1) and (2) ^[20] .

Earlier references to the works of Feusner can be found in the publications of J.E. Alderson ^[22] , G.J. Minty ^[23] , V.K. Chen ^[24] , F.T. Besha ^[25] , C.J. Colbourne , R.P. J. Day and L.D. Nela ^[26] .

Diakoptika Feusner

Feusner expressed some ideas of the diakoptic approach to the analysis of schemes ^[3] ^[17] long before the work of G. Crohn ^[27] . It was he who first introduced and used the concept of “subcircuit” (“partial circuit”) and proposed a method for dividing (bisecting) a circuit, which is based on bisection formulas for one (4) and two nodes (5), respectively:

\Delta =\Delta _{1}\cdot \Delta _{2}\qquad (4)

{\ displaystyle \ Delta = \ Delta _ {1} \ cdot \ Delta _ {2} \ qquad (4)}

\Delta =\Delta _{1}\cdot \Delta _{2}(a,b)+\Delta _{1}(a,b)\cdot \Delta _{2}\qquad (5)

{\ displaystyle \ Delta = \ Delta _ {1} \ cdot \ Delta _ {2} (a, b) + \ Delta _ {1} (a, b) \ cdot \ Delta _ {2} \ qquad (5) }

Where $\Delta _{1}$ ${\ displaystyle \ Delta _ {1}}$ and $\Delta _{2}$ ${\ displaystyle \ Delta _ {2}}$ - determinants of the first and second subcircuits of which the circuit consists; $\Delta _{1}(a,b)$ ${\ displaystyle \ Delta _ {1} (a, b)}$ and $\Delta _{2}(a,b)$ ${\ displaystyle \ Delta _ {2} (a, b)}$ - determinants of circuits formed respectively from the first and second subcircuits as a result of combining common nodes. Formulas (4) and (5) are clearly illustrated in Fig. 3 and fig. 4 respectively.

Methods for decomposing circuit identifiers

In addition to the method for extracting parameters by formulas (1) and (2) considered above, Feunser proposed and proved methods for expanding the determinant of a Z-scheme (Y-scheme) in a Z-circuit (Y-node) and in a Z-node (Y-circuit ) The formulations of these Feusner methods deserve to be given in full ^[3] ^[17] (the headings of the statements and their numbering do not belong to the original).

If a $h\geqslant \mu$ ${\ displaystyle h \ geqslant \ mu}$ then form combinations $h,h-1,\ldots ,1$ ${\ displaystyle h, h-1, \ ldots, 1}$ ; if a $h<\mu$ ${\ displaystyle h <\ mu}$ , then - combinations of $\mu ,\mu -1,\ldots ,1$ ${\ displaystyle \ mu, \ mu -1, \ ldots, 1}$ from the resistances of the branches of the circuit with the exception of those combinations of branches, when removed, the circuit breaks up into parts. Each such product of resistances is multiplied by the determinant of the circuit, which is obtained from the original circuit as a result of removing the contour branches and combining the nodes that are connected by contour branches that are not included in the combination. The sum of the indicated works is the identifier sought.
Decomposition of the determinant of a Y-scheme by a node. If a node with p Y-branches ending in any nodes of the original scheme is added to the Y-scheme, then the determinant of the new Y-scheme is the sum whose terms consist of all combinations in $p,p-1,\ldots ,1$ ${\ displaystyle p, p-1, \ ldots, 1}$ from the conductivities of the new branches, and each such product of conductivities is multiplied by the determinant of the circuit obtained from the original circuit as a result of combining the final nodes of the branches that are in this combination.
Decomposition of the determinant of a Z-scheme by a node. If a node with p z-branches ending in some nodes of the original scheme is added to the Z-scheme, then the determinant of the new Z-scheme is the sum whose terms consist of all combinations in $p-1,p-2,\ldots ,0$ ${\ displaystyle p-1, p-2, \ ldots, 0}$ from the resistances of the new branches, and each such product of resistances is multiplied by the determinant of the circuit obtained from the original circuit as a result of combining the final nodes of the added branches that are absent in this combination.
Expansion of the determinant of a Y-scheme with independent contours along a contour containing $p-1,p-2,\ldots ,0$ ${\ displaystyle p-1, p-2, \ ldots, 0}$ branches. If a $h\leqslant \mu$ ${\ displaystyle h \ leqslant \ mu}$ then form combinations $h-1,h-2,\ldots ,0$ ${\ displaystyle h-1, h-2, \ ldots, 0}$ ; if a $h>\mu$ ${\ displaystyle h> \ mu}$ , then - combinations of $h-1,h-2,\ldots ,h-\mu$ ${\ displaystyle h-1, h-2, \ ldots, h- \ mu}$ from the conductivities of the branches of the circuit with the exception of those combinations of branches, upon removal of which the circuit breaks up into disconnected parts. Each such product of conductivities is multiplied by the determinant of the circuit, which is obtained from the original circuit by removing the branches of the contour and combining the nodes that are connected by the branches that are in combination. The sum of these works is the required determinant.

Statements 1, 2, 3 are superior to modern formulations ^[28] ^[29] in generality and clarity. Statement 4, which, apparently, was not given in later sources, supplements the previous statements. As a result, we have a complete group of statements regarding the decomposition of the determinant of a circuit into a node and a contour. V. Feusner gives a rule ^[3] , which allows one to take into account the presence of multiple z-branches in the determinant expression obtained for a simplified scheme formed as a result of the formal replacement of multiple branches by single ones. This provides a significant reduction in the complexity of the calculation of complex electrical circuits .

Topological Transfer Formula

In 1847, two years after the publication of his laws, G.R. Kirchhoff tried to make the process of obtaining a decision more visual. His method of analyzing z-circuits without control connections uses the circuit equivalent circuit directly and does not require preliminary preparation of its equations. The dual result for y-circuits was published by Maxwell ^[2] in 1873. The literature on this subject is usually called 1892 - the date of the third edition of the famous treatise ^[30] ^[31] . Maxwell introduces a relation (later called a circuit function and SSF)

H=\Delta N/\Delta D\qquad (6)

{\ displaystyle H = \ Delta N / \ Delta D \ qquad (6)}

Where $\Delta N$ ${\ displaystyle \ Delta N}$ and $\Delta D$ ${\ displaystyle \ Delta D}$ - respectively, the numerator and denominator of the SSF, in which the parameters of all elements of the circuit are represented by symbols.

In 1902 , V. Feusner drew attention to the difficulties of constructing SSFs using the topological formulas of Kirchhoff and Maxwell . The formation of the SSF according to Foysner provides for the decomposition of the determinants of the original circuit and its derivatives from the circuits according to expressions (1) - (2) without drawing up the chain equations. It is important that at each step of the calculation it is necessary to deal with a circuit that is less complex than the original circuit, and not with abstract combinations of branches of the original circuit.

To simplify the finding of the SSF numerator of both the Z- and Y-schemes (compared with Kirchhoff and Maxwell 's formulas), Voisner obtained a formula in which the terms due to the contribution to the sum of the terms of the numerator of each circuit circuit passing through the voltage source and branch with the desired current ^[32] . The topological transfer formula proposed by Feusner allows one to find the SSF numerator by listing the transfer circuits between an independent source and a branch with the desired response:

\Delta N=\sum _{i\subset q}P_{i}\Delta _{i}\qquad (7)

{\ displaystyle \ Delta N = \ sum _ {i \ subset q} P_ {i} \ Delta _ {i} \ qquad (7)}

Where $q$ ${\ displaystyle q}$ - the number of transmission loops, $P_{i}$ ${\ displaystyle P_ {i}}$ Is the product of the conductivities included in $i{\text{-й}}$ ${\ displaystyle i {\ text {th}}}$ transmission circuit taken with the corresponding sign; $\Delta _{i}$ ${\ displaystyle \ Delta _ {i}}$ - the determinant of the circuit when tightening all the branches of the i- th circuit.

In a schematic form, the topological transfer formula is presented in the figure. The very idea of finding loops containing both a generator and a receiver for obtaining the numerators of circuit functions belongs to Feusner.

The topological formula of the Feisner transfer in a schematic form

Using the complete schema as a template

The first to use the complete circuit as a test circuit when developing circuit theory methods was Feusner's teacher, Kirchhoff . It was a complete four-node circuit proposed by Wheatstone ^[6] . It was also used by Maxwell , and in our time, experts still use the full four-node circuit as a basic test for modern computer systems for circuit simulation.

Feusner drew attention to the complexity of the analysis of the complete circuit, introduced by Maxwell , and considered a topological approach to the analysis of electrical circuits, in which the complete circuit is used as a template. In fact, Feusner introduced complete circuits with an arbitrary number of nodes into electrical engineering and developed methods for studying them that were effective for their time.

He proposed using for analysis of a circuit with the number of nodes equal to n, the well-known determinant of a complete circuit at n nodes, in which terms including parameters of missing branches in the circuits being analyzed were equal to zero. So, below is a complete Z-scheme at five nodes (Fig. A) and its determinant (8), calculated according to (1).

\Delta =(R_{1}(R_{10}(R_{3}((R_{2}+R_{6})((R_{5}+R_{7})(R_{4}+R_{8}+R_{9})+R_{4}(R_{8}+R_{9}))+R_{5}((R_{4}+R_{8})(R_{7}+R_{9})+R_{7}R_{9})+R_{8}(R_{4}(R_{7}+R_{9})+R_{7}R_{9}))+

{\ displaystyle \ Delta = (R_ {1} (R_ {10} (R_ {3} ((R_ {2} + R_ {6}) ((R_ {5} + R_ {7}) (R_ {4} + R_ {8} + R_ {9}) + R_ {4} (R_ {8} + R_ {9})) + R_ {5} ((R_ {4} + R_ {8}) (R_ {7} + R_ {9}) + R_ {7} R_ {9}) + R_ {8} (R_ {4} (R_ {7} + R_ {9}) + R_ {7} R_ {9})) +}

+(R_{4}(R_{2}(R_{8}+R_{9})+R_{8}R_{9}))(R_{6}+R_{7})+((R_{4}+R_{8})(R_{2}+R_{9})+R_{2}R_{9})(R_{5}(R_{6}+R_{7})+R_{6}R_{7}))+(R_{3}(R_{2}(R_{8}+R_{9})+R_{8}R_{9}))*

{\ displaystyle + (R_ {4} (R_ {2} (R_ {8} + R_ {9}) + R_ {8} R_ {9})) (R_ {6} + R_ {7}) + (( R_ {4} + R_ {8}) (R_ {2} + R_ {9}) + R_ {2} R_ {9}) (R_ {5} (R_ {6} + R_ {7}) + R_ { 6} R_ {7})) + (R_ {3} (R_ {2} (R_ {8} + R_ {9}) + R_ {8} R_ {9})) *}

*((R_{4}+R_{7})(R_{5}+R_{6})+R_{5}R_{6})+((R_{3}+R_{9})(R_{2}+R_{8})+R_{2}R_{8})(R_{4}(R_{5}(R_{6}+R_{7})+R_{6}R_{7})))+(R_{10}(R_{3}(R_{5}((R_{4}+R_{8})(R_{7}+R_{9})+R_{7}R_{9})+

{\ displaystyle * ((R_ {4} + R_ {7}) (R_ {5} + R_ {6}) + R_ {5} R_ {6}) + ((R_ {3} + R_ {9}) (R_ {2} + R_ {8}) + R_ {2} R_ {8}) (R_ {4} (R_ {5} (R_ {6} + R_ {7}) + R_ {6} R_ {7 }))) + (R_ {10} (R_ {3} (R_ {5} ((R_ {4} + R_ {8}) (R_ {7} + R_ {9}) + R_ {7} R_ { 9}) +}

+R_{8}(R_{4}(R_{7}+R_{9})+R_{7}R_{9}))+R_{7}R_{9}(R_{4}(R_{5}+R_{8})+R_{5}R_{8}))+R_{5}R_{8}(R_{3}(R_{4}(R_{7}+R_{9})+R_{7}R_{9})+R_{4}R_{7}R_{9}))(R_{2}+R_{6})+((R_{10}+R_{3})*

{\ displaystyle + R_ {8} (R_ {4} (R_ {7} + R_ {9}) + R_ {7} R_ {9})) + R_ {7} R_ {9} (R_ {4} ( R_ {5} + R_ {8}) + R_ {5} R_ {8})) + R_ {5} R_ {8} (R_ {3} (R_ {4} (R_ {7} + R_ {9} ) + R_ {7} R_ {9}) + R_ {4} R_ {7} R_ {9})) (R_ {2} + R_ {6}) + ((R_ {10} + R_ {3}) *}

*(R_{5}((R_{4}+R_{8})(R_{7}+R_{9})+R_{7}R_{9})+R_{8}(R_{4}(R_{7}+R_{9})+R_{7}R_{9}))+R_{4}(R_{5}(R_{7}(R_{8}+R_{9})+R_{8}R_{9})+R_{7}R_{8}R_{9}))(R_{2}R_{6}))\qquad (8)

{\ displaystyle * (R_ {5} ((R_ {4} + R_ {8}) (R_ {7} + R_ {9}) + R_ {7} R_ {9}) + R_ {8} (R_ { 4} (R_ {7} + R_ {9}) + R_ {7} R_ {9})) + R_ {4} (R_ {5} (R_ {7} (R_ {8} + R_ {9}) + R_ {8} R_ {9}) + R_ {7} R_ {8} R_ {9})) (R_ {2} R_ {6})) \ qquad (8)}

Illustration of the application of the full outline template method

To analyze the scheme in Figure b, it is enough to remove from the formula (8) all terms that include the parameters of the missing elements. As a result, we get:

\Delta =(R_{1}+R_{2})((R_{3}+R_{4})(R_{10}+R_{5}+R_{6})+R_{10}(R_{5}+R_{6}))+R_{6}((R_{3}+R_{4})(R_{10}+R_{5})+R_{10}R_{5})\qquad (9)

{\ displaystyle \ Delta = (R_ {1} + R_ {2}) ((R_ {3} + R_ {4}) (R_ {10} + R_ {5} + R_ {6}) + R_ {10} (R_ {5} + R_ {6})) + R_ {6} ((R_ {3} + R_ {4}) (R_ {10} + R_ {5}) + R_ {10} R_ {5}) \ qquad (9)}

Many years later, methods were developed that implement this approach for analysis ^[33] ^[34] and synthesis ^[31] ^{[35] of} RLC circuits. It is important that Feusner formulated all of his results for both Z and Y circuits, one of the first to use the duality principle ^[15] . After 56 years, the mathematician Clark, in a journal of the London Mathematical Society, re-examined one method of Feusner extension to prove Cayley 's formula on the number of trees T in a complete graph ^[36] . Cayley's formula

T=q^{2}-2\qquad (10)

{\ displaystyle T = q ^ {2} -2 \ qquad (10)}

where q are the nodes of the circuit (graph), Feusner received independently of this mathematician, who laid the foundations of graph theory .

Topological proof of the principle of reciprocity

In the work of Feusner ^[3], the principle of reciprocity is investigated and its topological proof is given. Moreover, Feusner presents this evidence only as a by-product, noting that Kirchhoff himself could have done it.

As is known, the reciprocity principle based on the reciprocity theorem states: if the EMF $E$ ${\ displaystyle E}$ acting in a branch of the circuit that does not contain other sources, causes a current in another branch $I$ ${\ displaystyle I}$ , then brought into this branch of EMF $E$ ${\ displaystyle E}$ will cause the same current in the first branch $I$ ${\ displaystyle I}$ .

We denote the conductor in which the emf source is located, by $a$ ${\ displaystyle a}$ therefore the SSF numerator $Z(N)$ ${\ displaystyle Z (N)}$ (6) which is multiplied by $E$ ${\ displaystyle E}$ and gives current $I_{a}$ ${\ displaystyle I_ {a}}$ this branch is equal $\Delta N_{a}$ ${\ displaystyle \ Delta N_ {a}}$ .

To find the numerator of an expression for current $i_{k}$ ${\ displaystyle i_ {k}}$ in another branch $b$ ${\ displaystyle b}$ , proceed as follows. Suppose that each individual conductor A forms closed circuits $K_{1},K_{2},\ldots ,K_{p}$ ${\ displaystyle K_ {1}, K_ {2}, \ ldots, K_ {p}}$ with constant intensity currents $I_{1},I_{2},\ldots ,I_{p}$ ${\ displaystyle I_ {1}, I_ {2}, \ ldots, I_ {p}}$ in the direction of passing through $a$ ${\ displaystyle a}$ . Obviously, the first law of Kirchhoff $\Sigma I_{n}=0$ ${\ displaystyle \ Sigma I_ {n} = 0}$ with respect to the branch point will be performed for the totality of these currents at any values $I$ ${\ displaystyle I}$ . Suppose that in each conductor of the circuit, the sum of the currents flowing through it gives the resulting current $i$ ${\ displaystyle i}$ , then the condition for each distribution of resistances in the circuit should be satisfied:

\sum I=i_{a}\qquad (11)

{\ displaystyle \ sum I = i_ {a} \ qquad (11)}

We assume that $I_{f}=Z_{f}\,E/\Delta N$ ${\ displaystyle I_ {f} = Z_ {f} \, E / \ Delta N}$ and $\Sigma Z_{f}=\Delta N_{a}$ ${\ displaystyle \ Sigma Z_ {f} = \ Delta N_ {a}}$ . Consequently, $Z_{f}$ ${\ displaystyle Z_ {f}}$ composed of members $\Delta N_{a}$ ${\ displaystyle \ Delta N_ {a}}$ . To get a way of making a current distribution possible, remember that removing some branch of the circuit $K$ ${\ displaystyle K}$ leads to its rupture and, therefore, the intensity of the current flowing through it $I$ ${\ displaystyle I}$ will be zero. Wherein $I_{f}$ ${\ displaystyle I_ {f}}$ , $Z_{f}$ ${\ displaystyle Z_ {f}}$ cannot contain resistance $R$ ${\ displaystyle R}$ conductors forming a circuit. Therefore, if $E$ ${\ displaystyle E}$ is in $a$ ${\ displaystyle a}$ then to get the numerator $i_{k}$ ${\ displaystyle i_ {k}}$ both conductors are used simultaneously $a$ ${\ displaystyle a}$ and $k$ ${\ displaystyle k}$ . Should take a sequence of members from $\Delta N_{a}$ ${\ displaystyle \ Delta N_ {a}}$ which does not occur $R$ ${\ displaystyle R}$ conductors contained in $K_{1}$ ${\ displaystyle K_ {1}}$ , join members that do not contain $R$ ${\ displaystyle R}$ of $K_{2}$ ${\ displaystyle K_ {2}}$ , and so on before using all the contours $K_{1},K_{2},\ldots ,K_{g}$ ${\ displaystyle K_ {1}, K_ {2}, \ ldots, K_ {g}}$ .

To determine the sign, choose some direction of the conductor k as positive, then when the direction of the current coincides, a term with a positive sign is obtained, if it does not coincide with the negative.

Feusner formulates a rule according to which the numerator $i_{\lambda }$ ${\ displaystyle i _ {\ lambda}}$ is the sum of the combinations of $R_{1},R_{2},\ldots ,R_{n}$ ${\ displaystyle R_ {1}, R_ {2}, \ ldots, R_ {n}}$ by $\mu -1$ ${\ displaystyle \ mu -1}$ elements, after removing the conductors of which there remains one closed figure containing $\lambda$ ${\ displaystyle \ lambda}$ . Each combination is multiplied by the sum of the EMFs that belong to the closed figure. EMF are considered positive in the direction, if the current is positive in this direction $i_{j}$ ${\ displaystyle i_ {j}}$ . To determine the current in the conductor $b$ ${\ displaystyle b}$ if the emf is in $a$ ${\ displaystyle a}$ , a closed loop is used that passes through both of these conductors ( $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ ) The same closed loop is used to determine the current in $a$ ${\ displaystyle a}$ if the emf is in $b$ ${\ displaystyle b}$ . Then if in the circuit of the EMF conductors from the branch $a$ ${\ displaystyle a}$ without change is transferred to $k$ ${\ displaystyle k}$ then in $a$ ${\ displaystyle a}$ the same current that used to be in $k$ ${\ displaystyle k}$ .

Generalized method of circuit currents

Maxwell, according to John Ambrose Fleming ^[37] , the inventor of the first electron lamp, later called a diode, in his last university lecture showed a different type of current decomposition in a circuit with conductors. Judging by how Fleming describes it, the method is not generally applicable. It is assumed that the circuit thus lies on a plane that the conductors do not overlap anywhere. The circumference of each circuit, in which one direct current is assumed, flows in a certain direction (counterclockwise). Two currents of boundary circuits of opposite values flow through each conductor inside the circuit, and their difference is the current flowing in this conductor. It is clear that such a circuit arrangement on a plane is not always possible, as, for example, in a circuit obtained by connecting two opposite nodes in a Wheatstone bridge circuit.

In the work ^[3] , according to Feusner himself, a “small change” is given, which makes the method generally applicable. It is possible, as Kirchhoff showed, for each chain to take different systems $\mu =n-m+1$ ${\ displaystyle \ mu = n-m + 1}$ closed circuits from which all possible closed circuits in a circuit can be composed. Feusner proposes to consider such a system $k_{1},k_{2},\ldots ,k_{\mu }$ ${\ displaystyle k_ {1}, k_ {2}, \ ldots, k _ {\ mu}}$ , while in each circuit one direct current flows $I_{1},I_{2},\ldots ,I_{\mu }$ ${\ displaystyle I_ {1}, I_ {2}, \ ldots, I _ {\ mu}}$ . For each circuit and each conductor, a direction is established in which the current should be directed positively. Then, Kirchhoff's law should be applied to each such circuit, which will allow $\mu$ ${\ displaystyle \ mu}$ linear equations between $E$ ${\ displaystyle E}$ , circuit resistances and $I_{1},\ldots ,I_{\mu }$ ${\ displaystyle I_ {1}, \ ldots, I _ {\ mu}}$ where you can find the sought currents.

Feusner points out that the determinant, which can be obtained using the classical notation of Kirchhoff’s law , will be $n$ ${\ displaystyle n}$ -th order, and the determinant obtained by Maxwell, only $\mu$ ${\ displaystyle \ mu}$ th order. Thus, the advantages of the new method are not as great as we would like. Individual elements of the Kirchhoff form usually also have $\mu$ ${\ displaystyle \ mu}$ order due $(m-1)$ ${\ displaystyle (m-1)}$ multiple factors $\pm 1$ {\ displaystyle \ pm 1} . In addition, a significantly larger number of mutually annihilating members is formed in Maxwell, therefore, the technique proposed by Maxwell does not have significant advantages compared to the original Kirchhoff approach.

Notes

↑ Kirchhoff G. R. Selected Works. - M .: Nauka, 1988 .-- 428 p.
↑ ¹ ² Maxwell D.K. A treatise on electricity and magnetism. In 2 volumes of T. 1. - M .: Nauka, 1989 .-- 416 p.
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ Feussner W. Ueber Stromverzweigung in netzformigen Leitern // Annalen der Physik. - 1902. - Bd 9, N 13. - S. 1304-1329.
↑ Jungnickel S., McCormach R. Intellectual mastery of nature. Theoretical Physics from Ohm to Einstein (Vol. 2): The Now Mighty Theoretical Physics 1870-1925. - Chicago and London: The University of Chicago Press. - 1986.
↑ Schulze FA Friedrich Wilhelm Feussner // Nature. - 1930. - No. 126 (23 August 1930). - P. 286.
↑ ¹ ² Wheatstone C. Beschreibung verschiedener neuen Instrumente und Methoden zur Bestimmung der Constanten einer Volta'schen Kette // Annalen der Physik und Chemie. - Leipzig, 1844. - Bd 62. - S. 499-543.
↑ Feussner W. Ueber den Bumerang // Sitzungsberichte der Gesellschaft zur Beforderung der gesammten Naturwissenschaften zu Marburg. - Marburg, 1869. - N 1 (Januar). - S. 7-15.
↑ ¹ ² Schulze FA Wilhelm Feussner // Physik Zeitschrift. - 1930. - No. 31. - P. 513-514.
↑ Feussner W. Ueber die von Hrn. Sekulic beschriebene Interferenzerscheinung // Annalen der Physik und Chemie. - 1873. - Bd 9, N 8. - S. 561-564.
↑ Feussner W. Neuer Beweis der Unrichtigkeit der Emissionstheorie des Lichts // Annalen der Physik und Chemie. - 1877. - Bd 10, N 2. - S. 317-332.
↑ Feussner W. Ueber die Interferenzerscheinungen dünner Blättchen mit besonderer Reucksicht auf die Theorie der Newtonschen Ringe // Annalen der Physik und Chemie. - 1881. - Bd 14, N 12. - S. 545-571.
↑ Winkelmann A. Handbook of Physics. Griffiths Phil. Trans. - 1895. - Vol. 2., Pt. 2. 338 p.
↑ Gehrcke E. Handbuch der physikalischen Optik. - Iter Band, lte Halfte, und 2ter Band, lte Halfte. Leipzig, Barth, 1926-1927. 470 pp.
↑ Thomas P. Geschichte und Gegenwart der Physik an der Philipps-Universitat Marburg
↑ ¹ ² Feussner W. Ueber zwei Sätze der Elektrostatik (betr. Die potentielle Energie eines Leitersystems). - Festschrift L. Boltzmann gewidmet. - Leipzig, 1904. - S. 537-541.
↑ Feussner W. Ueber einen Interferenzapparat und einer damit von Herrn Dr. Schmitt ausgefeuhrte untersuchung // Sitzungsberichte der Gesellschaft zur Beforderung der gesammten Naturwissenschaften zu Marburg. - Marburg, 1907. - S. 128-134.
↑ ¹ ² ³ ⁴ Feussner W. Zur Berechnung der Stromstarke in netzformigen Leitern // Annalen der Physik. - 1904. - Bd 15, N 12. - S. 385-394.
↑ Barrows JT Extension of Feussner's method to active networks // IRE Transactions on circuit theory. - 1966. - Vol. CT-13, N 6. - P. 198-200.
↑ Braun J. Topological analysis of networks containing nullators and norators // Electronics letters. - 1966. - Vol. 2, No. 11. - P. 427-428.
↑ ¹ ² Minty GJ A simple algorithm for listing all trees of a graph // IEEE Transactions on circuit theory. - 1965. - Vol. CT-12, No. 1.
↑ Knuth DE The art of computer programming (Pre-fascicle 4). A draft of section 7.2.1.6: Generating all trees. - Addison-Wesley, Stanford University. - 2004. - Vol. 4. - 81 p.
↑ Alderson GE, Lin PM Computer generation of symbolic network functions - new theory and implementation // IEEE Transactions on circuit theory. - 1973. -Vol. CT-20, No. 1. - P. 48-56.
↑ Carlin HJ, Youla DC Network synthesis with negative resistors // Proceedings of the IRE. - 1961 (May). - P. 907-920.
↑ Chen WK Unified theory on topological analysis of linear systems // Proceedings of the Institution of Electrical Engineers. - London, 1967. - Vol. 114, No. 11.
↑ Boesch FT, Li X., Suffel C. On the existence of uniformly optimally reliable networks // Networks. - 1991. - Vol. 21, No. 2. - R. 181—194.
↑ Colbourn CJ, Day RPJ, Nel LD Unranking and ranking spanning-trees of a graph // Journal of algorithms. - 1989. - Vol. 10, No. 2. - R. 271-286.
↑ Kron G. The study of complex systems in parts - diakoptika. - M .: Nauka, 1972.- 544 p.
↑ Dolbnya V. T. Topological methods of analysis and synthesis of electric circuits and systems. - Kharkov: Publishing House of Vishcha School at Kharkiv. state un-te, 1974. - 145 p.
↑ Theoretical foundations of electrical engineering. T. 1 / P. A. Ionkin, A. I. Darevsky, E. S. Kukharkin, V. G. Mironov, N. A. Melnikov. - M.: Higher School, 1976.- 544 p.
↑ Seshu S., Reed M. B. Linear graphs and electric circuits.- M .: Higher. school, 1971. - 448 p.
↑ ¹ ² Bellert S., Wozniacki G. Analysis and synthesis of electrical circuits using the structural number method. - M .: Mir, 1972.- 334 p.
↑ Feussner W. Ueber Verzweigung elektrischer Strome // Sitzungsberichte der Gesellschaft zur Beforderung der gesammten Naturwissenschaften zu Marburg. - Marburg, 1902. - No. 8 (December) .- S. 105-115.
↑ Filaretov V.V. Recursive methods for expressing the determinant of an undirected graph // Theoret. Electrical Engineering.-Lviv, 1986.- Vol. 40.- S. 6-12.
↑ Filaretov V.V. Formation of coefficients of functions of the RLC-scheme of the complete topological structure // Electricity. - 1987. - No. 6. - S. 42-47.
↑ Optimal implementation of linear electronic RLC circuits / A. A. Lanne, E. D. Mikhailova, B. S. Sargsyan, Ya. N. Matviychuk. - Kiev: Naukova Dumka, 1981.
↑ Clarke LE On Cayley`s formula for counting trees // The journal of the London Mathematical Society. - 1958. - Vol. 33, part 4, No. 132. - R. 471-474.
↑ Fleming JA Phil. Mag. - 1885.- (5) No. 20.- p. 221.

Literature

Gorshkov K. S., Filaretov V.V. Wilhelm Feusner's schematic approach and the method of circuit determinants / V. Filaretov. - Ulyanovsk: UlSTU, 2009. - 189 p. - ISBN 978–5–9795–0510–7.

[1] Kirchhoff G. R. Selected Works. - M .: Nauka, 1988 .-- 428 p.

[autogenerated3-2] ¹ ² Maxwell D.K. A treatise on electricity and magnetism. In 2 volumes of T. 1. - M .: Nauka, 1989 .-- 416 p.

[autogenerated7-3] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ Feussner W. Ueber Stromverzweigung in netzformigen Leitern // Annalen der Physik. - 1902. - Bd 9, N 13. - S. 1304-1329.

[4] Jungnickel S., McCormach R. Intellectual mastery of nature. Theoretical Physics from Ohm to Einstein (Vol. 2): The Now Mighty Theoretical Physics 1870-1925. - Chicago and London: The University of Chicago Press. - 1986.

[5] Schulze FA Friedrich Wilhelm Feussner // Nature. - 1930. - No. 126 (23 August 1930). - P. 286.

[autogenerated4-6] ¹ ² Wheatstone C. Beschreibung verschiedener neuen Instrumente und Methoden zur Bestimmung der Constanten einer Volta'schen Kette // Annalen der Physik und Chemie. - Leipzig, 1844. - Bd 62. - S. 499-543.

[7] Feussner W. Ueber den Bumerang // Sitzungsberichte der Gesellschaft zur Beforderung der gesammten Naturwissenschaften zu Marburg. - Marburg, 1869. - N 1 (Januar). - S. 7-15.

[autogenerated1-8] ¹ ² Schulze FA Wilhelm Feussner // Physik Zeitschrift. - 1930. - No. 31. - P. 513-514.

[9] Feussner W. Ueber die von Hrn. Sekulic beschriebene Interferenzerscheinung // Annalen der Physik und Chemie. - 1873. - Bd 9, N 8. - S. 561-564.

[10] Feussner W. Neuer Beweis der Unrichtigkeit der Emissionstheorie des Lichts // Annalen der Physik und Chemie. - 1877. - Bd 10, N 2. - S. 317-332.

[11] Feussner W. Ueber die Interferenzerscheinungen dünner Blättchen mit besonderer Reucksicht auf die Theorie der Newtonschen Ringe // Annalen der Physik und Chemie. - 1881. - Bd 14, N 12. - S. 545-571.

[12] Winkelmann A. Handbook of Physics. Griffiths Phil. Trans. - 1895. - Vol. 2., Pt. 2. 338 p.

[13] Gehrcke E. Handbuch der physikalischen Optik. - Iter Band, lte Halfte, und 2ter Band, lte Halfte. Leipzig, Barth, 1926-1927. 470 pp.

[14] Thomas P. Geschichte und Gegenwart der Physik an der Philipps-Universitat Marburg

[autogenerated2-15] ¹ ² Feussner W. Ueber zwei Sätze der Elektrostatik (betr. Die potentielle Energie eines Leitersystems). - Festschrift L. Boltzmann gewidmet. - Leipzig, 1904. - S. 537-541.

[16] Feussner W. Ueber einen Interferenzapparat und einer damit von Herrn Dr. Schmitt ausgefeuhrte untersuchung // Sitzungsberichte der Gesellschaft zur Beforderung der gesammten Naturwissenschaften zu Marburg. - Marburg, 1907. - S. 128-134.

[autogenerated5-17] ¹ ² ³ ⁴ Feussner W. Zur Berechnung der Stromstarke in netzformigen Leitern // Annalen der Physik. - 1904. - Bd 15, N 12. - S. 385-394.

[18] Barrows JT Extension of Feussner's method to active networks // IRE Transactions on circuit theory. - 1966. - Vol. CT-13, N 6. - P. 198-200.

[19] Braun J. Topological analysis of networks containing nullators and norators // Electronics letters. - 1966. - Vol. 2, No. 11. - P. 427-428.

[autogenerated6-20] ¹ ² Minty GJ A simple algorithm for listing all trees of a graph // IEEE Transactions on circuit theory. - 1965. - Vol. CT-12, No. 1.

[21] Knuth DE The art of computer programming (Pre-fascicle 4). A draft of section 7.2.1.6: Generating all trees. - Addison-Wesley, Stanford University. - 2004. - Vol. 4. - 81 p.

[22] Alderson GE, Lin PM Computer generation of symbolic network functions - new theory and implementation // IEEE Transactions on circuit theory. - 1973. -Vol. CT-20, No. 1. - P. 48-56.

[23] Carlin HJ, Youla DC Network synthesis with negative resistors // Proceedings of the IRE. - 1961 (May). - P. 907-920.

[24] Chen WK Unified theory on topological analysis of linear systems // Proceedings of the Institution of Electrical Engineers. - London, 1967. - Vol. 114, No. 11.

[25] Boesch FT, Li X., Suffel C. On the existence of uniformly optimally reliable networks // Networks. - 1991. - Vol. 21, No. 2. - R. 181—194.

[26] Colbourn CJ, Day RPJ, Nel LD Unranking and ranking spanning-trees of a graph // Journal of algorithms. - 1989. - Vol. 10, No. 2. - R. 271-286.

[27] Kron G. The study of complex systems in parts - diakoptika. - M .: Nauka, 1972.- 544 p.

[28] Dolbnya V. T. Topological methods of analysis and synthesis of electric circuits and systems. - Kharkov: Publishing House of Vishcha School at Kharkiv. state un-te, 1974. - 145 p.

[29] Theoretical foundations of electrical engineering. T. 1 / P. A. Ionkin, A. I. Darevsky, E. S. Kukharkin, V. G. Mironov, N. A. Melnikov. - M.: Higher School, 1976.- 544 p.

[30] Seshu S., Reed M. B. Linear graphs and electric circuits.- M .: Higher. school, 1971. - 448 p.

[autogenerated8-31] ¹ ² Bellert S., Wozniacki G. Analysis and synthesis of electrical circuits using the structural number method. - M .: Mir, 1972.- 334 p.

[32] Feussner W. Ueber Verzweigung elektrischer Strome // Sitzungsberichte der Gesellschaft zur Beforderung der gesammten Naturwissenschaften zu Marburg. - Marburg, 1902. - No. 8 (December) .- S. 105-115.

[33] Filaretov V.V. Recursive methods for expressing the determinant of an undirected graph // Theoret. Electrical Engineering.-Lviv, 1986.- Vol. 40.- S. 6-12.

[34] Filaretov V.V. Formation of coefficients of functions of the RLC-scheme of the complete topological structure // Electricity. - 1987. - No. 6. - S. 42-47.

[35] Optimal implementation of linear electronic RLC circuits / A. A. Lanne, E. D. Mikhailova, B. S. Sargsyan, Ya. N. Matviychuk. - Kiev: Naukova Dumka, 1981.

[36] Clarke LE On Cayley`s formula for counting trees // The journal of the London Mathematical Society. - 1958. - Vol. 33, part 4, No. 132. - R. 471-474.

[37] Fleming JA Phil. Mag. - 1885.- (5) No. 20.- p. 221.