The parallel operator (pronounced "parallel", [1] following the parallel lines notation from geometry; [2] [3] also known as reduced sum, parallel sum or parallel addition) is a mathematical function which is used as a shorthand in electrical engineering, [4] [5] [6] [nb 1] but is also used in kinetics, fluid mechanics and financial mathematics. [7] [8] The name parallel comes from the use of the operator computing the combined resistance of resistors in parallel.
The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as the "reciprocal formula" or " harmonic sum") and is defined by: [9] [6] [10] [11]
where a, b, and are elements of the extended complex numbers [12] [13]
The operator gives half of the harmonic mean of two numbers a and b. [7] [8]
As a special case, for any number :
Further, for all distinct numbers :
with representing the absolute value of , and meaning the minimum (least element) among x and y.
If and are distinct positive real numbers then
The concept has been extended from a scalar operation to matrices [14] [15] [16] [17] [18] and further generalized. [19]
The operator was originally introduced as reduced sum by Sundaram Seshu in 1956,
[20]
[21]
[14] studied as operator ∗
by Kent E. Erickson in 1959,
[22]
[23]
[14] and popularized by
Richard James Duffin and William Niles Anderson, Jr. as parallel addition or parallel sum operator :
in
mathematics and
network theory since 1966.
[15]
[16]
[1] While some authors continue to use this symbol up to the present,
[7]
[8] for example, Sujit Kumar Mitra used ∙
as a symbol in 1970.
[14] In
applied electronics, a ∥
sign became more common as the operator's symbol around 1974.
[24]
[25]
[26]
[27]
[28]
[nb 1]
[nb 2] This was often written as doubled vertical line (||) available in most
character sets (sometimes italicized as //
[29]
[30]), but now can be represented using
Unicode character U+2225 ( ∥ ) for "parallel to". In
LaTeX and related markup languages, the macros \|
and \parallel
are often used (and rarely \smallparallel
is used) to denote the operator's symbol.
Let represent the extended complex plane excluding zero, and the bijective function from to such that One has identities
and
This implies immediately that is a field where the parallel operator takes the place of the addition, and that this field is isomorphic to
The following properties may be obtained by translating through the corresponding properties of the complex numbers.
As for any field, satisfies a variety of basic identities.
It is commutative under parallel and multiplication:
It is associative under parallel and multiplication: [12] [7] [8]
Both operations have an identity element; for parallel the identity is while for multiplication the identity is 1:
Every element of has an inverse under parallel, equal to the additive inverse under addition. (But 0 has no inverse under parallel.)
The identity element is its own inverse,
Every element of has a multiplicative inverse :
Multiplication is distributive over parallel: [1] [7] [8]
Repeated parallel is equivalent to division,
Or, multiplying both sides by n,
Unlike for repeated addition, this does not commute:
Using the distributive property twice, the product of two parallel binomials can be expanded as
The square of a binomial is
The cube of a binomial is
In general, the nth power of a binomial can be expanded using binomial coefficients which are the reciprocal of those under addition, resulting in an analog of the binomial formula:
The following identities hold:
A parallel function is one which commutes with the parallel operation:[ citation needed]
For example, is a parallel function, because
As with a polynomial under addition, a parallel polynomial with coefficients in (with ) can be factored into a product of monomials:
for some roots (possibly repeated) in
Analogous to polynomials under addition, the polynomial equation
implies that for some k.
A linear equation can be easily solved via the parallel inverse:
To solve a parallel quadratic equation, complete the square to obtain an analog of the quadratic formula
The extended complex numbers including zero, is no longer a field under parallel and multiplication, because 0 has no inverse under parallel. (This is analogous to the way is not a field because has no additive inverse.)
For every non-zero a,
The quantity can either be left undefined (see indeterminate form) or defined to equal 0.
In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction, similar to multiplication. [1] [31] [9] [10]
There are applications of the parallel operator in electronics, optics, and study of periodicity:
In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits. [nb 2] There is a duality between the usual (series) sum and the parallel sum. [7] [8]
For instance, the total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors.
Likewise for the total capacitance of serial capacitors. [nb 2]
In geometric optics the thin lens approximation to the lens maker's equation.
The time between conjunctions of two orbiting bodies is called the synodic period. If the period of the slower body is T2, and the period of the faster is T1, then the synodic period is
Question:
Answer:
Answer:
Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959, [22] the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008 [32] [33] [34] as well as on the WP 34C [35] and WP 43S since 2015, [36] [37] allowing to solve even cascaded problems with few keystrokes like 270↵ Enter180∥120∥.
Given a field F there are two embeddings of F into the projective line P(F): z → [z : 1] and z → [1 : z]. These embeddings overlap except for [0:1] and [1:0]. The parallel operator relates the addition operation between the embeddings. In fact, the homographies on the projective line are represented by 2 x 2 matrices M(2,F), and the field operations (+ and ×) are extended to homographies. Each embedding has its addition a + b represented by the following matrix multiplications in M(2,A):
The two matrix products show that there are two subgroups of M(2,F) isomorphic to (F,+), the additive group of F. Depending on which embedding is used, one operation is +, the other is
[…] To have a convenient short notation for the joint resistance of resistors connected in parallel let […] A:B = AB/(A+B) […] A:B may be regarded as a new operation termed parallel addition […] Parallel addition is defined for any nonnegative numbers. The network model shows that parallel addition is commutative and associative. Moreover, multiplication is distributive over this operation. Consider now an algebraic expression in the operations (+) and (:) operating on positive numbers A, B, C, etc. […] To give a network interpretation of such a polynomial read A + B as "A series B" and A : B as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […][1] [2] (206 pages)
§359. […] ∥ for parallel occurs in Oughtred's Opuscula mathematica hactenus inedita (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in Kersey, [A] Caswell, Jones, [B] Wilson, [C] Emerson, [D] Kambly, [E] and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often […] Hall and Stevens [F] use "par [F] or ∥" for parallel […] [A] John Kersey, Algebra (London, 1673), Book IV, p. 177. [B] W. Jones, Synopsis palmarioum matheseos (London, 1706). [C] John Wilson, Trigonometry (Edinburgh, 1714), characters explained. [D] W. Emerson, Elements of Geometry (London, 1763), p. 4. [E] L. Kambly , Die Elementar-Mathematik, Part 2: Planimetrie, 43. edition (Breslau, 1876), p. 8. […] [F] H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London, 1889), p. 10. […][3]
[…] This mathematical relationship comes up often enough that it actually has a name: the "parallel operator", denoted ∥. When we say x∥y, it means . Note that this is a mathematical operator and does not say anything about the actual configuration. In the case of resistors the parallel operator is used for parallel resistors, but for other components (like capacitors) this is not the case. […](16 pages)
[…] When resistors with resistance a and b are placed in series, their compound resistance is the usual sum (hereafter the series sum) of the resistances a + b. If the resistances are placed in parallel, their compound resistance is the parallel sum of the resistances, which is denoted by the full colon […][4] (271 pages)
The parallel sum of two positive real numbers x:y = [(1/x) + (1/y)]−1 arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a duality between the usual (series) sum and the parallel sum. […][5] (24 pages)
[…] One convenient way to indicate two resistors are in parallel is to put a ∥ between them. […]
Für die Berechnung des Ersatzwiderstands der Parallelschaltung wird […] gern die Kurzschreibweise ∥ benutzt.
The purpose of this communication is to extend the concept of the scalar operation Reduced Sum introduced by Seshu […] and later elaborated by Erickson […] to matrices, to outline some interesting properties of this new matrix operation, and to apply the matrix operation in the analysis of series and parallel n-port networks. Let A and B be two non-singular square matrices having inverses, A−1 and B−1 respectively. We define the operation ∙ as A ∙ B = (A−1 + B−1)−1 and the operation ⊙ as A ⊙ B = A ∙ (−B). The operation ∙ is commutative and associative and is also distributive with respect to multiplication. […](3 pages)
[…] we define the parallel sum of A and B by the formula A(A + B)+B and denote it by A : B. If A and B are nonsingular this reduces to A : B = (A−1 + B−1)−1 which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the Hermitian semi-definite matrices form a commutative partially ordered semigroup under the parallel sum operation. […][6]
[…] The operation ∗ is defined as A ∗ B = AB/A + B. The symbol ∗ has algebraic properties which simplify the formal solution of many series-parallel network problems. If the operation ∗ were included as a subroutine in a digital computer, it could simplify the programming of certain network calculations. […](3 pages) (NB. See comment.)
[…] Comments on the operation ∗ […] a∗b = ab/(a+b) […](1 page) (NB. Refers to previous reference.)
This textbook evolved from a one-semester introductory electronics course taught by the authors at the Massachusetts Institute of Technology. […] The course is used by many freshmen as a precursor to the MIT Electrical Engineering Core Program. […] The preparation of a book of this size has drawn on the contribution of many people. The concept of teaching network theory and electronics as a single unified subject derives from Professor Campbell Searle, who taught the introductory electronics course when one of us ( S.D.S.) was a first-year physics graduate student trying to learn electronics. In addition, Professor Searle has provided invaluable constructive criticism throughout the writing of this text. Several members of the MIT faculty and nearly 40 graduate technical assistants have participated in the teaching of this material over the past five years, many of whom have made important contributions through their suggestions and examples. Among these, we especially wish to thank O. R. Mitchell, Irvin Englander, George Lewis, Ernest Vincent, David James, Kenway Wong, Gim Hom, Tom Davis, James Kirtley, and Robert Donaghey. The chairman of the MIT Department of Electrical Engineering, Professor Louis D. Smullin, has provided support and encouragement during this project, as have many colleagues throughout the department. […] The first result […] states that the total voltage across the parallel combination of R1 and R2 is the same as that which occurs across a single resistance of value R1 R2 (R1 + R2). Because this expression for parallel resistance occurs so often, it is given a special notation (R1∥R2). That is, when R1 and R2 are in parallel, the equivalent resistance is […](xii+623+5 pages) (NB. A teacher's manual was available as well. Early print runs contains a considerable number of typographical errors. See also: Wedlock's 1978 book.) [7]
[…] Introduction to Electronics and Instrumentation is a new and contemporary approach to the introductory electronics course. Designed for students with no prior experience with electronics, it develops the skills and knowledge necessary to use and understand modern electronic systems. […] John W. McWane […](NB. The SET Project was a two-year post-secondary curriculum developed between 1974 and 1977 preparing technicians to use electronic instruments.)
[…] The Technical Curriculum Research and Development Program, sponsored by the Imperial Organization of Social Services of Iran, is entering the fourth year of a five-year contract. Curriculum development in electronics and mechanical engineering continues. […] Administered jointly by C.A.E.S. and the Department of Materials Science and Engineering, the Project is under the supervision of Professor Merton C. Flemings. It is directed by Dr. John W. McWane. […] Curriculum Materials Development. This is the principal activity of the project and is concerned with the development of innovative, state-of-the-art course materials in needed areas of engineering technology […] new introductory course in electronics […] is entitled Introduction to Electronics and Instrumentation and consists of eight […] modules […] dc Current, Voltage, and Resistance; Basic Circuit Networks; Time Varying Signals; Operational Amplifiers; Power Supplies; ac Current, Voltage, and Impedance; Digital Circuits; and Electronic Measurement and Control. This course represents a major change and updating of the way in which electronics is introduced, and should be of great value to STI as well as to many US programs. […]
[…] Bruce D. Wedlock […] was the principle contributing author to Part I, BASIC CIRCUIT NETWORKS including the design of the companion examples. […] Most of the development of the IEI program was undertaken as part of the Technical Curriculum Research and Development Project of the MIT Center of Advanced Engineering Study. […] shorthand notation […] shorthand symbol ∥ […](xiii+545 pages) (NB. In 1981, a 216-pages laboratory manual accompanying this book existed as well. The work grew out of an MIT course program " The MIT Technical Curriculum Development Project - Introduction to Electronics and Instrumentation" developed between 1974 and 1979. In 1986, a second edition of this book was published under the title "Introduction to Electronics Technology".)
[…] Bei abgekürzter Schreibweise achte man sorgfältig auf die Anwendung von Klammern. […] Das Parallelzeichen ∥ der Kurzschreibweise hat die gleiche Bedeutung wie ein Multiplikationszeichen. Deshalb können Klammern entfallen.(446 pages)
PARAMETER […] TYP […] UNIT […] INPUT IMPEDANCE […] Common mode […] 1013 ∥ 2 […] Ω ∥ pF […] Differential […] 1013 ∥ 4 […] Ω ∥ pF […](37 pages) (NB. Unusual usage of ∥ for both values and units.)
The parallel operator (pronounced "parallel", [1] following the parallel lines notation from geometry; [2] [3] also known as reduced sum, parallel sum or parallel addition) is a mathematical function which is used as a shorthand in electrical engineering, [4] [5] [6] [nb 1] but is also used in kinetics, fluid mechanics and financial mathematics. [7] [8] The name parallel comes from the use of the operator computing the combined resistance of resistors in parallel.
The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as the "reciprocal formula" or " harmonic sum") and is defined by: [9] [6] [10] [11]
where a, b, and are elements of the extended complex numbers [12] [13]
The operator gives half of the harmonic mean of two numbers a and b. [7] [8]
As a special case, for any number :
Further, for all distinct numbers :
with representing the absolute value of , and meaning the minimum (least element) among x and y.
If and are distinct positive real numbers then
The concept has been extended from a scalar operation to matrices [14] [15] [16] [17] [18] and further generalized. [19]
The operator was originally introduced as reduced sum by Sundaram Seshu in 1956,
[20]
[21]
[14] studied as operator ∗
by Kent E. Erickson in 1959,
[22]
[23]
[14] and popularized by
Richard James Duffin and William Niles Anderson, Jr. as parallel addition or parallel sum operator :
in
mathematics and
network theory since 1966.
[15]
[16]
[1] While some authors continue to use this symbol up to the present,
[7]
[8] for example, Sujit Kumar Mitra used ∙
as a symbol in 1970.
[14] In
applied electronics, a ∥
sign became more common as the operator's symbol around 1974.
[24]
[25]
[26]
[27]
[28]
[nb 1]
[nb 2] This was often written as doubled vertical line (||) available in most
character sets (sometimes italicized as //
[29]
[30]), but now can be represented using
Unicode character U+2225 ( ∥ ) for "parallel to". In
LaTeX and related markup languages, the macros \|
and \parallel
are often used (and rarely \smallparallel
is used) to denote the operator's symbol.
Let represent the extended complex plane excluding zero, and the bijective function from to such that One has identities
and
This implies immediately that is a field where the parallel operator takes the place of the addition, and that this field is isomorphic to
The following properties may be obtained by translating through the corresponding properties of the complex numbers.
As for any field, satisfies a variety of basic identities.
It is commutative under parallel and multiplication:
It is associative under parallel and multiplication: [12] [7] [8]
Both operations have an identity element; for parallel the identity is while for multiplication the identity is 1:
Every element of has an inverse under parallel, equal to the additive inverse under addition. (But 0 has no inverse under parallel.)
The identity element is its own inverse,
Every element of has a multiplicative inverse :
Multiplication is distributive over parallel: [1] [7] [8]
Repeated parallel is equivalent to division,
Or, multiplying both sides by n,
Unlike for repeated addition, this does not commute:
Using the distributive property twice, the product of two parallel binomials can be expanded as
The square of a binomial is
The cube of a binomial is
In general, the nth power of a binomial can be expanded using binomial coefficients which are the reciprocal of those under addition, resulting in an analog of the binomial formula:
The following identities hold:
A parallel function is one which commutes with the parallel operation:[ citation needed]
For example, is a parallel function, because
As with a polynomial under addition, a parallel polynomial with coefficients in (with ) can be factored into a product of monomials:
for some roots (possibly repeated) in
Analogous to polynomials under addition, the polynomial equation
implies that for some k.
A linear equation can be easily solved via the parallel inverse:
To solve a parallel quadratic equation, complete the square to obtain an analog of the quadratic formula
The extended complex numbers including zero, is no longer a field under parallel and multiplication, because 0 has no inverse under parallel. (This is analogous to the way is not a field because has no additive inverse.)
For every non-zero a,
The quantity can either be left undefined (see indeterminate form) or defined to equal 0.
In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction, similar to multiplication. [1] [31] [9] [10]
There are applications of the parallel operator in electronics, optics, and study of periodicity:
In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits. [nb 2] There is a duality between the usual (series) sum and the parallel sum. [7] [8]
For instance, the total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors.
Likewise for the total capacitance of serial capacitors. [nb 2]
In geometric optics the thin lens approximation to the lens maker's equation.
The time between conjunctions of two orbiting bodies is called the synodic period. If the period of the slower body is T2, and the period of the faster is T1, then the synodic period is
Question:
Answer:
Answer:
Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959, [22] the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008 [32] [33] [34] as well as on the WP 34C [35] and WP 43S since 2015, [36] [37] allowing to solve even cascaded problems with few keystrokes like 270↵ Enter180∥120∥.
Given a field F there are two embeddings of F into the projective line P(F): z → [z : 1] and z → [1 : z]. These embeddings overlap except for [0:1] and [1:0]. The parallel operator relates the addition operation between the embeddings. In fact, the homographies on the projective line are represented by 2 x 2 matrices M(2,F), and the field operations (+ and ×) are extended to homographies. Each embedding has its addition a + b represented by the following matrix multiplications in M(2,A):
The two matrix products show that there are two subgroups of M(2,F) isomorphic to (F,+), the additive group of F. Depending on which embedding is used, one operation is +, the other is
[…] To have a convenient short notation for the joint resistance of resistors connected in parallel let […] A:B = AB/(A+B) […] A:B may be regarded as a new operation termed parallel addition […] Parallel addition is defined for any nonnegative numbers. The network model shows that parallel addition is commutative and associative. Moreover, multiplication is distributive over this operation. Consider now an algebraic expression in the operations (+) and (:) operating on positive numbers A, B, C, etc. […] To give a network interpretation of such a polynomial read A + B as "A series B" and A : B as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […][1] [2] (206 pages)
§359. […] ∥ for parallel occurs in Oughtred's Opuscula mathematica hactenus inedita (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in Kersey, [A] Caswell, Jones, [B] Wilson, [C] Emerson, [D] Kambly, [E] and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often […] Hall and Stevens [F] use "par [F] or ∥" for parallel […] [A] John Kersey, Algebra (London, 1673), Book IV, p. 177. [B] W. Jones, Synopsis palmarioum matheseos (London, 1706). [C] John Wilson, Trigonometry (Edinburgh, 1714), characters explained. [D] W. Emerson, Elements of Geometry (London, 1763), p. 4. [E] L. Kambly , Die Elementar-Mathematik, Part 2: Planimetrie, 43. edition (Breslau, 1876), p. 8. […] [F] H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London, 1889), p. 10. […][3]
[…] This mathematical relationship comes up often enough that it actually has a name: the "parallel operator", denoted ∥. When we say x∥y, it means . Note that this is a mathematical operator and does not say anything about the actual configuration. In the case of resistors the parallel operator is used for parallel resistors, but for other components (like capacitors) this is not the case. […](16 pages)
[…] When resistors with resistance a and b are placed in series, their compound resistance is the usual sum (hereafter the series sum) of the resistances a + b. If the resistances are placed in parallel, their compound resistance is the parallel sum of the resistances, which is denoted by the full colon […][4] (271 pages)
The parallel sum of two positive real numbers x:y = [(1/x) + (1/y)]−1 arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a duality between the usual (series) sum and the parallel sum. […][5] (24 pages)
[…] One convenient way to indicate two resistors are in parallel is to put a ∥ between them. […]
Für die Berechnung des Ersatzwiderstands der Parallelschaltung wird […] gern die Kurzschreibweise ∥ benutzt.
The purpose of this communication is to extend the concept of the scalar operation Reduced Sum introduced by Seshu […] and later elaborated by Erickson […] to matrices, to outline some interesting properties of this new matrix operation, and to apply the matrix operation in the analysis of series and parallel n-port networks. Let A and B be two non-singular square matrices having inverses, A−1 and B−1 respectively. We define the operation ∙ as A ∙ B = (A−1 + B−1)−1 and the operation ⊙ as A ⊙ B = A ∙ (−B). The operation ∙ is commutative and associative and is also distributive with respect to multiplication. […](3 pages)
[…] we define the parallel sum of A and B by the formula A(A + B)+B and denote it by A : B. If A and B are nonsingular this reduces to A : B = (A−1 + B−1)−1 which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the Hermitian semi-definite matrices form a commutative partially ordered semigroup under the parallel sum operation. […][6]
[…] The operation ∗ is defined as A ∗ B = AB/A + B. The symbol ∗ has algebraic properties which simplify the formal solution of many series-parallel network problems. If the operation ∗ were included as a subroutine in a digital computer, it could simplify the programming of certain network calculations. […](3 pages) (NB. See comment.)
[…] Comments on the operation ∗ […] a∗b = ab/(a+b) […](1 page) (NB. Refers to previous reference.)
This textbook evolved from a one-semester introductory electronics course taught by the authors at the Massachusetts Institute of Technology. […] The course is used by many freshmen as a precursor to the MIT Electrical Engineering Core Program. […] The preparation of a book of this size has drawn on the contribution of many people. The concept of teaching network theory and electronics as a single unified subject derives from Professor Campbell Searle, who taught the introductory electronics course when one of us ( S.D.S.) was a first-year physics graduate student trying to learn electronics. In addition, Professor Searle has provided invaluable constructive criticism throughout the writing of this text. Several members of the MIT faculty and nearly 40 graduate technical assistants have participated in the teaching of this material over the past five years, many of whom have made important contributions through their suggestions and examples. Among these, we especially wish to thank O. R. Mitchell, Irvin Englander, George Lewis, Ernest Vincent, David James, Kenway Wong, Gim Hom, Tom Davis, James Kirtley, and Robert Donaghey. The chairman of the MIT Department of Electrical Engineering, Professor Louis D. Smullin, has provided support and encouragement during this project, as have many colleagues throughout the department. […] The first result […] states that the total voltage across the parallel combination of R1 and R2 is the same as that which occurs across a single resistance of value R1 R2 (R1 + R2). Because this expression for parallel resistance occurs so often, it is given a special notation (R1∥R2). That is, when R1 and R2 are in parallel, the equivalent resistance is […](xii+623+5 pages) (NB. A teacher's manual was available as well. Early print runs contains a considerable number of typographical errors. See also: Wedlock's 1978 book.) [7]
[…] Introduction to Electronics and Instrumentation is a new and contemporary approach to the introductory electronics course. Designed for students with no prior experience with electronics, it develops the skills and knowledge necessary to use and understand modern electronic systems. […] John W. McWane […](NB. The SET Project was a two-year post-secondary curriculum developed between 1974 and 1977 preparing technicians to use electronic instruments.)
[…] The Technical Curriculum Research and Development Program, sponsored by the Imperial Organization of Social Services of Iran, is entering the fourth year of a five-year contract. Curriculum development in electronics and mechanical engineering continues. […] Administered jointly by C.A.E.S. and the Department of Materials Science and Engineering, the Project is under the supervision of Professor Merton C. Flemings. It is directed by Dr. John W. McWane. […] Curriculum Materials Development. This is the principal activity of the project and is concerned with the development of innovative, state-of-the-art course materials in needed areas of engineering technology […] new introductory course in electronics […] is entitled Introduction to Electronics and Instrumentation and consists of eight […] modules […] dc Current, Voltage, and Resistance; Basic Circuit Networks; Time Varying Signals; Operational Amplifiers; Power Supplies; ac Current, Voltage, and Impedance; Digital Circuits; and Electronic Measurement and Control. This course represents a major change and updating of the way in which electronics is introduced, and should be of great value to STI as well as to many US programs. […]
[…] Bruce D. Wedlock […] was the principle contributing author to Part I, BASIC CIRCUIT NETWORKS including the design of the companion examples. […] Most of the development of the IEI program was undertaken as part of the Technical Curriculum Research and Development Project of the MIT Center of Advanced Engineering Study. […] shorthand notation […] shorthand symbol ∥ […](xiii+545 pages) (NB. In 1981, a 216-pages laboratory manual accompanying this book existed as well. The work grew out of an MIT course program " The MIT Technical Curriculum Development Project - Introduction to Electronics and Instrumentation" developed between 1974 and 1979. In 1986, a second edition of this book was published under the title "Introduction to Electronics Technology".)
[…] Bei abgekürzter Schreibweise achte man sorgfältig auf die Anwendung von Klammern. […] Das Parallelzeichen ∥ der Kurzschreibweise hat die gleiche Bedeutung wie ein Multiplikationszeichen. Deshalb können Klammern entfallen.(446 pages)
PARAMETER […] TYP […] UNIT […] INPUT IMPEDANCE […] Common mode […] 1013 ∥ 2 […] Ω ∥ pF […] Differential […] 1013 ∥ 4 […] Ω ∥ pF […](37 pages) (NB. Unusual usage of ∥ for both values and units.)