In electrical engineering and mathematics, the parallel sum is a commutative, associative binary operation which derives from the formula for the resistance of resistors in parallel. The parallel sum is represented by an infix operator, which in elementary electrical texts is written as a pair of parallel lines "||"[1][2][3][4] but in the academic literature more usually as a colon ":".[5] David Ellerman claims that the parallel sum is "just as good" as the series sum.
Motivation
editCan't think of any words at the moment
Definition
editThe parallel sum of two terms may be defined either as
or as
The former definition is more readily generalized to more than two terms
but the latter definition has the advantage of remaining valid when one of the two arguments is zero.
For matrix arguments
editThe parallel sum of two non-singular square matrices may be defined as
This definition may be extended to singular matrices by rewriting it as
provided only that the sum of the two matrices is non-singular. If the sum of the matrices is singular, then the parallel sum may still be defined by adopting a particular generalized inverse, such as the Moore-Penrose generalized inverse (Duffin). Alternatively, the two matrices may be considered to be parallel summable only when the result is the same irrespective of the choice generalized inverse[6]
For vector arguments
editAnderson and Trapp[7] For two vectors and with non-zero
If additionally and are each non-zero, this can be written as
That is, the (generalized) inverse of the vector is taken to be , the smallest vector satisfying .
Identities
editSerial sum | Parallel sum | Remarks |
---|---|---|
Repetition | ||
Definition in terms of the other (terms assumed non-zero where necessary) | ||
Commutative property | ||
Associative property (if each parallel sum is defined) | ||
Distribution | ||
Common numerator | ||
Mutual cancellation | ||
Addition of exponents | ||
Differentiation |
Inequalities
editWhereas the ordinary sum of any two non-negative real numbers is greater than or equal to each of them, the parallel sum of two such numbers is less than or equal to each.
Lehman's series-parallel inequality
editMore generally, for an array of m rows in series, by n parallel columns of resistors, the overall resistance is higher if series connections are made first.
and this is likewise true when the R are positive definite square matrices.[5].
Parity with the 'serial' sum
editIn electrical engineering, both resistance -voltage drop divided by current- and conductance -current divided by voltage- are used to quantify the propensity of a material body to pass a current. Either of these quantities
In electrical engineering, the propensity of a material body to pass an electrical current may be quantified by either the body's resistance -the voltage drop across the body divided by the current through it- or by its conductance -the current divided by the voltage. In the case of
David Ellerman argues that the parallel sum is "just as good" as the series sum[8] and offers a number of illustrations that the
Harmonic mean
Gaussian equation for thin lenses
Geometric construction
Moore-Penrose generalized inverse
positive semi-definite just as simple to consider Hermitian semi-definite matrices.
References
edit- ^ Patrick, Dale R.; Fardo, Stephen W. (2008). Electrical Distribution Systems (2nd ed.). The Fairmont Press Inc. p. 21. ISBN 9780881735994.
- ^ Glisson, Tildon H. (2011). Introduction to Circuit Analysis and Design. Springer. p. 143. ISBN 9789048194421.
- ^ Walton, Alan Keith (1987). Network Analysis and Practice. Cambridge University Press. p. 35. ISBN 9780521319034.
- ^ O'Malley, John (1992). Schaum's Outline of Basic Circuit Analysis. Schaum's Outline (2nd ed.). McGraw-Hill. p. 33. ISBN 9780070478244.
- ^ a b c Bapat, R. B.; Raghavan, T. E. S. (1997). Nonnegative Matrices and Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press. pp. 153–156. ISBN 9780521571678.
- ^ Mitra, Sujit Kumar; Puri, Madan Lal (1973). "On parallel sum and difference of matrices". Journal of Mathematical Analysis and Applications. 44 (1): 92–97. doi:10.1016/0022-247X(73)90027-9. ISSN 0022-247X.
- ^ Anderson, W. N.; Trapp, G. E. (1988), "Network Matrix Operations for Vectors and Quaternions", in Datta, Biswa Nath (ed.), Linear Algebra in Signals, Systems, and Control, Proceedings in Applied Mathematics Series, vol. 32, SIAM, pp. 3–10, ISBN 9780898712230
- ^ Ellerman, David P. (1995). Intellectual Trespassing As a Way of Life: Essays in Philosophy, Economics, and Mathematics (PDF). Worldly Philosophy. Rowman & Littlefield. p. 237. ISBN 9780847679324.