Series-parallel Duality

The parallel operator (also known as reduced sum, parallel sum or parallel addition) $\,$ (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used as a shorthand in electrical engineering, but is also used in kinetics, fluid mechanics and financial mathematics. The name ''parallel'' comes from the use of the operator computing the combined resistance of resistors in parallel.

Overview

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:

:$\begin \parallel: &&\overline \times \overline &\to \overline \\ &&(a, b) &\mapsto a \parallel b = \frac = \frac, \end$

with $\overline = \mathbb\cup\$ being the complex projective line (with corresponding rules).

The operator gives half of the harmonic mean of two numbers ''a'' and ''b''.

As a special case, for any number $a \in \overline$:
:$a \parallel a = \frac1 = \tfrac12a.$

Further, for all distinct numbers
:$\big, a \parallel b \big, > \tfrac12 \min\bigl(, a, , , b, \bigr),$

with $\big, a \parallel b \big,$ representing the absolute value of $a \parallel b$, and $\min(x, y)$ meaning the minimum (least element) among and .

If $a$ and $b$ are distinct positive real numbers then $\tfrac12 \min(a, b) < \big, a \parallel b \big, < \min(a, b).$

The concept has been extended from a scalar operation to matrices and further generalized.

Notation

The operator was originally introduced as reduced sum by Sundaram Seshu in 1956, studied as operator ∗ by Kent E. Erickson in 1959, 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. While some authors continue to use this symbol up to the present, for example, Sujit Kumar Mitra used ∙ as a symbol in 1970. In applied electronics, a ∥ sign became more common as the operator's symbol around 1974. This was often written as doubled vertical line () available in most character sets (sometimes italicized as //), 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.

Rules

For addition, the parallel operator follows the commutative law:

:$a \parallel b = b \parallel a$

and the associative law:

:$(a \parallel b) \parallel c = a \parallel (b \parallel c) = a \parallel b \parallel c = \frac = \frac.$

Multiplication is distributive over this operation:

:$k\bigl(a \parallel b\bigr) = (ka) \parallel (kb).$

Further, the parallel operator has $\infty$ as neutral element. For any number $a,$

:$a \parallel \infty = \frac1 = a.$

For any non-zero number , the number is its inverse element:

:$a \parallel (-a) = \frac1 = \frac10 = \infty.$

However, $\bigl(\overline, \bigr)$ is not an abelian group, as has no inverse element. For every non-zero , $a \parallel 0 = 0.$ The quantity $0 \parallel (-0) = 0 \parallel 0$ can either be left undefined (see indeterminate form) or defined to equal . (This is analogous to the way $\bigl(\overline, \bigr)$ is not an abelian group because $\infty$ has no additive inverse.)

In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction, similar to multiplication.

Applications

In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits.

For instance, the total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors.
:
:$\frac = \frac + \frac + \cdots + \frac$.

Likewise for the total capacitance of serial capacitors.

The same principle can be applied to various problems in other disciplines. For example, in geometric optics the thin lens approximation to the lens maker's equation.

There is a duality between the usual (series) sum and the parallel sum.

Examples

Question:
: Three resistors $R_1 = 270\,\mathrm$, $R_2 = 180\,\mathrm$ and $R_3 = 120\,\mathrm$ are connected in parallel. What is their resulting resistance?

Answer:
: $R_1 \parallel R_2 \parallel R_3 = 270\,\mathrm \parallel 180\,\mathrm \parallel 120\,\mathrm = \frac \approx 56.84 \,\mathrm$
: The effectively resulting resistance is ca. 57 k Ω.

Question:
: A construction worker raises a wall in 5 hours. Another worker would need 7 hours for the same work. How long does it take to build the wall if both worker work in parallel?

Answer:
: $t_1 \parallel t_2 = 5\,\mathrm h \parallel 7\,\mathrm h = \frac \approx 2.92\,\mathrm h$
: They will finish in close to 3 hours.

Implementation

Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959, the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008 as well as on the WP 34C and WP 43S since 2015, allowing to solve even cascaded problems with few keystrokes like .

Projective view

The parallel operator may be understood as a homography on the projective line over a ring. The reciprocation operation is usually singular on null vectors, but with projective geometry the reciprocal is completed with "points at infinity". In fact, the translations from finite points are complemented by "translations at infinity" as valid projectivities. The parallel operator is the composition of two such translations at infinity.

Notes

References

Further reading

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* (10 pages)
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* (33 pages)
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(19 pages)
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* {{cite book , title=TLV3201, TLV3202: TLV320x 40-ns, microPOWER, Push-Pull Output Comparators , chapter=7.5 Electrical Characteristics: VCC = 5 V / 7.6 Electrical Characteristics: VCC = 2.7 V / 9.1.2.1 Inverting Comparator with Hysteresis , publisher=Texas Instruments Incorporated , publication-place=Dallas, Texas, USA , version=Revision B , id=SBOS561B , date=2022-06-03 , orig-date=2016, 2012 , pages=5, 6, 13–14 3, url=https://www.ti.com/lit/ds/symlink/tlv3201.pdf?ts=1660718632803 , access-date=2022-08-18 , url-status=live , archive-url=https://web.archive.org/web/20220817185705/https://www.ti.com/lit/ds/symlink/tlv3201.pdf?ts=1660718632803 , archive-date=2022-08-17 , quote-page=5 , quote=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.)

External links

* https://github.com/microsoftarchive/edx-platform-1/blob/master/common/lib/calc/calc/calc.py

Abstract algebra
Elementary algebra
Multiplication