International Scholarly Research Notices

Research Article

An Approach to Conformal Transformation Using Symbolic Language Facilities: Application in Electrical Engineering

Algorithm 1

It is useful to recall that the pound sign ( # ) marks that the characters following this sign represent a comment; therefore the software Maple does not process the above mentioned characters. For simplicity, in the algorithms of this article, the pound sign has been omitted when a sentence continues on several lines, but for running the program the sign must exist on every line.
#   An automatic generation of the sigma coefficient formula will be presented.
#   The fluxes between the point of abscissa π‘₯ 𝑀 = βˆ’ 1 and any point of abscissa 𝑐 will be considered.
#   In the program, both indexed names and symbols which are not indexed have been used.
#   The point π‘₯ 𝑀 = βˆ’ 1 is on the slot axis; on the line 𝐴 𝑧 𝐡 𝑧 .
# 𝑏 0 β€”the aperture breadth of the slot divided by the minimum air-gap thickness.
#   The reference field strength is that for the constant minimum air-gap thickness.
#   The involved relative field strength is its actual value divided by the reference value.
#   σ—flux difference due to the existence of the slot.
#   For obtaining the formula, the commands concerning the input variables should be disabled.
#   The multiplication sign is, in several versions, optional, a small space between factors suffices.
𝑃 ∢ = 𝐩 𝐫 𝐨 𝐜 ( 𝑠 )
#   The considered transformation function will be:
 𝑓 ∢ = ( 1 / e v a l 𝑓 ( πœ‹ ) ) β‹… ( 𝑀 2 βˆ’ ( 𝑏 2 0 + 2 . ) β‹… 𝑀 + 1 . / ( 𝑀 β‹… ( 𝑀 βˆ’ 1 ) ) ) ;
𝑧 ∢ = i n t ( 𝑓 , 𝑀 ) ;
𝑧 ∢ = e v a l ( 𝑧 , 𝑀 = 𝑠 ) ;
end proc;
𝐹 ∢ = 𝑃 ( 𝑀 ) ;
# π‘₯ 𝑀 β€”abscissa 𝑀 = βˆ’1 in the 𝑀 -plane.
# 𝑧 1 β€”complex vector potential function at point 𝑀 ∢ = βˆ’ 1 . for the actual configuration.
# 𝑧 2 β€”complex vector potential function at point 𝑀 ∢ = 0 , hence a point 𝐢 𝑀 for the actual configuration.
# 𝑐 β€”abscissa of any point 𝑐 ∈ ( βˆ’ 1 , 0 ) , in particular 0.
#    𝑏 0 ∢ = 1 . 5 ;
# π‘₯ 𝑀 ∢ = βˆ’ 1 ;
#    𝑐 ∢ = βˆ’ 0 . 1 ;
𝑧 1 ∢ = e v a l ( 𝑃 ( π‘₯ 𝑀 ) ) ;
𝑧 1 ∢ = e v a l 𝑓 ( 𝑧 1 ) ;
𝑧 2 ∢ = 𝑃 ( 𝑐 ) ;
𝑧 2 ∢ = e v a l 𝑓 ( 𝑧 2 ) ;
𝑧 𝑑 ∢ = 𝑧 2 βˆ’ 𝑧 1 ;
𝑧 𝑑 ∢ = s i m p l i f y ( 𝑧 𝑑 ) ;
# 𝑧 𝑑 β€”gives the flux between the two points, calculated in the 𝑀 -plane, in the case of actual field distribution in the 𝑧 -plane.
𝑧 𝑑 1 ∢ = e v a l 𝑓 ( ( 1 / πœ‹ ) l n ( π‘₯ 𝑀 ) ) ;
𝑧 𝑑 2 ∢ = e v a l 𝑓 ( ( 1 / πœ‹ ) l n ( 𝑐 ) ) ;
𝑧 𝑑 ∢ = 𝑧 𝑑 1 βˆ’ 𝑧 𝑑 2 ;
# 𝑧 𝑑 β€”gives the flux between the two points, calculated in the 𝑀 -plane, in the case of a uniform field distribution in the 𝑑 -plane (zeta-plane).
𝜎 ∢ = 𝑧 𝑑 βˆ’ 𝑧 𝑑 ;
𝜎 ∢ = s i m p l i f y ( 𝜎 ) ;
#   The total value will be 𝜎 𝑇 ∢ = 2 𝜎 .
#   Result after running the program:
#    𝑏 0 ∢ = 1 . 5 ;
# π‘₯ 𝑀 ∢ = βˆ’ 1 ;
#    𝑐 ∢ = βˆ’ 0 . 1 ;
1 𝜎 ∢ = βˆ’ √ ( 1 . 0.3183098861 ( βˆ’ 1 . l n ( π‘₯ 𝑀 ) + l n ( 𝑐 ) + l n ( βˆ’ 0 . 5 0 0 0 0 0 0 0 0 0 𝑏 2 0  βˆ’ 1 . + 𝑐 + 𝑐 2 βˆ’ 1 . 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . )
     βˆ’ 1 . a r c t a n h ( ( 0 . 5 0 0 0 0 0 0 0 0 0 ( βˆ’ 2 . + 𝑐 𝑏 2 0  + 2 . 𝑐 ) ) / 𝑐 2 βˆ’ 1 . 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . )
     + 𝑏 2 0 l n ( ( βˆ’ 1 . 𝑏 2 0 βˆ’ 1 . 𝑐 𝑏 2 0  + 2 . βˆ’ 1 . 𝑏 2 0  𝑐 2 βˆ’ 1 . 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . ) / ( 𝑐 βˆ’ 1 . ) )
     /  βˆ’ 1 . 𝑏 2 0 βˆ’ 1 . l n ( βˆ’ 0 . 5 0 0 0 0 0 0 0 0 0 𝑏 2 0  βˆ’ 1 . + π‘₯ 𝑀 + π‘₯ 𝑀 2 βˆ’ 1 . π‘₯ 𝑀 𝑏 2 0 βˆ’ 2 . π‘₯ 𝑀 + 1 . )
     + a r c t a n h ( ( 0 . 5 0 0 0 0 0 0 0 0 0 ( βˆ’ 2 . + π‘₯ 𝑀 𝑏 2 0  + 2 . π‘₯ 𝑀 ) ) / π‘₯ 𝑀 2 βˆ’ 1 . π‘₯ 𝑀 𝑏 2 0 βˆ’ 2 . π‘₯ 𝑀 + 1 )
     βˆ’ 1 . 𝑏 2 0 l n ( ( βˆ’ 1 . 𝑏 2 0 βˆ’ 1 . π‘₯ 𝑀 𝑏 2 0  + 2 . βˆ’ 1 . 𝑏 2 0  π‘₯ 𝑀 2 βˆ’ 1 . π‘₯ 𝑀 𝑏 2 0  βˆ’ 2 . π‘₯ 𝑀 + 1 . ) / ( π‘₯ 𝑀 βˆ’ 1 . ) ) / βˆ’ 1 . 𝑏 2 0 ) ) ;
# The results yielded by the formula obtained above and the enabled input data above:
                  𝑏 0 ∢ = 1 . 5
                  π‘₯ 𝑀 ∢ = βˆ’ 1
                  𝑐 ∢ = βˆ’ 0 . 1
               𝜎 ∢ = βˆ’ 0 . 1 3 3 3 7 3 6 7 8 0 βˆ’ 0 . 𝐼               ( 1)