International Scholarly Research Notices

Research Article

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

Algorithm 3

Program for calculating the normal component of the field strength versus the abscissa π‘₯ , the origin of which is on the slot axis.
#   The symbols which do not occur in Algorithm 1, are explained below.
# 𝑀 π‘˜ β€”variable symbol in integral for allowing certain transformations.
# 𝑒 β€”abscissa of point 𝐢 𝑀 in the 𝑀 -plane.
#   assume ( 0 < π‘Ž < 1 < 𝑏 < ∞ ) ;
# 𝐻 π‘Ž π‘₯ β€”tangential component of the field strength.
# 𝐻 π‘Ž 𝑦 β€”normal component of the field strength.
#   Warning. In order to keep the usual notation for the components of the field strength we have used the indexed names 𝐻 π‘₯ and 𝐻 𝑦 on the figures, but in the program we are obliged to modify the indices by putting, for example, 𝐻 π‘Ž π‘₯ and 𝐻 π‘Ž 𝑦 as components along the axes, otherwise, if in the program the ordinate 𝑦 also exists, errors appear, and the obtained values have no more meaning. For avoiding this circumstance, it is also possible to use only symbols.
π‘Ž ∢ = 0 . 2 5 ;
𝑏 ∢ = 4 .;
𝑏 0 √ ∢ = ( 𝑏 βˆ’ 1 ) / 𝑏 ;
𝐻 𝑦 0 ∢ = 1 ;
𝐹 ∢ = 𝐩 𝐫 𝐨 𝐜 ( 𝑠 )
g e n e r a l 𝑧 , 𝑠 ;
#   The transformation function is:
 𝑓 ∢ = ( 1 / πœ‹ ) β‹… 𝑀 π‘˜ 2 βˆ’ ( 𝑏 2 0 + 2 . ) 𝑀 π‘˜ + 1 . / ( 𝑀 π‘˜ β‹… ( 𝑀 π‘˜ βˆ’ 1 . ) ) ;
𝑧 ∢ = i n t ( 𝑓 , 𝑀 π‘˜ ) ;
𝑧 ∢ = e v a l ( 𝑧 , 𝑀 π‘˜ = 𝑠 ) + 0 . 5 𝐼 ;
#   The last imaginary constant has been introduced for to be in accordance with the configuration of Figure 1.
end proc;
𝑒 ∢ = 𝐹 ( βˆ’ 1 ) ;
𝑒 ∢ = e v a l ( 𝑒 ) ;
#   Co-ordinate introduced for to have the value zero of the π‘₯ -abscissa at the point of the minimum of 𝐻 π‘Ž 𝑦 .
for   𝑀   fromβ€‰β€‰βˆ’0.01  byβ€‰β€‰βˆ’0.25  toβ€‰β€‰βˆ’1.25  do
p r i n ( 𝑀 ) ;
𝑧 ∢ = 𝐹 ( 𝑀 ) ;
π‘₯ ∢ = β„œ ( 𝑧 βˆ’ 𝑒 ) ;
𝑦 ∢ = 𝔍 ( 𝑧 ) ;
𝐻 π‘Ž π‘₯  ∢ = β„œ ( c o n j u g a t e ( ( ( 𝑀 βˆ’ 1 . ) / 𝑀 2 βˆ’ ( 𝑏 2 0 + 2 . ) 𝑀 + 1 . ) 𝐼 ) ) ;
𝐻 π‘Ž 𝑦  ∢ = βˆ’ β„‘ ( c o n j u g a t e ( ( ( 𝑀 βˆ’ 1 . ) / 𝑀 2 βˆ’ ( 𝑏 2 0 + 2 . ) 𝑀 + 1 . ) 𝐼 ) ) ;
#   The sign minus has been introduced because, according to the assumed conditions in 𝑑 -plane (zeta plane), the field strength is up-down.
end do;
# p l o t ( [ π‘₯ , 𝐻 π‘Ž 𝑦 , 𝑀 = βˆ’ 1 . . βˆ’ 1 0 0 ] ) ;
# p l o t ( [ π‘₯ , 𝐻 π‘Ž 𝑦 , 𝑀 = βˆ’ 1 0 0 . . βˆ’ 1 0 0 0 ] ) ;
# p l o t ( [ π‘₯ , 𝐻 π‘Ž 𝑦 , 𝑀 = βˆ’ 1 0 0 0 . . βˆ’ 1 0 0 0 0 0 ] ) ;
# p l o t ( [ π‘₯ , 𝐻 π‘Ž 𝑦 , 𝑀 = βˆ’ 1 . . βˆ’ 0 . 0 1 ] ) ;
# p l o t ( [ π‘₯ , 𝐻 π‘Ž 𝑦 , 𝑀 = βˆ’ 0 . 0 1 . . βˆ’ 0 . 0 0 1 ] ) ;
# p l o t ( [ π‘₯ , 𝐻 π‘Ž 𝑦 , 𝑀 = βˆ’ 0 . 0 0 1 . . βˆ’ 0 . 0 0 0 0 1 ] ) ;
#   For plotting by using the same program, it suffices to disable the lines for w from…till  do
by putting # and enabling the plot statements by removing # .
#   Results after running the program:
               π‘Ž ∢ = 0 . 2 5
               𝑏 ∢ = 4 .
               𝑏 0 ∢ = 1 . 5 0 0 0 0 0 0 0 0
               𝐻 𝑦 0 ∢ = 1
Warning,β€‰β€‰β€œf”  is  implicitly  declared  local  to  procedureβ€‰β€‰β€œF”
Warning,β€‰β€‰β€œz”  is  implicitly  declared  local  to  procedureβ€‰β€‰β€œF”
𝐹 ∢ = 𝐩 𝐫 𝐨 𝐜 ( 𝑠 )
 local 𝑓 , 𝑧 ;
 general* 𝑧 , 𝑠 ;
  𝑓 ∢ = s q r t ( 𝑀 π‘˜ 2 βˆ’ ( 𝑏 2 0 + 2 . ) βˆ— 𝑀 π‘˜ + 1 . ) / ( πœ‹ βˆ— 𝑀 π‘˜ βˆ— ( 𝑀 π‘˜ βˆ’ 1 . ) ) ;
  𝑧 ∢ = i n t ( 𝑓 , 𝑀 π‘˜ ) ;
  𝑧 ∢ = e v a l ( 𝑧 , 𝑀 π‘˜ = 𝑠 ) + 0 . 5 βˆ— 𝐼
end proc;
         𝑒 ∢ = 0 . 4 2 0 7 2 7 9 5 1 1 + 1 . 0 0 0 0 0 0 0 0 0 𝐼
         𝑒 ∢ = 0 . 4 2 0 7 2 7 9 5 1 1 + 1 . 0 0 0 0 0 0 0 0 0 𝐼
              p r i n ( βˆ’ 0 . 0 1 )
         𝑧 ∢ = 2 . 0 4 8 2 5 5 5 8 9 + 1 . 0 0 0 0 0 0 0 0 0 𝐼
             π‘₯ ∢ = 1 . 6 2 7 5 2 7 6 3 8
            𝑦 ∢ = 1 . 0 0 0 0 0 0 0 0 0
              𝐻 π‘Ž π‘₯ ∢ = 0 .
           𝐻 π‘Ž 𝑦 ∢ = βˆ’ 0 . 9 8 9 1 5 0 8 1 5 6
             p r i n ( βˆ’ 0 . 2 6 )
         𝑧 ∢ = 0 . 9 4 4 1 0 6 3 1 0 5 + 1 . 0 0 0 0 0 0 0 0 0 𝐼
            π‘₯ ∢ = 0 . 5 2 3 3 7 8 3 5 9 4
            𝑦 ∢ = 1 . 0 0 0 0 0 0 0 0 0
               𝐻 π‘Ž π‘₯ ∢ = 0 .
            𝐻 π‘Ž 𝑦 ∢ = βˆ’ 0 . 8 5 4 8 3 1 7 7 5 8
              p r i n ( βˆ’ 0 . 5 1 )
         𝑧 ∢ = 0 . 6 8 6 8 8 9 9 8 2 9 + 1 . 0 0 0 0 0 0 0 0 0 𝐼
            π‘₯ ∢ = 0 . 2 6 6 1 6 2 0 3 1 8
            𝑦 ∢ = 1 . 0 0 0 0 0 0 0 0 0
              𝐻 π‘Ž π‘₯ ∢ = 0 .
           𝐻 π‘Ž 𝑦 ∢ = βˆ’ 0 . 8 1 5 6 0 8 7 5 7 4
              p r i n ( βˆ’ 0 . 7 6 )
         𝑧 ∢ = 0 . 5 2 9 8 0 0 3 3 9 7 + 1 . 0 0 0 0 0 0 0 0 0 𝐼
            π‘₯ ∢ = 0 . 1 0 9 0 7 2 3 8 8 6
            𝑦 ∢ = 1 . 0 0 0 0 0 0 0 0 0
              𝐻 π‘Ž π‘₯ ∢ = 0 .
           𝐻 π‘Ž 𝑦 ∢ = βˆ’ 0 . 8 0 2 6 9 1 2 0 5 1
             p r i n ( βˆ’ 1 . 0 1 )
         𝑧 ∢ = 0 . 4 1 6 7 6 8 8 4 5 6 + 1 . 0 0 0 0 0 0 0 0 0 𝐼
           π‘₯ ∢ = βˆ’ 0 . 0 0 3 9 5 9 1 0 5 5
            𝑦 ∢ = 1 . 0 0 0 0 0 0 0 0 0
              𝐻 π‘Ž π‘₯ ∢ = 0 .
           𝐻 π‘Ž 𝑦 ∢ = βˆ’ 0 . 8 0 0 0 0 3 5 6 4 3           ( 1)