1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
|
#!/usr/bin/perl
use strict;
use warnings;
# Perl solving a physics / electrodynamics problem involving
# symbolic mathematics, derivatives and complex numbers:
use lib '../lib';
use Math::Symbolic qw/:all/;
use Math::Complex;
# Given the following simple circuit:
#
# ----|||||-----/\/\/\---- (R = resistor,
# | R L | L = solenoid,
# | | U = alternating voltage)
# ---------O ~ O----------
# U(t)
#
# Question: What's the current in this circuit?
#
# We'll need some physics before letting the computer do the
# math:
# Applying Kirchhoff's rules, one quickly ends up with the
# following differential equation for the current:
# (L * dI/dt) + (R * I) = U
my $left = parse_from_string('L * total_derivative(I(t), t) + R * I(t)');
my $right = parse_from_string('U(t)');
# If we understand current and voltage to be complex functions,
# we'll be able to derive. ("'" denoting complex here)
# I'(t) = I'_max * e^(i*omega*t)
# U'(t) = U_max * e^(i*omega*t)
# (Please note that omega is the frequency of the alternating voltage.
# For example, the voltage from German outlets has a frequency of 50Hz.)
my $argument = parse_from_string('e^(i*omega*t)');
my $current = parse_from_string('I_max') * $argument;
my $voltage = parse_from_string('U_max') * $argument;
# Putting it into the equation:
$left->implement( I => $current );
$right->implement( U => $voltage );
$left = $left->apply_derivatives()->simplify();
# Now, we can solve the equation to get a complex function for
# the current:
$left /= $argument;
$right /= $argument;
my $quotient = parse_from_string('R + i*omega*L');
$left /= $quotient;
$right /= $quotient;
# Now we have:
# $left = $right
# I_max(t) = U_max / (R + i*omega*L)
# But I_max(t) is still complex and so is the right-hand-side of the
# equation!
# Making the symbolic i a "literal" Math::Complex i
$right->implement(
e => Math::Symbolic::Constant->euler(),
i => Math::Symbolic::Constant->new(i), # Math::Complex magic
);
print <<'HERE';
Sample of complex maximum current with the following values:
U_max => 100
R => 10
L => 10
omega => 1
HERE
print "Computed to: "
. $right->value(
U_max => 100,
R => 10,
L => 10,
omega => 1,
),
"\n\n";
# Now, we're dealing with alternating current and voltage.
# So let's make a generator that generates nice current
# functions of time!
# I(t) = Re(I_max(t)) * cos(omega*t - phase);
# Usage: generate_current(U_Max, R, L, omega, phase)
sub generate_current {
my $current = $right->new(); # cloning
$current *= parse_from_string('cos(omega*t - phase)');
$current->implement(
U_max => $_[0],
R => $_[1],
L => $_[2],
omega => $_[3],
phase => $_[4],
);
$current = $current->simplify();
return sub { Re( $current->value( t => $_[0] ) ) };
}
print "Sample current function with: 230V, 2Ohms, 0.1H, 50Hz, PI/4\n";
my $current_of_time = generate_current( 230, 2, 0.1, 50, PI / 4 );
print "The current at 0 seconds: " . $current_of_time->(0) . "\n";
print "The current at 0.1 seconds: " . $current_of_time->(0.1) . "\n";
print "The current at 1 second: " . $current_of_time->(1) . "\n";
|
