I recently had a problem with code that had worked last semester.
the old code was
$fac1 = Formula("$mag*e^($phi*j)/(s-$w*j)")->reduce;
the entire problem follows
$fac2 = Formula("$mag*e^(-$phi*j)/(s+$w*j)")->reduce;
$fac3 = Formula("$c3/(s-$r3)")->reduce;
$fac3 = Formula("$c3/(s-$r3)")->reduce;
This resulted in
$fac1 = Formula("$mag*e^($phi*j)/(s-$w*j)")->reduce;
was dropping the j in the exponent
but it didn't happen in the second term
$fac2 = Formula("$mag*e^(-$phi*j)/(s+$w*j)")->reduce;
by experiment I found that using exp(.) worked but I don't understand why
the code now reads
$fac1 = Formula("$mag*exp($phi*j)/(s-$w*j)")->reduce;
$fac2 = Formula("$mag*exp(-$phi*j)/(s+$w*j)")->reduce;
$fac3 = Formula("$c3/(s-$r3)")->reduce;
$fac2 = Formula("$mag*exp(-$phi*j)/(s+$w*j)")->reduce;
$fac3 = Formula("$c3/(s-$r3)")->reduce;
any help understanding this?
the entire problem follows
# DESCRIPTION
# Problem from 'Mathematics: The Language of Electrical and Computer Engineering', Viniotis and Trussell, 3rd ed.
# WeBWorK problem written by Joel Trussell, <hjt@ncsu.edu>
# ENDDESCRIPTION
## DBsubject(Electrical Engineering)
## DBchapter(Laplace Transforms)
## DBsection(Problems)
## Institution(North Carolina State University)
## Author(H. J. Trussell)
## TitleText1('Mathematics: The Language of Electrical and Computer Engineering')
## AuthorText1('Viniotis and Trussell')
## EditionText1('3')
## Problem1('8.16-21')
##############################
# Initialization
DOCUMENT();
loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"AnswerFormatHelp.pl",
"unorderedAnswer.pl",
"parserMultiAnswer.pl",
"parserFunction.pl",
"unionLists.pl",
"answerHints.pl",
"PGcourse.pl",
);
TEXT(beginproblem());
#############################
# Setup
Context("Complex");
Context()->variables->are(
s=>"Real",
t=>"Real"
);
# Redefine the Heaviside function so the answer checker will take either 1 or #h(t).
#parserFunction("h(t)" => "1");
# skip above and use step - need to assign it
## required to use step function
Context()->functions->add(
step => {
class => 'Parser::Legacy::Numeric',
perl => 'Parser::Legacy::Numeric::do_step'
},
);
parserFunction("u(t)" => "step(t)");
Context()->flags->set(
tolerance => 0.001,
tolType => "absolute",
);
# create differential equation with roots below
# complex roots for the input, real for the system- 1st order
# this means swapping r3 and (r1 and r2)
# force pure cosine f0r this problem
# PFE of c1/(s+r1) + c2/(s-r2) + c3/(s-r3)
$a = 0; # real part - force pure cosine
$w = random(1,6,1); # imaginary part - frequency
$r1 = $a + $w*i;
$r2 = $a - $w*i; #complex conjugate
$r3 = -random(1,9,1); ; # third root - for system
$A = non_zero_random(-6,6,1); # magnitude of input cosine
# c1 and c2 are complex conjugates
# since magnitude and phase are not easy to determine as integers
# let's just work the problem directly
# solve for coefficients of diffEq y'-r3*y = A y_s
# where Y_s(s) = 2*A*s/(s^2+w^2)
# need to solve for A, y(0)
$y0 = random(1,6,1);
# constants for Partial Fraction
$b2 = $y0;
$b1 = $A;
$b0 = $y0*$w**2;
$wsqr = $w**2;
$frac = Formula(" ($b2 * s^2 + $b1*s + $b0 )/[(s-$r3)*(s^2 + $wsqr)] ")->reduce;
# compute c3 by usual PFE method
$c3 = ($b2*(($r3)**2) + $b1*$r3 + $b0 )/(($r3)**2 + $w**2);
$c2 = ($b2*(($w*i)**2) + $b1*$w*i + $b0)/(2*$w*i *($w*i-$r3));
$wsq = (($w*i)**2);
$b1wi = $b1*$w*i;
$wi2 = 2*$w*i;
$wimr3 = ($w*i-$r3);
$mag = abs($c2);
#$phi = arg($c2);
#$c2re = Re($c2);
$c2conj = conj($c2);
$c2im = -($c2-$c2conj)*i;
$c2re = $c2+$c2conj;
# compute phi in range [-pi,pi]
if ($c2re > 0 ) {$phi = atan($c2im/$c2re);}
elsif ($c2im > 0) {$phi = atan($c2im/$c2re) + pi;}
else {$phi = atan($c2im/$c2re) - pi;}
$fac1 = Formula("$mag*exp($phi*j)/(s-$w*j)")->reduce;
$fac2 = Formula("$mag*exp(-$phi*j)/(s+$w*j)")->reduce;
$fac3 = Formula("$c3/(s-$r3)")->reduce;
#$fac1 = Formula("$mag*e^($phi*j)/(s-$w*j)")->reduce;
$ft = Formula("(2*$mag*cos($w*t+$phi) + $c3*e^($r3*t))*u(t)")->reduce;
#############################
# Main text
Context()->texStrings;
BEGIN_TEXT
This problem is related to Problems 8.16-21 in the text.
$BR mag = $mag, phi = $phi
$PAR
Consider the differential equation \( y'(t) - $r3 y(t) = $A cos( $w t) u(t) \),
$BR with initial condition \( y(0) = $y0,\)
$PAR
Find the Laplace transform of the solution \( Y(s). \) Write the solution as a single fraction in \( s \)
$BR
\( Y(s) = \) \{ ans_rule(60)\} \{ AnswerFormatHelp("formulas") \}
$PAR
END_TEXT
ANS( $frac-> cmp());
BEGIN_TEXT
Find the partial fraction decomposition of \( Y(s) \). Enter all factors as first order terms in \( s \), that is, all terms should be of the form \( \frac{c}{s-p} \), where \( c \) is a constant and the root \( p \) is a constant. Both \( c \) and \( p \) may be complex.
$BR
$BR
\( \displaystyle Y(s) = \) \{ ans_rule(20)\} \(+\) \{ ans_rule(20)\} \(+\) \{ ans_rule(20)\}
END_TEXT
UNORDERED_ANS( $fac1->cmp(), $fac2->cmp(), $fac3->cmp() );
BEGIN_TEXT
$PAR Find the inverse Laplace transform of \( \displaystyle Y(s) \). The solution must consist of all real terms. (Remeber to use \( u(t).\))
$BR
$BR
\( \displaystyle y(t) = {\mathcal L}^{-1} \left\lbrace Y(s) \right\rbrace = \)
\{ ans_rule(50) \}
\{ AnswerFormatHelp("formulas") \}
END_TEXT
Context()->normalStrings;
##############################
# Answer evaluation
$showPartialCorrectAnswers = 1;
# note that ALL answers in a problem must be unordered!!
# in this case, the terms are unordered and checked as such, but it is
# highly unlikely anyone would enter the last answer in the wrong place
# but so what if they did
ANS( $ft->cmp() );
ENDDOCUMENT();
# Problem from 'Mathematics: The Language of Electrical and Computer Engineering', Viniotis and Trussell, 3rd ed.
# WeBWorK problem written by Joel Trussell, <hjt@ncsu.edu>
# ENDDESCRIPTION
## DBsubject(Electrical Engineering)
## DBchapter(Laplace Transforms)
## DBsection(Problems)
## Institution(North Carolina State University)
## Author(H. J. Trussell)
## TitleText1('Mathematics: The Language of Electrical and Computer Engineering')
## AuthorText1('Viniotis and Trussell')
## EditionText1('3')
## Problem1('8.16-21')
##############################
# Initialization
DOCUMENT();
loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"AnswerFormatHelp.pl",
"unorderedAnswer.pl",
"parserMultiAnswer.pl",
"parserFunction.pl",
"unionLists.pl",
"answerHints.pl",
"PGcourse.pl",
);
TEXT(beginproblem());
#############################
# Setup
Context("Complex");
Context()->variables->are(
s=>"Real",
t=>"Real"
);
# Redefine the Heaviside function so the answer checker will take either 1 or #h(t).
#parserFunction("h(t)" => "1");
# skip above and use step - need to assign it
## required to use step function
Context()->functions->add(
step => {
class => 'Parser::Legacy::Numeric',
perl => 'Parser::Legacy::Numeric::do_step'
},
);
parserFunction("u(t)" => "step(t)");
Context()->flags->set(
tolerance => 0.001,
tolType => "absolute",
);
# create differential equation with roots below
# complex roots for the input, real for the system- 1st order
# this means swapping r3 and (r1 and r2)
# force pure cosine f0r this problem
# PFE of c1/(s+r1) + c2/(s-r2) + c3/(s-r3)
$a = 0; # real part - force pure cosine
$w = random(1,6,1); # imaginary part - frequency
$r1 = $a + $w*i;
$r2 = $a - $w*i; #complex conjugate
$r3 = -random(1,9,1); ; # third root - for system
$A = non_zero_random(-6,6,1); # magnitude of input cosine
# c1 and c2 are complex conjugates
# since magnitude and phase are not easy to determine as integers
# let's just work the problem directly
# solve for coefficients of diffEq y'-r3*y = A y_s
# where Y_s(s) = 2*A*s/(s^2+w^2)
# need to solve for A, y(0)
$y0 = random(1,6,1);
# constants for Partial Fraction
$b2 = $y0;
$b1 = $A;
$b0 = $y0*$w**2;
$wsqr = $w**2;
$frac = Formula(" ($b2 * s^2 + $b1*s + $b0 )/[(s-$r3)*(s^2 + $wsqr)] ")->reduce;
# compute c3 by usual PFE method
$c3 = ($b2*(($r3)**2) + $b1*$r3 + $b0 )/(($r3)**2 + $w**2);
$c2 = ($b2*(($w*i)**2) + $b1*$w*i + $b0)/(2*$w*i *($w*i-$r3));
$wsq = (($w*i)**2);
$b1wi = $b1*$w*i;
$wi2 = 2*$w*i;
$wimr3 = ($w*i-$r3);
$mag = abs($c2);
#$phi = arg($c2);
#$c2re = Re($c2);
$c2conj = conj($c2);
$c2im = -($c2-$c2conj)*i;
$c2re = $c2+$c2conj;
# compute phi in range [-pi,pi]
if ($c2re > 0 ) {$phi = atan($c2im/$c2re);}
elsif ($c2im > 0) {$phi = atan($c2im/$c2re) + pi;}
else {$phi = atan($c2im/$c2re) - pi;}
$fac1 = Formula("$mag*exp($phi*j)/(s-$w*j)")->reduce;
$fac2 = Formula("$mag*exp(-$phi*j)/(s+$w*j)")->reduce;
$fac3 = Formula("$c3/(s-$r3)")->reduce;
#$fac1 = Formula("$mag*e^($phi*j)/(s-$w*j)")->reduce;
#$fac2 = Formula("$mag*e^(-$phi*j)/(s+$w*j)")->reduce;
#$fac3 = Formula("$c3/(s-$r3)")->reduce;
#$fac3 = Formula("$c3/(s-$r3)")->reduce;
$ft = Formula("(2*$mag*cos($w*t+$phi) + $c3*e^($r3*t))*u(t)")->reduce;
#############################
# Main text
Context()->texStrings;
BEGIN_TEXT
This problem is related to Problems 8.16-21 in the text.
$BR mag = $mag, phi = $phi
$PAR
Consider the differential equation \( y'(t) - $r3 y(t) = $A cos( $w t) u(t) \),
$BR with initial condition \( y(0) = $y0,\)
$PAR
Find the Laplace transform of the solution \( Y(s). \) Write the solution as a single fraction in \( s \)
$BR
\( Y(s) = \) \{ ans_rule(60)\} \{ AnswerFormatHelp("formulas") \}
$PAR
END_TEXT
ANS( $frac-> cmp());
BEGIN_TEXT
Find the partial fraction decomposition of \( Y(s) \). Enter all factors as first order terms in \( s \), that is, all terms should be of the form \( \frac{c}{s-p} \), where \( c \) is a constant and the root \( p \) is a constant. Both \( c \) and \( p \) may be complex.
$BR
$BR
\( \displaystyle Y(s) = \) \{ ans_rule(20)\} \(+\) \{ ans_rule(20)\} \(+\) \{ ans_rule(20)\}
END_TEXT
UNORDERED_ANS( $fac1->cmp(), $fac2->cmp(), $fac3->cmp() );
BEGIN_TEXT
$PAR Find the inverse Laplace transform of \( \displaystyle Y(s) \). The solution must consist of all real terms. (Remeber to use \( u(t).\))
$BR
$BR
\( \displaystyle y(t) = {\mathcal L}^{-1} \left\lbrace Y(s) \right\rbrace = \)
\{ ans_rule(50) \}
\{ AnswerFormatHelp("formulas") \}
END_TEXT
Context()->normalStrings;
##############################
# Answer evaluation
$showPartialCorrectAnswers = 1;
# note that ALL answers in a problem must be unordered!!
# in this case, the terms are unordered and checked as such, but it is
# highly unlikely anyone would enter the last answer in the wrong place
# but so what if they did
ANS( $ft->cmp() );
ENDDOCUMENT();