def laplace_transform_comparison(fTilde, F_of_s):
frequency_vector = [0.1, 1, 10, 100]
laplace_transform_fTilde = [float(0) for x in range(0,len(fTilde))]
for i in range(0,len(fTilde)):
laplace_transform_fTilde[i] = laplace_transform(fTilde[i], t, s)
print(laplace_transform_fTilde[0])
print(laplace_transform_fTilde[0].evalf(subs={s:1}))
Python Sympy is a package that has symbolic math functions.
A few of the notable ones that are useful for this material are the:
a. Laplace transform
b. Inverse Laplace transform
c. Partial Fraction expansion
d. Polynomial expansion
e. Polynomial roots
"""
import sympy as sym
from sympy.abc import s,t,x,y,z
from sympy.integrals import laplace_transform
from sympy.integrals import inverse_laplace_transform
# Laplace transform (t->s)
U = laplace_transform(5*t, t, s)
print('U')
print(U[0])
# Result: 5/s**2
# Inverse Laplace transform (s->t)
X = inverse_laplace_transform(U[0],s,t)
print('X')
print(X)
# Result: 5*t*Heaviside(t)
# Function
F = 5*(s+1)/(s+3)**2
print('F')
print(F)
# Result: (5*s + 5)/(s + 3)**2
#print('y1:',y1)
a = int(input('enter the coefficient of second deri of y:'))
b = int(input('enter the coefficient of first deri of y:'))
c = int(input('enter the coefficient of y:'))
f = (input('enter the RHS function:'))
d = int(input('enter the value of y(0):'))
e = int(input('enter the value of Dy(0):'))
f = sympify(f)
y1 = diff(y, t)
y2 = diff(y1, t)
eq = a * y2 + b * y1 + c * y - f
print('equation:', eq)
print(' ')
eqn = laplace_transform(eq, t, s)
print('after laplace equation:', eqn)
print(' ')
eqn = sympify(eqn)
#eqn=eqn.subs(t,0)
#eqn=eqn.subs(y(0),d)
print('after replacing:', eqn)
print(' ')
eqn = eqn.subs(sympify('LaplaceTransform(y(t), t, s)'), Y)
print('after replacing:', eqn)
print(' ')
eqn = eqn.subs(sympify(y1), e)
#%%
from sympy.integrals import laplace_transform
from sympy import *
from sympy.abc import t, s
Tau = 1.0
U = laplace_transform(Heaviside(t), t, s)
print(f"The Laplace transform of the unit step function, u(t) is:{U[0]}")
# Use U[0] becaie the result of the laplace_transform function is a tuple.
h = -1 / Tau * exp(-t / Tau)
print(f"The impulse response of a High Pass filter with Tau={Tau} is: {h}")
H = laplace_transform(h, t, s)
print(f"The Laplace transform of a High Pass filter with Tau={Tau} is: {h}")
Y = U[0] * H[0]
print(
f"The frequency response of a unit step input into a High Pass filter with Tau={Tau} is {Y}."
)
Y_PF = apart(Y)
print(
f"The partial fraction expansion of the frequency response of a unit step input into a High Pass filter with Tau={Tau} is: {Y_PF}"
)
y = inverse_laplace_transform(Y_PF, s, t).evalf().simplify()
print(f"Time domain step reponse of a High Pass filter with Tau={Tau} is {y}")
plot(y + 1, (t, 0, 5 * Tau),
title='Step Response High Pass',
xlabel='time(s)',
ylabel='Step response')
#%%
from sympy.integrals import laplace_transform
from sympy import *
def answers():
global Answ1, Answ2, Answ3, Answ4, Answ5
s, t, x, y, z = symbols('s t x y z')
A2.delete(0, END)
if E1.get() == '' or E1.get() == 0:
Answ1 = ''
else:
if E1.index(0) == '-':
Answ1 = sympify(laplace_transform(E1.get(), t, s, noconds=True))
sig = '-'
Answ1s = sig + Answ1
A2.insert(0, Answ1s)
else:
Answ1 = sympify(laplace_transform(E1.get(), t, s, noconds=True))
A2.insert(0, Answ1)
if E2.get() == '' or E2.get() == 0:
Answ2 = ''
else:
if E2.index(0) == '-':
Answ2 = sympify(laplace_transform(E2.get(), t, s, noconds=True))
sig = ' - '
Answ2s = sig + Answ2
A2.insert(END, Answ2s)
else:
Answ2 = sympify(laplace_transform(E2.get(), t, s, noconds=True))
A2.insert(END, ' + ')
A2.insert(END, Answ2)
if E3.get() == '' or E3.get() == 0:
Answ3 = ''
else:
if E3.index(0) == '-':
Answ3 = sympify(laplace_transform(E3.get(), t, s, noconds=True))
sig = ' - '
Answ3s = sig + Answ3
A2.insert(END, Answ3s)
else:
Answ3 = sympify(laplace_transform(E3.get(), t, s, noconds=True))
A2.insert(END, ' + ')
A2.insert(END, Answ3)
if E4.get() == '' or E4.get() == 0:
Answ4 = ''
else:
if E4.index(0) == '-':
Answ4 = sympify(laplace_transform(E4.get(), t, s, noconds=True))
sig = ' - '
Answ4s = sig + Answ4
A2.insert(END, Answ4s)
else:
Answ4 = sympify(laplace_transform(E4.get(), t, s, noconds=True))
A2.insert(END, ' + ')
A2.insert(END, Answ4)
if E5.get() == '' or E5.get() == 0:
Answ5 = ''
else:
if E5.index(0) == '-':
Answ5 = sympify(laplace_transform(E5.get(), t, s, noconds=True))
sig = ' - '
Answ5s = sig + Answ5
A2.insert(END, Answ5s)
else:
Answ5 = sympify(laplace_transform(E5.get(), t, s, noconds=True))
A2.insert(END, ' + ')
A2.insert(END, Answ5)
A2.config(font=("Helvetica", 15))
A2.config(state=DISABLED)
laplace.subs(s, s - a)
# In[15]:
import numpy as np
from sympy.integrals import laplace_transform
from sympy.abc import t, s, a, b
# In[16]:
f = cos(t)
listu = []
for i in np.arange(0, 8 * np.pi, 0.2):
listu.append(f.subs(t, i))
pt.plot(np.arange(0, 8 * np.pi, 0.2), listu, label="f(t) graph")
g = laplace_transform(f, t, s)
g = g[0]
listy = [] #g(s) values
for i in np.arange(0, 8 * np.pi, 0.2):
listy.append(g.subs(s, i))
pt.plot(np.arange(0, 8 * np.pi, 0.2), listy, label="f(s) graph")
pt.legend()
# In[17]:
f = exp(a * t) * sin(b * t)
#can solve this easily by first shifting property
#take f1
f1 = sin(b * t)
g1 = laplace_transform(f1, t, s)
g1 = g1[0]
def laplace_carson(f, t, s):
l = laplace_transform(f, t, s)
l_c = sy.simplify(s * l[0])
return l_c
from tool import * from sympy import * from sympy.abc import x, k, f from sympy.integrals import laplace_transform, fourier_transform f1 = fourier_transform(exp(-x**2), x, k) print(f1) f1 = laplace_transform(sin(x), x, f) print(f1) # help(laplace_transform) help(fourier_series)
a=[]
ti = np.linspace(0,30000,25)
#print ti
for i in range(0,25):
a.append(Heaviside(ti[i]))
#print a
#plotting input function
plt.figure(1)
plt.plot(ti, a)
plt.title('Input Function')
plt.xlabel('Time')
#plt.ylabel('')
plt.show()
Input_func_laplace = laplace_transform(Heaviside(t).rewrite(exp),t,s)
#print Input_func_laplace[0]
#transfer func as given in paper
Trans_func = k*((tau_p*s+1)/(tau_d*s+1))
Output_func_laplace = Input_func_laplace[0]*Trans_func
print Output_func_laplace
'''c=[]
plt.figure(3)
for i in range(0,25):
c.append(Output_func_laplace.subs(s,ti[i]))
plt.plot(ti,c)
plt.title('Laplace Function curve')
plt.show()'''