Description

Jineeth N [EE20B051]
1 Introduction
This assignemnt involves the analysis of filters using Laplace Transforms. Python’s symbolic solving library, Sympy is used to solve Modified Nodal Analysis equations. Scipy library is used to calculate output responses of the systems
Low Pass Filter
The low pass filter that we use gives the following matrix equation after simplification of the modified nodal equations.

Below is the code for Low Pass Filter
def lowpass(R1, R2, C1, C2, G, Vi):
s = symbols(“s”)
# Creating the matrices and solving them to get the output voltage. A = Matrix( [
[0, 0, 1, -1 / G],
[-1 / (1 + s * R2 * C2), 1, 0, 0],
[0, -G, G, 1],
[-1 / R1 – 1 / R2 – s * C1, 1 / R2, 0, s * C1],
]
)
b = Matrix([0, 0, 0, -Vi / R1]) V = A.inv() * b return A, b, V

Figure 1: Magnitude response of Low Pass filter
High Pass Filter
The high pass filter we use gives the following matrix equations after simplification of the modified nodal equations

Below is the code for High Pass Filter
def highpass(R1, R3, C1, C2, G, Vi):
s = symbols(“s”)
# Creating the matrices and solving them to get the output voltage. A = Matrix( [
[0, -1, 0, 1 / G],
[s * C2 * R3 / (s * C2 * R3 + 1), 0, -1, 0],
[0, G, -G, 1],
[-s * C2 – 1 / R1 – s * C1, 0, s * C2, 1 / R1],
]
)
b = Matrix([0, 0, 0, -Vi * s * C1]) V = A.inv() * b return A, b, V

Figure 2: Magnitude response of High Pass filter
2 Tasks
2.1 Step response Low Pass Filter
Output Voltage for unit step function as an input
# These lines of code will calculate the step response for the lowpass filter.
A1, b1, V1 = lowpass(10000, 10000, 1e-9, 1e-9, 1.586, 1 / s)
Vo1 = V1[3] H1 = sympyToCoeff(Vo1) t, y1 = sp.impulse(H1, None, p.linspace(0, 5e-3, 10000))
# The plot for step response of a lowpass filter. plt.figure(1) plt.plot(t, y1) plt.title(r”Step Response for low pass filter”, fontsize=12) plt.xlabel(r”$t$”, fontsize=10) plt.ylabel(r”$V_o(t)$”, fontsize=10) plt.grid(True)

Figure 3: Step response of Low Pass filter
2.2 Response to sum of sinusoids
Input function is,

Output response for the Low Pass filter can be found as shown in the code snippet.
# The response is also calculated for sum of sinusoids.
a = 2e3 b = 2e6 vi = p.sin(a * PI * t) + p.cos(b * PI * t) t, y2, svec = sp.lsim(H, vi, t)
# The plot for output response for sum of sinusoids of a lowpass filter. plt.figure(2) plt.plot(t, y2)
plt.title(r”Output voltage for sum of sinusoids”, fontsize=12) plt.xlabel(r”$t$”, fontsize=10)
plt.ylabel(r”$V_o(t)$”, fontsize=10) plt.grid(True)

Figure 4: Output Voltage for sum of sinusoids
2.3 Response to damped sinusoid
In this case we assign the input voltage as a damped sinusoid like, Low frequency,
Vi(t) = e−500t( cos(2000πt) )uo(t) V
High frequency,

2.3.1 High Pass filter response
Below is the code for calculating output of High Pass filter for damped sin functions
# The below piece of code will calculate the transfer function for the highpass filter.
A3, b3, V3 = highpass(10000, 10000, 1e-9, 1e-9, 1.586, 1)
Vo3 = V3[3]
H3 = sympyToCoeff(Vo3) hf3 = lambdify(s, Vo3, “numpy”) v3 = hf3(ss)
Output for Low and High frequency inputs

Figure 5: Output for Low frequency sinusoid

Figure 6: Output for High frequency sinusoid
2.3.2 Low Pass filter response
Below is the code for calculating output of Low Pass filter for damped sin functions
A4, b4, V4 = lowpass(10000, 10000, 1e-9, 1e-9, 1.586, 1)
Vo4 = V4[3] H4 = sympyToCoeff(Vo4) hf3 = lambdify(s, Vo4, “numpy”) v3 = hf3(ss)

Figure 7: Output for Low frequency sinusoid

Figure 8: Output for High frequency sinusoid
2.4 Step Response of High Pass filter
Step response can be calculated by substituting, V i(s) = 1/s.
A5, b5, V5 = highpass(10000, 10000, 1e-9, 1e-9, 1.586, 1 / s)
Vo5 = V5[3] H5 = sympyToCoeff(Vo5) t, y5 = sp.impulse(H5, None, p.linspace(0, 5e-3, 10000))

Figure 9: Output for High frequency sinusoid
3 Conclusion
The sympy module has allowed us to analyse circuits by analytically solving their node equations. And we used signal toolbox in python to find the time domain responses of the frequency responses