Series RLC Circuit with DC Source

Nerd Cafe

Circuit Description:

We have a series RLC circuit connected to a DC voltage source V₀​.

Initial conditions:

• V₀​: DC step voltage

• R: Resistance (Ω)

• L: Inductance (H)

• C: Capacitance (F)

s-Domain Circuit Representation

Apply KVL in the t-domain:

$$v_{0}=v_{R}+v_{L}+v_{C}=Ri(t)+L\frac{d i(t)}{dt}+\frac{1}{C}\int_{}^{}i(t)dt\\$$

Apply KVL in the s-domain:

$$\frac{V_{0}}{s}=RI(s)+LsI(s)+\frac{1}{sC}I(s)$$

Factor out 𝐼(𝑠):

$$\frac{V_{0}}{s}=\left( R+Ls+\frac{1}{sC} \right)I(s)$$

Solve for current 𝐼(𝑠):

$$I(s)=\frac{V_{0}}{s(R+Ls+\frac{1}{sC})}=\frac{V_{0}}{Ls^{2}+Rs+\frac{1}{C}}$$

4. Solving for Current in the s-Domain

Let’s define:

• a=L

• b=R

• c=1/C

Then:

$$I(s)=\frac{V_{0}}{as^{2}+bs+c}$$

Case Example:

Let’s assume:

• V0=10V

• R=2 Ω

• L=1 H

• C=0.25 F

Then:

$$I(s)=\frac{10}{s^{2}+2s+4}$$

This is a standard second-order system.

5. Inverse Laplace: Time-Domain Current

Use the known inverse Laplace pairs:

$$L^{-1}\left\{ \frac{\omega}{\left( s+\alpha \right)^{2}+\omega^{2}} \right\}=e^{-\alpha.t}sin\left( \omega.t \right)$$

We complete the square:

$$s^{2}+2s+4=(s+1)^{2}+3\Rightarrow \omega=\sqrt{3}\;\;and\;\;\alpha=1$$

So:

$$I(s)=\frac{10}{(s+1)^{2}+3}\Rightarrow i(t)=10e^{-t}\frac{sin\left( \sqrt{3}t \right)}{\sqrt{3}}$$

6. Python Simulation Using SymPy

Here’s the full Python code to simulate:

import sympy as sp
import matplotlib.pyplot as plt
import numpy as np

# Symbols
s, t = sp.symbols('s t')
V0 = 10
R = 2
L = 1
C = 0.25

# Define I(s)
Is = V0 / (L*s**2 + R*s + 1/C)

# Inverse Laplace Transform
it = sp.inverse_laplace_transform(Is, s, t)
it_simplified = sp.simplify(it)
print("i(t) =", it_simplified)

# Lambdify for plotting
i_func = sp.lambdify(t, it_simplified, modules=['numpy'])

# Plotting
time_vals = np.linspace(0, 10, 400)
current_vals = i_func(time_vals)

plt.figure(figsize=(10, 5))
plt.plot(time_vals, current_vals, label='i(t)', color='blue')
plt.title("Current Response i(t) in Series RLC Circuit with DC Source")
plt.xlabel("Time (s)")
plt.ylabel("Current (A)")
plt.grid(True)
plt.legend()
plt.show()

Output

Keywords

RLC circuit, series RLC, Laplace transform, s-domain analysis, DC source, transient response, differential equations, inverse Laplace, time domain, second-order system, underdamped circuit, damping factor, electrical engineering, current response, step input, Python simulation, SymPy, circuit analysis, signal processing, engineering math, nerd cafe