RL Circuit Equations in the s-Domain

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1. What Is an RL Circuit?

An RL circuit consists of a resistor (R) and an inductor (L) connected either in series or in parallel. These components resist and store energy in different ways.

2. Time-Domain Equations of an RL Series Circuit

Kirchhoff’s Voltage Law (KVL):

For a series RL circuit with input voltage v(t), resistor R, and inductor L:

$$v(t)=v_{R}(t)+v_{L}(t)$$

Using Ohm’s law:

$$v(t)=Ri(t)+L\frac{d i(t)}{dt}$$

3. Transform to s-Domain (Laplace Transform)

Laplace Transform Rules:

$$L\left\{ v(t) \right\}=L\left\{ Ri(t)+L\frac{d i(t)}{dt} \right\}$$

So,

$$V_{s}=RI(s)+L\left( sI(s)-i(0^{-}) \right)=I(s)\left( Ls+1 \right)-Li(0^{-})$$

If the initial current 𝑖 (0⁻) =0, this simplifies to:

$$V(s)=(R+sL)I(s)$$

Or:

$$I(s)=\frac{V(s)}{R+sL}$$

4. Example Problem (Step Input)

Let:

• R=10 Ω

• L=1 HL

• v(t)=5⋅u(t) (step function)

• i(0)=0

Laplace of 𝑣(𝑡) = 5𝑢( 𝑡 ):

$$V(s)=\frac{5}{s}$$

Solve for 𝐼(𝑠):

$$I(s)=\frac{\frac{5}{s}}{10+s.1}=\frac{5}{s\left( 10+s \right)}$$

Partial Fraction Decomposition:

$$\frac{5}{s\left( 10+s \right)}=\frac{A}{s}+\frac{B}{s+10}$$

Multiply both sides:

$$5=A(s+10)+Bs=(A+B)s+10A$$

Solve:

$$if\;\;s=0\Rightarrow A=0.5\\ if\;\;s=-10\Rightarrow B=-0.5$$

So:

$$i(t)=\frac{1}{2}-\frac{1}{2}e^{-10t}$$

5. Python Code Example

We’ll use sympy for symbolic math and matplotlib for plotting.

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

# Define symbols
t, s = sp.symbols('t s')
R = 10
L = 1
V_s = 5 / s  # Laplace of 5 * u(t)

# I(s) = V(s) / (R + sL)
I_s = V_s / (R + s * L)
I_s = sp.simplify(I_s)

# Inverse Laplace to get i(t)
i_t = sp.inverse_laplace_transform(I_s, s, t)
print("Current i(t):")
sp.pprint(i_t)

# Convert to numerical function
i_t_func = sp.lambdify(t, i_t, 'numpy')

# Plotting
time_vals = np.linspace(0, 1, 1000)
current_vals = i_t_func(time_vals)

plt.figure(figsize=(8, 4))
plt.plot(time_vals, current_vals, label='i(t)', color='blue')
plt.title("Current Response in RL Circuit (Step Input)")
plt.xlabel("Time (s)")
plt.ylabel("Current (A)")
plt.grid(True)
plt.legend()
plt.show()

Output

Kewords

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