LC Circuit Equations in the s-Domain

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What is an LC Circuit?

An LC circuit (also known as a resonant or tank circuit) is a simple electrical circuit consisting of:

• an Inductor L (H)

• a Capacitor C (F)

It's used in:

• Oscillators

• Filters

• Tuned circuits (like radios)

Step 1: Time-Domain Equations of LC Circuit

Let’s start from the KVL (Kirchhoff's Voltage Law) for the series LC circuit:

$$v_{L}(t)+v_{C}(t)=0$$

Where:

$$V_{L}(t)=L\frac{d i(t)}{dt}$$

and

$$V_{C}(t)=\frac{1}{C}\int_{}^{}i(t)dt$$

Differentiate both sides:

$$L\frac{d^{2} i}{dt^{2}}+\frac{1}{C}i(t)=0$$

Step 2: Convert to s-domain using Laplace Transform

Apply the Laplace Transform assuming zero initial conditions:

$$L\left\{ L\frac{d^{2} i}{dt^{2}}+\frac{1}{C}i(t) \right\}=Ls^{2}I(s)+\frac{1}{C}I(s)=I(s)\left( Ls^{2}+\frac{1}{C} \right)=0$$

Final Equation in the s-domain:

$$Ls^{2}+\frac{1}{C}=0\Rightarrow s^{2}=-\frac{1}{LC}\Rightarrow s=\pm j\omega_{0}\;\;where\;\;\omega_{0}=\frac{1}{\sqrt{LC}}$$

Step 3: Python Implementation

Let’s use sympy to model this in Python.

Installation:

pip install sympy matplotlib

Python Code:

import sympy as sp

# Define symbols
s, L, C = sp.symbols('s L C', real=True, positive=True)

# Define the equation in s-domain
Z_total = L*s**2 + 1/C

# Solve for s
s_solutions = sp.solve(Z_total, s)
print("Roots of the LC circuit equation (s-domain):")
print(s_solutions)

# Resonant frequency
omega_0 = sp.sqrt(1/(L*C))
print("\nResonant Frequency ω₀ =")
sp.pprint(omega_0)

Output:

Roots of the LC circuit equation (s-domain):
[]

Resonant Frequency ω₀ =
  1  
─────
√C⋅√L

Step 4: Example with Real Values

Let's say:

• L=10 mH=10×10⁻³

• C=100 nF=100×10⁻⁹C

Python Code:

# Numerical evaluation
L_val = 10e-3  # 10 mH
C_val = 100e-9  # 100 nF

omega_0_num = omega_0.subs({L: L_val, C: C_val})
f_0 = omega_0_num / (2 * sp.pi)
print(f"\nResonant frequency (Hz): {f_0.evalf():.2f}")

Output

Resonant frequency (Hz): 5032.92

Keywords

LC circuit, s-domain, Laplace transform, inductor, capacitor, resonant frequency, transfer function, electrical engineering, differential equations, impedance, resonance, oscillation, symbolic math, Python, SymPy, frequency response, current waveform, undamped oscillation, time-domain analysis, circuit analysis, nerd cafe