RC Circuit Equations in the s-Domain

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1. What is the s-domain?

The s-domain is the domain after applying the Laplace Transform. It’s widely used in circuit analysis because it converts differential equations to algebraic equations, which are easier to handle.

The Laplace variable s is defined as:

$$s=\sigma+j\omega$$

2. RC Circuit Basics in Time Domain

Components:

• R = Resistance (Ohms)

• C = Capacitance (Farads)

Time-domain differential equation:

For a series RC circuit with input V_in and output across the capacitor V_out:

$$V_{in}(t)=V_{R}(t)+V_{C}(t) \\ \\ V_{R}(t)=R.i(t)\;\;and\;\;V_{C}(t)=\frac{1}{C}\int_{}^{}i(t)dt$$

Using:

$$i(t)=C\frac{dV_{C}(t)}{dt}$$

We get:

$$V_{in}=RC\frac{dV_{C}(t) }{dt}+V_{C}(t)$$

3. Laplace Transform of RC Circuit

Apply Laplace Transform:

$$L\left\{ V_{in} \right\}=L\left\{ RC\frac{dV_{C}(t) }{dt}+V_{C}(t) \right\}=RC\times L\left\{ \frac{d V_{C}(t)}{dt} \right\}+L\left\{ V_{C}(t) \right\}$$

So:

$$V_{in}(s)=RC\left( sV_{C}(s)-V_{C}(0) \right)+V_{C}(s)$$

If initial capacitor voltage 𝑉_𝐶(0) = 0 :

$$V_{in}(s)=(RCs+1)V_{C}(s)$$

Transfer Function:

$$H(s)=\frac{V_{out}(s)}{V_{in}(s)}=\frac{1}{RCs+1}$$

4. Solving RC Circuit in s-domain

Given any input V_in(s), the output is:

$$V_{out}(s)=H(s)V_{in}(s)$$

Use inverse Laplace to get 𝑉_𝑜𝑢𝑡( 𝑡 ).

5. Python Implementation (Symbolic & Numeric)

We’ll use SymPy for symbolic analysis and SciPy/Matplotlib for numerical/graphical output.

import sympy as sp

# Define symbols
s, R, C = sp.symbols('s R C', positive=True)
Vin = sp.Function('Vin')(s)
Vout = sp.Function('Vout')(s)

# Transfer function H(s)
H = 1 / (1 + R * C * s)

# Output in s-domain
Vout = H * Vin
Vout

Output

$$\frac{V_{in}(s)}{RCs+1}$$

6. Example : Step Response of RC Circuit

Input:

$$V_{in}(t)=u(t)\Rightarrow u(t)=\frac{1}{s}$$

Output:

$$V_{out}(s)=\frac{1}{RCs+1}\times \frac{1}{s}=\frac{1}{s(RCs+1)}$$

Inverse Laplace:

Use partial fractions:

$$\frac{1}{s(RCs+1)}=\frac{1}{s}-\frac{1}{s+\frac{1}{RC}}\Rightarrow V_{out}(t)=1-e^{-\frac{t}{RC}} \;\;for \; \;t\ge 0$$

Python Code for Step Response (Numeric)

import numpy as np
import matplotlib.pyplot as plt

# RC values
R = 1e3  # 1k Ohm
C = 1e-6 # 1uF
RC = R * C

# Time vector
t = np.linspace(0, 0.01, 1000)  # 0 to 10ms

# Step response
v_out = 1 - np.exp(-t / RC)

# Plot
plt.plot(t * 1000, v_out, label="Vout(t)")
plt.xlabel('Time (ms)')
plt.ylabel('Voltage (V)')
plt.title('Step Response of RC Circuit')
plt.grid(True)
plt.legend()
plt.show()

Output

Keywords

RC circuit, s-domain, Laplace transform, transfer function, step response, sinusoidal input, time domain, frequency domain, capacitor voltage, resistor current, symbolic math, SymPy, SciPy, Python simulation, inverse Laplace, first-order circuit, circuit analysis, transient response, electrical engineering, control systems, nerd cafe