Program 1

Half Wave Rectifier with Capacitor Filter

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt

# Parameters
R = 5000  # Resistance in ohms
C = 5e-6  # Capacitance in farads
f = 50    # Frequency of AC supply in Hz
V_peak = 10  # Peak value of AC voltage in volts
t_max = 0.2 # Time duration for simulation in seconds
dt = 1e-5    # Time step for Euler method in seconds

# Number of steps
n_steps = int(t_max / dt)

# Time array
time = np.linspace(0, t_max, n_steps)

# Rectified input voltage
V_in = V_peak * np.abs(np.sin(2 * np.pi * f * time))

# Voltage across capacitor
V_cap = np.zeros(n_steps)

# Euler method for solving the ODE
for i in range(1, n_steps):
    if V_in[i] > V_cap[i-1]:  # Charging phase
        dV = (V_in[i] - V_cap[i-1]) / (R * C)
    else:  # Discharging phase
        dV = -V_cap[i-1] / (R * C)
    V_cap[i] = V_cap[i-1] + dV * dt

# Plotting the results
plt.figure(figsize=(10, 6))

plt.plot(time, V_in, label='Rectified Input Voltage (V_in)', color='orange')
plt.plot(time, V_cap, label='Capacitor Voltage (V_cap)', color='blue')

plt.title('Half-Wave Rectifier with Capacitor Filter')
plt.xlabel('Time (s)')
plt.ylabel('Voltage (V)')
plt.legend()
plt.grid(True)
plt.savefig('plot.png')
plt.show()

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Theory of Capacitor in Half-Wave Rectifier Circuit

1. Half-Wave Rectifier with Capacitor Filter

A half-wave rectifier converts an alternating current (AC) signal into a pulsating direct current (DC) signal by allowing only the positive half-cycles of the AC signal to pass through while blocking the negative half-cycles.

To smooth out the pulsating DC signal, a capacitor is placed across the load. The capacitor charges when the rectified voltage is higher than the voltage across it and discharges when the rectified voltage drops. This creates a more stable DC output.

2. Mathematical Model

The voltage V(t) across the capacitor is governed by different ordinary differential equations (ODEs) during the charging and discharging phases:

Charging Phase:

dV(t)/dt = (1/RC) * (V_in(t) - V(t))

This equation describes how the capacitor charges when the rectified input voltage V_in(t) is greater than the current capacitor voltage V(t).

Discharging Phase:

dV(t)/dt = -V(t)/(RC)

This equation describes the discharging process when V_in(t) is less than V(t), causing the capacitor to release its stored energy to maintain the load voltage.

Theoretical Principles & Computational Physics Guide

Half-Wave Rectifier with Capacitor Filter: Nonlinear AC Circuit Simulation

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

AC to DC Conversion: Exploits unidirectional diode conduction combined with capacitive charge storage to produce smooth DC voltage.

Piecewise Conduction: When $V_{\text{in}} > V_C + V_D$, diode conducts; when $V_{\text{in}}$ drops, the diode cuts off and the capacitor discharges through the load.

Ripple Voltage: Peak-to-peak ripple is inversely proportional to load resistance and filter capacitance: $V_r \approx V_m / (f R_L C)$.

2. Mathematical Formulation & Algorithmic Pipeline

Rectifier conduction criteria and peak-to-peak ripple formula:

$$ V_r \approx \frac{V_m}{f R_L C}, \quad V_{\text{out}}(t + \Delta t) = \begin{cases} V_{\text{in}}(t) - V_D, & V_{\text{in}} > V_C + V_D \ V_C(t) e^{-\Delta t / (R_L C)}, & \text{diode OFF} \end{cases} $$

3. Numerical Mechanics & Implementation Logic

Input Generation: Generate sinusoidal supply $V_{\text{in}}(t) = V_m \sin(\omega t)$.
Diode State Evaluation: If $V_{\text{in}} > V_{\text{out}} + V_D$, set $V_{\text{out}} = V_{\text{in}} - V_D$ (charging phase).
Discharge Phase: Else, update via exponential decay: $V_{\text{out}} \leftarrow V_{\text{out}}(1 - \Delta t / (R_L C))$.
Metric Evaluation: Calculate average DC voltage and ripple factor $\gamma = V_r / V_{\text{dc}}$.

4. Relevant Computational Practice Exercises

Exercise 1

Simulate rectifier output for filter capacitances $C = 10\,\mu\text{F}, 100\,\mu\text{F}, 1000\,\mu\text{F}$ and observe ripple reduction.

Exercise 2

Measure the ripple factor and compare with the theoretical prediction $V_r = V_m / (f R_L C)$.

Exercise 3

Extend the simulation to model a full-wave bridge rectifier topology.

Standard Literature & References:

Paul Horowitz and Winfield Hill, "The Art of Electronics", Cambridge University Press (2015).

Ch. 6.4 Half Wave Rectifier with Capacitor Filter