Half Wave Rectifier with Capacitor Filter: Program 1

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.

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')

Ordinary Differential Equation-1

Half Wave Rectifier with Capacitor Filter: Program 1

Theory of Capacitor in Half-Wave Rectifier Circuit

1. Half-Wave Rectifier with Capacitor Filter

2. Mathematical Model

Output 1