Instructions: Answer all questions. Show all necessary steps.

Derive the expression for ripple for the circuit FWR with inductor filter.

The ripple factor is a measure of the AC component present in the rectified output. For a Full Wave Rectifier (FWR) with an inductor filter, the inductor is placed in series with the load to oppose the change in current, thereby smoothing the output voltage.

Consider a full-wave rectifier circuit with an inductor filter connected in series with the load resistance R_L. The output voltage of the FWR without filter is a series of half-sine waves. When an inductor (L) is added in series, it opposes the rapid changes in current. The output current and voltage will be smoother than without the filter.

The output voltage of an unfiltered FWR can be represented by a Fourier series:

For a full-wave rectified sine wave, the Fourier series is:

Where:

• V_dc = 2V_mπ is the DC component.
• V_m is the peak voltage of the AC input.
• ω₀ = 2πf is the fundamental angular frequency.

The AC component (ripple voltage) is given by:

The RMS value of the ripple voltage (V_r,rms) is calculated from the coefficients of the AC components:

The primary component of the ripple is the fundamental frequency (n=1). However, the formula for n=1 has a denominator of (1-1²)=0, which implies an infinite amplitude if the formula were strictly applied. This indicates that the simple Fourier series expansion of a perfect full-wave rectified sine wave does not accurately represent the behavior of a filtered circuit where the inductor limits the current. In a practical filtered circuit, the output is not a pure sine wave but a smoothed waveform.

For a practical inductor filter, the output voltage and current are approximated as having a DC component and a small AC ripple component. The ripple voltage (v_r) is the difference between the actual output voltage and the DC component.

The inductor's impedance is Z_L = jωL. At the ripple frequency, the inductor offers significant opposition to the AC component, reducing its amplitude. The load resistance is R_L. The total impedance to the ripple current is Z = R_L + jωL.

A common approximation for the ripple voltage in an FWR with an inductor filter assumes the output current is nearly constant DC with a small sinusoidal ripple. The ripple voltage across the load is then approximately the ripple current multiplied by the load resistance.

The RMS value of the ripple voltage (V_r,rms) is often approximated by considering only the fundamental component of the ripple, but more accurately derived by considering the impedance. A simplified approach for the ripple factor (γ) is based on the ratio of the RMS ripple voltage to the DC voltage:

For an inductor filter, the ripple factor is significantly reduced compared to an unfiltered rectifier. A common approximation for the ripple factor of an FWR with an inductor filter is:

Where ω is the fundamental ripple frequency (2ω₀ for FWR, so ω = 4πf). This formula highlights that the ripple factor decreases significantly with increasing inductor value (L) and load resistance (R_L). The term ωL is the inductive reactance at the ripple frequency.

A more detailed derivation involves analyzing the current waveform. Assuming the current is approximately sinusoidal with a DC offset, the ripple voltage across the load is the ripple current multiplied by R_L. The inductor's impedance at the ripple frequency is X_L = ωL. The total impedance to the ripple current is Z = sqrt(R_L² + X_L²). The ripple factor is then related to the ratio of R_L to Z.

The ripple factor for an FWR with an inductor filter is generally given by:

This simplified expression assumes that the inductive reactance ωL is much larger than R_L, which is the condition for effective filtering. The actual derivation involves complex analysis of the current and voltage waveforms under load conditions.

Instructions: Answer all questions. Show all necessary steps.

Explain the working of semiconductor photo diode.

A semiconductor photodiode is a semiconductor device that converts light into an electrical current. It is essentially a p-n junction diode that is specifically designed to be sensitive to light. When photons with sufficient energy strike the photodiode, they create electron-hole pairs in the depletion region, which are then swept across the junction by the electric field, generating a current.

Construction:

A photodiode is typically constructed from semiconductor materials like silicon (Si) or germanium (Ge). It consists of a p-n junction formed by doping a semiconductor material. The junction is usually exposed to light through a transparent window or a lens. The device has two terminals, an anode (+) connected to the p-type material and a cathode (-) connected to the n-type material.

Working Principle:

The operation of a photodiode can be understood in three modes:

1. Photovoltaic Mode (Zero Bias):
   When no external voltage is applied, the photodiode acts like a solar cell. Incident photons generate electron-hole pairs in the depletion region. The built-in electric field across the depletion region separates these charge carriers, with electrons moving to the n-side and holes to the p-side. This separation creates a small voltage across the diode, known as the photovoltage. If an external circuit is connected, a current flows.
2. Photoconductive Mode (Reverse Bias):
   This is the most common mode for detection. The photodiode is operated under reverse bias. In this state, the depletion region widens, and the electric field across it becomes stronger. When photons strike the depletion region, they generate electron-hole pairs. The strong electric field quickly sweeps these carriers across the junction, resulting in a reverse current called the photocurrent. This photocurrent is directly proportional to the intensity of the incident light. Even in complete darkness, a small reverse current, known as the dark current, flows due to thermally generated carriers. The total reverse current is the sum of the photocurrent and the dark current. The photoconductive mode offers faster response times and a wider linear range compared to the photovoltaic mode.
3. Avalanche Mode (Strong Reverse Bias):
   When the reverse bias voltage is increased to a very high level, close to the breakdown voltage, the photodiode enters the avalanche mode. The strong electric field accelerates the generated carriers to high energies. These energetic carriers collide with the semiconductor lattice, generating more electron-hole pairs through impact ionization. This process leads to an avalanche effect, where a small number of initially generated carriers can produce a large number of secondary carriers, resulting in internal gain. This mode provides very high sensitivity but is prone to noise and requires careful control of the bias voltage.

Photocurrent Generation:

The energy of an incident photon must be greater than or equal to the bandgap energy (E_g) of the semiconductor material to create an electron-hole pair. The energy of a photon is given by E = hf = hc/λ, where h is Planck's constant, f is the frequency, c is the speed of light, and λ is the wavelength.

For silicon, E_g is approximately 1.1 eV. Photons with wavelengths shorter than approximately 1.1 µm can generate electron-hole pairs in silicon.

In reverse bias, the generated electron-hole pairs are separated by the electric field. Electrons are swept towards the n-side, and holes are swept towards the p-side. This movement of charge carriers constitutes the photocurrent.

Characteristics:

The key characteristic of a photodiode is its response to light. The relationship between the reverse current and the incident light intensity is approximately linear in the photoconductive mode. The dark current is a critical parameter, as it sets the noise floor for detecting weak light signals.

Applications:

Photodiodes are used in a wide variety of applications, including:

• Light sensors (e.g., in cameras, automatic lighting control)
• Optical communication receivers
• Medical imaging (e.g., CT scanners)
• Bar code scanners
• Remote controls
• Solar cells (in photovoltaic mode)