Question

The input signal to a second-order low-pass Figure \(P\) Butterworth filter is a full-wave rectified sine wave with an amplitude of \(2.5 \pi \mathrm{V}\) and a fundamental frequency of 5000 rad/s. The corner frequency of the filter is \(1 \mathrm{krad} / \mathrm{s}\). Write the first two terms in the Fourier series that represents the steady-state output voltage of the filter.

Short answer

The first two terms in the output Fourier series are the DC component (2.5 V) and negligible due to attenuation.

Step-by-step solution

Understand the Input Signal

The input signal is a full-wave rectified sine wave with an amplitude of \(2.5 \pi \mathrm{V}\) and a fundamental frequency of 5000 rad/s. A full-wave rectified sine wave can be represented using its Fourier series, which consists of a DC term and a series of harmonics of the fundamental frequency.

Calculate the Fourier Series Coefficients for the Input

The Fourier series for the full-wave rectified sine wave can be approximated using the first few terms. The DC component (average value) is \(a_0 = \frac{2 \times 2.5 \pi}{\pi} = 5\). The first harmonic term has a coefficient \(a_1\) that diminishes due to rectification. Harmonics are generally given by \(a_n = \frac{4A}{n^2 \pi}\), where \(A\) is the amplitude of the unrectified sine wave. For \(n = 1\), this becomes \(a_1 = 4 \times 2.5/\pi \), recognizing the symmetry and periodicity of the rectified wave.

Apply Low-Pass Filter Characteristics

A second-order Butterworth filter with a corner frequency of 1 krad/s significantly attenuates frequencies above this point. The critical aspect of solving for the output is knowing that attenuation for a frequency \(\omega\) is given by \(H(\omega) = \frac{1}{\sqrt{1 + (\omega/\omega_c)^4}}\), where \(\omega_c = 1000 \, \, \text{rad/s}\). Only harmonics with \(\omega < \omega_c\) will significantly pass through.

Determine Output Fourier Series Terms

The DC component \(a_0/2\), which is 2.5, will remain unchanged as the filter does not affect DC. For frequencies \(5000 \,\text{rad/s}\) (the fundamental), the attenuation \(H(5000)\) is significant, effectively reducing its magnitude in the output. Compute \(H(5000) = \frac{1}{\sqrt{1 + (5)^4}} \approx 0.012\), which means the output of the first harmonic is negligible.

Key concepts

Butterworth Filter

A Butterworth Filter is a type of linear filter designed for signal processing that allows signals with a frequency lower than a certain cutoff frequency to pass through while attenuating frequencies higher than this cutoff. The Butterworth filter is especially renowned for having a very flat frequency response in the passband. This flat response means it doesn't introduce ripples, in contrast to other types of filters, such as the Chebyshev filter.

There are important characteristics to keep in mind about Butterworth filters:

• Order: The order of the filter determines how sharp the cutoff is. A higher order indicates a steeper cutoff.
• Cutoff Frequency: This is the frequency at which the filter starts to attenuate the input signal.
• Poles: Butterworth filters have poles that are evenly spaced on a semicircle in the s-plane, ensuring optimal smoothness in the passband.

Understanding the nature of a Butterworth filter is crucial when dealing with applications requiring minimal distortion and a smooth response, such as in audio processing.

Low-pass Filter

A Low-pass Filter is a key element in signal processing. It allows signals with frequencies lower than a specified cutoff frequency to pass through while suppressing higher frequency signals.

Here's what you need to know about low-pass filters:

• They can be implemented using electronic components like resistors, capacitors, and inductors in analog circuits.
• In digital signal processing, low-pass filters are used to reduce the high-frequency components of a signal, which can include noise.
• The degree of attenuation for frequencies above the cutoff is determined by the filter's design. For instance, a second-order Butterworth low-pass filter will attenuate the signal more as the frequency increases beyond the cutoff.

Low-pass filters are widely used in various applications, including audio processing to smoothen the sound, image processing to remove noise, and in communication systems to isolate desired signal frequencies.

Signal Processing

Signal Processing involves analyzing, modifying, and synthesizing signals like sound, images, and scientific measurements. It is a vast field with numerous applications, marrying the world of mathematics, electrical engineering, and computer science.

Key areas of signal processing include:

• Filtering: Removing unwanted components or features from a signal.
• Fourier Transform: A mathematical transform that changes a signal from its time domain to a frequency domain representation.
• Compression: Reducing the amount of data required to represent a signal without significant loss of information.

Fourier Transform plays a significant role in signal processing, particularly in the context of frequency analysis. By utilizing Fourier transforms, we can pick out the frequency components of a signal, which is fundamental for operations like filtering and compression.

Rectified Sine Wave

A Rectified Sine Wave is a type of waveform commonly used in power electronics and signal processing. It refers to a sine wave that has been modified to contain only positive or only negative values. Specifically, a full-wave rectified sine wave takes a sinusoidal wave and flips the negative half upwards, resulting in a series of arches or humps.

Key properties of a rectified sine wave include:

• Non-zero average value: Unlike a typical sine wave, a full-wave rectified sine wave has a non-zero mean, meaning it has a DC component.
• Harmonics: A rectified sine wave can be decomposed into a Fourier series, which includes a DC component and a set of harmonics.
• Applications: They are often used in DC power supplies where conversion from AC to a stabilized DC current is necessary.

The transformation to a rectified sine wave introduces harmonics that weren't present in the original sinusoidal signal. These harmonics can affect the operation of other connected circuits, requiring careful management in design.