Chapter 16: Problem 53
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
Step by step solution
Understand the Input Signal
Calculate the Fourier Series Coefficients for the Input
Apply Low-Pass Filter Characteristics
Determine Output Fourier Series Terms
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Butterworth 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.
Low-pass Filter
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.
Signal Processing
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.
Rectified Sine Wave
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.