HW7.1 (A DTFS Puzzle) Consider a discrete-time signal x : Z → R having the following the properties: • • • a[n+4l] = x[n], VI, n∈Z . 2 Σα[n] = 2. n=-1 Σ(−1)"x[n] = 4. n=-1 2 • Σ x[n] cos [ẩn] = 2 Σ x[n] sin [n] = = 0. n=-1 n=-1 (a) Determine the complex exponential Fourier series coefficients X-1, X0, X1, and X2 for the signal x. From the coefficients, determine and provide a well- labeled plot for the signal x. (b) Based on your results from part (a), determine the fundamental period p and fundamental frequency wo of the signal x. Express x in terms of its DTFS co- efficients and complex exponentials of appropriate frequencies, and identify each coefficient and its corresponding harmonic frequency.

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Fault Analysis

In an electrical engineering system, a fault is an abnormal electric current flowing through the system, and fault analysis is the study of these faults. For instance, a short-circuit is a fault where the live wire touches the ground or neutral wire. If a circuit is interrupted by a failure of a current-carrying wire then an open circuit fault occurs. The open-circuit faults are also called series faults. One or more phases or ground may be involved in a three-phase fault or may occur only between two phases. Current flows into the earth in case of a ground fault or earth fault. For most situations, the prospective short circuit current of a fault that is predictable can be calculated. To limit the loss of service because of a failure, the protective devices can detect fault conditions and operate circuit breakers and also other devices. All phases equally can be affected in a poly-phase system which is known as symmetric electrical fault. The asymmetric fault becomes more complicated to analyze if only some phases are affected. By using methods such as symmetrical components, the power system analysis of these types of faults is often simplified. The main objective of power system protection is to design, detect, and interrupt power system faults.