A Problem-Solving Approach to Electric Circuits

In our study of electric circuits so far, we have primarily focused on circuits driven by sinusoidal sources. However, many real-world signals encountered in electrical engineering are non-sinusoidal, yet often exhibit periodic behavior. This chapter introduces the powerful tool of Fourier Series, which allows us to represent such periodic signals as a sum of sinusoids of different frequencies. Building upon this foundation, we will then explore the Fourier Transform, a technique that extends this concept to analyze non-periodic signals in the frequency domain. Understanding these concepts is crucial for analyzing the behavior of circuits subjected to complex waveforms and for designing systems that operate across a spectrum of frequencies.

The natural frequencies of a circuit are the values of complex frequency s that characterize the circuit’s zero-input response. Specifically, if a voltage or current in the circuit has a zero-input response of the form \( {k}_1{e}^{s_1t}+{k}_2{e}^{s_2t}+{k}_3{e}^{s_3t}+\dots \) where k_i are non-zero constant determined by the initial conditions, then that values of s_i are the natural frequencies of the circuit.