This chapter provides a comprehensive analysis of electrical circuits in transient regimes and their response to alternating currents (AC). It begins by introducing the quasi-static approximation, a fundamental assumption in circuit theory that simplifies analysis by neglecting wave propagation effects, allowing for the application of Kirchhoff’s laws to time-varying circuits. The chapter then characterizes the behavior of passive circuit elements—resistors, capacitors, and inductors—in the quasi-static regime. It derives their fundamental current-voltage (I-V) relationships and the expressions for energy stored in capacitors (electric field) and inductors (magnetic field). The analysis of first-order circuits, specifically RC and RL circuits in series, demonstrates their transient responses, including exponential charging/discharging and the concept of time constants ( \(\tau _{RC} =RC\) and \(\tau _{RL} =L/R\) ). The energy transfer and dissipation in these circuits are also thoroughly examined. The chapter then extends to second-order circuits, focusing on the series LC circuit as an ideal oscillator with a characteristic resonant frequency ( \(\omega _0 =1/\sqrt{LC}\) )), and the more general RLC circuit, which exhibits overdamped, critically damped, and underdamped oscillatory responses depending on the damping factor and resonant frequency. The second part of the chapter transitions to the forced sinusoidal regime (AC currents). It introduces the powerful complex representation of AC circuit components, defining impedance (Z) as a complex quantity that generalizes resistance to include inductive and capacitive reactances. The rules for series and parallel connection of impedances are derived. The concept of a transfer function ( \(H(j\omega )=V_{\text {out}} /V_{\text {in}}\) )) is central to analyzing the sinusoidal response of linear networks, quantifying the gain ( \(G(\omega )\) ) and phase shift ( \(\phi (\omega )\) ) as functions of frequency. Bode plots are introduced as standard graphical tools for visualizing these frequency responses. Finally, the chapter applies these concepts to the design and analysis of electronic filters (low-pass, high-pass, band-pass, and band-stop filters), demonstrating how different RLC circuit configurations can achieve specific filtering characteristics. The quality factor (Q-factor) is introduced as a dimensionless parameter characterizing the sharpness of resonance and the selectivity of filters. The chapter concludes with a discussion of power in AC circuits, distinguishing between active power (useful work) and reactive power (energy oscillating between source and load), and introducing the concept of power factor.

Circuits in Transient Regimes

章节
发表日期：

2025年

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Fabian Cadiz,
Arnaud Couairon

摘要

This chapter provides a comprehensive analysis of electrical circuits in transient regimes and their response to alternating currents (AC). It begins by introducing the quasi-static approximation, a fundamental assumption in circuit theory that simplifies analysis by neglecting wave propagation effects, allowing for the application of Kirchhoff’s laws to time-varying circuits. The chapter then characterizes the behavior of passive circuit elements—resistors, capacitors, and inductors—in the quasi-static regime. It derives their fundamental current-voltage (I-V) relationships and the expressions for energy stored in capacitors (electric field) and inductors (magnetic field). The analysis of first-order circuits, specifically RC and RL circuits in series, demonstrates their transient responses, including exponential charging/discharging and the concept of time constants ( \(\tau _{RC} =RC\) and \(\tau _{RL} =L/R\) ). The energy transfer and dissipation in these circuits are also thoroughly examined. The chapter then extends to second-order circuits, focusing on the series LC circuit as an ideal oscillator with a characteristic resonant frequency ( \(\omega _0 =1/\sqrt{LC}\) )), and the more general RLC circuit, which exhibits overdamped, critically damped, and underdamped oscillatory responses depending on the damping factor and resonant frequency. The second part of the chapter transitions to the forced sinusoidal regime (AC currents). It introduces the powerful complex representation of AC circuit components, defining impedance (Z) as a complex quantity that generalizes resistance to include inductive and capacitive reactances. The rules for series and parallel connection of impedances are derived. The concept of a transfer function ( \(H(j\omega )=V_{\text {out}} /V_{\text {in}}\) )) is central to analyzing the sinusoidal response of linear networks, quantifying the gain ( \(G(\omega )\) ) and phase shift ( \(\phi (\omega )\) ) as functions of frequency. Bode plots are introduced as standard graphical tools for visualizing these frequency responses. Finally, the chapter applies these concepts to the design and analysis of electronic filters (low-pass, high-pass, band-pass, and band-stop filters), demonstrating how different RLC circuit configurations can achieve specific filtering characteristics. The quality factor (Q-factor) is introduced as a dimensionless parameter characterizing the sharpness of resonance and the selectivity of filters. The chapter concludes with a discussion of power in AC circuits, distinguishing between active power (useful work) and reactive power (energy oscillating between source and load), and introducing the concept of power factor.