A sinusoidal voltage Δv(t)=(40.0 V) sin(100 t) is applied to a series R L C circuit with L=160 m

step 1 We are given that delta V as a function of T is 40 volts, sign of 100T. And in the circuit, L is

A sinusoidal voltage Δv(t)=(40.0 V) sin(100 t) is applied to...

A sinusoidal voltage $\Delta v(t)=(40.0 \mathrm{V}) \sin (100 t)$ is applied to a series $R L C$ circuit with $L=160 \mathrm{mH}, C=99.0 \mu \mathrm{F},$ and $R=68.0 \Omega .$ (a) What is the impedance of the circuit? (b) What is the maximum current? (c) Determine the numerical values for $I_{\max }, \omega,$ and $\phi$ in the equation $i(t)=$ $I_{\max } \sin (\omega t-\phi)$

Key Concepts

Impedance

Impedance is the total opposition that a circuit presents to the flow of alternating current. In an AC circuit, it combines the resistive effects (resistance) and the reactive effects (due to inductors and capacitors) into a complex quantity, often represented as a magnitude and a phase angle. In a series RLC circuit, the impedance is calculated by taking the square root of the sum of the resistance squared and the net reactance squared (where the net reactance is the difference between the inductive reactance and the capacitive reactance).

Inductive and Capacitive Reactance

Inductive reactance and capacitive reactance are the components of impedance that arise from inductors and capacitors, respectively. Inductive reactance is proportional to the angular frequency and the inductance, causing the current to lag the voltage, while capacitive reactance is inversely proportional to the angular frequency and the capacitance, causing the current to lead the voltage. Their difference determines the net reactive effect in the circuit, which in turn influences both the magnitude of the impedance and the phase shift between voltage and current.

Phasor Representation

Phasor representation is a technique used to simplify the analysis of sinusoidal functions in AC circuits by representing them as complex numbers (phasors) in the complex plane. This method converts differential equations into algebraic equations by accounting for the magnitude and phase angles of sinusoidal quantities, making it easier to analyze circuit behavior such as current, voltage, impedance, and phase relationships.

Maximum Current in AC Circuits

The maximum current, or peak current, in an alternating current circuit is determined by dividing the peak voltage by the magnitude of the impedance. This is a direct consequence of Ohm’s Law in the AC domain, and it provides insight into the amplitude of the sinusoidal current response given the applied sinusoidal voltage and the circuit's overall opposition to current flow.

Phase Angle in AC Circuits

The phase angle in an AC circuit represents the time shift between the sinusoidal voltage and current waveforms. It is determined by the ratio of the net reactance (inductive minus capacitive reactance) to the resistance in the circuit. This phase difference is significant for understanding the timing relationship between current and voltage, which affects power delivery and circuit resonance characteristics.

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