Determine the frequency so that the current Ii and the...

Determine the frequency so that the current $I_i$ and the voltage $V_o$ in the circuit of of Figure P4.35 are in phase.

$$
\begin{aligned}
Z_n & =13.000+j \omega 3 \Omega \\
R & =120 \Omega \\
L & =19 \mathrm{mH} \quad C=220 \mathrm{pF}
\end{aligned}
$$
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Key Concepts

Complex Impedance

Complex impedance is a representation of the opposition a circuit element presents to alternating current. It combines both resistive (real) components and reactive (imaginary) components into a single quantity (Z = R + jX), enabling analysis of magnitude and phase effects in AC circuits.

Reactive Components

Inductors and capacitors offer opposition to changes in current and voltage through their reactances, which are frequency-dependent. The inductive reactance (X_L = ?L) increases with frequency while the capacitive reactance (X_C = 1/(?C)) decreases with frequency, causing phase shifts between voltage and current.

Resonance in RLC Circuits

Resonance occurs in RLC circuits when the inductive and capacitive reactances are equal in magnitude but opposite in sign, resulting in their cancellation. This condition eliminates the net reactive component, leaving the circuit's impedance purely resistive at a specific frequency known as the resonant frequency.

Phase Relationship in AC Circuits

The phase relationship between current and voltage in AC circuits depends on the circuit's overall impedance. When the net reactive components cancel each other, the voltage and current are in phase, meaning there is no phase shift between them. This in-phase condition is crucial in power transfer and signal integrity applications.

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