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Impedance is a crucial concept in AC circuits. It essentially measures the total opposition that a circuit presents to the flow of alternating current. Unlike simple resistance, impedance combines both resistive and reactive elements. This is because, in AC circuits, both resistors and components like inductors or capacitors influence the current.
In this exercise, the impedance of a circuit containing a resistor and an inductor in series is calculated. The key is using the impedance formula: \[ Z = \sqrt{R^2 + (X_L)^2} \]where:

• R = resistance in ohms (\(\Omega\))
• \(X_L\) = inductive reactance in ohms (\(\Omega\))

This formula helps determine how much the resistor and the inductor together impede the flow of AC. It's important to use the correct values of \( R \) and \( X_L \) specific to your circuit conditions.

A resistor's role in any electrical circuit is to limit the current flowing through the circuit. In AC circuits, a resistor's operation remains straightforward; it provides a constant resistance (\( R \)) opposing the flow of current.
For this exercise, we deal with a 36-k\(\Omega\) resistor. Because resistors affect both AC and DC currents alike, they are often crucial in controlling circuit current levels.
When calculating impedance in AC circuits, the resistor contributes the resistive part of the impedance which is purely real and unaffected by frequency, unlike reactance.

An inductor is a passive electrical component that stores energy in a magnetic field when electric current flows through it. It's commonly used to manage AC currents by affecting their phase and amplitude.
In this exercise, the inductor has an inductance \( L \) of 55 mH. The inductor induces a reactive component called inductive reactance in an AC circuit. This reactance depends on the frequency of the AC signal passing through the inductor.
Inductors oppose changes in current because of their nature to store and release energy, making their behavior frequency-dependent. This characteristic is quantified as inductive reactance, which we'll explore next.

Inductive reactance, symbolized as \( X_L \), is a measure of an inductor's opposition to changes in current in an AC circuit. The faster the current alternates, the greater the inductor opposes it.
The reactance is calculated using:\[ X_L = 2\pi f L \]Here, \( f \) represents the frequency in hertz (Hz), and \( L \) is inductance in henries (H).

• Lower frequencies mean smaller \( X_L \), allowing more current through.
• Higher frequencies result in a larger \( X_L \), reducing current flow.

For example, at 50 Hz, the reactance is about 17.28 \(\Omega\), and at 30,000 Hz, it's 10,362.26 \(\Omega\). This shows that inductive reactance increases with frequency, influencing the overall impedance.