91Ó°ÊÓ

Inductive reactance, often denoted as \(X_L\), is a fundamental concept in analyzing AC circuits containing inductors. It describes how an inductor opposes changes in current, impeding its flow at certain frequencies. This opposition results from the inductor generating a back EMF (Electromotive Force) proportional to the rate of change of current. The key formula for inductive reactance is
\[ X_L = 2\pi f L \]where \(f\) is the frequency and \(L\) is the inductance.

• At higher frequencies, \(X_L\) increases, meaning inductors oppose change more strongly.
• In AC circuits, \(X_L\) influences the phase and amplitude of current relative to voltage.

In the problem, using the voltage \(\left| \mathbf{V}_L \right| = 20\,\text{V}\), and the calculated current \(|\mathbf{I}| = 0.04\,\text{A}\), \(X_L\) is found to be \(500\,\Omega\). This value of inductive reactance occurs at the resonant frequency and plays a crucial part when matching it with capacitive reactance.

Capacitive reactance, or \(X_C\), measures how a capacitor resists the flow of AC. Instead of impeding the current, like resistors, capacitors allow more current to pass as frequency increases. The capacitive reactance is given by the formula
\[ X_C = \frac{1}{2\pi f C} \]where \(C\) is the capacitance.

• \(X_C\) decreases as the frequency increases, expanding current flow through the capacitor.
• At resonance in a series circuit, \(X_C\) matches \(X_L\), balancing the circuit's reactive components.

In the example, at the given resonant frequency of \(1 \,\text{MHz}\), \(X_C\) is calculated equal to \(X_L\) at \(500 \, \Omega\). This equality is a distinctive feature of resonance, making use of both reactance types indispensable in tuning applications.

The resonant frequency, \(f_0\), is a unique frequency at which a system's oscillations naturally favor a maximum response. In a series resonant circuit, this is where inductive and capacitive reactances cancel each other out. The resonant frequency is defined by the formula
\[f_0 = \frac{1}{2\pi\sqrt{LC}}\]This shows the inverse relationship between the frequency and the square roots of both inductance \(L\) and capacitance \(C\).

• At resonance, the impedance is purely resistive, equating to the circuit's resistance \(R\).
• Energy shifts seamlessly between the inductor and capacitor, minimizing overall reactance.

Calculating \(L\) and \(C\) in the exercise, where \(f_0 = 1 \,\text{MHz}\), results in \(79.58 \,\mu H\) for inductance and \(318.31 \,\text{pF}\) for capacitance, perfect for achieving this harmonious response.

Impedance, denoted as \(Z\), is the total opposition a circuit offers to the flow of alternating current. It encompasses both resistance \(R\) and reactance (both inductive \(X_L\) and capacitive \(X_C\)). Impedance is a complex quantity, expressed as
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

• At resonance, \(X_L = X_C\), so the reactive part cancels out, simplifying \(Z\) to just \(R\).
• It affects both the amplitude and phase of the current relating to voltage in AC circuits.

In the resonant circuit, the impedance becomes simple, just \(50 \,\Omega\), due to the cancellation effect, making it easier to analyze and calculate parameters like the voltage across each component, such as the capacitor voltage \(\left| \mathbf{V}_C \right|\), which equals the inductor voltage \(20 \,\text{V}\).