An inductor and capacitor are connected in series across an AC generator, and the voltage across the inductor is higher than across the capacitor. Is the generator frequency above or below resonance?

When studying AC circuits, it's essential to understand the behavior of inductors and capacitors. Both these components exhibit reactance, which is their resistance to an alternating current (AC). However, they act differently over various frequencies. An inductor's reactance, called inductive reactance, increases with the frequency of the AC. This is shown by the formula \( XL = 2\pi f L \), where \( XL \) is the inductive reactance, \( f \) is the frequency, and \( L \) is the inductance.

In contrast, a capacitor's reactance, known as capacitive reactance, decreases with an increase in frequency. The formula for capacitive reactance is \( XC = \frac{1}{2\pi f C} \), where \( XC \) is the capacitive reactance, and \( C \) is the capacitance.

• High frequency: Inductor has more reactance, Capacitor has less.
• Low frequency: Inductor has less reactance, Capacitor has more.

Understanding these concepts helps us analyze AC circuits more effectively, especially when tuning a circuit to a specific frequency or analyzing its performance across a spectrum of frequencies.

A fascinating phenomenon occurs when we connect an inductor and a capacitor in series in an AC circuit—this setup can achieve a condition known as series resonance. At resonance, the inductive and capacitive reactances are equal in magnitude but opposite in phase, effectively canceling each other out. As a result, the circuit’s overall impedance, which is the total opposition to AC, is minimized, and the circuit behaves as if it were solely resistive.

The frequency at which this phenomenon occurs is called the resonance frequency or natural frequency of the circuit, calculated using the formula \( f_r = \frac{1}{2\pi\sqrt{LC}} \), where \( f_r \) is the resonance frequency, \( L \) is the inductance, and \( C \) is the capacitance. At this frequency, the circuit allows the maximum current to flow and can be used to filter or select specific frequencies in applications such as radios and other communication devices.

An AC generator, also called an alternator, converts mechanical energy into electrical energy to produce alternating current. The frequency of the generated AC is proportional to the speed of the generator’s rotation and the number of poles in the generator’s construction. This frequency is crucial because it needs to match the desired operating conditions of the circuits it supplies.

In the context of resonance, the generator frequency is the variable that we adjust to reach the condition where inductive and capacitive reactances equalize in a series circuit. If we have a fixed inductance and capacitance, altering the generator's frequency can lead us to that sweet spot known as the resonance frequency. This principle is widely applied in tuning circuits to specific frequencies, such as in radio receivers to hone in on a particular station.

The concept of impedance is critical in understanding AC circuits. Impedance, symbolized as \( Z \), represents the total opposition that a circuit presents to alternating current, incorporating both resistance \( R \) and reactance \( X \). The impedance of a circuit is a complex quantity, consisting of a real part (resistance) and an imaginary part (reactance), often represented as \( Z = R + jX \), where \( j \) is the imaginary unit.

At series resonance, the impedance is purely resistive since the reactances cancel each other out. This results in the minimum impedance and maximum power transfer through the circuit. Analyzing and controlling impedance is vital for the efficient operation of electrical devices, from your smartphone to large industrial motors, as it affects how much current flows and how much power is used.