Question

A resistance \(R\), capacitance \(C\), and inductance \(L\) are connected in series to a voltage source with amplitude \(V\) and variable angular frequency \(\omega\). If \(\omega\) = \(\omega$$_0\) , the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of \(R\), \(C\), \(L\), and \(V\).

Short answer

(a) \( \frac{V}{R} \); (b) \( \frac{V}{\omega_0C} \); (c) \( \frac{V \cdot L}{R} \omega_0 \); (d) \( \frac{V^2}{2\omega_0^2R^2C} \); (e) \( \frac{LV^2}{2R^2} \).

Step-by-step solution

Identify the Resonance Frequency

In an RLC series circuit, the resonance angular frequency \( \omega_0 \) is given by \( \omega_0 = \frac{1}{\sqrt{LC}} \). At resonance, the reactive components cancel each other out, and the impedance is purely resistive, equal to the resistance \( R \).

Calculate Maximum Current in the Resistor

At resonance, the impedance \( Z \) is equal to \( R \). Thus, the maximum current \( I \) through the resistor at resonance is \( I = \frac{V}{R} \), where \( V \) is the voltage of the source.

Calculate Maximum Voltage Across the Capacitor

The maximum voltage across the capacitor is given by \( V_C = \frac{I}{\omega_0 C} \). Substituting \( I \) from the previous step, \( V_C = \frac{V}{\omega_0 RC} \).

Calculate Maximum Voltage Across the Inductor

The maximum voltage across the inductor is given by \( V_L = I \cdot \omega_0 L \). Substituting \( I \) from the second step, \( V_L = \frac{V \cdot L}{R} \cdot \omega_0 \).

Calculate Maximum Energy Stored in the Capacitor

The energy stored in the capacitor is given by \( U_C = \frac{1}{2} C V_C^2 \). Substitute \( V_C = \frac{V}{\omega_0 RC} \) from the third step to find the maximum stored energy.

Calculate Maximum Energy Stored in the Inductor

The energy stored in the inductor is given by \( U_L = \frac{1}{2} L I^2 \). Substitute \( I = \frac{V}{R} \) from the second step to find the maximum stored energy.

Key concepts

Resonance Frequency

In the analysis of an RLC circuit, understanding resonance frequency is key. This is the frequency at which the circuit naturally oscillates without external force. In a series RLC circuit, the resonance angular frequency, denoted as \( \omega_0 \), is determined by the formula \( \omega_0 = \frac{1}{\sqrt{LC}} \), where \( L \) is the inductance and \( C \) is the capacitance.

At this specific frequency:

• The inductive reactance and capacitive reactance offset each other, effectively canceling out.
• The circuit's total impedance is minimized to the resistance \( R \) alone.
• Maximum current flows through the circuit as impedance is at its lowest.

Resonance plays a crucial role in circuits, aiding in understanding how they respond to different frequencies.

Impedance

Impedance in an RLC circuit represents the total resistance to the flow of alternating current. It combines the effects of resistance \( R \), capacitive reactance \( X_C = \frac{1}{\omega C} \), and inductive reactance \( X_L = \omega L \).

At the resonance frequency:

• The reactive components (inductive and capacitive) become equal in magnitude but opposite in sign, thus they cancel out each other.
• The impedance simplifies to the resistance \( R \), as \( Z = R \).
• This simplification maximizes the current flow since there's no additional reactance to impede it.

The concept of impedance is vital for analyzing how an RLC circuit behaves under different frequency conditions.

Energy Storage

Energy storage in an RLC circuit is primarily carried out by the capacitor and the inductor. Each stores energy in a unique manner:

• A capacitor stores energy in an electric field. The energy is given by \( U_C = \frac{1}{2} C V_C^2 \), where \( V_C \) is the voltage across the capacitor.
• An inductor stores energy in a magnetic field. The energy is expressed as \( U_L = \frac{1}{2} L I^2 \), where \( I \) is the current through the inductor.

At resonance, these energy components are at their maximum due to the large current flowing through the circuit. Even though the energy storage does not contribute to energy transfer to the load, it is crucial for sustaining the oscillations of the circuit.

Capacitance

Capacitance is the circuit's ability to store charge per unit voltage across its plates. In the formula \( C = \frac{Q}{V} \), \( C \) represents capacitance, \( Q \) the charge, and \( V \) the voltage.

For an RLC circuit, the capacitance affects how quickly the capacitor can charge and discharge.

• Higher capacitance means more energy can be stored at the same voltage level.
• It also influences the resonance frequency since \( \omega_0 = \frac{1}{\sqrt{LC}} \), showing a direct relationship with both capacitance and inductance.

Understanding capacitance helps predict the behavior of the circuit under alternating current and how voltage fluctuates across the capacitor.

Inductance

Inductance is the property of a circuit due to its inductors, to oppose changes in current flow. It is measured in henrys (H).

The key role of inductance is seen in its contribution to magnetic field energy storage.

• It plays a part in determining the circuit’s resonance frequency alongside capacitance.
• Higher inductance means the circuit can handle more current oscillation before reaching saturation.
• Inductors, when subjected to a changing current, generate an electromotive force (EMF) that opposes the change, aligning with Lenz's Law.

In an RLC circuit, managing inductance allows control over how quickly the circuit reaches the resonance frequency and how robust the current and voltage oscillations become.