Understanding Power Dissipation in AC Circuits

In alternating current (AC) circuits, energy is supplied by a source and can be stored or dissipated by circuit components. Understanding where and how power is dissipated is crucial for analyzing these circuits. Let's look at how different components handle power:

• Resistors (R): Resistors dissipate electrical energy as heat. The average power dissipated in a resistor in an AC circuit is given by \(P_{avg} = I_{rms}^2 R\), where \(I_{rms}\) is the root-mean-square current through the resistor.
• Ideal Inductors (L): Ideal inductors store energy in a magnetic field during one part of the cycle and return it to the circuit during another part. The average power dissipated by an ideal inductor over a complete cycle is zero.
• Ideal Capacitors (C): Ideal capacitors store energy in an electric field and return it to the circuit. The average power dissipated by an ideal capacitor over a complete cycle is zero.

Power dissipation in AC circuits only occurs in resistive components.

Analyzing AC Circuit Power Characteristics: Matching List I and List II

We need to match different types of AC circuits from List I with their power dissipation characteristics described in List II.

Analysis of List I - List II Matches:

Let's analyze each item in List I and find its corresponding match in List II based on the principles of AC circuit power.

(A) Resistive Circuit and (III) Power dissipated only in resistor

A purely resistive AC circuit contains only resistors. As discussed, resistors are the components that dissipate energy as heat. Therefore, in a resistive circuit, power is dissipated exclusively in the resistors present.

This is a direct match based on the definition of power dissipation in AC components.

(B) Purely inductive or capacitive circuit and (I) No power dissipation

A purely inductive circuit contains only an ideal inductor, and a purely capacitive circuit contains only an ideal capacitor. Ideal inductors and capacitors do not dissipate average power over a cycle; they only store and release energy. Thus, in purely inductive or capacitive circuits, the average power dissipation is zero.

This aligns perfectly with the characteristic "No power dissipation".

(D) Power dissipated at resonance in LCR circuit and (II) Maximum power dissipation because \(X_L=X_C\)

An LCR series circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series. Resonance occurs in an LCR series circuit when the inductive reactance (\(X_L\)) equals the capacitive reactance (\(X_C\)).

At resonance, \(X_L = X_C\). The total impedance (\(Z\)) of the series LCR circuit is given by:

At resonance, since \(X_L - X_C = 0\), the impedance becomes minimum and is equal to the resistance \(Z = R\). This minimum impedance leads to maximum current in the circuit for a given applied voltage (\(I_{rms} = V_{rms} / Z\)).

The power dissipated in the circuit is \(P = I_{rms}^2 R\). Since the current is maximum at resonance, and R is constant, the power dissipated (\(I_{rms}^2 R\)) is also maximum at resonance.

Therefore, the power dissipated at resonance in an LCR circuit is maximum, and this condition occurs because \(X_L = X_C\).

This directly matches the characteristic "Maximum power dissipation because \(X_L=X_C\)".

(C) LCR series circuit and (IV) Maximum power dissipation

An LCR series circuit contains resistance, inductance, and capacitance. Power is dissipated in the resistor (R). While a general LCR circuit does not always dissipate maximum power (unless it is at resonance), among the given options for LCR circuits, "Maximum power dissipation" (IV) is the best fit when considering the peak performance of such a circuit, which occurs at resonance. Option (II) is specifically tied to the *reason* for maximum power (because \(X_L=X_C\)) which is linked to resonance. Option (IV) simply states "Maximum power dissipation". Given that a general LCR circuit *can achieve* maximum power dissipation at resonance, and option (D) explicitly covers the resonance condition with reason, option (IV) seems to represent the potential for maximum power that exists within the LCR circuit framework, particularly when contrasted with circuits that dissipate no power (purely reactive) or circuits where power is limited only by resistance.

Thus, (C) LCR series circuit is matched with (IV) Maximum power dissipation, representing the circuit type where maximum power can be attained.

Summary of Matches

Based on our analysis of power dissipation in different AC circuit types and the resonance condition, the correct matches are:

Revision Table: AC Circuit Power Dissipation

This table summarizes the power characteristics of basic AC circuit components and types.

Additional Information: Key Concepts in AC Circuits and Resonance

• Impedance (\(Z\)): In AC circuits, impedance is the total opposition to current flow, combining resistance and reactance. For a series LCR circuit, \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).
• Reactance (\(X_L, X_C\)): Reactance is the opposition to current flow offered by inductors (\(X_L\)) and capacitors (\(X_C\)) due to energy storage. \(X_L = \omega L\) and \(X_C = \frac{1}{\omega C}\), where \(\omega\) is the angular frequency.
• Resonance Condition (\(X_L = X_C\)): This condition occurs at a specific frequency (\(\omega_0 = \frac{1}{\sqrt{LC}}\)) in a series LCR circuit, leading to minimum impedance (\(Z=R\)) and maximum current for a given voltage.
• Power Factor (\(\cos \phi\)): The power factor of an AC circuit is the cosine of the phase angle (\(\phi\)) between the voltage and current. It represents the fraction of the total power that is actually dissipated as useful power. \(P_{avg} = V_{rms} I_{rms} \cos \phi\). For a purely resistive circuit, \(\phi=0\), \(\cos \phi=1\) (power factor = 1). For purely reactive circuits, \(\phi = \pm 90^\circ\), \(\cos \phi = 0\) (power factor = 0). At resonance in an LCR circuit, \(\phi=0\), \(\cos \phi=1\) (power factor = 1, maximum power).