Question

In general in an alternating current circuit. (a) The average value of current is zero. (b) The average value of square of current is zero. (c) Average power dissipation is zero. (d) The phase difference between voltage and current is zero.

Short answer

In summary, (a) The average value of current is zero, which is correct for AC circuits. (d) The phase difference between voltage and current is zero, which is correct for purely resistive AC circuits but not for general AC circuits with inductors and/or capacitors. However, statements (b) and (c) are incorrect: the average value of square of current is not zero, and the average power dissipation is not zero in an AC circuit.

Step-by-step solution

Statement (a) - Average value of current is zero.

This statement is correct for AC circuits. In an AC circuit, current alternates its direction periodically, and as a result, the average value of the current over a complete cycle is zero. The current waveform is usually sinusoidal, and in one cycle, it has equal positive and negative parts.

Statement (b) - The average value of square of current is zero.

This statement is incorrect for AC circuits. The average value of the square of current is not zero. Since squaring a value always gives a positive result, squaring the current will yield a positive value during both the positive and negative halves of the AC cycle, so the average squared current over a cycle will always be greater than zero.

Statement (c) - Average power dissipation is zero.

This statement is incorrect for AC circuits. In an AC circuit, the average power dissipation is not zero. While the average voltage and current are zero over a complete cycle, the power dissipation is related to the product of the instantaneous voltage and current, which is not zero in general. In a resistive load, the power follows a quadratic relation, i.e., $P=VI$, where both voltage and current have equal phase angles. In such a case, the average power will have a positive value.

Statement (d) - The phase difference between voltage and current is zero.

This statement is correct when considering purely resistive AC circuits. In a resistive circuit, the voltage and current waveforms are in phase. However, for general AC circuits containing inductors and/or capacitors, the phase difference between voltage and current can be non-zero; such circuits are called reactive circuits.

Key concepts

Average Current

In alternating current (AC) circuits, the concept of average current refers to the average value of the current over one complete cycle. AC current periodically changes direction, which means that it alternates between positive and negative values.
When calculated over a full cycle including both positive and negative halves, the average current is zero.
This is due to the symmetrical nature of the sinusoidal waveform. Some key points about average current include:

• AC current changes direction periodically.
• The average current over a full cycle is zero.
• This occurs because the positive and negative halves of the waveform cancel each other out.

Understanding the average current is fundamental because it highlights how AC circuits differ from direct current (DC) circuits, where the current flows in only one direction.

Average Power Dissipation

Average power dissipation in AC circuits is an essential concept for understanding how energy is consumed over time. Unlike the average current, the average power dissipation is not zero in an AC circuit.
This is because power dissipation is calculated by the product of instantaneous voltage and current. Let's break down some important points:

• Power dissipation relates to how much power a circuit element uses.
• In resistive circuits, the power dissipation averages to a positive value.
• Mathematically, it involves the root mean square (RMS) values of voltage and current.
• Energy is exchanged over time, but real power results in energy being consumed.

For many real-world applications, knowing the average power dissipation helps in evaluating energy efficiency and safety in electrical systems.

Phase Difference

Phase difference refers to the difference in phase angle between the voltage and current waveforms in AC circuits. This concept is pivotal in analyzing AC circuits, especially when they include reactive elements like inductors or capacitors.
For purely resistive AC circuits, the phase difference is zero, meaning voltage and current are perfectly in phase. Here are some crucial aspects of phase difference:

• In reactive circuits (with inductors or capacitors), phase differences can be non-zero.
• Phase difference determines reactive and real power.
• Zero phase difference implies that the peaks of voltage and current align perfectly.
• Electricity in buildings often involves circuits with phase differences due to complex loads.

Understanding phase difference helps design and troubleshoot efficient AC systems.

Sinusoidal Waveform

The sinusoidal waveform is a cornerstone of AC circuit analysis. It is the most common waveform for AC signals and represents how voltage or current varies over time in a periodic manner.
The sinusoidal waveform is characterized by its amplitude, frequency, and phase.Key attributes of sinusoidal waveforms include:

• They exhibit symmetric properties, making analysis straightforward.
• Beyond electrical applications, sinusoidal patterns appear in acoustics and light waves.
• Mathematical representation is typically: $I(t)=I_{max}\sin (\omega t+\varphi )$, where $I_{max}$ is peak current, $\omega$ is angular frequency, and $\varphi$ is phase angle.
• In AC circuits, this waveform ensures predictable power delivery and system balance.

Recognizing and working with sinusoidal waveforms is crucial for anyone dealing with AC circuits and electrical engineering.