Can the instantaneous power output of an ac source ever be negative ? Can the average power output be negative ?

Explanation

To answer the question regarding the instantaneous and average power output of an AC source, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Understanding Instantaneous Power**:
The instantaneous power \( P \) of an AC source can be expressed using the formula:
\[ P = V_0 I_0 \sin(\omega t) \sin(\omega t + \phi) \]
where \( V_0 \) is the peak voltage, \( I_0 \) is the peak current, \( \omega \) is the angular frequency, and \( \phi \) is the phase difference.

**Hint**: Remember that instantaneous power varies with time and depends on the sine functions of the voltage and current.

2. **Using Trigonometric Identities**:
We can use the product-to-sum identities to simplify the expression for instantaneous power:
\[ P = \frac{V_0 I_0}{2} \left[ \cos(\phi) - \cos(2\omega t + \phi) \right] \]

**Hint**: Familiarize yourself with trigonometric identities, especially the product-to-sum formulas.

3. **Analyzing the Terms**:
In the expression \( \cos(\phi) - \cos(2\omega t + \phi) \), we note that \( \cos(\phi) \) is a constant value, while \( \cos(2\omega t + \phi) \) oscillates between -1 and 1.

**Hint**: Consider the range of cosine functions and how they affect the overall expression.

4. **Determining Conditions for Negative Instantaneous Power**:
Since \( \cos(2\omega t + \phi) \) can take values from -1 to 1, there will be instances when:
\[ \cos(\phi) < \cos(2\omega t + \phi) \]
This means that the term \( \cos(\phi) - \cos(2\omega t + \phi) \) can become negative, leading to:
\[ P < 0 \]
Therefore, the instantaneous power can indeed be negative.

**Hint**: Think about how the oscillating nature of AC affects power calculations.

5. **Understanding Average Power**:
The average power \( P_{avg} \) over one complete cycle is given by:
\[ P_{avg} = \frac{V_0}{\sqrt{2}} \cdot \frac{I_0}{\sqrt{2}} \cdot \cos(\phi) = V_{rms} I_{rms} \cos(\phi) \]
where \( V_{rms} \) and \( I_{rms} \) are the root mean square values of voltage and current, respectively.

**Hint**: Average power is calculated over a full cycle, which smooths out the instantaneous variations.

6. **Analyzing Average Power**:
The term \( \cos(\phi) \) (the power factor) is always between -1 and 1, but since it represents the cosine of the phase difference, it will always be non-negative when considering resistive loads. Thus, \( P_{avg} \) cannot be negative.

**Hint**: Remember that average power reflects the effective power consumed by the circuit, which cannot be negative.

### Conclusion:
- The **instantaneous power** of an AC source can be negative due to the oscillating nature of the sine functions involved.
- The **average power** output of an AC source cannot be negative as it is always dependent on the non-negative power factor \( \cos(\phi) \).