Calculating Instantaneous Power in AC Circuits

To find the instantaneous power in a resistive circuit, we use the relationship between voltage, resistance, and power. Instantaneous power is the power at a specific moment in time in an AC (Alternating Current) circuit. This calculation is crucial for understanding energy dissipation in electrical components.

Understanding Instantaneous Power

Instantaneous power (\(p\)) in an electrical circuit is the product of instantaneous voltage (\(v\)) and instantaneous current (\(i\)). It can also be expressed in terms of voltage and resistance, or current and resistance. For a purely resistive circuit, such as a 15-ohm resistance, the instantaneous voltage and instantaneous current are in phase, meaning they reach their peak and zero values at the same time.

• Voltage (\(v\)): The electrical potential difference across the resistance, given as \(105 \sin 377t\) (V), which varies sinusoidally with time in an AC circuit.
• Resistance (\(R\)): The opposition to the flow of electric current, provided as 15 Ω.
• Instantaneous Power (\(p\)): The rate at which electrical energy is consumed or dissipated by the 15-ohm resistance at any given instant.

Given Electrical Parameters for Power Calculation

We are provided with the following values for determining the instantaneous power expression:

• Resistance (\(R\)): 15 Ω
• Instantaneous Voltage (\(v\)): \(105 \sin 377t\) (V)

Formula for Instantaneous Power Expression

The instantaneous power (\(p\)) dissipated by a resistance (\(R\)) when an instantaneous voltage (\(v\)) is applied across it can be calculated using the formula derived from Ohm's Law and the basic power formula.

The fundamental formula for power is \(P = VI\).

According to Ohm's Law, the current \(I\) through a resistor is given by \(I = \frac{V}{R}\).

Substituting the expression for \(I\) into the power formula, we get:

$$p = v \times i = v \times \frac{v}{R} = \frac{v^2}{R}$$

This formula is ideal for calculating instantaneous power when the instantaneous voltage and resistance are known.

Step-by-Step Instantaneous Power Calculation

Let's calculate the instantaneous power expression using the given instantaneous voltage and resistance with the formula \(p = \frac{v^2}{R}\).

1. Identify the given instantaneous voltage expression: The instantaneous voltage is \(v = 105 \sin 377t\) (V).
2. Identify the given resistance value: The resistance is \(R = 15\) Ω.
3. Substitute these values into the instantaneous power formula: $$p = \frac{v^2}{R}$$ $$p = \frac{(105 \sin 377t)^2}{15}$$
4. Square the voltage term: First, square the amplitude (105) and the sinusoidal function: $$105^2 = 11025$$ $$(\sin 377t)^2 = \sin^2 377t$$ So, the expression becomes: $$p = \frac{11025 \sin^2 377t}{15}$$
5. Perform the division to simplify the expression: $$p = \frac{11025}{15} \sin^2 377t$$ $$p = 735 \sin^2 377t$$

Final Instantaneous Power Expression

The expression for instantaneous power for the given 15-ohm resistance with the voltage \(v = 105 \sin 377t\) (V) is:

$$p = 735 \sin^2 377t \text{ (W)}$$

This result represents the instantaneous power dissipated by the resistance at any given time \(t\), measured in Watts (W).