Why in capacitance across AC circuit we use Sin(wt +pi/2) and not Sin(pi/2 -wt) ?

In AC circuits, the behavior of capacitors is often described using sinusoidal functions, particularly when analyzing how voltage and current interact. The choice between using Sin(wt + π/2) and Sin(π/2 - wt) relates to the phase relationship between the current and voltage in a capacitive circuit. Let's break this down step by step.

Understanding Phase Relationships

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees, or π/2 radians. This means that when the voltage reaches its maximum value, the current is already at its maximum value, but it occurs a quarter cycle earlier.

Why Sin(wt + π/2)?

The expression Sin(wt + π/2) represents a sine wave that is shifted to the left by 90 degrees. This shift indicates that the current waveform reaches its peak before the voltage waveform does. In mathematical terms, if we let:

• V(t) = V₀ * Sin(wt) (voltage as a function of time)
• I(t) = I₀ * Sin(wt + π/2) (current as a function of time)

Here, V₀ and I₀ are the peak values of voltage and current, respectively. The current I(t) is at its maximum when wt = π/2, which corresponds to the voltage being zero. This is the essence of the leading phase relationship in capacitors.

What About Sin(π/2 - wt)?

On the other hand, the expression Sin(π/2 - wt) can be rewritten using the co-function identity:

• Sin(π/2 - x) = Cos(x)

Thus, Sin(π/2 - wt) translates to Cos(wt), which represents a waveform that peaks at a different time compared to the sine function. In this case, the current would lag behind the voltage, which is not the behavior we observe in a capacitive circuit.

Visualizing the Waveforms

To visualize this, imagine two sine waves on a graph. The voltage wave (Sin(wt)) oscillates up and down, while the current wave (Sin(wt + π/2)) reaches its peak earlier. If you were to plot these functions, you would see that the current wave leads the voltage wave by a quarter of a cycle, confirming the phase relationship.

Practical Implications

This phase difference is crucial in AC circuit analysis, especially when calculating impedance and power factor. In capacitive circuits, the leading current indicates that capacitors store energy in the electric field, releasing it back into the circuit, which is a fundamental aspect of their operation.

Summary

In summary, we use Sin(wt + π/2) for current in a capacitive AC circuit because it accurately reflects the leading phase relationship between current and voltage. Using Sin(π/2 - wt) would incorrectly suggest that the current lags behind the voltage, which does not align with the physical behavior of capacitors in AC circuits. Understanding these relationships is essential for analyzing and designing effective electrical systems.