AC Real Power
Real power with a phase angle between V and I.
Formula
Variables
How it works
In an AC circuit, voltage and current may not rise and fall together; the angle \(\phi\) measures that phase shift. Only the in-phase part of the current, \(I\cos\phi\), does net work over a cycle, while the out-of-phase part alternately stores and returns energy in inductors and capacitors, so the average real power is \(P = VI\cos\phi\) using RMS values. This comes from averaging the instantaneous power \(p(t)=v(t)i(t)\) over one full AC period.
When to use it
Use this when you need the actual power converted into heat, motion, light, or other useful output in an AC circuit or load. It is especially important for motors, transformers, and any load with a non-unity power factor.
Worked example, step by step
A 230 V RMS supply delivers 8.5 A RMS to a load with phase angle \(\phi = 36^\circ\). Step 1: Write the formula: \(P = VI\cos\phi\) Step 2: Substitute the values: \(P = 230 \times 8.5 \times \cos(36^\circ)\) Step 3: Calculate the cosine: \(\cos(36^\circ) \approx 0.809\) Step 4: Multiply: \(P \approx 230 \times 8.5 \times 0.809\) Step 5: Compute the result: \(P \approx 1582\ \text{W}\) Final answer: \(P \approx 1.58\ \text{kW}\)
Common mistakes to avoid
- Using peak voltage and peak current instead of RMS values
- Treating \(VI\) as real power even when voltage and current are out of phase
- Forgetting that \(\cos\phi\) can reduce power dramatically for inductive or capacitive loads
Did You Know?
A poor power factor makes utilities deliver more current than needed, which is why factories are billed to correct it.
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