AC foundations
Instantaneous sine-wave voltage
- where
- instantaneous voltage (V)
- peak voltage (V)
- angle (°)
Explain this equation
Gives the voltage at one particular moment on a pure sine wave, from its peak value and how far through the cycle you are. Useful for understanding what a scope shows or the voltage at a switching point.
Vₚ is the peak voltage, the highest value reached in each half-cycle. θ is the angle through the cycle in degrees, measured from where the wave crosses zero heading positive; one full cycle is 360°. v is the voltage at that instant, and a negative answer means the polarity has reversed.
A sine wave is what a coil turning steadily in a magnetic field produces, so the voltage follows sin θ as the angle advances. sin θ swings between −1 and +1, scaling the peak up and down.
At 30° the voltage is half the peak, at 90° it equals the peak, and at 180° it is back to zero. A 230 V RMS supply peaks at about 325 V, so at 30° you would see about 163 V.
Watch out: enter the peak, not the RMS value. For a 230 V supply, multiply by √2 first.
RMS, peak and instantaneous values answer different questions. At a specific angle, multiply the peak by the sine of that angle.
Negative values mean the opposite polarity. An instantaneous zero does not mean the RMS supply is zero.
Enter your values to see the result.
Common mistake
Putting RMS voltage in place of peak.
Limits
Pure sine wave with zero phase at its rising zero crossing; degree mode.
Sources
The tools calculate the displayed equation. They do not automatically choose cable sizes, approve installations, supply current regulatory tables, certify equipment or replace inspection and testing.