Calculate instantaneous voltage v(t) from peak voltage and angle or time using v(t) = Vmax · sin(ωt) for AC sine waves and circuits.

Zero-phase sine wave: v = Vₘ sin(θ), with θ = ωt. Enter peak voltage, not RMS; time and phase may be signed.

Instantaneous Voltage Formula

Instantaneous voltage gives the voltage of an ideal AC sinusoid at a specified moment in time. The full form includes a phase offset:

v(t) = Vₘₐₓ × sin(ω t + varphi)
Symbol Variable Unit
v(t)Instantaneous voltage at time tV
VmaxPeak (maximum) voltageV
ωAngular frequencyrad/s
tTimes
φPhase angle (initial offset)rad

The calculator uses φ = 0 (zero phase offset). Choose “At an angle” to enter the waveform angle ωt, or “At a time” to enter frequency and time; select the voltage result unit. Peak voltage and frequency are nonnegative. Angular frequency, regular frequency, and period relate by:

ω = 2π f = (2π) / (T)

AC Power Standards: Real-World Instantaneous Voltage

For an ideal sinusoidal supply, peak voltage is RMS voltage multiplied by √2. The table gives illustrative nominal examples; actual voltages, waveforms and frequencies can vary. Industrial and transmission voltage examples use the stated voltage basis (for example, line-to-line); do not interchange it with line-to-neutral voltage.

Region Nominal (RMS) Vmax (Peak) Frequency ω (rad/s) Period (ms)
US / Canada120 V169.7 V60 Hz376.9916.67
EU / UK / AU230 V325.3 V50 Hz314.1620.00
Japan100 V141.4 V50 / 60 Hz314.16 / 376.9920.0 / 16.7
US Industrial (3-phase)480 V678.8 V60 Hz376.9916.67
High-Voltage Transmission115,000 V162,635 V60 Hz376.9916.67

Vmax = Vrms × √2. For US household current the complete equation is: v(t) = 169.7 × sin(376.99t) volts.

Voltage Type Comparison

Type Formula 120 V RMS Example Primary Use
InstantaneousVmax × sin(ωt)-169.7 to +169.7 V (varies)Waveform analysis, transient response, oscilloscope readings
PeakVrms × √2169.7 VInsulation ratings, capacitor voltage ratings
RMSVmax / √2 (0.707 × Vmax)120 VPower calculations, utility billing, appliance ratings
Full-wave rectified average(2/π) × Vmax (0.637 × Vmax)108.1 VRectifier and DC power supply design

Key Waveform Positions

For a zero-phase sinusoid, critical voltage values occur at predictable fractions of the period (T = 1/f):

Condition Occurs when Time (t) At 60 Hz At 50 Hz
Zero crossing (rising)ωt = 0, 2π, 4π...0, T, 2T...0, 16.67 ms...0, 20.00 ms...
Positive peak (+Vmax)ωt = π/2T/44.17 ms5.00 ms
Zero crossing (falling)ωt = πT/28.33 ms10.00 ms
Negative peak (-Vmax)ωt = 3π/23T/412.50 ms15.00 ms