When is instantaneous voltage maximum?

At 90° (or π/2 radians) in the cycle, sin = 1, so v = Vpeak. For UK mains at 50Hz, this occurs at t = 5ms into each cycle.

When is instantaneous voltage zero?

At 0° and 180° (the zero crossings). For 50Hz mains, zero crossings occur every 10ms.

What is the phase offset φ?

The phase offset shifts the waveform in time. φ = 0 means the wave starts at zero and rises. φ = 90° means the wave starts at its peak (cosine wave). It is used when comparing multiple signals.

Can I enter time in degrees instead of milliseconds?

Yes. Select ° (angle) from the unit dropdown. One full cycle is 360° regardless of frequency. This is useful when thinking in terms of waveform position rather than absolute time.

How do I find when the voltage reaches a specific value?

Use the reverse solve tab. Enter peak voltage, frequency and target voltage. The calculator shows both times per cycle when that voltage occurs (rising and falling).

Instantaneous Voltage Calculator

v(t) = V_pksin(2πft + φ)

ω = 2πf

v(t) = V_pksin(ωt + φ)

Enter Waveform Parameters

Vp Peak Voltage

Or enter RMS × √2

f Frequency

t Time or Angle

Enter time or select degrees/radians

φ Phase Offset

Initial phase (default 0°)

Enter peak voltage, frequency and time

Instantaneous Voltage

v(t) Instantaneous

—

Effective Angle

—

v(t)
—
V

ωt + φ

—

degrees

% of Peak

—

V_rms

—

Period T

—

rad/s

Find Time for a Target Voltage

When does v(t) first reach the target value?

Vp Peak Voltage

f Frequency

v Target Voltage

Must be between −Vpk and +Vpk

Reverse Solve Result

First Occurrence (rising)

—

Second Occurrence (falling)

—

Figure 1: A sine wave cycle has four key points: zero at 0°, positive peak at 90°, zero at 180°, and negative peak at 270°. The instantaneous voltage at any other angle is V_pk × sin(angle).

Table of Contents

Fundamentals

What Is Instantaneous Voltage?
The Formula

Worked Examples

UK Mains at 5 ms
Audio Signal at 45°
Finding Zero Crossings

Deep Dive

Key Points on the Cycle
Relationship to RMS
Phase Offset

Reference

Frequently Asked Questions
Related AC Circuit Calculators

What Is Instantaneous Voltage?

Instantaneous voltage is the exact voltage at a specific moment in time. Unlike RMS or peak voltage, which describe the overall waveform, instantaneous voltage tells you what the voltage is right now. For a sine wave it varies continuously between +V_pk and −V_pk, passing through zero twice per cycle.

Knowing the instantaneous voltage is essential for understanding zero-crossing detection, switching timing in power electronics, ADC sampling, and safety analysis (e.g. what voltage is present when a fuse blows at a specific point in the cycle). The RMS Voltage Calculator converts between RMS and peak values.

The Formula

v(t) = V_pk × sin(2πft + φ)
Where V_pk = peak voltage, f = frequency, t = time, φ = phase offset.
Alternatively: v(t) = V_pk × sin(ωt + φ) where ω = 2πf

Worked Example — UK Mains at 5 ms

Given: V_rms = 230 V → V_pk = 325.3 V, f = 50 Hz, t = 5 ms, φ = 0

Step 1 — ω = 2π × 50 = 314.16 rad/s

Step 2 — ωt = 314.16 × 0.005 = 1.5708 rad = 90°

Step 3 — v(5ms) = 325.3 × sin(90°) = 325.3 × 1 = 325.3 V

At t = 5 ms (quarter cycle), the voltage reaches its positive peak. This is when the capacitors in a power supply see maximum charging current and when insulation stress is highest.

Worked Example — Audio Signal at 45°

Given: V_pk = 1.096 V (0.775 V RMS line level), f = 1 kHz, angle = 45°

Step 1 — v = 1.096 × sin(45°) = 1.096 × 0.7071 = 0.775 V

At 45°, the instantaneous voltage of this particular signal equals its RMS value. This is a mathematical coincidence unique to sin(45°) = 1/√2 = the RMS/peak ratio for sine waves.

Worked Example — Finding Zero Crossings

Given: UK mains 50 Hz. When is v(t) = 0?

Step 1 — sin(ωt) = 0 when ωt = 0, π, 2π, ...

Step 2 — t = 0, T/2, T, ... = 0 ms, 10 ms, 20 ms, ...

Zero crossings occur every half-period (10 ms for 50 Hz). Zero-crossing detectors are used in dimmer circuits, synchronised switching, and phase angle measurement.

Key Points on the Cycle

There are four critical points in every sine wave cycle: the positive peak at 90° (v = +V_pk), the first zero crossing at 180° (v = 0, falling), the negative peak at 270° (v = −V_pk), and the second zero crossing at 360° (v = 0, rising). For UK mains at 50 Hz these occur at 5 ms, 10 ms, 15 ms and 20 ms respectively.

Relationship to RMS

RMS voltage is derived from instantaneous voltage by squaring every point, averaging over a full cycle, then taking the square root. For a sine wave this gives V_rms = V_pk/√2. The Impedance Calculator uses RMS values for all its power calculations, while this calculator works with peak and instantaneous values.

Phase Offset

The phase offset φ shifts the entire waveform in time. A cosine wave is simply a sine wave with φ = 90°. In power systems, generators connected in parallel must have matching phase — even a small phase difference creates large circulating currents. The Voltage Phasor Calculator handles multi-source phase relationships.

Frequently Asked Questions

Can instantaneous voltage be negative?

Yes. During the negative half-cycle, v(t) is negative. For UK mains, v ranges from +325 V to −325 V. The sign indicates the polarity reversal that defines alternating current.

Is instantaneous voltage the same as peak voltage?

Only at 90° and 270° in the cycle. At all other points, |v(t)| < V_pk. Peak voltage is the maximum instantaneous voltage.

Why do I need to know instantaneous voltage?

For switching timing (SCR/TRIAC firing angle), capacitor voltage stress analysis, ADC sample timing, zero-crossing detection, and understanding what happens at the moment a circuit breaker trips or a fuse blows.

How does frequency affect instantaneous voltage?

Frequency does not change the peak value, but it changes how quickly the voltage cycles. Higher frequency means the waveform completes more cycles per second, so any given angle is reached sooner in absolute time.

What if my waveform is not a sine wave?

This calculator assumes a pure sine wave. Non-sinusoidal waveforms can be decomposed into harmonics using Fourier analysis, then the instantaneous voltage of each harmonic can be calculated and summed.

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Last updated: March 2026