In AC circuit theory, current and voltage are said to be “in phase” when their time-varying waveforms cross zero together and reach positive and negative peaks at the same instants (same phase angle), regardless of amplitude. Which statement captures this condition most precisely?

Introduction / Context:
In alternating-current (AC) analysis, “in phase” describes a precise timing relationship between two sinusoidal quantities such as voltage and current. When two sinusoids are in phase, their instantaneous values rise and fall together with no time lead or lag. This is fundamental for understanding power factor, reactive elements, and resonance.

Given Data / Assumptions:

• Both signals are periodic sinusoids at the same frequency.
• We are comparing the timing of their zero crossings and peaks.
• Amplitude equality is not required for phase equality.

Concept / Approach:
Phase angle describes the shift between two sinusoids of the same frequency. A phase angle of 0 degrees means the waveforms reach corresponding points (zero crossings, peaks, troughs) at the same instants. Different amplitudes do not change phase; a larger amplitude simply scales the waveform vertically. Likewise, being both AC or both DC does not define phase; DC has no phase in the sinusoidal sense.

Step-by-Step Solution:
Define in-phase: phase angle = 0 degrees.Identify practical test: simultaneous zero crossings and simultaneous positive/negative peaks.Note that amplitude equality is unnecessary; only timing matters.Select the statement that states the timing criterion explicitly.

Verification / Alternative check:
Express sinusoids as v = Vpsin(ωt) and i = Ipsin(ωt + φ). If φ = 0, v and i reach all characteristic points simultaneously. Oscilloscope Lissajous figures for in-phase signals show a straight line with positive slope when scales match.

Why Other Options Are Wrong:
Both AC or both DC: does not define phase alignment.Same amplitude and frequency: amplitude is irrelevant to phase; frequency must match but that alone does not guarantee zero phase shift.All of the above: includes incorrect conditions, therefore false.

Common Pitfalls:
Confusing equal amplitude with equal phase; forgetting that equal frequency without specifying phase could still mean a lead/lag.

Final Answer:
their waveforms cross through zero and reach positive and negative peaks at the same time