In a single-phase half-wave AC regulator, the average current over one full cycle is:

Introduction / Context:
A half-wave AC regulator controls conduction in only one half-cycle (depending on device orientation). This inherently introduces a DC component unless special measures are taken. The sign of the average current depends on which half-wave is controlled and the circuit orientation.

Given Data / Assumptions:

• Single thyristor (or controlled device) in a half-wave regulator arrangement.
• Purely resistive load assumed for simplicity.
• Firing occurs in either positive or negative half-cycles depending on connection.

Concept / Approach:

Over a full cycle, conduction in only one half-cycle creates a net average that is nonzero. If the controlled device conducts on positive halves, the average is positive; if it conducts on negative halves, the average is negative. Therefore the sign is not fixed a priori—it depends on connection and control polarity.

Step-by-Step Reasoning:

Verification / Alternative check:

Integrate i(t) = V_msin(ωt)/R over the conduction interval in one half-cycle; the other half is zero. The average sign matches the conducting half-cycle.

Why Other Options Are Wrong:

• Always positive/negative: depends on device orientation.
• Always zero: incorrect for half-wave regulation; that is true only for symmetrical full-wave control with equal angles.

Common Pitfalls:

• Confusing half-wave regulators with full-wave regulators where DC cancels.
• Ignoring that wiring orientation selects which half-cycle is controlled.

Final Answer:

may be positive or negative