Phasors: ac current and voltage as rotating vectors

What is a phasor?

In plain words: picture the hand of a clock turning the wrong way, anticlockwise, once per cycle. The height of its tip above the centre line rises and falls exactly like a sine wave, just as the projection of uniform circular motion is simple harmonic. Each ac quantity gets its own hand; the angle between two hands is their phase difference.

Conventions: this page uses peak-length rotating phasors. If a diagram instead uses rms lengths, multiply its projections by √2 to recover instantaneous values. Voltage and current arrows can share a diagram to compare phase, but have separate unit scales and cannot be added to each other. Pure L/C phase statements refer to zero-offset sinusoidal operation.

Phasor diagram

• A phasor rotates anticlockwise about the origin at angular speed ω. The length of the voltage phasor V is the peak voltage v_m, and of the current phasor I the peak current i_m (NCERT page 181, Figure 7.4).
• Their projections on the vertical axis, v_m sin ωt and i_m sin ωt, are the values of the voltage and current at that instant. As the phasors rotate, these projections trace out the curves of v and i against ωt (NCERT page 181, Figure 7.4(b)).
• For a resistor, V and I point in the same direction at all times: the phase angle between voltage and current is zero (NCERT page 181).
• Phasors are not vectors in space. Voltage and current are scalars. NCERT explains that the amplitudes and phases of sinusoidally varying scalars combine in the same way as the projections of rotating vectors, so phasors give a simple way to add them using a rule we already know (NCERT footnote, page 181).

Reading phase from a phasor diagram

• The angle between two phasors is the phase difference between the quantities. Since phasors rotate anticlockwise, a phasor that is behind another in the rotation lags it (reasoned from NCERT page 181).
• For an inductor, the current phasor is π/2 behind the voltage phasor: the current lags the voltage by a quarter cycle (NCERT Section 7.4, page 182). For a capacitor, the current phasor is π/2 ahead: the current leads (NCERT Section 7.5, pages 184–185).
• If the current phasor is an angle φ ahead of the voltage phasor, then v = v_m sin ωt and i = i_m sin(ωt + φ); if it is behind, φ is negative (reasoned).
• A cosine is a sine a quarter turn ahead: i_m cos ωt = i_m sin(ωt + π/2), so its phasor is π/2 ahead (NCERT uses this step in Section 7.5, page 184).
• Adding two sinusoidal quantities of the same frequency is done by adding their phasors like vectors: the resultant phasor’s length is the amplitude of the sum, and its angle gives its phase. NCERT uses this for the series LCR circuit (NCERT Section 7.6, page 187).

• Phasors are often drawn with rms lengths instead of peak lengths. Dividing every length by √2 changes no angle, so rms meter readings add as phasors too: 30 V across a resistor and 40 V across an inductor in series give √(30² + 40²) = 50 V across the source, not 70 V (reasoned; NCERT Section 7.6, page 187).

Solved examples

Common mistakes to avoid

Quick revision summary

• A phasor is a vector rotating anticlockwise at angular speed ω; its length is the peak value of an ac quantity.
• The projection of a phasor on the vertical axis gives the instantaneous value, such as v = v_m sin ωt.
• The angle between two phasors is the phase difference; for a resistor, V and I point the same way.
• Current lags voltage by π/2 in an inductor and leads it by π/2 in a capacitor.
• Sinusoids of the same frequency add like their phasors; voltage and current themselves are scalars.

Questions students ask

What is the difference between a phasor and a vector?

A vector stands for a quantity with a direction in space, such as a force, while a phasor is an arrow in a diagram standing for a quantity that varies sinusoidally in time, such as an ac voltage: its length is the amplitude, its angle from the horizontal axis is the phase ωt + φ, and its rotation at ω is simply time passing (NCERT page 181 and its footnote). So the angle between two phasors is a phase difference, not an angle in space, and turning the entire diagram preserves relative phases but advances the represented time. At a fixed clock time the instantaneous values do change; alternatively, a common phase shift can represent a new choice of time origin. What phasors take from vectors is the rule of addition, which works for quantities of one frequency and the same physical kind: voltages add to voltages, currents to currents.

Why can two phasors be added like vectors?

Two phasors can be added like vectors because the vertical projection of a vector sum is the sum of the projections, so at every instant the resultant’s projection is v₁ + v₂; and if both quantities have the same ω, the phasors keep a fixed angle and the parallelogram turns rigidly, so the resultant is a phasor of fixed length, a sinusoid of that frequency (reasoned from NCERT page 181 and its footnote). With different frequencies the angle between the phasors keeps changing, the resultant’s length rises and falls, and the sum is not a sinusoid. So one phasor diagram holds only quantities of one frequency, such as the voltages round a circuit driven by one source.

Why do phasors rotate anticlockwise?

Phasors rotate anticlockwise by convention, the one trigonometry uses for positive angles from the +x-axis, so a phasor turning at ω makes the angle ωt + φ and its vertical projection is A sin(ωt + φ) (NCERT pages 182 and 185, where the phasors turn counter-clockwise). Nothing physical depends on the choice, but it must be kept throughout: with the sense reversed, a phasor that leads would appear to be behind, and every lead would be read as a lag. NCERT’s reference circle for SHM turns the same way (NCERT Class 11 Physics Part II, page 264); the clockwise case is worked under clockwise motion.

How do you find the phase difference between voltage and current from the time between their peaks?

The phase difference between voltage and current is the time between matching points of the two waves as a fraction of the period, times 2π: φ = 2πΔt/T = ωΔt in radians, or 360° × Δt/T (reasoned from ω = 2π/T; NCERT page 183 uses it for the quarter-period lag in an inductor; see phase in SHM). On 50 Hz mains, with T = 20 ms, a current that peaks 4.0 ms after the voltage lags it by 360° × 4.0/20 = 72°, about 1.26 rad. The wave that peaks first leads; if the gap is more than half a period, it is simpler to say that the other wave leads by T − Δt.

How is a phasor diagram related to the reference circle of SHM?

A phasor diagram is the reference circle of simple harmonic motion applied to ac: in both, an arrow as long as the amplitude turns anticlockwise at ω, and its projection on a fixed line gives the oscillating quantity at each instant (NCERT Class 11 Physics Part II, pages 264–265; NCERT page 181). Class 11 projects on the x-axis, x = A cos(ωt + φ), and NCERT’s phasors on the vertical axis, v = v_m sin ωt; either diameter works. A phasor diagram adds several arrows of one frequency to one picture, such as a voltage and a current, so their phase difference is read as an angle.

Is the phasor diagram in the JEE Main and NEET syllabus?

The phasor diagram is not named in either syllabus, but both list the topics that NCERT solves with it: Unit 14 of the JEE Main syllabus has “reactance and impedance, LCR series circuit, resonance, power in AC circuits”, and Unit 14 of the NEET (UG) 2026 syllabus has “reactance and impedance: LCR series circuit, resonance: power in AC circuits”. NCERT introduces phasors in Section 7.3 (page 181) and finds the current, impedance and phase angle of the series LCR circuit from a phasor diagram in Section 7.6 (pages 186–188), so the phasor diagram is the standard tool for those topics in both exams.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. What does the vertical projection of a voltage phasor represent?

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   The value of the voltage at that instant, v = v_m sin ωt (NCERT page 181).

2. Q2. What is the angle between the voltage and current phasors for a pure resistor?

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   Zero: they point in the same direction at all times (NCERT page 181).

3. Q3. Two sinusoidal currents of the same frequency have amplitudes 6 A and 8 A and are 90° out of phase. What is the amplitude of their sum?

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   √(6² + 8²) = 10 A (reasoned with the phasor rule, NCERT page 181).