Login Free Test Start Mock

Physics · Alternating Current

Phasors — Representation of AC

When a circuit element other than a pure resistor is driven by alternating voltage, the current is no longer in step with the voltage. NCERT §7.3 introduces the phasor — a rotating vector whose vertical projection reproduces a sinusoid — to capture this phase relationship at a glance. For NEET, the phasor diagram is the single tool that converts every reactance, lead–lag and LCR impedance problem into simple geometry.

Why we need phasors

For a pure resistor, NCERT shows that the voltage and the current reach zero, minimum and maximum values at the same respective times — they are in phase. The instantaneous voltage $v = v_m\sin\omega t$ and the current $i = i_m\sin\omega t$ track each other exactly.

This simple agreement breaks down the moment an inductor or a capacitor enters the circuit. There the current no longer rises and falls together with the voltage; one quantity is shifted ahead of or behind the other. To display this phase relationship cleanly, NCERT §7.3 introduces the notion of phasors.

ElementPhase relation (current vs voltage)Phasor picture
ResistorIn phase, $\phi = 0$$I$ parallel to $V$
InductorCurrent lags voltage by $\pi/2$$I$ is $\pi/2$ behind $V$
CapacitorCurrent leads voltage by $\pi/2$$I$ is $\pi/2$ ahead of $V$

What a phasor is

A phasor is a vector that rotates about the origin with angular speed $\omega$. The magnitude of the phasor represents the amplitude, or peak value, of the oscillating quantity — $v_m$ for voltage and $i_m$ for current. The phasors $V$ and $I$ rotate counter-clockwise, and as they turn they generate the sinusoidal voltage and current curves.

NCERT is careful on one subtle point: although voltage and current in an AC circuit are represented by phasors, they are not vectors themselves — they are scalar quantities. The amplitudes and phases of harmonically varying scalars happen to combine in exactly the same way as the projections of rotating vectors of matching magnitude and direction. The rotating-vector picture is therefore only a convenient device for adding sinusoids using a rule we already know.

Figure 1 · The rotating phasor x vertical axis V (length = v₄) v₄ sinωt ωt

The phasor $V$ has length $v_m$. At an angle $\omega t$ from the horizontal, its projection on the vertical axis is $v_m\sin\omega t$ — the instantaneous voltage. As $V$ rotates, this projection sweeps out the AC waveform.

Projection gives the instantaneous value

The defining rule of the diagram: the projection of a phasor on the vertical axis gives the instantaneous value of the quantity at that moment. For the voltage and current phasors of a resistor circuit, these projections are $v_m\sin\omega t$ and $i_m\sin\omega t$ respectively. As the phasors rotate with frequency $\omega$, the projections trace out the familiar sinusoidal curves of $v$ and $i$ against $\omega t$.

This is why the length must be the peak value. The vertical projection of a vector of length $v_m$ tilted at $\omega t$ is $v_m\sin\omega t$, which equals the instantaneous voltage by construction. If the diagram were drawn with the RMS length $v_{rms}=v_m/\sqrt{2}$, the projection would no longer reproduce the correct instantaneous waveform.

NEET Trap

Phasor length is the peak value, not the RMS value

A phasor's length stands for the amplitude $v_m$ or $i_m$ — the peak value. Students who instinctively reach for the RMS value when drawing the diagram break the projection rule, because $v_m\sin\omega t$ is what must appear on the vertical axis. RMS values are used when computing power and reading meters, not for setting the geometric length of a phasor.

Length $=$ peak $v_m$ (so projection $= v_m\sin\omega t$). Recall $v_m = \sqrt{2}\,v_{rms}$.

The angle is the phase

The instantaneous angle the phasor makes with the horizontal axis is the phase of the sinusoid, $\omega t$. Adding a phase constant simply rotates the starting orientation: a current written as $i = i_m\sin(\omega t + \phi)$ corresponds to a phasor that sits an angle $\phi$ ahead of a reference phasor drawn at angle $\omega t$.

Because both the voltage and current phasors rotate at the same angular speed $\omega$, the angle between them never changes as the diagram spins. That fixed angle is precisely the phase difference between voltage and current — a constant of the circuit, independent of the instant chosen.

↻
Build on this

See the phasor's first application where current and voltage stay perfectly aligned in AC Applied to a Resistor.

Reading phase difference off the diagram

For a pure resistor, the phasors $V$ and $I$ point in the same direction at all times. The angle between them is zero, so the phase difference is zero and the current is in phase with the voltage. The diagram collapses to a single line carrying two parallel arrows.

For reactive elements the phasors separate. With a counter-clockwise rotation convention, a phasor that sits ahead in the direction of rotation leads; one that sits behind lags. NCERT records that in a purely inductive circuit the current phasor $I$ is $\pi/2$ behind the voltage phasor $V$ (current lags), while in a purely capacitive circuit $I$ is $\pi/2$ ahead of $V$ (current leads).

Figure 2 · Two phasors with phase angle V I Resistor: φ = 0 V I φ = π/2 Capacitor: I leads V

Left: a resistor's phasors are parallel, so $\phi=0$. Right: a capacitor's current phasor is $\pi/2$ ahead of the voltage phasor, so the current leads. The constant angle between the arrows is the phase difference.

Phasors in the series LCR circuit

The real payoff of phasors appears in the series LCR circuit (NCERT §7.6.1). Because the resistor, inductor and capacitor are in series, the current is the same in each element at any instant, with one common amplitude and phase, $i = i_m\sin(\omega t + \phi)$. Taking $I$ as the reference phasor, the element voltages have fixed orientations: $V_R$ is parallel to $I$, $V_C$ is $\pi/2$ behind $I$, and $V_L$ is $\pi/2$ ahead of $I$.

The lengths of these voltage phasors are $v_{Rm}=i_m R$, $v_{Cm}=i_m X_C$ and $v_{Lm}=i_m X_L$. The loop equation $v_L + v_R + v_C = v$ is the vertical-component statement of the phasor sum $\mathbf{V_L}+\mathbf{V_R}+\mathbf{V_C}=\mathbf{V}$. Since $V_L$ and $V_C$ lie along the same line in opposite directions, they combine into a single phasor of magnitude $|v_{Cm}-v_{Lm}|$, and $V$ becomes the hypotenuse of a right triangle with legs $V_R$ and $(V_C+V_L)$.

Worked Example

A purely capacitive circuit is driven by $v = v_m\sin\omega t$. Using the phasor picture, write the current and state its phase relative to the voltage.

In the phasor diagram the current phasor $I$ is $\pi/2$ ahead of the voltage phasor $V$. Projecting onto the vertical axis, $i = i_m\sin\!\left(\omega t + \dfrac{\pi}{2}\right)$. The current therefore leads the voltage by $\pi/2$, reaching its peak one-quarter period earlier than the voltage — consistent with the rightmost diagram in Figure 2.

Quick Recap

Phasors in one screen

  • A phasor is a vector rotating about the origin at angular speed $\omega$; its length is the peak value ($v_m$ or $i_m$).
  • The projection on the vertical axis gives the instantaneous value, e.g. $v_m\sin\omega t$.
  • The phasor's angle with the horizontal is the phase $\omega t$; a constant $\phi$ rotates the start orientation.
  • Phase difference is the fixed angle between two phasors, since both rotate at the same $\omega$.
  • Resistor: $\phi=0$ (parallel). Inductor: current lags by $\pi/2$. Capacitor: current leads by $\pi/2$.
  • Phasors are a tool only — voltage and current are scalars, not true vectors.

NEET PYQ Snapshot — Phasors — Representation of AC

Real NEET questions where reading lead, lag and phase angle from the phasor picture decides the answer.

NEET 2016

A small signal voltage $V(t)=V_0\sin\omega t$ is applied across an ideal capacitor $C$:

  1. Over a full cycle the capacitor $C$ does not consume any energy from the voltage source
  2. Current $I(t)$ is in phase with voltage $V(t)$
  3. Current $I(t)$ leads voltage $V(t)$ by 180°
  4. Current $I(t)$ lags voltage $V(t)$ by 90°
Answer: (1)

In an ideal capacitor the current phasor leads the voltage phasor by 90°, and over a full cycle the capacitor consumes no energy. The phasor picture rules out options (2)–(4), and the wattless-current property fixes (1) as correct.

NEET 2020

A series LCR circuit is connected to a voltage source. When $L$ is removed, the phase difference between current and voltage is $\pi/3$. If instead $C$ is removed, the phase difference is again $\pi/3$. The power factor of the circuit is:

  1. 0.5
  2. 1.0
  3. −1.0
  4. zero
Answer: (2)

Removing $L$ gives $\tan\phi = X_C/R$; removing $C$ gives $\tan\phi = X_L/R$. Equal phase angles force $X_L = X_C$, i.e. resonance. With the reactances cancelling on the phasor diagram, $Z=R$, so $\cos\phi = R/Z = 1$.

NEET 2025

To a 220 V, 50 Hz AC supply, a resistor of 20 Ω, a capacitor of reactance 25 Ω and an inductor of reactance 45 Ω are connected in series. The current in the circuit and the phase angle between current and voltage are, respectively:

  1. 15.6 A and 45°
  2. 7.8 A and 30°
  3. 7.8 A and 45°
  4. 15.6 A and 30°
Answer: (3)

On the phasor diagram the net reactance is $X_L - X_C = 45-25 = 20\ \Omega$, equal to $R$. Then $\tan\phi = (X_L-X_C)/R = 20/20 = 1$, giving $\phi = 45°$, and $I = 220/\sqrt{20^2+20^2} = 220/(20\sqrt2) \approx 7.8\ \text{A}$.

FAQs — Phasors — Representation of AC

Common doubts on phasor length, projection and phase difference, answered from NCERT §7.3.

What is a phasor?
A phasor is a vector that rotates about the origin with angular speed ω. Its magnitude equals the peak value (amplitude) of the sinusoidal quantity it represents, and its vertical-axis projection at any instant gives the instantaneous value of that voltage or current. Voltage and current in an AC circuit are represented by phasors, but they are scalar quantities, not vectors themselves.
Does the length of a phasor represent the RMS value or the peak value?
The length of a phasor represents the peak value (amplitude), such as vm or im, not the RMS value. The instantaneous value is obtained by projecting this peak-length phasor onto the vertical axis. Using the RMS value as the length is a common error; the projection of an RMS-length phasor would not reproduce the correct vm sin ωt waveform.
How is the instantaneous value obtained from a phasor?
The instantaneous value is the projection of the phasor onto the vertical axis. For a voltage phasor of length vm tilted at angle ωt from the horizontal, the vertical projection is vm sin ωt, which equals the instantaneous voltage v at that moment. As the phasor rotates with frequency ω, this projection traces out the sinusoidal waveform.
How does a phasor diagram show phase difference?
The phase difference is the fixed angle between the two phasors. Both phasors rotate together at the same angular speed ω, so the angle between them stays constant. If the current phasor is ahead of the voltage phasor in the counter-clockwise direction, the current leads; if it is behind, the current lags. For a pure resistor the two phasors are parallel, so the phase angle is zero.
Why do current and voltage have zero phase difference in a resistor?
For a pure resistor the voltage and current phasors point in the same direction at all times. Their vertical projections, vm sin ωt and im sin ωt, reach zero, minimum and maximum at the same instants. Because the phasors are always parallel, the phase angle between voltage and current is zero, meaning they are in phase.
Are phasors actually vectors?
No. According to NCERT, voltage and current in an AC circuit are represented by phasors, but they are not vectors themselves; they are scalar quantities. Harmonically varying scalars combine mathematically in the same way as the projections of rotating vectors of corresponding magnitudes and directions, so the rotating-vector picture is only a convenient tool for adding sinusoids.
Related Topics
Alternating Current Back to the full chapter hub. AC Applied to a Resistor The in-phase case: parallel phasors, φ = 0. AC Applied to an Inductor Current lags voltage by π/2 on the diagram. AC Applied to a Capacitor Current leads voltage by π/2 on the diagram. Series LCR Circuit & Resonance Phasor addition gives impedance and resonance. Power in AC Circuit & Power Factor Phase angle φ sets cos φ and power. Transformers Stepping AC voltage up and down.