Phasors

When circuits are driven by sinusoidal sources operating at a fixed frequency, analyzing them using time-domain differential equations becomes unnecessarily complex. The concept of phasors provides a powerful algebraic alternative by representing sinusoidal quantities as complex numbers, reducing differential equations into simple algebraic ones. This is the foundation of AC circuit analysis and is heavily tested in GATE Network Analysis.

Core Concept Explanation

A sinusoidal signal of the form v(t) = Vm cos(wt + phi) has three characteristics: amplitude Vm, angular frequency w (in rad/s), and phase angle phi (in degrees or radians). When all sources in a circuit operate at the same frequency w, the frequency information becomes redundant in calculations. The only information that changes from element to element is the amplitude and phase. A phasor captures exactly this: it is a complex number of the form V = Vm angle phi that encodes amplitude and phase while discarding the time-varying factor.

The mathematical basis for phasors is Euler's formula, which states that e^(jwt) = cos(wt) + j sin(wt). A sinusoidal signal v(t) = Vm cos(wt + phi) can be written as the real part of Vm e^(j(wt+phi)), which equals the real part of (Vm e^(jphi)) e^(jwt). The factor e^(jwt) is common to all quantities in a single-frequency circuit, so it is suppressed. What remains is the phasor V = Vm e^(jphi) = Vm angle phi.

In rectangular form, the same phasor is written as V = Vm cos(phi) + j Vm sin(phi). The real part represents the in-phase component and the imaginary part represents the quadrature component. Addition and subtraction of phasors follow the same rules as complex number addition, which is far simpler than adding two trigonometric functions in the time domain.

A critical convention to remember is that phasors are typically defined using the cosine reference. If a signal is given in sine form, convert it to cosine first using sin(wt+phi) = cos(wt+phi-90) before extracting the phasor. Mixing sine and cosine references is one of the most common errors in phasor problems.

Mathematical Expression

The formal transformation between time domain and phasor domain is as follows. Given a sinusoid x(t) = Xm cos(wt + phi), the corresponding phasor in polar form is X = Xm angle phi and in rectangular form is X = Xm cos(phi) + j Xm sin(phi). The inverse transformation recovers the time-domain signal as x(t) = Re[X e^(jwt)] = Xm cos(wt + phi).

Differentiation in the time domain corresponds to multiplication by jw in the phasor domain: d/dt corresponds to jw. This is the key property that converts differential equations for capacitors and inductors into algebraic equations. For integration, the correspondence is division by jw: integral corresponds to 1/jw. These transformations make phasor analysis algebraically tractable.

Practical Understanding

Phasors are not just a mathematical convenience. They directly encode what matters in AC power systems: how much voltage or current is present (amplitude) and how it aligns in time relative to a reference (phase). In a purely resistive circuit, voltage and current phasors point in the same direction (same phase). In an inductive circuit, the current phasor lags behind the voltage phasor by 90 degrees. In a capacitive circuit, the current phasor leads the voltage phasor by 90 degrees.

The concept of phase difference between two phasors determines energy flow and power factor in AC systems. It also determines how voltages combine across series elements, which is not a simple algebraic sum in the time domain but becomes straightforward vector addition in the phasor domain. Phasor diagrams are routinely used by engineers to visualize and correct power factor in industrial systems.

Solved Numerical Example

Two sinusoidal voltages in a series circuit are given as v1(t) = 10 cos(100t + 30) V and v2(t) = 8 cos(100t - 45) V. The total voltage is the sum v(t) = v1(t) + v2(t). In the time domain, adding these directly is tedious. Using phasors, V1 and V2 are converted to rectangular form, added as complex numbers, and then converted back to polar form to recover the amplitude and phase of the resultant sinusoid.

Given:
v1(t) = 10 cos(100t + 30°) V
v2(t) = 8 cos(100t - 45°) V
Frequency w = 100 rad/s

Why this formula applies:
Both signals share the same frequency. Phasors can be added as complex numbers.

Formula:
V = V1 + V2 (complex addition in rectangular form)

Substitution:
V1 = 10 cos(30°) + j 10 sin(30°) = 8.66 + j5.00
V2 = 8 cos(-45°) + j 8 sin(-45°) = 5.66 - j5.66

Calculation:
V = (8.66 + 5.66) + j(5.00 - 5.66)
V = 14.32 + j(-0.66)
|V| = sqrt(14.32² + 0.66²) = sqrt(205.06 + 0.44) = sqrt(205.50) ≈ 14.34
angle(V) = arctan(-0.66 / 14.32) ≈ -2.64°

Final Answer with units:
v(t) = 14.34 cos(100t - 2.64°) V

Exam Tip: Always convert sine-referenced sinusoids to cosine form before finding phasors. Use sin(wt+phi) = cos(wt+phi-90°). In GATE, many problems mix sine and cosine references intentionally to test this conversion.

• A phasor is a complex number representing a sinusoid's amplitude and phase at a fixed frequency, discarding the time-varying e^(jwt) factor.
• Polar form: X = Xm angle phi. Rectangular form: X = Xm cos(phi) + j Xm sin(phi).
• Differentiation in time domain corresponds to multiplication by jw in phasor domain, converting differential equations to algebraic ones.
• Phasors can be added, subtracted, multiplied, and divided using standard complex number arithmetic.
• Always use the cosine reference. Convert sine-form signals using sin(wt+phi) = cos(wt+phi-90) before extracting phasors.

Quick Revision

• Phasor of v(t) = Vm cos(wt+phi) is V = Vm angle phi (polar) or Vm e^(jphi) (exponential).
• d/dt in time domain = multiply by jw in phasor domain. Integration = divide by jw.
• For resistor: V = IR (phasors in phase). For inductor: V = jwLI. For capacitor: V = I/(jwC).
• Phasor addition requires rectangular form. Phasor magnitude and angle use polar form.
• ELI the ICE man: In inductor (L), E leads I; In capacitor (C), I leads E.
• GATE trap: Mixing sine and cosine reference without converting leads to 90 degree phase error.
• Phasor analysis is valid only when all sources operate at the same frequency.