Phasor Analysis

Why it matters

Phasor analysis is essential for simplifying the analysis of AC circuits, which are prevalent in power systems and electronic devices. By converting sinusoidal functions into phasors, engineers can easily perform calculations involving impedance, voltage, and current, leading to more efficient circuit design and troubleshooting.

Key ideas

• Phasors: A phasor is a complex number representing the magnitude and phase of a sinusoidal function. It simplifies the analysis of AC circuits by transforming differential equations into algebraic ones.
• Complex Numbers: Phasors use complex numbers, where use the convention v(t) = √2 Re{V exp(jωt)} for RMS phasor V. Thus V = a+jb gives √2[a cosωt − b sinωt]; the sign follows the chosen exponential convention.
• Impedance: In phasor analysis, impedance (Z) is a complex quantity combining resistance (R) and reactance (X), expressed as Z = R + jX.
• Ohm’s Law in Phasors: The phasor form of Ohm’s Law is V = IZ, where V and I are phasors representing voltage and current, respectively.
• Phasor Diagrams: These are graphical representations of phasors, showing the relationships between different sinusoidal quantities in a circuit.

Formulas

• V = IZ
  • V: Voltage (Volts)
  • I: Current (Amperes)
  • Z: Impedance (Ohms)
• Z = R + jX
  • R: Resistance (Ohms)
  • X: Reactance (Ohms)
  • j: Imaginary unit
• I = V/Z
  • I: Current (Amperes)
  • V: Voltage (Volts)
  • Z: Impedance (Ohms)

Phasors below are RMS values in single-frequency sinusoidal steady state. They do not include switching transients.

Worked example

Given:

• Voltage, V = 230∠30° V
• Impedance, Z = 10 + j5 Ω

Steps:

1. Convert the given voltage into a phasor: V = 230∠30°
2. Use Ohm’s Law in phasor form: I = V/Z
3. Substitute the values: I = (230∠30°) / (10 + j5)
4. Calculate the magnitude of Z: |Z| = √(10² + 5²) = √125 = 11.18 Ω
5. Calculate the phase angle of Z: θ = tan⁻¹(5/10) = 26.57°
6. Calculate the current phasor: I = 230∠30° / 11.18∠26.57°
7. Simplify: I = 20.57∠3.43° A

Final Answer: 20.57∠3.43° A

Common mistakes

• Confusing the angle of a phasor with its magnitude.
• Forgetting to convert angles from degrees to radians when necessary.
• Incorrectly adding or subtracting phasors without considering their angles.

For GATE EE

Phasor analysis questions often involve calculating current, voltage, or impedance in AC circuits. Practice converting between time-domain and phasor-domain representations, and solving circuit problems using phasor diagrams and complex algebra.

Quick check

1. What is a phasor?
2. How is impedance represented in phasor analysis?
3. What is the phasor form of Ohm’s Law?

Answers: 1. A complex number representing magnitude and phase of a sinusoidal function. 2. As a complex number Z = R + jX. 3. V = IZ.