Time Domain vs. Phasor Domain

• Time Domain Signal:

  $$ v(t) = V_m \cos(\omega t + \phi) $$
  • Represents instantaneous voltage as a function of time.
  • Harder to manipulate analytically for AC circuit analysis.

• Phasor Domain:

  $$ V = V_m e^{j\phi} \quad \text{or} \quad V = V_m (\cos(\phi) + j \sin(\phi)) $$
  • Represents a complex number encoding amplitude and phase.
  • Omits time dependency (only valid for steady-state sinusoidal signals).
  • Based on Euler’s formula: $$ e^{j\phi} = \cos(\phi) + j \sin(\phi) $$
  • Shorthand notation is $V_m \angle \phi$

Conversion Between Phasor and Time Domain

• From Phasor to Time Domain: Multiply phasor $V$ by $e^{j\omega t}$, then take the real part

  $$ v(t) = \Re\{ V e^{j\omega t} \} = V_m \cos(\omega t + \phi) $$

• From Time Domain to Phasor Domain: Drop the time-dependent $e^{j\omega t}$ and keep only the amplitude and phase

  $$ v(t) = V_m \cos(\omega t + \phi) \quad \Rightarrow \quad V = V_m e^{j\phi} $$

Phasor Arithmetic

Let $V_1 = A e^{j\theta_1}$, $V_2 = B e^{j\theta_2}$:

• Addition/Subtraction:

  • Convert to rectangular form and add real/imaginary parts separately $$ V = V_{m1}(\cos\phi_1 + j\sin\phi_1) \pm V_{m2}(\cos\phi_2 + j\sin\phi_2) $$

• Multiplication: Multiply magnitudes, add angles

  $$ V_1 \cdot V_2 = AB e^{j(\theta_1 + \theta_2)} $$

• Division: Divide magnitudes, subtract angles

  $$ \frac{V_1}{V_2} = \frac{A}{B} e^{j(\theta_1 - \theta_2)} $$

Sinusoidal Steady State Analysis

RC Circuit Example

RL Circuit Example

• Should have $\cos$ at the end

Phasor Arithmetic with Multiple Frequencies

• Cannot be done, have to use Superposition instead

References

• Course handout 2: phasor and time domains