Phasor Analysis, Complex Representation, and AC Resistor Response

Fundamentals of Phasors and Complex Representation

• Phasors are abstract vector representations of sinusoidal signals that combine both signal magnitude and phase angle into a unified complex value.
• Phasors behave as geometric vectors in the complex domain.
• Multiplication by the imaginary unit $j$:
  • Multiplying a voltage signal by $j$ does not alter its fundamental magnitude relative to the origin.
  • A multiplication by $j$ introduces an exact $90^\rho$ phase shift (a counterclockwise rotation of $90^\rho$ in the complex plane).
• Polar phasor notation structure:
  • A polar phasor vector is denoted as $\mathbf{V}_a = 20 \angle -30^\circ$
  • The scalar value $20$ represents the magnitude, specifying the precise physical length of the vector.
  • The angular value $-30^\circ$ represents the phase angle, defining the orientation of the vector relative to the horizontal real axis.

Polar to Rectangular Form Conversion

• The phase angle explicitly defines the spatial rotation between the vector and the horizontal real axis.
• Converting a polar phasor $\mathbf{V} = V_m \angle \theta$ into rectangular form requires projecting the magnitude onto the real and imaginary coordinate axes:
  • Real axis component: $V_{real} = V_m \times \cos(\theta)$
  • Imaginary axis component: $V_{imag} = V_m \times \sin(\theta)$
• For the specific vector magnitude $20$ at a phase angle of $-30^\circ$:
  • Real component formula: $20 \times \cos(-30^\circ) = 20 \times \frac{\sqrt{3}}{2} \approx 17.32$
  • Imaginary component formula: $20 \times \sin(-30^\circ) = 20 \times \left(-\frac{1}{2}\right) = -10$
  • Rectangular form solution: $\mathbf{V}_a = 17.32 - j 10$
• Calculating the magnitude from rectangular components follows the Pythagorean distance relation: $V_m = \sqrt{(V_{real})^2 + (V_{imag})^2}$

Signal Frequency and RMS Calculations

• Impact of Frequency Doubling on Phasor Behavior:
  • Doubling the operating frequency ($f \rightarrow 2f$) doubles the angular velocity ($\omega = 2 \pi f \rightarrow 2\omega$) at which the vector rotates in the complex plane.
  • In static time-independent phasor notation, the phasor magnitude and initial reference phase angle remain unchanged, but its dynamic rotation rate in time-domain animations is twice as fast.
• Root Mean Square (RMS) Value Calculation:
  • Peak voltage value ($V_{peak}$ or $V_m$): $339$
  • The formula relating RMS voltage to peak voltage for a sinusoidal signal is: $V_{rms} = \frac{V_{peak}}{\sqrt{2}}$
  • Substituting the peak value gives: $V_{rms} = \frac{339}{\sqrt{2}} \approx 239.71$

AC Resistor Response and Phasor Ohm's Law

• Application of Ohm's Law to AC Circuit Elements:
  • Ohm's Law applies directly to phasors in alternating current (AC) circuit analysis.
  • Phasor current relation: $\mathbf{I} = \frac{\mathbf{V}}{R}$
  • Phasor voltage relation: $\mathbf{V} = \mathbf{I} R$
• Phase Relationship across Resistive Components:
  • Resistance $R$ is a purely real scalar quantity with zero imaginary component ($R \angle 0^\circ$).
  • The voltage phasor $\mathbf{V}$ and current phasor $\mathbf{I}$ across a resistor pass through the circuit strictly in phase with one another.
  • Zero angular displacement exists between current and voltage for a resistive load in the phasor view.