Question E5C08:
What coordinate system is often used to display the phase angle of a circuit containing resistance, inductive, and/or capacitive reactance?  

Choices:  
A: Maidenhead grid  
B: Faraday grid  
C: Elliptical coordinates  
D: Polar coordinates    

Correct Answer is  D: Polar coordinates  

To display phase angle in circuits with resistance and reactance, we use phasors. Phasors are naturally represented in polar coordinates.

Why D (Polar coordinates) is correct:
- Impedance and phasors are often expressed as a magnitude and an angle:  
  $$Z = R + jX \quad \rightarrow \quad |Z|\angle\theta$$
- The magnitude is $$|Z| = \sqrt{R^2 + X^2}$$ and the phase angle is $$\theta = \tan^{-1}\!\left(\frac{X}{R}\right)$$
- Phasor diagrams plot these as vectors in a polar plane, where the angle represents the phase difference between voltage and current.

Why the other choices are incorrect:
- A: Maidenhead grid — a geographic location system used in amateur radio, not for circuit phase plots.
- B: Faraday grid — not a standard coordinate system for circuit analysis or phasors.
- C: Elliptical coordinates — a mathematical system used in certain differential equation solutions, not typical for circuit phase/impedance representation.

Useful formulas and rules of thumb:
- Rectangular to polar:  
  $$Z = R + jX \;\Rightarrow\; |Z| = \sqrt{R^2 + X^2},\quad \theta = \tan^{-1}\!\left(\frac{X}{R}\right)$$
- Sign convention:  
  - Inductive reactance: $X_L = +\omega L$ ⇒ current lags voltage (positive phase angle).  
  - Capacitive reactance: $X_C = -\frac{1}{\omega C}$ ⇒ current leads voltage (negative phase angle).
- Power factor:  
  $$\text{PF} = \cos\theta = \frac{R}{|Z|}$$

These relationships are visualized cleanly on a polar plot, hence polar coordinates are used.