Question E9B10:
What is the principle of a Method of Moments analysis?  

Choices:  
A: A wire is modeled as a series of segments, each having a uniform value of current  
B: A wire is modeled as a single sine-wave current generator  
C: A wire is modeled as a single sine-wave voltage source  
D: A wire is modeled as a series of segments, each having a distinct value of voltage across it    

Correct Answer is  A: A wire is modeled as a series of segments, each having a uniform value of current  

To analyze wire antennas with the Method of Moments (MoM), we convert a continuous current distribution into a finite set of unknowns by segmenting the wire and assuming a simple current shape on each segment.

Why A is correct
- MoM discretizes the wire into short segments and assumes a simple basis function for current on each segment, most simply a piecewise-constant (uniform) current:
  $$I(z) \approx \sum_{n=1}^{N} I_n f_n(z)$$
  where $f_n(z)$ is nonzero only on segment $n$, and $I_n$ is the (uniform) current on that segment.
- Applying the boundary condition (e.g., tangential electric field equals zero on a PEC) leads to a matrix equation:
  $$\mathbf{Z}\,\mathbf{I} = \mathbf{V}$$
  which is solved for the segment currents $\mathbf{I}$.
- This is the core idea behind MoM/NEC wire modeling.

Why the other choices are incorrect
- B: “Single sine-wave current generator” — MoM does not force a single sinusoidal current along the entire wire; it solves for many segment currents to capture standing-wave variations and discontinuities.
- C: “Single sine-wave voltage source” — The unknowns in MoM are currents distributed along the wire, not a single lumped voltage source.
- D: “Distinct value of voltage across segments” — MoM solves for current coefficients, not segment voltages. For a good conductor, local voltage drop along a short segment is not the primary unknown; fields from segment currents enforce the boundary condition.

Useful formulas and tips
- Electric-field integral equation (EFIE) form (conceptual):
  $$\hat{\mathbf t}\cdot \mathbf{E}^{inc}(\mathbf r) + j\omega\mu \int_{L} G(\mathbf r,\mathbf r')\, I(\mathbf r')\, d\ell' = 0$$
  Expand $I(\mathbf r)$ in basis functions and test to form:
  $$Z_{mn} I_n = V_m$$
- Rule of thumb for segmentation:
  - Segment length ≲ $0.05\lambda$ to $0.1\lambda$
  - Use more, shorter segments near feed points, bends, and discontinuities
  - Typical: 10–20 segments per half-wavelength element

Note: While some implementations use piecewise-linear (“rooftop”) or other basis functions, the exam emphasizes the principle: modeling as a series of short segments each with an assumed uniform current.