• Home
  • Textbooks
  • Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures
  • Fundamentals of Electromagnetic Theory

Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures

Cam Nguyen

Chapter 2

Fundamentals of Electromagnetic Theory - all with Video Answers

Educators

Chapter Questions

03:31

Problem 1

Derive the boundary conditions (2.15) between two different media as shown in Fig. 2.2.

Arun Bana
Arun Bana
Numerade Educator

Problem 2

Using the Poynting vector, prove that the average power density of a signal propagating in a waveguide is given by Eq. (2.20).

Check back soon!
04:37

Problem 3

Show that TE modes can be characterized only by the magnetic scalar potential $\psi^h(x, y)$.

Ameer Said
Ameer Said
Numerade Educator

Problem 4

Derive Eqs. (2.51)-(2.54).

Check back soon!
04:37

Problem 5

Show that TM modes can be characterized only by the electric scalar potential $\psi^c(x, y)$.

Ameer Said
Ameer Said
Numerade Educator
01:44

Problem 6

Prove that, in any waveguides, both $E_z$ and $H_z$ cannot be even or odd simultaneously.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 7

Consider a general waveguide with perfectly conducting walls as shown in Fig. 2.3. Derive the following boundary conditions along the surface of the conductor for both TE and TM modes:

TE Modes: $\quad \frac{\partial \psi^h}{\partial n}=0 \quad$ (Neumann's Condition)
TM Modes: $\quad \psi^e=0 \quad$ (Dirichlet's Condition)

Check back soon!

Problem 8

Verify the power orthogonality relationship for nondegenerate modes in a lossless circular waveguide.

Check back soon!

Problem 9

Verify the power orthogonality relationship for nondegenerate modes in a lossless rectangular waveguide.

Check back soon!
09:09

Problem 10

The electric and magnetic fields of the eigenmodes existing in planar transmission lines are normally determined approximately. Show that, under this condition, the orthogonality relationship (2.100) does not hold. Show also that, if these fields were calculated with a good accuracy, the orthogonality relation is well satisfied.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator

Problem 11

Derive the constant $C$ in Eq. (2.109) that describes the fields of a new set of modes in terms of the fields of nonorthogonal degenerate modes, such that the new modes satisfy the power orthogonality (2.108).

Check back soon!

Problem 12

Derive the orthogonality relations (2.84b) and (2.85) for TM and TE modes, respectively.

Check back soon!

Problem 13

Derive the orthogonality relations (2.86) for hybrid modes.

Check back soon!

Problem 14

Derive the orthogonality relation (2.101) and (2.103) for a lossless waveguide.

Check back soon!

Problem 15

In general, there exist many degenerate modes in a waveguide. Some are coupled together while others are not. Prove that the mode coupling does not take place between the two degenerate $\mathrm{TE}_{m n}$ and $\mathrm{TM}_{m n}$ modes in a rectangular waveguide: that is, prove that the power-interaction terms

$$
P_{\mathrm{TE}}^{\mathrm{TM}}=\frac{1}{2} \iint_S \mathbf{E}_{\mathrm{TE}_{\operatorname{man}}} \times \mathbf{H}_{\mathrm{TM}_{\operatorname{men}}}^* \cdot d S=0
$$

and

$$
P_{\mathrm{TM}}^{\mathrm{TE}}=\frac{1}{2} \iint_S E_{\mathrm{TM}_{\mathrm{men}}} \times \mathbf{H}_{\mathrm{TE}}^* * \cdot d S=0
$$

Check back soon!

Problem 16

Prove that, for a lossless waveguide, the orthogonality relationship (2.101) holds for nondegenerate modes.

Check back soon!