Chapter 8: Problem 11

(a) From $(8.7 .10)$ and $(8.7 .13)$, show that the guided ave impedance $\left(E_{x}{ }^{2}+E_{y}{ }^{2}\right)^{1 / 2} /$ $\left(H_{x}{ }^{2}+H_{z}{ }^{2}\right)^{1 / 2}$ is $Z_{\mathrm{TE}}=\frac{Z_{0}}{\left[1-\left(\lambda_{0} / \lambda_{e}\right)^{2}\right]^{1 / 2}} \quad$ TE modes $Z_{\text {T? }}=Z_{0}\left[1-\left(\lambda_{0} / \lambda_{c}\right)^{2}\right]^{1 / 2} \quad$ TM modes, where $Z_{0}$ is the unbounded wave impedance $(8.3 .10)$ or, more generally, $(8.3 .12) .(b)$ For the TE o dominant mode in rectangular waveguide, show that the peak potential difference between opposite points in the cross section is $$ V_{0}=\left[\int_{0}^{b} E_{y}\left(x=\frac{1}{2} a\right) d y\right]_{p e s k}=b E_{0} $$ and that the peak axial current flowing in the top wall is $$ I_{0} \equiv\left[\frac{1}{\mu_{0}} \int_{0}^{a} B_{x}(y=b) d x\right]_{\text {pak }}=\frac{2 a E_{0}}{\pi Z_{\mathrm{TE}}} $$ Since the result of Prob. 8.7.10b can be written $$ \bar{P}=\frac{a b}{4} \frac{E_{0}{ }^{2}}{Z_{\mathrm{TB}}} $$ we can define three other (mode-dependent) waveguide impedances as follows: $$ \begin{aligned} &Z_{V, I}=\frac{V_{0}}{I_{0}}=\frac{\pi}{2}\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \\ &Z_{P, V}=\frac{V_{0}{ }^{2}}{2 P}=2\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \\\ &Z_{P, I}=\frac{2 \bar{P}}{I_{0}{ }^{2}}=\frac{\pi^{2}}{8}\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \end{aligned} $$ which differ by small numerical factors. Only systems supporting a TEM mode (e.g., Sec. 8.1), have a unique impedance.

Short Answer

Expert verified

The guided wave impedance in a waveguide depends on its operating mode. For TE and TM modes, we use equations (8.7 .10) and (8.7 .13) to derive the magnitudes ratios of electric field to magnetic field, which provide expressions for guided wave impedance in these modes. In the TE dominant mode of a rectangular waveguide, the peak potential difference (across width) can be found by integrating the electric field (E_y) over the width (b) of the waveguide. The peak axial current flowing in the top wall can be found by integrating the magnetic field (B_x) over the width (a) and then dividing by the permeability (μ_0). Three mode-dependent waveguide impedances can be defined and their relationships to each other and to Z_TE can be derived: 1. Z_V, I = V_0/I_0, relating the peak potential difference to the peak axial current. 2. Z_P, V = V_0^2/(2P), relating the square of the peak potential difference to the power. 3. Z_P, I = 2 * P/I_0^2, relating the power to the square of the peak axial current. These impedances differ only by small numerical factors, emphasizing that only systems supporting a TEM mode have a unique impedance.

Step by step solution

To derive the guided wave impedance, we first look for the relations between electric field ($E_x, E_y$) and magnetic field ($H_x, H_z$) using equations $(8.7 .10)$ and $(8.7 .13)$. By calculating the magnitude ratios of electric field to magnetic field, we can obtain the expressions for guided wave impedance for TE and TM modes. #Step 2: Find the peak potential difference across the cross-section of the rectangular waveguide for TE dominant mode#

For the TE dominant mode, we can find the peak potential difference across the cross-section by integrating the electric field $E_y$ over the width of the rectangular waveguide (width $b$) with $x=\frac{1}{2}a$. This gives us the expression for $V_0$ as shown in the exercise. #Step 3: Find the peak axial current flowing in the top wall of the rectangular waveguide for the TE dominant mode#

We can find the peak axial current flowing in the top wall by integrating the magnetic field $B_x$ over the width of the rectangular waveguide (width $a$) with $y=b$ and then dividing it by the permeability $\mu_0$. This will result in the expression for $I_0$ shown in the exercise. #Step 4: Define the three mode-dependent waveguide impedances and their relationships#

The three mode-dependent waveguide impedances can be defined as follows: $Z_{V, I} = \frac{V_0}{I_0}$, $Z_{P, V} = \frac{V_0^{2}}{2P}$, and $Z_{P, I} = \frac{2 \bar{P}}{I_0^{2}}$. We can then derive their relationships to each other and to $Z_{\mathrm{TE}}$, which are presented in the exercise. Notice that they differ only by small numerical factors. This indicates that only systems supporting a TEM mode have a unique impedance.