# Solution to Problem 3.10: Partially Filled Parallel Plate Waveguide Let's analyze the partially filled parallel plate waveguide as shown in the figure. --- ## **Step 1: Structure Description** - **Plate separation:** \( d \) - **Width of dielectric slab:** \( W \) - **Relative permittivity in slab:** \( \epsilon_r \) - **Permittivity elsewhere:** \( \epsilon_ \) The dielectric slab of width \( W \) and permittivity \( \epsilon_r \epsilon_ \) is sandwiched symmetrically between two regions of air (\( \epsilon_ \)). --- ## **Step 2: TE Mode Fields** For the **TE mode** (Transverse Electric), \( E_z = \), and the nonzero field component is \( H_z \). Let the propagation be along the \( x \)-direction, and the plates are at \( y = \) and \( y = d \). Assume \( H_z(x, y, t) = H_z(x, y)e^{j\omega t} \). We solve for the **lowest order (TE\(_1\))** mode. --- ### **Wave Equation for \( H_z \)** The general wave equation for \( H_z \) is: \[ \left( \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial x^2} + k^2 \right) H_z(x, y) = \] where \( k^2 = \omega^2 \mu_ \epsilon \) (depends on region). Assume propagation: \( H_z(x, y) = \psi(x) \sin\left(\frac{\pi y}{d}\right) \) (lowest order TE mode). --- ### **Separation of Regions** - **Region 1:** \( x < -W/2 \), \( \epsilon = \epsilon_ \) - **Region 2:** \( |x| < W/2 \), \( \epsilon = \epsilon_r \epsilon_ \) - **Region 3:** \( x > W/2 \), \( \epsilon = \epsilon_ \) --- ### **Field Solutions** #### **General Form** For each region, the equation for \( \psi(x) \) is: \[ \frac{d^2 \psi}{dx^2} + (k^2 - (\pi/d)^2) \psi = \beta^2 \psi \] where \( \beta \) is the propagation constant. Define: - \( k_^2 = \omega^2 \mu_ \epsilon_ \) - \( k_2^2 = \omega^2 \mu_ \epsilon_r \epsilon_ \) Let \( \gamma_1^2 = (\pi/d)^2 - k_^2 + \beta^2 \) in Region 1 and 3 Let \( \gamma_2^2 = (\pi/d)^2 - k_2^2 + \beta^2 \) in Region 2 But for propagating modes, \( \beta^2 < (\pi/d)^2 \), so usually \( \gamma \) is real (evanescent in x). So, - **Region 1/3 (air):** \[ \psi_1(x) = A e^{\alpha(x+W/2)} \quad (x < -W/2) \] \[ \psi_3(x) = B e^{-\alpha(x-W/2)} \quad (x > W/2) \] where \( \alpha = \sqrt{\left(\frac{\pi}{d}\right)^2 - k_^2 + \beta^2} \) - **Region 2 (dielectric):** \[ \psi_2(x) = C \cos(q x) + D \sin(q x) \quad (|x| < W/2) \] where \( q = \sqrt{k_2^2 - \left(\frac{\pi}{d}\right)^2 - \beta^2} \) --- ### **Boundary Conditions** At \( x = \pm W/2 \): - Continuity of \( H_z \) - Continuity of \( \frac{1}{\epsilon} \frac{\partial H_z}{\partial x} \) Assume **even symmetry** (lowest mode), so \( D = \): \[ \psi_2(x) = C \cos(q x) \] At \( x = W/2 \): \[ \begin{align*} \psi_2(W/2) &= \psi_3(W/2) \\ C \cos(q W/2) &= B\\ \end{align*} \] \[ \frac{1}{\epsilon_r \epsilon_} \left. \frac{d\psi_2}{dx} \right|_{x=W/2} = \frac{1}{\epsilon_} \left. \frac{d\psi_3}{dx} \right|_{x=W/2} \] \[ -\frac{q}{\epsilon_r \epsilon_} C \sin(q W/2) = -\frac{\alpha}{\epsilon_} B \] Substitute \( B = C \cos(q W/2) \): \[ -\frac{q}{\epsilon_r} \sin(q W/2) = -\alpha \cos(q W/2) \] \[ \frac{q}{\epsilon_r \alpha} \tan(q W/2) = 1 \] --- ## **Step 3: Cutoff Frequency** At **cutoff**, \( \beta = \): - \( \alpha = \sqrt{(\pi/d)^2 - k_^2} \) - \( q = \sqrt{k_2^2 - (\pi/d)^2} \) Plug into the transcendental equation: \[ \frac{q}{\epsilon_r \alpha} \tan(q W/2) = 1 \] **Solve this equation for the cutoff frequency \( f_c \).** --- ## **Step 4: TEM Wave Possibility** A **TEM mode** requires both electric and magnetic fields to be entirely transverse, meaning there can be no longitudinal (in \( x \), the direction of propagation) electric or magnetic field. For a parallel plate waveguide **with a single dielectric interface**, TEM is possible. However, for **multiple dielectrics** (as here), boundary conditions at the interface (discontinuity of \( E_x \)) *prevent* a pure TEM mode. **Conclusion:** **A TEM wave cannot propagate in this structure** because the dielectric interface violates the necessary conditions for the existence of a TEM mode. --- ## **Summary** ### **Lowest Order TE Mode:** - Field: \( H_z(x, y) = \psi(x) \sin(\pi y/d) \) - \( \psi(x) \) as above in each region - **Cutoff frequency** is determined by: \[ \frac{q}{\epsilon_r \alpha} \tan(q W/2) = 1 \] where - \( q = \sqrt{k_2^2 - (\pi/d)^2} \) - \( \alpha = \sqrt{(\pi/d)^2 - k_^2} \) ### **TEM Mode:** - **Cannot propagate** in this structure. --- Let me know if you need the explicit expressions for the **electric and magnetic field components**!