Microwave Engineering (EC-702 (A)) - Unit 2 Short Notes

Rectangular Waveguides

TE and TM Modes

• TE_mn (Transverse Electric): E_z = 0, H_z ≠ 0.

• TM_mn (Transverse Magnetic): H_z = 0, E_z ≠ 0.

• Dominant mode: TE₁₀ (lowest cutoff frequency).

• Field expressions derived from wave equation with boundary conditions (E_tangential = 0, H_normal = 0 on walls).

Cutoff Wavelength and Frequency

For TE_mn/TM_mn in rectangular waveguide (dimensions a × b, a > b):

$$ \lambda_c = \frac{2}{\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}} $$

$$ f_c = \frac{c}{2\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}} \quad \text{(for air-filled)} $$

• TE₁₀: $$\displaystyle \lambda_{c10} = 2a $$, $$\displaystyle f_{c10} = \frac{c}{2a} $$.

Guided Wavelength

$$ \lambda_g = \frac{\lambda}{\sqrt{1 - \left(\frac{\lambda}{\lambda_c}\right)^2}} = \frac{\lambda}{\sqrt{1 - \left(\frac{f_c}{f}\right)^2}} $$

• Phase constant: $$\displaystyle \beta = \frac{2\pi}{\lambda_g} = \sqrt{k^2 - k_c^2} $$, where $$\displaystyle k = \frac{2\pi}{\lambda} $$, $$\displaystyle k_c = \frac{2\pi}{\lambda_c} $$.

Characteristic Wave Impedance

• TE modes:

$$ Z_{TE} = \frac{\eta}{\sqrt{1 - \left(\frac{f_c}{f}\right)^2}} $$

• TM modes:

$$ Z_{TM} = \eta \sqrt{1 - \left(\frac{f_c}{f}\right)^2} $$

where $$\displaystyle \eta = \sqrt{\mu_0/\varepsilon_0} $$ (intrinsic impedance of free space).

[!TIP]

For TM₁₁ in rectangular waveguide (a = 3 cm, b = 2 cm, f = 10 GHz):

$$\displaystyle f_{c11} = \frac{c}{2}\sqrt{(1/0.03)^2 + (1/0.02)^2} \approx 5.42 $$ GHz.

$$\displaystyle Z_{TM11} = 120\pi \sqrt{1 - (5.42/10)^2} \approx 328\ \Omega $$.

Circular Waveguides

Dominant Mode: TE₁₁

• Cutoff wavelength:

$$ \lambda_{c11} = \frac{2\pi a}{p'_{11}} \approx 1.706 \cdot 2a = 3.412a $$

where $$\displaystyle p'_{11} \approx 1.841 $$ (first root of J'₁(x)=0), a = radius.

• Cutoff frequency: $$\displaystyle f_{c11} = \frac{c}{\lambda_{c11}} = \frac{c}{3.412a} $$.

• Guided wavelength:

$$ \lambda_g = \frac{\lambda}{\sqrt{1 - \left(\frac{\lambda}{\lambda_{c11}}\right)^2}} $$

[!EXAMPLE]

For air-filled circular waveguide, diameter = 4 cm → a = 2 cm:

$$\displaystyle \lambda_{c11} \approx 3.412 \times 0.02 = 0.06824 $$ m = 6.824 cm,

$$\displaystyle f_{c11} \approx \frac{3\times10^8}{0.06824} \approx 4.4 $$ GHz.

Determination of Propagating Modes at Given Frequency

1. Compute $$\displaystyle k = \frac{2\pi f}{c} $$.

2. For each TE_mn/TM_mn, find $$\displaystyle k_{cmn} = \frac{x'_{mn}}{a} $$ (TE) or $$\displaystyle \frac{x_{mn}}{a} $$ (TM), where $$\displaystyle x'_{mn} $$, $$\displaystyle x_{mn} $$ are roots of J_m'(x)=0 (TE) or J_m(x)=0 (TM).

3. Mode propagates if $$\displaystyle k > k_{cmn} $$ ⇔ $$\displaystyle f > f_{cmn} $$.

[!EXAMPLE]

Circular waveguide, a = 2 cm, f = 10 GHz:

$$\displaystyle k = \frac{2\pi \times 10\times10^9}{3\times10^8} \approx 209.4 $$ rad/m.

$$\displaystyle k_{c11} = \frac{1.841}{0.02} = 92.05 $$ rad/m → TE₁₁ propagates.

Check TE₂₁: $$\displaystyle x'_{21} \approx 3.054 $$, $$\displaystyle k_{c21} = 152.7 $$ rad/m → propagates.

TE₀₁: $$\displaystyle x'_{01} \approx 3.832 $$, $$\displaystyle k_{c01} = 191.6 $$ rad/m → propagates.

TM₀₁: $$\displaystyle x_{01} \approx 2.405 $$, $$\displaystyle k_{c01} = 120.25 $$ rad/m → propagates.

List all modes with $$\displaystyle k_{cmn} < 209.4 $$.

Key Formulas Summary

[!EXAM TIP]

• Always verify mode type (TE/TM) before using impedance formula.

• For circular waveguides, remember TE₁₁ is dominant, not TE₀₁.

• In mode determination, compute $$\displaystyle f_{cmn} $$ for all low-order modes and compare with operating frequency.

• For TM modes in rectangular waveguides, $$\displaystyle Z_{TM} < \eta $$; for TE, $$\displaystyle Z_{TE} > \eta $$ (above cutoff).