Intrinsic Impedance in TM Mode MCQ Quiz - Objective Question with Answer for Intrinsic Impedance in TM Mode - Download Free PDF

E_z = 20 sin 40π x sin 50πy sin (π × 10¹¹t - βz) v/m

The wave is traveling in z-direction and there is electric field is in z direction. Therefore the mode is TM. Given dimensions of waveguide as 5 cm × 2 cm.

comparing with $E_z=E_0\sin \left(\frac{m\pi }{a}\right)\sin \left(\frac{n\pi }{b}\right)\sin \left(\omega t-\beta z\right)$

⇒ m = 2, n = 1

⇒ Mode is TM₂₁, ω = π × 10¹¹ rad/s

Cut-off frequency $f_c=\frac{c}{2\sqrt{{\mu }_r{\epsilon }_r}}\sqrt{{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2}$

≃ 3.2 GHz

Operating frequency f = 50 GHz

$\begin{array}{l}\Rightarrow {\eta }_{T{M}_{21}}=\eta \sqrt{1-{\left(\frac{f_c}{f_c}\right)}^2}\\ =\frac{377}{\sqrt{9}}\sqrt{1-{\left(\frac{3.2}{50}\right)}^2}\\ \left(\eta =\frac{{\eta }_0}{\sqrt{{}_r}}\right)\end{array}$

≃ 125.41

E_z = 20 sin 40π x sin 50πy sin (π × 10¹¹t - βz) v/m

The wave is traveling in z-direction and there is electric field is in z direction. Therefore the mode is TM. Given dimensions of waveguide as 5 cm × 2 cm.

comparing with $E_z=E_0\sin \left(\frac{m\pi }{a}\right)\sin \left(\frac{n\pi }{b}\right)\sin \left(\omega t-\beta z\right)$

⇒ m = 2, n = 1

⇒ Mode is TM₂₁, ω = π × 10¹¹ rad/s

Cut-off frequency $f_c=\frac{c}{2\sqrt{{\mu }_r{\epsilon }_r}}\sqrt{{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2}$

≃ 3.2 GHz

Operating frequency f = 50 GHz

$\begin{array}{l}\Rightarrow {\eta }_{T{M}_{21}}=\eta \sqrt{1-{\left(\frac{f_c}{f_c}\right)}^2}\\ =\frac{377}{\sqrt{9}}\sqrt{1-{\left(\frac{3.2}{50}\right)}^2}\\ \left(\eta =\frac{{\eta }_0}{\sqrt{{}_r}}\right)\end{array}$

≃ 125.41

Comparing with the standard equation

$\frac{m\pi }{a}=\frac{2\pi }{a}$

m = 2

$\frac{n\pi }{b}=\frac{3\pi }{b}$

n = 3

Wave is propogating in z-direction and electric field exists in z-direction

So mode is TM

TM_mn = TM₂₃

Cutoff frequency TM₂₃

$=\frac{c}{2}\sqrt{{\left(\frac{2}{6}\right)}^2+{\left(\frac{3}{3}\right)}^2}$

$=\frac{3\times {10}^{10}}{2}\sqrt{\frac{1}{9}+1}$

$f_c=\frac{3\times {10}^{10}}{2}\times \frac{1}{3}\sqrt{10}$

$f_c=\frac{10\sqrt{10}}{2}GHz$

f_c = 15.81 GHz

Given f = 10¹¹ Hz = 100 GHz

$\cos \theta =\sqrt{1-{\left(\frac{f_c}{f}\right)}^2}$

$=\sqrt{1-{\left(\frac{15.81}{100}\right)}^2}$

= 0.987

Intrinsic impedance for TM modes

= η cos θ

= 377 × 0.987