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Chapter 12: Problem 11

A \(150-\mathrm{MHz}\) uniform plane wave is normally incident from air onto \(\mathrm{a}\) material whose intrinsic impedance is unknown. Measurements yield a standing wave ratio of 3 and the appearance of an electric field minimum at \(0.3\) wavelengths in front of the interface. Determine the impedance of the unknown material.

Short Answer

Expert verified
Based on the given frequency of the plane wave, the standing wave ratio, and the location of the electric field minimum relative to the interface, the intrinsic impedance of the unknown material that the wave is incident upon is found to be 1125 Ω.

Step by step solution

01

Recall the relationship between SWR and reflection coefficient

The standing wave ratio (SWR) is related to the reflection coefficient \(\Gamma\) by the following formula: $$SWR = \frac{1+|\Gamma|}{1-|\Gamma|}$$ In this exercise, we are given the value of SWR to be 3. Plug this value into the formula to solve for the reflection coefficient: $$3 = \frac{1+|\Gamma|}{1-|\Gamma|}$$
02

Find the Reflection Coefficient

From Step 1, we have: $$3 = \frac{1+|\Gamma|}{1-|\Gamma|}$$ By rearranging the equation, we can solve for the reflection coefficient \(|\Gamma|\): $$|\Gamma|=\frac{SWR-1}{SWR+1}=\frac{3-1}{3+1}=\frac{2}{4}=0.5$$
03

Determine the intrinsic impedance of the unknown material

The reflection coefficient is related to the intrinsic impedance of the unknown material \(Z_2\) and the intrinsic impedance of the air \(Z_1\) by the following formula: $$\Gamma = \frac{Z_2 - Z_1}{Z_2 + Z_1}$$ We know that the intrinsic impedance of the air (\(Z_1\)) is \(377\ \Omega\). Also, we have the magnitude of the reflection coefficient \(|\Gamma|=0.5\). To find the intrinsic impedance of the unknown material, we can rearrange the equation above: $$Z_2 = \frac{Z_1(1+\Gamma)}{1-\Gamma}$$ Plug in the values: $$Z_2 = \frac{377 (1+0.5)}{1-0.5} = \frac{377 \times 1.5}{0.5} = 1125\ \Omega$$
04

Conclusion

The intrinsic impedance of the unknown material is \(1125\ \Omega\).

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Most popular questions from this chapter

Region 1, \(z<0\), and region \(2, z>0\), are both perfect dielectrics ( \(\mu=\mu_{0}\), \(\epsilon^{\prime \prime}=0\) ). A uniform plane wave traveling in the \(\mathbf{a}_{z}\) direction has a radian frequency of \(3 \times 10^{10} \mathrm{rad} / \mathrm{s}\). Its wavelengths in the two regions are \(\lambda_{1}=\) \(5 \mathrm{~cm}\) and \(\lambda_{2}=3 \mathrm{~cm}\). What percentage of the energy incident on the boundary is \((a)\) reflected; \((b)\) transmitted? \((c)\) What is the standing wave ratio in region \(1 ?\)

A uniform plane wave in region 1 is normally incident on the planar boundary separating regions 1 and 2. If \(\epsilon_{1}^{\prime \prime}=\epsilon_{2}^{\prime \prime}=0\), while \(\epsilon_{r 1}^{\prime}=\mu_{r 1}^{3}\) and \(\epsilon_{r 2}^{\prime}=\mu_{r 2}^{3}\), find the ratio \(\epsilon_{r 2}^{\prime} / \epsilon_{r 1}^{\prime}\) if \(20 \%\) of the energy in the incident wave is reflected at the boundary. There are two possible answers.

A left-circularly polarized plane wave is normally incident onto the surface of a perfect conductor. (a) Construct the superposition of the incident and reflected waves in phasor form. (b) Determine the real instantaneous form of the result of part ( \(a\) ). \((c)\) Describe the wave that is formed.

You are given four slabs of lossless dielectric, all with the same intrinsic impedance, \(\eta\), known to be different from that of free space. The thickness of each slab is \(\lambda / 4\), where \(\lambda\) is the wavelength as measured in the slab material. The slabs are to be positioned parallel to one another, and the combination lies in the path of a uniform plane wave, normally incident. The slabs are to be arranged such that the airspaces between them are either zero, one-quarter wavelength, or one-half wavelength in thickness. Specify an arrangement of slabs and airspaces such that \((a)\) the wave is totally transmitted through the stack, and \((b)\) the stack presents the highest reflectivity to the incident wave. Several answers may exist.

A \(50-\mathrm{MHz}\) uniform plane wave is normally incident from air onto the surface of a calm ocean. For seawater, \(\sigma=4 \mathrm{~S} / \mathrm{m}\), and \(\epsilon_{r}^{\prime}=78 .(a)\) Determine the fractions of the incident power that are reflected and transmitted. (b) Qualitatively, how (if at all) will these answers change as the frequency is increased?
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