Question

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

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

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: $$3 = \frac{1+|\Gamma|}{1-|\Gamma|}$$

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 \(|\Gamma|\): $$|\Gamma|=\frac{SWR-1}{SWR+1}=\frac{3-1}{3+1}=\frac{2}{4}=0.5$$

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: $$Z_2 = \frac{377 (1+0.5)}{1-0.5} = \frac{377 \times 1.5}{0.5} = 1125\ \Omega$$

Conclusion

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

Key concepts

Intrinsic Impedance

Intrinsic impedance refers to a fundamental property of a medium through which an electromagnetic wave propagates. It is denoted by the symbol \(Z\) and is defined as the ratio of the electric field \(E\) to the magnetic field \(H\) in the medium. It essentially characterizes how the wave interacts with the material in terms of transmission and reflection.
For electromagnetic waves in free space, the intrinsic impedance is approximately \(377\ \Omega\). This is a crucial value because it acts as a baseline when comparing the intrinsic impedance of other materials. Knowing the intrinsic impedance of a material helps us predict how much of an incoming electromagnetic wave will be reflected back, as opposed to being transmitted into the material itself.
In the context of the given exercise, we are asked to calculate the intrinsic impedance of an unknown material when an electromagnetic wave is incident from air, which has a known intrinsic impedance of \(377\ \Omega\). By calculating the reflection coefficient, one can determine the value of the unknown material's intrinsic impedance. This allows engineers to design better systems in telecommunications and signal processing.

Reflection Coefficient

The reflection coefficient \(\Gamma\) is a measure that represents how much of an electromagnetic wave is reflected by an impedance discontinuity in the transmission medium. Its value ranges from \(-1\) to \(1\), where a value of \(0\) means no reflection, and \(1\) means total reflection.
The reflection coefficient is intimately connected with both the intrinsic impedance of materials and the standing wave ratio. In mathematical terms, it is defined by the equation: \(\Gamma = \frac{Z_2 - Z_1}{Z_2 + Z_1}\), where \(Z_1\) and \(Z_2\) represent the intrinsic impedances of two different media.
When solving the original exercise, we calculated \(|\Gamma| = 0.5\) by using the standing wave ratio (SWR) given. This tells us that half of the wave's amplitude is reflected back from the material interface. Understanding \(\Gamma\) is essential in optics, acoustics, and electrical engineering because it helps in the design of systems to minimize unwanted reflections, thereby improving performance.

Standing Wave Ratio (SWR)

The Standing Wave Ratio (SWR) is a critical parameter that represents the ratio of the amplitude of partially standing waves on a transmission line. The SWR indicates the efficiency of power transfer from a transmission line to a load. An SWR of 1 means complete power transfer with no reflections, while higher values indicate inefficiencies caused by mismatch between the line and the load.
SWR is derived using the reflection coefficient \(\Gamma\), with the relation: \[ SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} \]This relation was essential in the step-by-step solution where an SWR of \(3\) was given. By plugging this value into the formula, we derived \(|\Gamma| = 0.5\). With this information, the intrinsic impedance of the unknown material was determined.
The SWR is significant in any field involving wave propagation, like radio broadcasting and microwave communications. Engineers use it to optimize the design of antennas and transmission lines, ensuring that energy is efficiently transmitted and received.