Question

Two microstrip lines are fabricated end-to-end on a \(2-\mathrm{mm}\) -thick wafer of lithium niobate \(\left(\epsilon_{r}^{\prime}=4.8\right)\). Line 1 is of \(4 \mathrm{~mm}\) width; line 2 (unfortunately) has been fabricated with a \(5 \mathrm{~mm}\) width. Determine the power loss in \(\mathrm{dB}\) for waves transmitted through the junction.

Short answer

Question: Calculate the power loss at the junction of two microstrip lines with different widths on a 2mm-thick lithium niobate wafer, given that the relative dielectric constant is 4.8, the width of line 1 is 4mm, and the width of line 2 is 5mm. Express your answer in decibels (dB). Solution: 1. Calculate the impedance of each microstrip line: - For line 1: \(Z_{01}=\frac{120\pi}{\sqrt{\varepsilon_{1}}}\frac{1}{4+1.393+0.667\frac{2}{4}}\) - For line 2: \(Z_{02}=\frac{120\pi}{\sqrt{\varepsilon_{2}}}\frac{1}{5+1.393+0.667\frac{2}{5}}\) 2. Calculate the reflection coefficient: - \(\Gamma=\frac{Z_{02}-Z_{01}}{Z_{02}+Z_{01}}\) 3. Calculate the power loss in dB: - \(PowerLoss_{dB}=-20\log_{10}\left(1-|\Gamma|^2\right)\)

Step-by-step solution

Calculate the impedance of each microstrip line

To determine the impedance \(\left(Z_{0}\right)\) of each microstrip line, we must first calculate the effective dielectric constant \(\left(\varepsilon_{r}\right)\). We will use the following formula for effective dielectric constant: \(\varepsilon_{r}=\frac{\varepsilon_{r}^{\prime}+1}{2}+\frac{\varepsilon_{r}^{\prime}-1}{2}\cdot\frac{1}{\sqrt{1+12\frac{h}{w}}}\) For line 1: \(\varepsilon_{1}=\frac{4.8+1}{2}+\frac{4.8-1}{2}\cdot\frac{1}{\sqrt{1+12\frac{2}{4}}}\) For line 2: \(\varepsilon_{2}=\frac{4.8+1}{2}+\frac{4.8-1}{2}\cdot\frac{1}{\sqrt{1+12\frac{2}{5}}}\) Now, we will use the following formula for impedance of the microstrip line: \(Z_{0}=\frac{120\pi}{\sqrt{\varepsilon_{r}}}\frac{1}{w+1.393+0.667\frac{h}{w}}\) For line 1: \(Z_{01}=\frac{120\pi}{\sqrt{\varepsilon_{1}}}\frac{1}{4+1.393+0.667\frac{2}{4}}\) For line 2: \(Z_{02}=\frac{120\pi}{\sqrt{\varepsilon_{2}}}\frac{1}{5+1.393+0.667\frac{2}{5}}\)

Calculate the reflection coefficient

Given the calculated impedances, we will now find the reflection coefficient \(\Gamma\) using the following formula: \(\Gamma=\frac{Z_{02}-Z_{01}}{Z_{02}+Z_{01}}\)

Calculate the power loss in dB

Finally, we can find the power loss in decibels using the reflection coefficient: \(PowerLoss_{dB}=-20\log_{10}\left(1-|\Gamma|^2\right)\)

Key concepts

Microwave Engineering

Microwave engineering is a specialized area of electrical engineering focused on the design, analysis, and application of devices that operate within the microwave frequency range, typically from 1 GHz to 300 GHz. These devices include antennas, transmitters, receivers, and the circuits that connect them, like microstrip lines. Microstrip lines are a form of transmission line used to carry high-frequency signals, commonly in microwave communication systems.

Understanding microstrip line impedance is vital in microwave engineering since impedance mismatch can lead to signal reflection that causes power loss and signal degradation. Hence, accurate calculation and impedance matching are essential for efficient microwave transmission.

Electromagnetic Waves

Electromagnetic waves are waves composed of oscillating electric and magnetic fields and they propagate at the speed of light. These waves can travel through a vacuum and do not require a medium. In the context of a microstrip line which is a component often dealt with in microwave engineering, electromagnetic waves carry the signals that must be transmitted with minimal losses.

When engineering these devices, considerations of how electromagnetic waves interact with the materials and shapes of different components are crucial. For instance, wave reflection at a junction is an instance of wave-interaction that can lead to a reduction of the transmitted electromagnetic wave's intensity.

Dielectric Constant

The dielectric constant, symbolized as \(\varepsilon_{r}\), is a measure of a material's ability to store electrical potential energy under the influence of an electric field. It is also known as the relative permittivity of a material, indicating how much the material can 'permittivate' or permit the establishment of an electric field within it relative to vacuum.

In the context of microstrip lines, the dielectric constant of the substrate impacts the effective dielectric constant \(\varepsilon_{r}\) which affects the impedance of the line. The correct calculation of the effective dielectric constant is critical because it influences signal propagation speed and the characteristic impedance of the microstrip line.

Reflection Coefficient

The reflection coefficient, represented by \(\Gamma\), is a parameter that quantifies how much of an electromagnetic wave is reflected back due to an impedance mismatch. It is a crucial concept in microstrip line design because it directly impacts how much power is lost at junctions or discontinuities. The coefficient's magnitude ranges from 0 (no reflection) to 1 (total reflection).

For semi-reflective boundaries, the reflection coefficient is between 0 and 1, and it is expressed in terms of the impedance of the two lines at the junction. This parameter is pivotal when assessing transmission efficiency, and in optimizing the design to minimize power loss through careful impedance matching.

Power Loss in dB

Power loss in dB (decibels) is a logarithmic measure of the ratio of power entering a component to the power coming out of it. It is utilized in the field of microwave engineering to represent losses in a transmission line due to various factors, including reflection. The decibel scale is used because it allows a wide range of power ratios to be represented in a more manageable range of numbers.

When a microstrip line has an impedance mismatch, some power is reflected back, and the power loss can be calculated using the reflection coefficient. The equation \(PowerLoss_{dB}=-20\log_{10}(1-|\Gamma|^2)\) gives the power loss in dB due to reflection. Engineers strive to design lines with low reflection coefficients to reduce power loss, which is essential for efficient signal transmission.