Question

The parameters of a certain transmission line operating at \(\omega=6 \times 10^{8} \mathrm{rad} / \mathrm{s}\) are \(L=0.35 \mu \mathrm{H} / \mathrm{m}, C=40 \mathrm{pF} / \mathrm{m}, G=75 \mu \mathrm{S} / \mathrm{m}\), and \(R=17 \Omega / \mathrm{m}\). Find \(\gamma, \alpha, \beta, \lambda\), and \(Z_{0}\)

Short answer

Question: Determine the propagation constant, attenuation constant, phase constant, wavelength, and characteristic impedance of a transmission line operating at a frequency of 600 MHz, with resistance 17 Ω/m, inductance 0.35 μH/m, conductance 75 μS/m, and capacitance 40 pF/m. Answer: The calculated values are as follows: 1. Propagation constant, γ = 2.98 + j5.00 2. Attenuation constant, α = 2.98 Np/m 3. Phase constant, β = 5.00 rad/m 4. Wavelength, λ = 1.257 m 5. Characteristic impedance, Z₀ = 74.57 + j147.17

Step-by-step solution

Calculate Propagation Constant, γ

Using the provided parameters and the formula mentioned in the analysis, we have: $$\gamma = \sqrt{[(17 + j (6 \times 10^8)(0.35 \times 10^{-6}))(75 \times 10^{-6} + j (6 \times 10^8)(40 \times 10^{-12})]}$$ Calculate the terms inside the square root and simplify: $$\gamma = \sqrt{(17 + j126)(75\times10^{-6} +j120)}$$ Calculate the product and take the square root: $$\gamma = 2.98 + j5.00$$

Calculate Attenuation Constant, α and Phase Constant, β

We can obtain the attenuation constant, α as the real part of the propagation constant, γ, and the phase constant, β as the imaginary part of γ: $$α = Re(\gamma) = 2.98 \, \mathrm{Np/m}$$ $$β = Im(\gamma) = 5.00 \, \mathrm{rad/m}$$

Calculate Wavelength, λ

To find the wavelength, λ, use the formula: $$λ = \frac{2\pi}{\beta}$$ Insert the calculated value of β: $$λ = \frac{2\pi}{5.00}$$ Finally, find the wavelength: $$λ = 1.257 \, \mathrm{m}$$

Calculate Characteristic Impedance, Z₀

Using the provided parameters and the formula mentioned in the analysis, we have: $$Z₀ = \sqrt{\frac{(17 + j (6 \times 10^8)(0.35 \times 10^{-6}))}{(75 \times 10^{-6} + j (6 \times 10^8)(40 \times 10^{-12})})}$$ Compute the ratio inside the square root and simplify: $$Z₀ = \sqrt{\frac{17 + j126}{75\times10^{-6} +j120}}$$ Calculate the square root: $$Z₀ = 74.57 + j147.17$$ The obtained results are: 1. Propagation constant, γ = 2.98 + j5.00 2. Attenuation constant, α = 2.98 Np/m 3. Phase constant, β = 5.00 rad/m 4. Wavelength, λ = 1.257 m 5. Characteristic impedance, Z₀ = 74.57 + j147.17

Key concepts

Propagation Constant (γ)

Understanding the propagation constant, denoted as \( \gamma \), is crucial in the study of transmission lines. It's a complex number, composed of both the attenuation constant (\(\alpha\)) and the phase constant (\(\beta\)). This parameter effectively describes how the amplitude and phase of a signal vary as it travels along a transmission line.

In the given exercise, we calculated \( \gamma \) using the intrinsic parameters of the transmission line like resistance \((R)\), inductance \((L)\), conductance \((G)\), and capacitance \((C)\). This calculation reflects the transmission line's ability to propagate an electromagnetic wave, combining both the losses (resistive and dielectric) and the phase shift introduced per unit length of the line.

Attenuation Constant (α)

The attenuation constant, represented by \(\alpha\), is a measure of the signal's power loss as it travels down the transmission line. It's given in nepers per meter (Np/m) and is found by taking the real part of the propagation constant \(\gamma\).

In the provided exercise, after calculating the propagation constant, we identified \(\alpha\) to be 2.98 Np/m. This information is vital for understanding how quickly the signal's power diminishes. High attenuation means that the signal will not travel far before it becomes too weak to be useful, which is a crucial consideration in designing communication systems.

Phase Constant (β)

The phase constant, \(\beta\), part of the propagation constant \(\gamma\), indicates how the phase of the signal changes along the transmission line. It is expressed in radians per meter (rad/m) and is determined by the imaginary part of \(\gamma\).

In our step-by-step solution, we found \(\beta\) to be 5.00 rad/m. Together, \(\alpha\) and \(\beta\) completely describe the propagation constant and, hence, the behavior of a signal transmitted through a line. Recognizing the phase shift that occurs can help us to understand and control the timing and synchronization in the transmission of data signals.

Wavelength (λ)

Wavelength, symbolized as \(\lambda\), is another fundamental concept in transmission line theory. It defines the physical length of one cycle of a wave and is inversely related to the frequency. For signals on transmission lines, it's particularly important because it affects how the line must be managed at different frequencies.

The formula \(\lambda = \frac{2\pi}{\beta}\) provides the wavelength when the phase constant is known. From the calculation in the exercise, \(\lambda\) is found to be 1.257 m. Understanding wavelength is key for designing transmission lines to properly fit the frequencies they will carry, especially when dealing with high-frequency signals.

Characteristic Impedance (Z₀)

Characteristic impedance, represented by \(Z_0\), is a vital parameter that describes the relationship between the voltage and current in a transmission line. It's calculated using the intrinsic parameters of the transmission line and is expressed as a complex number reflecting both resistance (real part) and reactance (imaginary part).

In our exercise, we determined \(Z_0\) to be 74.57 + j147.17. A transmission line's characteristic impedance is critical for ensuring maximum power transfer and preventing reflections, as it must match the load impedance to avoid discontinuities. Therefore, a precise computation of \(Z_0\) is essential for the efficient design and operation of any transmission system.