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Solved A lossy parallel-wire transmission line consist of | Çℎḙḡḡ.com

 

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a lossy parallel-wire transmission line consists of no. 12 wire awg (d = 81 mils). the distance between the wire centers is 10 inches. if the lossy characteristic impedance is almost equal to the lossless characteristic impedance, what is the attenuation coefficient in db per 100 ft of the line at 400 mhz?

 

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a lossy parallel-wire transmission line consists of no. 12 wire awg (d = 81 mils). the distance between the wire centers is 10 inches. if the lossy characteristic impedance is almost equal to the lossless characteristic impedance, what is the attenuation coefficient in db per 100 ft of the line at 400 mhz?

To calculate the attenuation coefficient in dB per 100 ft of the line at 400 MHz, we need to consider the characteristics of the lossy parallel-wire transmission line.

Given data:

• Wire diameter (d) = 81 mils
• Distance between wire centers = 10 inches
• Frequency = 400 MHz

First, let's calculate the characteristic impedance (Z0) of the lossless parallel-wire transmission line using the formula:
\[ Z_0 = \frac{276}{\sqrt{\varepsilon_r}} \cdot \ln\left(\frac{2h}{d}\right) \]
where:

• \( \varepsilon_r \) is the relative permittivity of the dielectric material between the wires (assumed to be air, so \( \varepsilon_r = 1 \))
• h is the distance between the wire centers

Plugging in the values:
\[ Z_0 = \frac{276}{\sqrt{1}} \cdot \ln\left(\frac{2 \times 10\,\text{inches}}{81\,\text{mils}}\right) \]
\[ Z_0 = 276 \cdot \ln\left(\frac{20}{81}\right) \]
\[ Z_0 \approx 276 \cdot \ln(0.2469) \]
\[ Z_0 \approx 276 \cdot (-1.4019) \]
\[ Z_0 \approx -387.15\,\Omega \]

Since the lossy characteristic impedance is almost equal to the lossless characteristic impedance, we can assume the characteristic impedance of the lossy line is also approximately -387.15 Ω.

Next, let's calculate the attenuation coefficient (α) in Np/m at 400 MHz using the formula:
\[ \alpha = \frac{2\pi f \cdot \sqrt{\mu \cdot \varepsilon}}{v_p} \cdot \sqrt{\frac{\text{R}}{2} + j\omega L} \]
where:

• f is the frequency in Hz
• μ is the permeability of the medium (assumed to be the permeability of air, so μ = 4π x 10^(-7) H/m)
• ε is the permittivity of the medium (assumed to be the permittivity of air, so ε = 8.854 x 10^(-12) F/m)
• vp is the phase velocity of the signal in the transmission line
• R is the resistance per unit length of the transmission line
• L is the inductance per unit length of the transmission line

Given that the lossy characteristic impedance is approximately equal to the lossless characteristic impedance, we can assume that the series resistance per unit length is the main cause of attenuation. Therefore, we can simplify the attenuation coefficient formula to:
\[ \alpha = \frac{R}{2} \]

Now, we can calculate the attenuation coefficient in dB per 100 ft of the line at 400 MHz:
\[ \alpha_{\text{dB/ft}} = 10 \cdot \log\left(e^{\alpha \cdot \frac{100}{0.3048}}\right) \]
\[ \alpha_{\text{dB/ft}} = 10 \cdot \log\left(e^{-387.15/2 \cdot \frac{100}{0.3048}}\right) \]
\[ \alpha_{\text{dB/ft}} = 10 \cdot \log\left(e^{-6348.5}\right) \]
\[ \alpha_{\text{dB/ft}} = 10 \cdot (-6348.5) \]
\[ \alpha_{\text{dB/ft}} = -63485 \, \text{dB/ft} \]

Therefore, the attenuation coefficient in dB per 100 ft of the line at 400 MHz is approximately -63485 dB/ft.

 

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