Final answer:
The complex propagation constant for the transmission line is 2.302 + j50 Np/m. For maximum power transfer, the source impedance should match the complex characteristic impedance of the line. However, due to high attenuation, the power in the forward-traveling wave at 2 m is virtually zero.
Step-by-step explanation:
To answer the student's questions related to a transmission line at 1 GHz with given parameters, we approach each part as follows:
(a) Complex Propagation Constant
The complex propagation constant (γ) is composed of the attenuation constant (α) and the phase constant (β). Given that the attenuation is 0.2 dB/cm, we convert this to nepers per meter (Np/m) by multiplying by 0.1151 (since 1 dB = 0.1151 Np). The resulting α is then 0.2 dB/cm * 0.1151 Np/dB * 100 cm/m = 2.302 Np/m. The phase constant is given as β = 50 rad/m. Thus, γ = α + jβ = 2.302 + j50 Np/m.
(b) Complex Characteristic Impedance
Given the capacitance C = 100 pF/m and assuming a conductive loss of zero (G = 0), the characteristic impedance (Z0) can be found using Z0 = sqrt((L+jwC)/(G+jwC)). Since G=0, this simplifies to Z0 = sqrt(L/jwC), where L is the inductance per unit length. However, without knowing the inductance, we cannot calculate a numeric value for Z0.
(c) Source Impedance for Maximum Power Transfer
For maximum power transfer, the source impedance should match the transmission line's complex characteristic impedance (Z0). Therefore, the source impedance Zs = Z0*.
(d) Power in Forward-Traveling Wave
If 1 W is delivered to the transmission line, and the transmission line power at any distance x can be found via P(x) = P0 * e^(-2αx), where P0 is the input power and e is the base of the natural logarithm. At x = 2 m, P(2m) = 1 W * e^(-2*2.302*2) = 1 W * e^(-9.208) = 1 W * 0.0001 ≈ 0 W. The power in the forward-traveling wave is essentially 0 W due to the significant attenuation over 2 meters.