This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
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Here's how to solve question 3b: Step 1: List the given parameters and calculate the angular frequency (). • Frequency, f = 10 MHz = 10 × 10^6 Hz • Inductance, L = 95.3 nH/m = 95.3 × 10^-9 H/m • Capacitance, C = 68.5 pF/m = 68.5 × 10^-12 F/m • Resistance, R = 0.685 /m • Conductance, G = 0 The angular frequency is: = 2 f = 2 (10 × 10^6 Hz) = 62.83185 × 10^6 rad/s Now, calculate the complex impedance per unit length (Z) and admittance per unit length (Y): Z = R + j L = 0.685 + j(62.83185 × 10^6)(95.3 × 10^-9) = 0.685 + j5.9880 /m Y = G + j C = 0 + j(62.83185 × 10^6)(68.5 × 10^-12) = j4.3039 × 10^-3 S/m Step 2: Calculate the characteristics impedance (Z_0). The characteristic impedance is given by: Z_0 = sqrt((Z)/(Y)) = sqrt((R + j L)/(G + j C)) Z_0 = sqrt((0.685 + j5.9880)/(j4.3039 × 10^-3)) To simplify, multiply the numerator and denominator inside the square root by -j: Z_0 = sqrt(((0.685 + j5.9880)(-j))/(j4.3039 × 10^-3)(-j)) = sqrt((5.9880 - j0.685)/(4.3039 × 10^-3)) Z_0 = sqrt(1391.26 - j159.15) Convert the complex number to polar form: Magnitude: sqrt(1391.26^2 + (-159.15)^2) = 1400.33 Angle: ((-159.15)/(1391.26)) = -6.51^ So, 1391.26 - j159.15 = 1400.33 -6.51^. Z_0 = sqrt(1400.33 -6.51^) = sqrt(1400.33) ((-6.51^)/(2)) Z_0 = 37.42 -3.26^ In rectangular form: Z_0 = 37.42 (-3.26^) + j37.42 (-3.26^) = 37.36 - j2.12 The characteristics impedance is: Z_0 = 37.36 - j2.12 Step 3: Calculate the propagation constant ( = + j). The propagation constant is given by: = sqrt(Z · Y) = sqrt((R + j L)(G + j C)) = sqrt((0.685 + j5.9880)(j4.3039 × 10^-3)) = sqrt(j(0.685 × 4.3039 × 10^-3)) + j^2(5.9880 × 4.3039 × 10^-3) = sqrt(j(2.9481 × 10^-3)) - (25.770 × 10^-3) = sqrt(-0.025770 + j0.0029481) Convert the complex number to polar form: Magnitude: sqrt((-0.025770)^2 + (0.0029481)^2) = 0.025938 Angle: ((0.0029481)/(-0.025770)) = -6.51^. Since the real part is negative and the imaginary part is positive, the angle is in the second quadrant: 180^ - 6.51^ = 173.49^. So, -0.025770 + j0.0029481 = 0.025938 173.49^. = sqrt(0.025938 173.49^) = sqrt(0.025938) ((173.49^)/(2)) = 0.16105 86.745^ In rectangular form: = 0.16105 (86.745^) + j0.16105 (86.745^) = 0.00913 + j0.1608 Step 4: Determine the attenuation constant () and phase constant (). From = + j: The attenuation constant is = Re() = 0.00913 Np/m. The phase constant (space shift constant) is = Im() = 0.1608 rad/m. Convert the attenuation constant to dB/m: _dB = 8.686 × = 8.686 × 0.00913 = 0.0793 dB/m The attenuation constant is: = 0.00913 Np/m or 0.0793 dB/m The space shift constant is: = 0.1608 rad/m Step 5: Calculate the velocity of propagation (v_p). The velocity of propagation is given by: v_p = ()/() v_p = 62.83185 × 10^6 rad/s0.1608 rad/m v_p = 390.74 × 10^6 m/s v_p = 3.91 × 10^8 m/s The velocity of propagation is: v_p = 3.91 × 10^8 m/s Last free one today — make it count tomorrow, or type /upgrade for unlimited.