Transmission lines hw
ECE 391: Transmission Lines Winter Term 2024 Homework Assignment #2 due Tuesday, Jan. 30, 6pm 1. A transmission line trace on a PCB has a total inductance of 160 nH and a total capacitance of 25 pF. The trace is 20 cm long. Determine (a) the per-unit-length parameters L and C of the transmission line; (b) the characteristic impedance of the trace; (c) the velocity of propagation on the trace; (d) the time delay of a signal traveling through the trace. Make sure to include the proper units. 2. You have a 5 meter long coaxial cable of unknown characteristics. With a capacitance meter at hand you were able to measure the total capacitance of the cable as 600 pF. Using a TDR instrument, you were able to determine the flight time from one end of the cable to the other as 45 nsec. Determine the velocity of propagation on and the characteristic impedance of the cable. 3. A transmission line is probed at two locations z1 and z2 > z1 and the voltage waveforms are observed on the oscilloscope. The voltage waveform at z1 is found as v1 (t) = 2V · u(t − 6nsec) and at z2 as v2 (t) = 2V · u(t − 4nsec) where u(t) is the unit step function. Determine (a) the direction in which the wave is propagating on the transmission line; (b) the velocity of propagation on the line if z2 − z1 = 36 cm. 4. Design a coaxial cable (specify physical parameters) with characteristic impedance Z0 = 75Ω using a polyethylene dielectric (ϵr = 2.3) between the inner and outer conductor. (The freespace permittivity and magnetic permeability are ϵ0 ≈ 8.854·10−12 F/m and µ0 = 4π ·10−7 H/m, respectively.) The cable should provide a signal delay of 4 nsec. What other design considerations should be included? Now, you are asked to design a coaxial cable with characteristic impedance Z0 = 400Ω. Comment on the practicality of your design. 5. Consider the transmission line circuit given below. Let Z0 = 50Ω, RS = 250Ω and RL = ∞. At time t = 0 a 10V battery is connected at the near end of the transmission line (vS (t) = 10 V · u(t)). Draw a lattice diagram and include the numerical values for voltage and current of the first four traveling waves. Sketch v(t) and i(t) at z = 0, z = 0.2l and z = l. Determine the steady-state response for v(t) and i(t) for t → ∞. Verify your results by simulation in ADS assuming td = 10 nsec. Include a screenshot of your ADS schematic with your ADS results. Hint: In your ADS circuit you can create an accessible node at z = 0.2l with two cascaded lines of appropriated delays and with the total delay equalling td = 10 nsec. 6. A rectangular pulse of 10V amplitude and 1 nsec duration is applied through a 25Ω series resistance to the input terminals of a lossless line having characteristic impedance Z0 = 50Ω. The line is 40 cm long and is short-circuited at the far end. Draw a lattice diagram and determine the voltage and current responses at the midpoint of the line as a function of time up to 8 nsec. Assume that the dielectric constant of the material surrounding the conductors is 2.25. Verify your results by simulation in ADS. Include a screenshot of your ADS schematic with your ADS results. How do your results change if the pulse duration is increased to 5 nsec?
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