Q1. Phase shift due to finite length of circuit conductors. On a particular type of circuit board, assume a signal travels at 2/3 of the speed of light. Suppose a high-weed system using this type of board can tolerate a phase shift in a sine-wave signal due to propagation delay of only 15^o before failing to work. Calculate the maximum length l_MAX (in m, or a suitable subdivision such as cm or mm) that a circuit-board trace carrying this signal can have at each of the following frequencies without encountering exercise phase shift due to the propagation delay along the trace: (a) f = 10 MHz, (b) f = 30 MHz, (c) f = 500 MHz, and (d) f = 2.5 GHz.

Q2. Impedance-matching problem with pi network. Suppose the source of RF energy in a circuit is a transistor whose output impedance Z_OUT at 25 MHz is a small resistance in series with a capacitive reactance: Z_OUT = 5 - j20Ω. Following the steps described in the following, design a pi-network matching circuit to transform the transistor's output impedance to perfectly match a real load impedance of Z₀ = 50Ω. Use the transistor's output capacitance as the input capacitor for your pi network. This will establish the Q or your circuit and leads to a unique solution.

(a) Mathematically convert the transistor's output impedance Z_OUT to an output admittance Y_OUT = 1/Z_OUT = G_OUT +jB_OUT.

(b) Find the transistor output circuit's Q with the formula Q = B_OUT/G_OUT.

(c) Using the L-network design formulas for R_L < R_S, set R_S = 1/G_OUT and Q_D = Q and solve for the intermediate resistance R_L = R_Hr.

(d) Design an L network with X₁ = 1/B_OUT and X₂ = begin an inductive reactance. This network should mach the transistor output to the resistance R_Hr.

(e) Design formulas for R_L > R_S, design a second L network with a series inductance (reactance X₃) and shunt output capacitor (reactance X₄) to transform intermediate resistance R_IP, up to the output resistance R_L = Z_OUT = 50Ω.

(f) Finally, combine the inductive reactance's X₂ and X₃ into one reactance, and draw a diagram of your final circuit. It should look like a single L network with a series inductor followed by a shunt output capacitor. But the input capacitor of the circuit is provided by the transistor, making it actually a pi network.