(a) A discrete memoryless source generates three independent message symbols m_1, m_2 and m_3 with probabilities 0·9, 0·08 and…

(a) (i) Use the Huffman method for a discrete memoryless source. The probabilities are 9/10, 2/25 and 1/50. The two least probable symbols are m₃ and m₂, so combine 0.02+0.08=0.10. The remaining probabilities are 0.90 and 0.10; combine them to 1.00. The tree gives depths 1, 2, 2. Assign 0 to m₁ as required:

• m₁=0, m₂=10, m₃=11. It satisfies Kraft equality: 1/2+1/4+1/4=1. The code is prefix-free, hence uniquely decodable. Average code length L=0.9(1)+0.08(2)+0.02(2)=0.90+0.16+0.04=11/10=1.10 bits/symbol. Entropy H=−Σp log₂p. For this source, H=0.9 log₂(10/9)+0.08 log₂(12.5)+0.02 log₂(50)=0.1368+0.2915+0.1129=0.5412 bits/symbol. Taking the uncompressed representation as the 2-bit fixed-length code for three symbols, compression ratio CR=2/(11/10)=20/11=1.818. Coding efficiency η=H/L=0.5412/1.10=0.492=49.2%. The efficiency is below 100% because L is still above H. Average length = 1.10 bits/symbol, CR = 1.818, efficiency = 49.2%.

(ii) For a first-order extension, group two independent symbols. The pair probabilities are products: 11=0.81, 12=0.072, 21=0.072, 13=0.018, 31=0.018, 22=0.0064, 23=0.0016, 32=0.0016, 33=0.0004. Their sum is 1.0000. Apply Huffman merging to these nine probabilities: 33+23=0.0020; 32+0.0020=0.0036; 22+0.0036=0.0100; 13+0.0100=0.0280; 31+0.0280=0.0460; 12+0.0460=0.1180; 21+0.1180=0.1900; 11+0.1900=1.0000. The resulting depths are 11=1, 21=2, 12=3, 31=4, 13=5, 22=6, 32=7, 33=8, 23=8. One valid prefix code is:

• 11=0, 21=10, 12=110, 31=1110, 13=11110, 22=111111, 32=1111100, 33=11111010, 23=11111011. No codeword is a prefix of another, so the extended code is uniquely decodable. Average length per extended symbol, i.e. per pair, is L₂=0.81(1)+0.072(2)+0.072(3)+0.018(4)+0.018(5)+0.0064(6)+0.0016(7)+0.0004(8)+0.0016(8) =0.81+0.144+0.216+0.072+0.090+0.0384+0.0112+0.0032+0.0128=1747/1250=1.3976 bits/pair. The value is below 2 bits/pair because the dominant pair 11 has length 1. Per original source symbol this is L₂/2=1747/2500=0.6988 bits/symbol. This is well below the original 1.10 bits/symbol. The entropy of the first-order extension is H₂=2H=1.0824 bits/pair. A fixed-length binary code for nine pair symbols needs 4 bits/pair. Hence CR=4/(1747/1250)=5000/1747=2.862, equivalently 2/0.6988=2.862. Efficiency η=H₂/L₂=1.0824/1.3976=0.774=77.4%. The extension improves efficiency because the pair distribution is more peaked than the single-symbol distribution. Average length = 1.3976 bits/pair, i.e. 0.6988 bits/symbol, CR = 2.862, efficiency = 77.4%.

(b) For sinusoidal steady state, constant impedances, and an ideal relay whose trip threshold is absorbed into the constants, the universal relay torque equation is T=K₁VI cos(θ−φ)+K₂V²+K₃I². Here T is torque in N·m for consistent units, V and I are rms phasors, θ is the angle of V with respect to I, φ is the relay phase angle, and K₁, K₂, K₃ are relay constants. The three terms are cross torque, voltage torque, and current torque. Let the fault impedance seen by the relay be Zf=V/I=R+jX. Then V=I(R+jX), so VI cos(θ−φ)=I²(R cosφ+X sinφ), V²=I²(R²+X²). The operating boundary is obtained by setting T=0: K₁(R cosφ+X sinφ)+K₂(R²+X²)+K₃=0. For K₂≠0, divide by K₂ and put A=K₁/K₂, B=K₃/K₂: R²+X²+A(R cosφ+X sinφ)+B=0. Completing the square gives (R+A/2 cosφ)²+(X+A/2 sinφ)²=A²/4−B. This is the general circle form in the R–X plane; the radius is real only when A²/4−B≥0. The side that operates is selected by the signs of the constants; the usual settings are stated below.

• Impedance relay: choose K₁=0 and B=−Zset². The linear terms vanish, giving R²+X²=Zset². It is a circle centred at the origin with radius Zset. The operating zone is the interior of the circle; the no-operation zone is the exterior.
• Reactance relay: choose K₂=0, φ=90°, and K₃=−K₁Xset. Then cosφ=0 and sinφ=1, so the boundary is X=Xset. It is a straight horizontal line parallel to the R axis. With the usual forward setting, the operating zone is the side containing the origin, X<Xset; the no-operation zone is X>Xset.
• Mho relay: choose K₃=0, A=−Zset, and φ=φset. The characteristic is R²+X²−Zset(R cosφset+X sinφset)=0. This is a circle through the origin with diameter Zset∠φset. Its centre is (Zset/2 cosφset, Zset/2 sinφset) and its radius is Zset/2. The operating zone is the interior; the no-operation zone is the exterior. Drawing: draw the R axis horizontal to the right and the X axis vertical upward. For the impedance relay, draw a circle centred at the origin with radius Zset. For the reactance relay, draw a horizontal line at X=Xset and shade the side containing the origin. For the mho relay, plot the point (Zset cosφset, Zset sinφset), draw the circle having the line from the origin to that point as its diameter, and shade the interior. Mark the unshaded exterior or upper region as no operation.

(c) Assume sinusoidal steady state and constant section impedances. Take the far-end voltage at C as reference: V_C=240∠0 V. The load power factors are lagging with respect to V_C, so the current angles are θ_C=−cos⁻¹0.8=−36.87°, θ_B=−cos⁻¹0.6=−53.13°. Thus I_C=150∠−36.87°=150(0.8−j0.6)=120−j90 A, I_B=150∠−53.13°=150(0.6−j0.8)=90−j120 A. Section BC carries only the C load, so I_BC=I_C. Section AB carries the B load plus the current feeding C, so I_AB=I_B+I_C=210−j210 A. Each section impedance is Z=0.15+j0.25 Ω. Voltage drop in BC: I_CZ=(120−j90)(0.15+j0.25)=(120×0.15+90×0.25)+j(120×0.25−90×0.15)=40.5+j16.5 V. Therefore V_B=V_C+I_CZ=240+40.5+j16.5=280.5+j16.5 V. Voltage drop in AB: I_ABZ=(210−j210)(0.15+j0.25)=(210×0.15+210×0.25)+j(210×0.25−210×0.15)=84+j21 V. Therefore V_A=V_B+I_ABZ=280.5+j16.5+84+j21=364.5+j37.5 V. As a check, V_A=V_C+I_BZ+2I_CZ; I_BZ=43.5+j4.5 V and 2I_CZ=81+j33 V, giving the same 364.5+j37.5 V. The supply voltage magnitude is |V_A|=√(364.5²+37.5²)=√134266.5=366.4 V. The phase angle of V_A with respect to V_C is tan⁻¹(37.5/364.5)=tan⁻¹(25/243)=5.87°. All voltages are rms values, and the positive angle means the supply voltage leads the far-end voltage. Supply voltage = 366.4 V, leading the far-end voltage by 5.87°.