EECE 210 Homework 3 with Solution, Fall 2014

10 k 30 mA 2 k 4 k 3 k 1.5 k 600  IO a b c Homework 3 P3.1.16 Determine each of the resistors of the equivalent delta between terminals ‘a’, ‘b’, and ‘c in Figure P3.1.16, expressed in S. Solution: The inner  is transformed to three 2/3  resistors connected in Y. To each is added 1/3  in series to give three 1  resistors connected in Y. These are transformed to three 1/3 S resistors connected in . Each of the resistors of the equivalent  will be (1/3 + 1/6) = 1/2 S. P3.1.18 Determine Rin in Figure P3.1.18. Solution: Apply a test voltage source VT; Rin = VT/IT. The current in R/2 is 2VT/R. From KVL around the mesh in the middle, starting at node ‘b’ and going CCW: +2VT – Vca – VT = 0, which gives Vca = VT. The current through R in the middle branch is VT/R directed upwards. From KVL around the outer loop, starting at node ‘b’ and going CCW: -2VX + VX – VT = 0, which gives VX = -VT. The current in R on the RHS is also VT/R directed upwards. From KCL at node ‘a’: IT+ VT/R + VT/R – 2VT/R. This makes IT = 0, so that Rin = VT/IT  . P3.2.11 Determine IO by 1-3 2  3 S 3 S 1/6 S a b c 1/6 S 1/6 S Figure P3.1.16 R/2 Figure P3.1.18 a b Rin + – R 2V + – + – V VX R 2VX – + R/2 a b + – R 2VT + – + – VT VX R 2VX – + IT 2VT /R + – VT VT /R VT /R c d + – VT 5 A 3  Ix IO 3Ix 2  7  24 0.25IX successive current division at nodes ‘c’, ‘b’, and then ‘a’ in Figure P3.2.11. Solution: The resistance between nodes ‘c’ and ‘d’ is 0.6||3 = 0.5 k; From Ohm’s law 0.5Ibc = 0.6IO so that Ibc = 1.2IO. The resistance to the right of node ‘b’ is (1.5 + 0.5) = 2 k. Hence, Iab = 2Ibc = 2.4IO. The resistance to the right of node ‘a’ is (4 + 2||2) = 5 k, which is one-half of 10 k. Hence, Iab = (2/3)30 mA = 20 mA. equating this to 2.4IO, gives IO = 20/2.4 = 200/24 = 25/3 mA. P3.2.15 Determine IO in Figure P3.2.15 using source transformation. Solution: The 10 V source in series with 2  is transformed to a 5 A source in parallel with 2 . The 3IX source in series with 5  is transformed to a 0.6IX current source in parallel with 5 . The 5  is combined in parallel with 2  to give a parallel resistance of 10/7 . The 0.6IX current source in parallel 10/7  is transformed to a voltage source of 6IX/7 in series with 10/7 . The 10/7  in series with 2  is a resistance of 24/7 . The 6IX/7 voltage source in series with 24/7  is transformed to a current source of 0.25IX in parallel with 24/7 . The circuit becomes as shown. From KCL: 5 + 3IX/2 + IX + 7IX/8 = 0.25IX, or 40 + 12IX + 8IX + 7IX = 2 IX. This gives IX = -40/25 = -1.6 A; IO = 7IX/8 – 0.25IX = 0.625IX = -1 A. P3.2.16 Determine VY in Figure P3.2.16. Solution: Initialize. All given values and the required VY are entered. 2-3 10 k 30 mA 2 k 4 k 3 k 1.5 k 600  IO Figure P3.2.11 a b c Ibc Iab d + – 10 V 2  2  3  2  – + 5  3IX IX IO Figure P3.2.15 Figure P3.2.16 VY IXIX + – VSRC 4  8  10  Ix Ix 6  + – Vy IX IX 4 V VY Deduce. Step 1: From KCL the current between the 4  and 8  resistors is zero. Step 2: This means that VY can be obtained by voltage division as: V. P3.3.1 Determine ISRC in Figure P3.3.1 so that no current flows in Solution: Initialize. The given values are entered, the nodes are labelled, and zero current is assumed to flow. Deduce. Step 1: Because of the zero current, the same current passes through the 3 k and 1 k resistors. Hence, voltage division applies, and V. Step 2: Vbc = 0, since no current flows in RL. Step 3: From KCL at node ‘c’, ISRC flows through the 3 k resistor, so that Vcd = 3ISRC. Step4: Applying KVL around the mesh ‘dabc’ starting at node ‘d’ and going CW: 3 + 6 + 0 – 3ISRC = 0, which gives ISRC = 3 mA. P3.3.7 Determine VX in Figure P3.3.7. Solution: Initialize. All given values and the required VX are entered. The nodes are labelled. Simplify. No meaningful simplifications can be made. Deduce. Step 1: If the upper mesh is enclosed by a, 9 A enter this figure from the left, and 3 A leave the figure through the source. It follows from KCL that 6 A should flow 3-3 + – 4  8  6  10  Ix Ix 4 V + – VY I = 0 3 k RL 1 k 3 k 6 V 12 V ISRC Figure P3.3.1 + – + – 3 k RL 1 k 3 k 6 V 12 V ISRC + – + – a b c d 0 0 9 A 3 A 30 V 2  4  3  + – VX Figure P3.3.7 + – 9 A 3 A 30 V 2  4  3  + + – 6 A + 18 V I 9 – I a b through the 3  resistor. Step 2: voltage drop across the 3  resistor is 18 V in the direction of the 6A. Explore. To derive VX, the voltage across the 4  resistor is needed. If an unknown current I is assigned through this resistor, the current in the upper branch is, from KCL at node ‘a’: (9 – I); I can then be found from KVL around the mesh. Starting at node ‘a’ and going CW; -30 -2(9 – I) + 4I = 0. This gives I = 8 A. It follows that VX = 48 + 18 = 50 V. Check. The 30 V source in series with 2  is transformed to a 15 A source in parallel with 2 ; the parallel combination of this resistor and the 4  resistor is a resistor of . From KCL at node ‘a’, the current in the 4/3  resistor is 24 A, and the Vab = 32 V. Adding this to 18 V gives VX = 50 V, as before. P3.3.8 Determine VO in Figure P3.3.8. Solution: Initialize. All given values and the required VO are entered. The nodes are labelled. Deduce. Step 1: Vbc = 4IO. Step 2: Current in 4  resistor is (4IO – 12)/4 = (IO – 3) A. Step 3: Current entering node ‘b’ from the 2  resistor in the middle is (2IO – 3) A. Step 4: Current entering node ‘a’ from the 2  resistor on the left is (2IO – 3 – 3IO) = -(IO + 3) A. Step 5: IO can be determined from KVL around the outer loop. Starting at node ‘c’ and going CW: +24 + 2(IO +3) – 2(2IO – 3) – 4IO = 0, or 36 – 6IO = 0, which gives IO = 6 A, and VO = 12 V. 4-3 9 A 3 A 15 A 4/3  3  Vx 24 A 9 A 6 A 32 V 18 V – + – + – + a b c 4  2  2  2  + – VO 2  12 V IO 3IO 24 V Figure P3.3.8 + – + – 4  2  2  2  + – VO 2  12 V IO 3IO 24 V + – + – a b c 4IO (IO – 3) A (2IO – 3) A -(IO + 3) A Check. The circuit can be reduced to a two-essential-node circuit by source transformation. The 24 V source in series with 2  is transformed to a 6 A source in parallel with 2 . The 12 A source is combined with the 3IO source to give a source of (3IO + 12) = 3(IO + 4) A in parallel with 2 . The current source is transformed to a voltage source 6(IO + 4) V and the two 2  sources combined into a 4  resistor. The circuit becomes a shown. KCL at node ‘b’: , or 3IO + 12 – 2IO = 4IO – 6, or 3IO = 18, and IO = 6 A, as before. P3.3.11 Determine VY in Figure P3.3.11. Solution: Initialize. All given values and the required VY are entered. The nodes are labelled. It is seen that this is a two-essential-node circuit. Simplify. The 30  and 20  resistors are combined into a 50  resistor. The 30  and 15  resistors are combined into a 45  resistor. The circuit is redrawn to show it more clearly as a two-essential-node circuit. Deduce. Vab = 50IX; The current in the 30 V and 45  branch is: A. From KCL at node ‘a’: 10 – IX + 2 = , or 108 – 9IX = 10IX – 6, or 19IX = 114, which gives: IX = 6 A. It follows that Vab = 506 = 300 V = VY – 402 + 406, or VY = 300 – 160 = 140 V. 5-3 4  2  4  + – VO 2  12 V IO 6(IO + 4) V + – + – b c 4IO (IO – 3) A + – 10 A 30  30  20  15  30 V – + Ix 40Ix 40  2 A + – VY Figure P3.3.11 IX 40IX + – 10 A 45  50  30 V Ix 2 A + – VY 40IX 40  – + (10 – Ix) 50Ix a b 9 10  6 IX