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University of California, Berkeley EE 42/100 Spring 2013 Prof. K. Pister Homework 1 Solution 1. 2 hours (b) W = 1.8 x 3600 = 6.48 (kJ) (c) Ampere-hours = = 0.2 (Ah) 2. (a) 32 Ω 8Ω a Req 4Ω 16 Ω 8Ω b 32 Ω a Req 4Ω 16 Ω b 32 Ω a Req 4Ω 8Ω 4Ω 40 Ω b a Req b a Req 4 Ω // 40 Ω = 3.63 Ω b 16 Ω (b) 3. I1 = 3 A 4. Start by arbitrarily labeling the all the currents in the branches and all the voltages across each resistor 12 Ω IX d + V _ 5 Loop 2 3Ω I2 c I1 5Ω + V1 _ 3Ω a + V2 _ + V3 + V4 _ 6Ω _ Loop 1 15 V 10 V b a) Applying KVL to Loop 1: -10 – V1 + V2 +V3 +15 = 0 Furthermore, by using Ohm’s Law, the voltages can be expressed in terms of the associated currents and resistances. The KVL Loop1 equation can be re-written as: -10 – (I1 * 5) + (I2 * 3) + (I2 * 3) + 15 = 0 – (5 * I1) + (6 * I2) + 5 = 0 (1) Applying KVL to Loop 2: -V3 – V2 + V5 + V4 = 0 -( I2*3) – (I2*3) + (-IX * 12) + (-IX * 6) = 0 ; Note the negative sign associated with IX - 6*I2 – 18*IX = 0 Applying KCL at node C: I1 + I2 – IX = 0 (2) IX = I1 + I2 (3) Substitute (3) into (2): - 6*I2 – 18*(I1 + I2) = 0 -18*I1 – 24*I2 = 0 (4) Simultaneously solve (4) and (1): 10 I1 = 19 = 0.5263 A Plug I1 back into (4) yields: I2 = - 0.3947 A Using the values of I1 and I2 in (3): IX = 0.1315 A b) To find Vab, you can apply KVL to the following loop: 12 Ω d 6Ω 3Ω I2 c I1 5Ω 10 V 3Ω + V2 _ +a + V3 + V1 _ _ Vab Loop _ b 15 V -10 - V1 + V2 + Vab = 0 Vab = 10 + V1 – V2 = 10 + (I1 * 5) - (-I2 * 3) = 13.82 V c) 12 Ω d + IX + V4 _ Vda 3Ω I2 c I1 5Ω _ 6Ω 3Ω + V2 _ a + V3 _ + V1 _ 15 V 10 V b Applying KVL, we have: - V3 - Vda + V4 = 0 Vda = V4 - V3 = (- IX * 6) – (I2 * 3) = 0.39 V d) Vdb = Vda + Vab = 14.21 V e) Power associated with 10V source is P10V = I1 * V = (0.5263 A) * (10V) = 5.263 W. Power is being delivered to the 10V source since the value of P10V is positive. (The 15V source is actually the one supplying the power to the circuit) 5. 6. a) Wire length L = 10 m Wire cross-sectional area = 2mm2 = 2 x 10-6 m2 Resistivity ρ = 1.72 x 10-8 (Ω m) 𝐿 10 RCord = ρ x 𝐴 = (1.72 x 10-8) x 2𝑥10−6 = 0.086 Ω b) RCube RCable RCord Supply RCar RCube RCable RCord c) When there’s no current flowing, the measured voltage on the line is 124 V. This implies that the supply voltage is also equal to 124V. I = 12 A RCube + + Supply RCable RCord VCord/Cable _ VCube/Cord _ RCube RCord RCable VCube/Cord = 114 V, VCube/Cord = 111 V, VCar = 110 V, RCar= VCar/I = 110V / 12A = 9.167 Ω To find RCable, we can focus on the small portion of the circuit and apply Ohm’s Law: + VCar _ RCar I = 12 A RCable + + VCord/Cable _ VCar _ RCar RCable VCord/Cable = I x (RCable + RCar + RCable) = I x (2RCable + RCar) 111 V = 12A x (2RCable + 9.167) RCable = 0.0415 Ω To find RCord, same methodology as above can be applied: I = 12 A RCable RCord + + VCord/Cable _ VCube/Cord _ RCord + VCar _ RCar RCable VCube/Cord = I x (2RCord + 2RCable + RCar) = I x 2RCord + I x (2RCable + RCar) = I x 2RCord + VCord/Cable 114V = 12A x 2RCord + 111V RCord = 0.125 Ω The measured RCord is 45% greater than what we initially calculated. The increased resistance value could be due to the rising temperature of the wire when it conducts current. As the wire heats up (due to Joule heating), electrons undergo more collisions, causing the resistance of the wire to also increase. For copper, this causes a resistance increase of about 0.4% per Kelvin. In addition, the connection between the plug and the receptacle is not perfect, and introduces some contact resistance. To find RCube, we have: I = 12 A RCube + + VCord/Cable _ VCube/Cord _ Supply RCable RCord RCube RCord + VCar _ RCar RCable VSupply = I x (2RCube + 2RCord + 2RCable + RCar) = I x 2RCube + I x (2RCord + 2RCable + RCar) = I x 2RCube + VCube/Cord 124V = 12A x 2RCube + 114 RCube = 0.4166 Ω c) P= IV; V=IR P = I2R PCable = I2R = (12A)2 x (2 x RCable) = 11.95 W PCord = I2R = (12A)2 x (2 x RCord) = 36 W PCube = I2R = (12A)2 x (2 x RCube) = 120 W 7. Standard units are: Power = Watt [W] Current = Ampere [A] Voltage = Volt [V] Energy = Joules [J] kWh is a unit of energy. Power = I*V = Energy/time 𝑃𝑜𝑤𝑒𝑟 = 𝐸𝑛𝑒𝑟𝑔𝑦 𝑇𝑖𝑚𝑒 𝑃𝑜𝑤𝑒𝑟 ∙ 𝑇𝑖𝑚𝑒 = 𝐸𝑛𝑒𝑟𝑔𝑦 Assuming a 24kWh battery capacity of Nissan Leaf, the required charging time with a 120V output at 12A is: 𝑇𝑖𝑚𝑒 = 𝐸𝑛𝑒𝑟𝑔𝑦 𝐸𝑛𝑒𝑟𝑔𝑦 24 ∗ 1000 𝑊 ∗ 1 ℎ𝑟 = = = 16.67 ℎ𝑜𝑢𝑟𝑠 𝑃𝑜𝑤𝑒𝑟 𝐼∗𝑉 12 𝐴 ∗ 120 𝑉 With a 208V outlet at 30A, the required charging time is: 𝑇𝑖𝑚𝑒 = 24 ∗ 1000 𝑊 ∗ 1 ℎ𝑟 = 3.85 ℎ𝑜𝑢𝑟𝑠 30 𝐴 ∗ 208 𝑉 8. The power delivered by a 350V 255A charger is (350V*255A)=89.25kW The time required to charge an 85kWh battery with a 350V outlet at 255A is: 𝑇𝑖𝑚𝑒 = 85 ∗ 1000 𝑊 ∗ 1 ℎ𝑟 = 0.95 ℎ𝑜𝑢𝑟𝑠 ≈ 57 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 255 𝐴 ∗ 350 𝑉 9. With a 20% efficiency, the maximum power that solar panel can generate is 0.02 * 1 kW/m2 = 20 W/m2. With output voltage of 20V, the output current of one square meter of panel is: P = I* V I = P/V = 20 W / 20V = 1A