Lecture #12 EGR 272 – Circuit Theory II Read: Chapter 11 in Electric Circuits, 6th Edition by Nilsson Generator and line impedances Sometimes impedances in the generator and the lines are considered in 3-phase circuits. This will result in power losses in each line and in reduced load voltages. Example: The 4-wire Y-Y system below has a balanced generator with 720 V RMS and an acb phase sequence. Determine IaA, Van, VAN, and the power loss in each line. c C j3 + 1 - 2 j10 j2 n 5 A N b B a Generator Line Load

Lecture #12 EGR 272 – Circuit Theory II - system Recall for a  generator that VL = Vp. So analyzing the - system is similar to analyzing the Y- system except that the line voltages are more easily found. Example: Determine the three line currents for a - system that has a balanced generator with 240 V RMS and a negative phase sequence. The loads are as follows: ZAB = 3+j4, ZBC = 3-j4, and ZCA = 2+j2.

Lecture #12 EGR 272 – Circuit Theory II  Load with line impedances Including line impedances with a  load makes the analysis more difficult. A good way to approach the problem is to use a -Y conversion to change the  load into a Y load. Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations In EGR 271 -Y and Y- transformations were used with resistive circuits. These transformations can also be used with circuits consisting of AC impedances. Recall that the transformation equations are derived based on a specific labeling of the impedances, so the equations below are somewhat useless without the corresponding figures.

Lecture #12 EGR 272 – Circuit Theory II Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations Special case: If the load is balanced, these equations reduce to:

Lecture #12 EGR 272 – Circuit Theory II Example: Determine the three line currents in a 3 Y- circuit described as follows: The Y generator is balanced with an abc phase sequence and Van = 480 V Each of the 3 lines (between source and load) has an impedance of 2 + j4 ohms The  load is balanced where each of the three loads have an impedance of 60 + j90 ohms

Lecture #12 EGR 272 – Circuit Theory II Power Calculations in 3 Circuits Total power delivered = (power delivered to each phase) Or Note: Power can be calculated, as it would be for any AC circuit. For example, total power could be found by finding the power to the resistive portion to each load.

Lecture #12 EGR 272 – Circuit Theory II Example: Find the total power delivered to the Y-Y circuit analyzed last class (4-wire Y-Y system has a balanced generator with Van = 480 V and a positive phase sequence with ZAN = ZBN = 2 + j2 and ZCN = 2 - j2).

Lecture #12 EGR 272 – Circuit Theory II Example: Find the total power delivered to the Y-Y circuit analyzed last class (balanced system with Van = 240 V, a negative phase sequence, and with impedances as follows: ZAB = 6 + j8, ZBCN = 6 – j8, and ZCA = 6).

Lecture #12 EGR 272 – Circuit Theory II Measuring Power with Wattmeters: A wattmeter is a piece of equipment that measures average power, P, in watts. A wattmeter has connections for both current and voltage, as shown below on the left (Electric Circuits, 9th Ed., by Nilsson). Note that the positive side of the current coil and the positive side of the voltage coil are labeled + or +. The wattmeter shown below on the right shows how a wattmeter might be connected in a circuit. n + - a W1 Van IaA

Lecture #12 EGR 272 – Circuit Theory II Wattmeter reading: + W1 I V - The 2-wattmeter method and the 3-wattmeter method: Two common methods for measuring power are the 2-wattmeter method and the 3-wattmeter method. In the 3-wattmeter method, all negative voltage connection on each of the wattmeters is common (typically on the neutral line). In the 2-wattmeter method, the positive voltage terminal on two wattmeters is connected to any two of the lines and both negative terminals are connected to the third line. It can be proven that total power is the sum of the wattmeter readings in either method. Both methods are illustrated on the following page.

Lecture #12 EGR 272 – Circuit Theory II The 3-wattmeter method: c C + - n A N b B a W1 W2 W3 Vbn Vcn Van ZAN ZBN ZCN IaA IbB IcC The 2-wattmeter method: c C + - n A N b B a WA WB Vbn Vcn Van ZAN ZBN ZCN IaA IbB Vac Vbc

Lecture #12 EGR 272 – Circuit Theory II Example: Determine the reading for each wattmeter below and the total power absorbed by the load if the circuit has a balanced generator with Van = 480V, a positive phase sequence, and impedances ZAN = 6+j8, ZBN = 8+j6, and ZCN = 5-j5. c C + - n A N b B a WA WB Vbn Vcn Van ZAN ZBN ZCN IaA IbB Vac Vbc