CHAPTER 11 Balanced Three-Phase Circuits
11.1 Balanced Three-Phase Voltages Three sinusoidal voltages Identical amplitudes and frequencies Out of phase 120 with each other by exactly As the a-phase voltage, the b-phase voltage, and the c-phase voltage. abc (or positive) phase sequence acb (or negative) phase sequence
Figure 11.1 A basic three-phase circuit.
abc (or positive) phase sequence Figure 11.2 Phasor diagrams of a balanced set of three-phase voltages. (a) The abc (positive) sequence.
acb (or negative) phase sequence Figure 11.2 Phasor diagrams of a balanced set of three-phase voltages. (b) The acb (negative) sequence.
11.2 Three-Phase Voltage Sources Figure 11.3 A sketch of a threephase voltage source.
There are two ways of interconnecting the separate phase windings to form a three-phase source: in either a wye (Y) or a delta (Δ) n in Fig. 11.4(a), is called the neutral terminal of the source. Figure 11.4 The two basic connections of an ideal three-phase source. (a) A Y-connected source. (b) A -connected source.
Three-phase sources and loads can be either Y- connected or Δ-connected
Three-phase source with winding impedance Figure 11.5 A model of a three-phase source with winding impedance: (a) a Y-connected source; and (b) a -connected source.
11.3 Analysis of the Wye-Wye Circuit Figure 11.6 A three-phase Y-Y system.
Conditions for a balanced three-phase circuit 1. The voltage sources form a set of balanced threephase voltages. In Fig. 11.6, this means that V V and are a set of balanced three-phase voltages. Vc,n,, a,n b,n 2. The impedance of each phase of the voltage source is the same. In Fig. 11.6, this means that Z ga = Z gb = Z gc. 3. The impedance of each line (or phase) conductor is the same. In Fig. 11.6, this means that Z 1a = Z 1b = Z 1c. 4. The impedance of each phase of the load is the same. In Fig. 11.6, this means that Z A = Z B = Z C.
where a balanced three phase circuit,
When the system is balanced, the three line currents are
Single-phase equivalent circuit The current in the a-phase conductor line is simply the voltage generated in the a-phase winding of the generator divided by the total impedance in the a- phase of the circuit. Figure 11.7 A single-phase equivalent circuit.
Line-to-line and line-to-neutral voltages Figure 11.8 Line-to-line and line-to-neutral voltages.
Relationship between line-to-line and line-to-neutral voltages 1. The magnitude of the line-to-line voltage is 3 times the magnitude of the line-to-neutral voltage. 2. The line-to-line voltages form a balanced threephase set of voltages. 3. The set of line-to-line voltages leads the set of lineto-neutral voltages by 30.
Phasor diagrams Figure 11.9 Phasor diagrams showing the relationship between line-to-line and line-to-neutral voltages in a balanced system. (a) The abc sequence. (b) The acb sequence.
Line voltage v.s. Phase voltage Line voltage refers to the voltage across any pair of lines; phase voltage refers to the voltage across a single phase. Line current refers to the current in a single line; phase current refers to current in a single phase. In a Δ connection, line voltage and phase voltage are identical. In a Y connection, line current and phase current are identical.
Example 11.1 A balanced three-phase Y-connected generator with positive sequence has an impedance of 0.2 + j0.5ω/f and an internal voltage of 120 V/f The generator feeds a balanced three-phase Y-connected load having an impedance of 39 + j28ω/f. The impedance of the line connecting the generator to the load is 0.8 + j1.5ω/f. The a-phase internal voltage of the generator is specified as the reference phasor. a) Construct the a-phase equivalent circuit of the system. b) Calculate the three line currents I aa, I bb, and I cc.
Example 11.1 c) Calculate the three phase voltages at the load, V AN, V BN, and V CN. d) Calculate the line voltages V AB, V BC, and V CA at the terminals of the load. e) Calculate the phase voltages at the terminals of the generator, I an, I bn, and I cn. f) Calculate the line voltages I ab, I bc, and I ca and at the terminals of the generator. g) Repeat (a) (f) for a negative phase sequence.
Example 11.1 Figure 11.10 The single-phase equivalent circuit for Example 11.1.
Example 11.1
Example 11.1
Example 11.1
Example 11.1
Example 11.1
11.4 Analysis of the Wye-Delta Circuit When the load is balanced, the impedance of each leg of the wye is one third the impedance of each leg of the delta. Relationship between three-phase delta-connected and wye-connected impedance
After the D load has been replaced by its Y equivalent, the a-phase can be modeled by the single phase equivalent circuit shown in Fig. 11.11. Figure 11.11 A single-phase equivalent circuit.
When a load (or source) is connected in a delta, the current in each leg of the delta is the phase current, and the voltage across each leg is the phase voltage. Figure 11.12 A circuit used to establish the relationship between line currents and phase currents in a balanced load.
To demonstrate the relationship between the phase currents and line currents, we assume a positive phase sequence and let I f represent the magnitude of the phase current
Figure 11.13 Phasor diagrams showing the relationship between line currents and phase currents in a -connected load. (a) The positive sequence. (b) The negative sequence.
Example 11.2 The Y-connected source in Example 11.1 feeds a - connected load through a distribution line having an impedance of 0.3 + j0.9ω/f. The load impedance is 118.5 + j85.5ω/f.use the a-phase internal voltage of the generator as the reference. a) Construct a single-phase equivalent circuit of the three-phase system. b) Calculate the line currents I aa, I bb, and I cc. c) Calculate the phase voltages at the load terminals. d) Calculate the phase currents of the load. e) Calculate the line voltages at the source terminals.
Example 11.2 Figure 11.14 The single-phase equivalent circuit for Example 11.2.
Example 11.2
Example 11.2
Example 11.2
Example 11.2
11.5 Power Calculations in Balanced Three-Phase Circuits Average Power in a Balanced Wye Load Figure 11.15 A balanced Y load used to introduce average power calculations in three-phase circuits.
Average Power in a Balanced Wye Load
Total real power in a balanced threephase load
Complex Power in a Balanced Wye Load Total reactive power in a balanced three-phase load Total complex power in a balanced threephase load
Power Calculations in a Balanced Delta Load Figure 11.16 A -connected load used to discuss power calculations.
The total power delivered to a balanced Δ-connected load
Instantaneous Power in Three-Phase Circuits The total instantaneous power is the sum of the instantaneous phase powers, which reduces to 1.5V m I m cosθ f ; that is, Note this result is consistent with Eq. 11.35 since V 2 and I 2. m V f m I f
Example 11.3 a) Calculate the average power per phase delivered to the Y-connected load of Example 11.1. b) Calculate the total average power delivered to the load. c) Calculate the total average power lost in the line. d) Calculate the total average power lost in the generator. e) Calculate the total number of magnetizing vars absorbed by the load. f) Calculate the total complex power delivered by the source.
Example 11.3
Example 11.3
Example 11.3
Example 11.4 a) Calculate the total complex power delivered to the D- connected load of Example 11.2. b) What percentage of the average power at the sending end of the line is delivered to the load?
Example 11.4
Example 11.5 A balanced three-phase load requires 480 kw at a lagging power factor of 0.8. The load is fed from a line having an impedance of 0.005 + j0.025ω/f. The line voltage at the terminals of the load is 600V. a) Construct a single-phase equivalent circuit of the system. b) Calculate the magnitude of the line current. c) Calculate the magnitude of the line voltage at the sending end of the line. d) Calculate the power factor at the sending end of the line.
Example 11.5 Figure 11.17 The single-phase equivalent circuit for Example 11.5.
Example 11.5
Example 11.5
Example 11.5
Example 11.5
11.6 Measuring Average Power in Three-Phase Circuits Electrodynamometer wattmeter: current coil, potential coil. The wattmeter deflects upscale when (1) the polarity-marked terminal of the current coil is toward the source, and (2) the polarity-marked terminal of the potential coil is connected to the same line in which the current coil has been inserted. Figure 11.18 The key features of the electrodynamometer wattmeter.
The Two-Wattmeter Method To measure the total power at the terminals of the box, we need to know n 1currents and voltages. Figure 11.19 A general circuit whose power is supplied by n conductors.
Figure 11.20 A circuit used to analyze the two-wattmeter method of measuring average power delivered to a balanced load.
For a positive phase sequence, To find the total power, we add W 1 and W 2 thus
Readings of the two wattmeters A closer look at Eqs. 11.58 and 11.59 reveals the following about the readings of the two wattmeters: 1. If the power factor is greater than 0.5, both wattmeters read positive. 2. If the power factor equals 0.5, one wattmeter reads zero. 3. If the power factor is less than 0.5, one wattmeter reads negative. 4. Reversing the phase sequence will interchange the readings on the two wattmeters.
Example 11.6 Calculate the reading of each wattmeter in the circuit in Fig. 11.20 if the phase voltage at the load is 120 V and (a) Z 8 j6 ;(b) Z 8 j6 ; f (c) Z j and (d) Z f 10 75. (e) Verify for (a) (d) that the sum of the wattmeter readings equals the total power delivered to the load f 5 f 5 3 ;
Example 11.6
Example 11.6
Example 11.6
Example 11.6