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3.3 Power and Power Triangles in AC Circuits

This discussion will focus on single-phase ac circuits and develop the definitions for ac powers: complex power, active (real) power, and reactive power. In addition, the graphical method for ac powers, the power triangles, will be presented.

The instantaneous power delivered to a load can be expressed as

Equation 3.26


The instantaneous power may be positive or negative depending on the sign of ν(t) and i(t), which is related to the sign of the signal at a given time. A positive power means that power flows from the supply to the load, and a negative value indicates that power flows from the load to the supply.

In the case of sine wave voltage and current, using trigonometric identities, the instantaneous power may be expressed as the sum of two sinusoids of twice the frequency.

Equation 3.27


Equation 3.28


Equation 3.29


Equation 3.30


In equations 3.27 through 3.30, the phase angle θ can be any value varying from −90° to +90° for general resistance, inductance, and capacitance loads (RLC).

The first term on the right-hand side of the power equation is known as instantaneous average power, real power, or active power, and is measured in watts (W), kW, or MW.

The second term on the right-hand side is called instantaneous reactive power, and its average value is zero. The maximum value of this term is known as the reactive power, and it is measured in volt-ampere reactive (VAR), kVAR, or MVAR.

The instantaneous powers corresponding to various loads are graphically illustrated in Fig. 3-10. Note that if a single phase of the system is concerned, our previous discussions can be applied to the three-phase balanced ac circuits as well.

03fig10.gifFigure 3-10. The waveforms showing the features of the instantaneous powers: (a) pure resistance load, (b) pure inductance load, and (c) pure capacitance load.

Hence, the active power and the reactive power are given by

Equation 3.31


Equation 3.32


As studied earlier, the cosine of the phase angle θ between the voltage and the current is called power factor.

The apparent power S can be calculated from P and Q as

Equation 3.33


The apparent power is measured in volt-amperes (VA), kVA, or MVA.

The complex power in ac circuits can be given as

Equation 3.34


Here S indicates a complex number. As indicated, the real part of the complex power equals the active power P, and the imaginary part is the reactive power Q.

Hence, from these definitions, the equations associated with the active, reactive, and apparent power can be developed geometrically on a right triangle called a power triangle. The power triangle is shown in Fig. 3-11 for an inductive and a capacitive load.

03fig11.gifFigure 3-11. Derivation of power triangles in ac circuits: (a) impedance triangles, (b) power triangles in inductive loads, and (c) in capacitive loads.

The impedance angle θ is also called a power angle. If the current lags the voltage, the load is inductive, the angle θ is positive, and the case is said to have a lagging power factor. Conversely, if the current leads voltage, the load is capacitive, the angle θ is negative, and the case is said to have a leading power factor. Remember that the power factor cos θ does not actually lag or lead; the current lags or leads the voltage.

3.3.1 Virtual Instrument Panel

The power triangle using the phasors is illustrated on the front panel of the VI in Fig. 3-12. In the phasor graph, the horizontal axis represents the active power and the vertical axis represents the reactive power.

03fig12.jpgFigure 3-12. The front panel and brief user guide of Power Triangles.vi.

3.3.2 Self-Study Questions

Open and run the custom-written VI named Power Triangles.vi in the Chapter 3 folder, and investigate the following questions.


Set Voltage Amplitude = 339 V, Base Voltage = 339 V, Base Current = 10 A, Rload = 10 Ω, Xload = 10 Ω, f = 50 Hz, and observe the waveforms of the voltage, the current, and the power triangle graph. What are the values of the active, reactive, and apparent powers and the power factor of the load? Verify the displayed values analytically, and estimate the power values by using the power triangle graph.


Keep the identical settings as in question 1. Then vary the values of the impedance as


respectively, and observe the waveforms of the voltage, the current, the power, and the power phasors.


A single-phase motor winding has a resistance of 10 Ω and an inductive reactance of 25 Ω at 50 Hz. Calculate the current, the phase angle, the power factor, the apparent power, the active power, and the reactive power.


Answers: 9.28 A, 68.13°, 0.371, 2320 VA, 860 W, 2155 VAR


A load connected to an ac supply consists of a lamp load of 10 kW at a unity power factor, a motor load of 80 kVA at a power factor of 0.8 lagging, and another motor load of 40 kVA at a power factor of 0.7 leading. What are the values of the total active power supplied by the source and the power factor of the source?

Hints: To find the total active power you may consider each load separately and sum its active powers and reactive powers. The power factor of the source is equal to the combined power factors of the loads.


Answers: Ptotal = 102 kW, PF = 0.98

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