- Fundamental Definitions
- AC Circuit Analysis
- Power and Power Triangles in AC Circuits
- Power Factor Correction
- Star-Delta and Delta-Star Conversion in Three-Phase AC Circuits
- Voltage and Currents in Star- and Delta-Connected Loads
- Voltage and Current Phasors in Three-Phase Systems
- Power in Three-Phase AC Circuits
- Three-Phase Power Measurement and Data Logging
- References

## 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

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.

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.

**Figure 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

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

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

The complex power in ac circuits can be given as

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.

**Figure 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.

**Figure 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.