## Phase Angle and Power Factor

The instantaneous voltage and current are given by v = Vp sin (fflt), i = Ip sin (at + 0)

FIGURE 7.4 Current and voltage across a resistor, capacitor, and inductor, showing the phase relationship between the voltage and current. Voltage, dashed line; current, solid line.

where Vp and Ip are the peak values; a, rad/s, is the angular velocity (which is 2n times the frequency); and the angle, 0, is the difference in degrees between the instantaneous voltage and current (the sine wave for voltage and the sine wave for current). For a resistor the voltage and current are in phase and the average power over one cycle is

P = V*I = Vp sin (at) * Ip sin (at) = 0.5 Vp Ip (7.12)

For capacitors and inductors, the voltage and current are 90° out of phase and the average power is zero:

P = Vp sin (at) * Ip sin (at + 90) = Vp sin (at) * Ip cos (at) = 0

In all real circuits there is inductance, capacitance, and resistance, so the current and voltage will not be completely in phase (Figure 7.5). The instantaneous power to an arbitrary AC circuit oscillates because both the voltage and current oscillate:

Volts

Current

Power

Volts

Current

Power

Time

FIGURE 7.5 Instantaneous power in an arbitrary AC circuit.

Time

FIGURE 7.5 Instantaneous power in an arbitrary AC circuit.

The average power is found by integrating over one cycle:

Vplp cos0 2

However, what is measured for AC circuits are average values of current and voltage, which are given by the root mean square values:

V = (v* v)0 5 = {Vp sin (ffli)* Vp sin (ffli)j 0 5 = Vp/205 = 0.707 Vp

However, that is for single-phase. For three phases, the measured current is reduced by 305. So, for a three-phase transfer of power, each leg is transferring a current equal to the (coil current)/1.73, and therefore the wire size needed is smaller. The real power generated or consumed is given by

where cos <p is the power factor. Adding a number of induction generators to the utility line can change the power factor and reduce the actual power delivered, a concern of the utility company, since the utility grid supplies the reactive power for the induction generators. Therefore, some wind turbines and most wind plants have capacitors added to the wind turbine or to the electric substation. There are a number of electrical conversion systems for wind turbines [1-8].

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

Get My Free Ebook