(1) |

, where φ is an angle between these vectors.

Let's start with electric force. If a point charge Q is placed in electric field it will experience a force, which is proportional to the amount of Q. This force per unit charge is called electric field:

(2) |

Combining (1) and (2) yields the expression for work in electric field to move a charge Q from a point A to a point B:

(3) |

From (3) the

, | (4) |

The linear integral in (4) is called

(5) |

P=V×i, | (6) |

Note that generally, any linear integral is a function of the path from A to B. In electrostatic fields however, the integral (5) does not depend on the path and is the function only of the electric field and coordinates of points A and B. Likewise, work in electrostatic field does not depend on the path. Particularly, the work to move a charge around any closed loop (i.e. when A=B) is zero. Fields in which work does not depend on path are called

Electrostatic fields are generated by electric charges that remain in rest or are moving with a constant speed. Therefore, such fields present mainly academic interest since no electronic circuit would work without accelerating the charges. Only changing fields are usable in electrical systems. In such fields integral (5) along a closed loop is no longer zero due to the presence of variable magnetic field:

, | (7) |

As the result, the linear integral in (5) generally depends on the pass from a point A to a point B. Therefore the voltage between any two points A and B is no longer defined. Strictly speaking, in variable electric fields we may only talk about voltage between points A and B along a given path. For different paths between the same points A and B the voltage (5) may be different. If magnetic flux is the same through entire surface Ac formed by the loop, then (7) can be simplified:

, | (8) |

We see from (8) that in order to minimize the effects of changing magnetic field we need to reduce the rate of change of magnetic field dB/dt and/or the area Ac of the loop. If their product is small enough we may neglect it and consider the electric field quasi-potential with the familiar definition of voltage.

In AC circuits all quantities in (4), (5) and (6) are continuously varying and are functions of time

, | (9) |

The AC values are often stated as root-mean-square (RMS). The RMS value of any variable X(t) is generally defined by

(10) |

The ratio between watts and apparent power is called

(11) |

(12) |

(13) |

(14) |

For a sinusoidal voltage we derive from (14):

, | (15) |

It can be derived from (10) that for any pure sine wave signal X(t): .

Then the expression for active power (15) can be rewritten as:

, | (16) |

By comparing (11) and (16) we derive the power factor equation for a sinusoidal voltage:

, | (17) |

In practice, the phase shift between voltage and current is caused by capacitors and inductances, while higher current harmonics are caused by non-linear components like rectifiers. The ratio between apparent power associated with higher order harmonics and apparent power associated with fundamental harmonic is called

, | (18) |

For a periodic current from (12):

, |

(20) |

(21) |

If you wonder how to place right-align equation numbers in a document, check my post on numbering equations with chapter numbers.

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©2004, 2016 Lazar Rozenblat