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Leading and Lagging Current There are three options in a circuit for current. It can be leading, lagging, or be in phase with voltage. These can all be seen when you map current and voltage of an alternating circuits, also known as a AC Circuits, against time. The only time that the voltage and circuit are in phase together is when they are both oscillating at equal frequencies. There does become a point in the phase shift that if the current leads the voltage by more than 90 degrees, it then can be stated that the current now lags that voltage by 180 degrees minus the phase shift. 90 degrees phase shift is the determining point if the current is either leading or lagging the voltage ^[1]

Each of the main components of a circuit ( Resistor, Capacitor, and Inductors) can be seen as a resistor. All of them produce resistance in either fractional or exponential ways. Here are their complex number forms for each of them.

• Resistor, R = R ^[2]
• Capacitor, Zc = 1/jωC ^[3]
• Inductor. Zl = jωL ^[4]
• ω=2πf ^[5]

Formal definition of lagging current is lagging current can be formally defined as “an alternating current that reaches its maximum value up to 90° behind the voltage that produces it” ^[6]. This can also be stated as that the voltage and current are out of phase. In an inductive circuit, current will be at its maximum phase shift and lagging the voltage.

Leading current can be formally defined as “an alternating current that reaches its maximum value up to 90° ahead of the voltage that produces it” ^[7]. They are both out of phase from each other. In a purely capacitor circuit, current will be at its maximum phase shift and leading the voltage.

For leading and lagging current it is very simple to tell right away when using Angle notation.

 ${\displaystyle A\angle \theta .}$ ^[8]

Just by looking at the equation, the value of theta is the important factor for Leading and Lagging Current. By applying this theta value to a Angle notation. It is then possible to construct a simple Phasor Diagram (this can be seen on the Phasor page). Using complex numbers is a way to simplify calculating certain components in RLC circuits. Also it is one of the quickest ways to notice right away if the current is leading or lagging in the circuit. For example it is easy very easy to convert these different notations from one set of coordinates to another. Starting from the Angle Notation  ${\displaystyle \angle \theta }$ can represent either the vector  ${\displaystyle (\cos \theta ,\sin \theta )\,}$  or the complex number  ${\displaystyle \cos \theta +j\sin \theta =e^{j\theta },\,}$  both of which have magnitudes of 1.

After taking your theta value and plugging it into the vector form so that you get something that looks like this  ${\displaystyle (\cos \theta ,\sin \theta )\,}$ . It can then be applied on a graph where you us a two dimensional Cartesian coordinate system. This then lets allows a visual of the vector that was just established. Since the magnitude of the vector has a maximum of 1, the ranges in the x direction are -1 to 1. For the y direction it is also -1 to 1.

The earliest source of data that has surfaced so far, that I have found, is an article from the American Academy of Arts and Sciences in their 1911 issued journal. A.E. Kennelly uses traditional methods in solving vector diagrams for oscillating circuits, which can also include alternating current circuits as well. The math goes way beyond the simplification of how people today mathematically solve for different components using vector math for circuits.

• Alternating current
• Angle notation
• Complex numbers
• Impedence
• Ohm's Law
• Phase Shift
• Phasor
• RLC circuits

References:

• Bowick, Chris, John Blyler, and Cheryl J. Ajluni. RF Circuit Design. 2nd ed. Amsterdam: Newnes/Elsevier, 2008. Print.

• Gaydecki, Patrick. Foundations of Digital Signal Processing: Theory, Algorithms and Hardware Design. 2nd ed. London: Institution of Electrical Engineers, 2004. Print

• Gilmore, Rowan, and Les Besser. Passive Circuits and Systems. Boston [u.a.: Artech House, 2003. Print.

• Hayt, W. H., and J. E. Kemmerly. Engineering Circuit Analysis. 2nd ed. New York: McGraw-Hill, 1971. Print.

• Kennelly, A. E. "Vector-Diagrams of Oscillating-Current Circuits." American Academy of Arts & Sciences 46.17 (1911): 373-421. Jstor. ITHAKA. Web. 1 May 2012. < http://www.jstor.org/stable/20022665>.

• "Lagging Current." TheFreeDictionary.com. Web. 1 May 2012. < http://encyclopedia2.thefreedictionary.com/lagging_current>.

• "Leading Current." TheFreeDictionary.com. Web. 1 May 2012. < http://encyclopedia2.thefreedictionary.com/leading_current>.

• Nilsson, James William; Riedel, Susan A. (2008). Electric circuits (8th ed.). Prentice Hall. p. 338. ISBN  0-13-198925-1., Chapter 9, page 338

• Smith, Ralph J. Circuit Devices and Systems. 3rd ed. New York: John Wiley & Sons, 1976. Print.