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Leading and lagging current

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Leading and lagging current

There are three options in a circuit for current. It can be leading, lagging, or in phase with voltage. These can all be seen when one maps current and voltage of alternating (AC) circuits against time. The only time that the voltage and current are in phase together is when the load is resistive. If at some point in the phase shift the current leads the voltage by more than 90 degrees, it can then be stated that the current lags that voltage by 180 degrees minus the phase shift. Ninety 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 inductor) can be seen as an impedance. All of them produce resistance in either fractional or exponential ways. Here are their complex number forms:

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


Angle notation

Angle notation can easily determine leading and lagging current:

 A \ang \theta.[6]

In this equation, the value of theta is the important factor for leading and lagging current. Using complex numbers is a way to simplify analyzing certain components in RLC circuits. It is also one of the quickest ways to notice right away if the current is leading or lagging in the circuit. For example, it is very easy to convert these between polar and rectangular coordinates. Starting from the polar notation,  \ang \theta can represent either the vector  (\cos \theta, \sin \theta)\,  or the rectangular notation  \cos \theta + j \sin \theta = e^{j\theta},\,  both of which have magnitudes of 1.

Lagging Current

 A \ang \theta =A \ang \delta - \beta

Lagging current can be formally defined as “an alternating current that reaches its maximum value up to 90° behind the voltage that produces it”.[7] This means that current lags the voltage when  \beta is less than  \delta. This can also be stated as the voltage and current being out of phase.

For an inductor, current lags the voltage. This characteristic carries over into circuits with primarily inductive loads, in which current lags the voltage.

Leading Current

 A \ang \theta =A \ang \delta - \beta

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”.[8] This means that the current leads the voltage when  \beta is greater than  \delta. They are both out of phase from each other.

In a purely capacitive circuit, current will be at its maximum phase shift and leading the voltage.

Visualizing Leading and Lagging Current

A simple phasor diagram with a two dimensional Cartesian coordinate system and phasors can be used to visualize leading and lagging current.

Historical Documents Regarding Leading and Lagging Currents

An early source of data is an article from the 1911 American Academy of Arts and Sciences by A.E. Kennelly. 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.

For a capacitor, current leads the voltage. This characteristic carries over into circuits with primarily capacitive loads, in which current leads the voltage.

See also

Notes

References:
  1. ^ Gilmore, Besser p. 19
  2. ^ Bowick, Blyler, Ajluni 2008, pg. 25
  3. ^ Bowick, Blyler, Ajluni 2008, pg. 25
  4. ^ Bowick, Blyler, Ajluni 2008, pg. 25
  5. ^ Bowick, Blyler, Ajluni 2008, pg. 25
  6. ^ Nilsson p. 338
  7. ^ "Lagging Current".
  8. ^ "Leading Current".

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. .
  • "Lagging Current." TheFreeDictionary.com. Web. 1 May 2012. .
  • "Leading Current." TheFreeDictionary.com. Web. 1 May 2012. .
  • 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.
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