Reactance (electronics)

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For other uses, see Reactance (disambiguation). $\scriptstyle{\Chi}$ $\scriptstyle{\Chi}$

$\tilde{Z} = R + j\Chi$ $\scriptstyle{|\tilde{Z}|}$

$|\tilde{Z}| = \sqrt{ZZ^*} = \sqrt{R^2 + \Chi^2}$

$\theta = \arctan{\left({\Chi \over R}\right)}$

The magnitude is the ratio of the voltage and current amplitudes, while the phase is the voltage–current phase difference.

$\scriptstyle{\Chi > 0}$

$\scriptstyle{\Chi = 0}$

$\scriptstyle{\Chi < 0}$

The reciprocal of reactance is susceptance.

1 Physical significance
2 Capacitive reactance
3 Inductive reactance
4 Phase relationship
5 References
6 See also
7 External links

Physical significance

Determining the voltage-current relationship requires knowledge of both the resistance and the reactance. The reactance on its own gives only limited physical information about an electrical component or network.

A positive reactance implies that the circuit is inductive, where phase of the voltage leads the phase of the current; while a negative reactance implies that the circuit is capacitive, where phase of the voltage lags the phase of the current
A reactance of zero implies the current and voltage are in phase and conversely if the reactance is non-zero then there is a phase difference between the voltage and current

There are certain specific effects that depend on the reactance alone, for example; resonance in an series RLC circuit occurs when the reactances X_C and X_L are equal but opposite, and the impedance has a phase angle of zero.

Capacitive reactance

Main article: Capacitance

$\scriptstyle{\Chi_C}$

$\Chi_C = -\frac {1} {\omega C} = -\frac {1} {2\pi f C}\quad$

A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

At low frequencies a capacitor is open circuit, as no current flows in the dielectric. A DC voltage applied across a capacitor causes charge to accumulate on one side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

Inductive reactance

Main article: Inductance

$\scriptstyle{\Chi_L}$

$X_L = \omega L = 2\pi f L\quad$ $\scriptstyle{\mathcal{E}}$

$\mathcal{E} = -{{d\Phi_B} \over dt}\quad$

For an inductor consisting of a coil with N loops this gives.

$\mathcal{E} = -N{d\Phi_B \over dt}\quad$

The back-emf is the source of the opposition to current flow. A constant direct current has a zero rate-of-change, and sees an inductor as a short-circuit (it is typically made from a material with a low resistivity). An alternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

Phase relationship

$\scriptstyle{\pi/2}$

The origin of the different signs for capacitive and inductive reactance is the phase factor in the impedance.

$\tilde{Z}_C = {1 \over \omega C}e^{j(-{\pi \over 2})} = j\left(-{1 \over \omega C}\right) = j\Chi_C\quad$

$\tilde{Z}_L = \omega Le^{j{\pi \over 2}} = j\omega L = j\Chi_L\quad$ $\scriptstyle{\pi/2}$

References

Pohl R. W. Elektrizitätslehre. – Berlin-Gottingen-Heidelberg: Springer-Verlag, 1960.
Popov V. P. The Principles of Theory of Circuits. – M.: Higher School, 1985, 496 p. (In Russian).
Küpfmüller K. Einführung in die theoretische Elektrotechnik, Springer-Verlag, 1959.
Young, Hugh D.; Roger A. Goodman and A. Lewis Ford [1949] (2004). Sears and Zemansky's University Physics, 11 ed, San Francisco: Addison Wesley. ISBN 0-8053-9179-7.