Definitions

Reactance (electronics)

Reactance is a circuit element's opposition to an alternating current, caused by the build up of electric or magnetic fields in the element due to the current. Both fields act to produce counter emf that is proportional to either the rate of change (time derivative), or accumulation (time integral) of the current. In vector analysis, Reactance is the imaginary part of electrical impedance, used to compute amplitude and phase changes of sinusoidal alternating current going through the circuit element. It is denoted by the symbol $scriptstyle\left\{Chi\right\}$. The SI unit of reactance is the ohm.

Both reactance $scriptstyle\left\{Chi\right\}$ and resistance $scriptstyle\left\{R\right\}$ are required to calculate the impedance $scriptstyle\left\{tilde\left\{Z\right\}\right\}$, although in some circuits one of these may dominate: an approximate knowledge of the minor component is useful to determine if it may be neglected.

$tilde\left\{Z\right\} = R + jChi$

Both the magnitude $scriptstyle$

> and the phase $scriptstyle\left\{theta\right\}$ of the impedance depend on both the resistance and the reactance.

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

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

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

• If $scriptstyle\left\{Chi > 0\right\}$, the reactance is said to be inductive
• If $scriptstyle\left\{Chi = 0\right\}$, then the impedance is purely resistive
• If $scriptstyle\left\{Chi < 0\right\}$, the reactance is said to be capacitive

The reciprocal of reactance is susceptance.

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.

1. 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
2. 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 XC and XL are equal but opposite, and the impedance has a phase angle of zero.

Capacitive reactance

Capacitive reactance $scriptstyle\left\{Chi_C\right\}$ is inversely proportional to the signal frequency $scriptstyle\left\{f\right\}$ and the capacitance $scriptstyle\left\{C\right\}$.

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

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

Inductive reactance $scriptstyle\left\{Chi_L\right\}$ is proportional to the signal frequency $scriptstyle\left\{f\right\}$ and the inductance $scriptstyle\left\{L\right\}$.

$X_L = omega L = 2pi f Lquad$

An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf $scriptstyle\left\{mathcal\left\{E\right\}\right\}$ (voltage opposing current) due to a rate-of-change of magnetic flux density $scriptstyle\left\{B\right\}$ through a current loop.

$mathcal\left\{E\right\} = -\left\{\left\{dPhi_B\right\} over dt\right\}quad$

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

$mathcal\left\{E\right\} = -N\left\{dPhi_B over dt\right\}quad$

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

The phase of the voltage across a purely reactive device (a device with a resistance of zero) lags the current by $scriptstyle\left\{pi/2\right\}$ radians for a capacitive reactance and leads the current by $scriptstyle\left\{pi/2\right\}$ radians for an inductive reactance. Note that without knowledge of both the resistance and reactance we cannot determine the voltage--current relationships.

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

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

$tilde\left\{Z\right\}_L = omega Le^\left\{j\left\{pi over 2\right\}\right\} = jomega L = jChi_Lquad$

For a reactive component the sinusoidal voltage across the component is in quadrature (a $scriptstyle\left\{pi/2\right\}$ phase difference) with the sinusoidal current through the component. The component alternately absorbs energy from the circuit and then returns energy to the circuit, thus a pure reactance does not dissipate power.

References

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