Homological algebra

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In electrical and electronic systems, reactance is the opposition of a circuit element to a change of electric current or voltage, due to that element's inductance or capacitance. A built-up electric field resists the change of voltage on the element, while a magnetic field resists the change of current. The notion of reactance is similar to electrical resistance, but they differ in several respects.

An ideal resistor has zero reactance, while ideal inductors and capacitors consist entirely of reactance. The magnitude of the reactance of an inductor is proportional to frequency, while the magnitude of the reactance of a capacitor is inversely proportional to frequency.

Analysis

In phasor analysis, reactance is used to compute amplitude and phase changes of sinusoidal alternating current going through the circuit element. It is denoted by the symbol $\scriptstyle {X}$.

Both reactance $\scriptstyle {X}$ and resistance $\scriptstyle {R}$ are components of impedance $\scriptstyle {Z}$.

$Z=R+jX\,$

where

• $\scriptstyle {Z}$ is the impedance, measured in ohms.
• $\scriptstyle {R}$ is the resistance, measured in ohms.
• $\scriptstyle {X}$ is the reactance, measured in ohms.
• $\scriptstyle j\;=\;{\sqrt {-1}}$

Both capacitive reactance $\scriptstyle {X_{C}}$ and inductive reactance $\scriptstyle {X_{L}}$ contribute to the total reactance $\scriptstyle {X}$.

${X=X_{L}-X_{C}=\omega L-{\frac {1}{\omega C}}}$

where

• $\scriptstyle {X_{C}}$ is the capacitive reactance, measured in ohms
• $\scriptstyle {X_{L}}$ is the inductive reactance, measured in ohms

Although $\scriptstyle {X_{L}}$ and $\scriptstyle {X_{C}}$ are both positive by convention, the capacitive reactance $\scriptstyle {X_{C}}$ makes a negative contribution to total reactance.

Hence,

• If $\scriptstyle X\;>\;0$, the reactance is said to be inductive.
• If $\scriptstyle X\;=\;0$, then the impedance is purely resistive.
• If $\scriptstyle X\;<\;0$, the reactance is said to be capacitive

Capacitive reactance

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Capacitive reactance is an opposition to the change of voltage across an element. Capacitive reactance $\scriptstyle {X_{C}}$ is inversely proportional to the signal frequency $\scriptstyle {f}$ (or angular frequency ω) and the capacitance $\scriptstyle {C}$.

$X_{C}={\frac {1}{\omega C}}={\frac {1}{2\pi fC}}$

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 positive charge to accumulate on one side and negative charge to accumulate on the other 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 polarity and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

Inductive reactance

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Inductive reactance is an opposition to the change of current through an element. Inductive reactance $\scriptstyle {X_{L}}$ is proportional to the sinusoidal signal frequency $\scriptstyle {f}$ and the inductance $\scriptstyle {L}$.

$X_{L}=\omega L=2\pi fL$

The average current flowing in an inductance $\scriptstyle {L}$ in series with a sinusoidal AC voltage source of RMS amplitude $\scriptstyle {A}$ and frequency $\scriptstyle {f}$ is equal to:

$I_{L}={A \over \omega L}={A \over 2\pi fL}.$

The average current flowing in an inductance $\scriptstyle {L}$ in series with a square wave AC voltage source of RMS amplitude $\scriptstyle {A}$ and frequency $\scriptstyle {f}$ is equal to:

$I_{L}={A\pi ^{2} \over 8\omega L}={A\pi \over 16fL}$ making it appear as if the inductive reactance to a square wave was $X_{L}={16 \over \pi }fL$

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

${\mathcal {E}}=-{{d\Phi _{B}} \over dt}$

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

${\mathcal {E}}=-N{d\Phi _{B} \over dt}$

The counter-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 capacitor with an infinite resistance or an inductor with a resistance of zero) lags the current by $\scriptstyle {\pi /2}$ radians for a capacitive reactance and leads the current by $\scriptstyle {\pi /2}$ radians for an inductive reactance. Note that without knowledge of both the resistance and reactance the relationship between voltage and current cannot be determined.

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

${\begin{aligned}{\tilde {Z}}_{C}&={1 \over \omega C}e^{j(-{\pi \over 2})}=-j\left({\frac {1}{\omega C}}\right)=-jX_{C}\\{\tilde {Z}}_{L}&=\omega Le^{j{\pi \over 2}}=j\omega L=jX_{L}\quad \end{aligned}}$

For a reactive component the sinusoidal voltage across the component is in quadrature (a $\scriptstyle {\pi /2}$ 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.

See also

• Electrical measurements
• Susceptance
• Magnetic reactance

References

1. Pohl R. W. Elektrizitätslehre. – Berlin-Göttingen-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.
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External links

• Interactive Java Tutorial on Inductive Reactance National High Magnetic Field Laboratory