Electrical reactance

Capacitive reactance

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

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

There are two choices in the literature for defining reactance for a capacitor. One is to use a uniform notion of reactance as the imaginary part of impedance, in which case the reactance of a capacitor is the negative number, $${\displaystyle X_C=-\frac{1}{\omega C}=-\frac{1}{2\pi fC}.}$$

Another choice is to define capacitive reactance as a positive number, $${\displaystyle X_C=\frac{1}{\omega C}=\frac{1}{2\pi fC}.}$$

In this case however one needs to remember to add a negative sign for the impedance of a capacitor, i.e. ⁠${\displaystyle Z_c=-j{X}_c}$⁠.

At ⁠${\displaystyle f=0}$⁠, the magnitude of the capacitor's reactance is infinite, behaving like an open circuit (preventing any current from flowing through the dielectric). As frequency increases, the magnitude of reactance decreases, allowing more current to flow. As ${\displaystyle f}$ approaches ⁠${\displaystyle \mathrm{\infty }}$⁠, the capacitor's reactance approaches ⁠${\displaystyle 0}$⁠, behaving like a short circuit.

The application of a DC voltage 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 (ideal AC current source), a capacitor will only accumulate a limited amount of charge before the potential difference changes polarity and the charge is returned to the source. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

Inductive reactance

Inductive reactance is a property exhibited by an inductor, and inductive reactance exists based on the fact that an electric current produces a magnetic field around it. In the context of an AC circuit (although this concept applies any time current is changing), this magnetic field is constantly changing as a result of current that oscillates back and forth. It is this change in magnetic field that induces another electric current to flow in the same wire – called counter-electromotive force (counter-EMF) – in a direction such as to oppose the flow of the current originally responsible for producing the magnetic field (known as Lenz's law). Hence, inductive reactance is an opposition to the change of current through an element.

For an ideal inductor in an AC circuit, the inhibitive effect on change in current flow results in a delay, or a phase shift, of the alternating current with respect to alternating voltage. Specifically, an ideal inductor (with no resistance) will cause the current to lag the voltage by a quarter cycle, or 90°.

In electric power systems, inductive reactance (and capacitive reactance, however inductive reactance is more common) can limit the power capacity of an AC transmission line, because power is not completely transferred when voltage and current are out-of-phase (detailed above). That is, current will flow for an out-of-phase system, however real power at certain times will not be transferred, because there will be points during which instantaneous current is positive while instantaneous voltage is negative, or vice versa, implying negative power transfer. Hence, real work is not performed when power transfer is "negative". However, current still flows even when a system is out-of-phase, which causes transmission lines to heat up due to current flow. Consequently, transmission lines can only heat up so much (or else they would physically sag too much, due to the heat expanding the metal transmission lines), so transmission line operators have a "ceiling" on the amount of current that can flow through a given line, and excessive inductive reactance can limit the power capacity of a line. Power providers utilize capacitors to shift the phase and minimize the losses, based on usage patterns.

Inductive reactance ${\displaystyle X_L}$ is proportional to the sinusoidal signal frequency ${\displaystyle f}$ and the inductance ⁠${\displaystyle L}$⁠, which depends on the physical shape of the inductor: $${\displaystyle X_L=\omega L=2\pi fL.}$$

The average current flowing through an inductance ${\displaystyle L}$ in series with a sinusoidal AC voltage source of RMS amplitude ${\displaystyle A}$ and frequency ${\displaystyle f}$ is equal to: $${\displaystyle I_L=\frac{A}{\omega L}=\frac{A}{2\pi fL}.}$$

Because a square wave has multiple amplitudes at sinusoidal harmonics, the average current flowing through an inductance ${\displaystyle L}$ in series with a square wave AC voltage source of RMS amplitude ${\displaystyle A}$ and frequency ${\displaystyle f}$ is equal to: $${\displaystyle I_L=\frac{A{\pi }^2}{8\omega L}=\frac{A\pi }{16fL},}$$ making it appear as if the inductive reactance to a square wave was about 19% smaller ${\displaystyle X_L=\frac{16}{\pi }fL}$ than the reactance to the AC sine wave.

Any conductor of finite dimensions has inductance; the inductance is made larger by the multiple turns in an electromagnetic coil. Faraday's law of electromagnetic induction gives the counter-emf ${\displaystyle \mathcal{E}}$ (voltage opposing current) due to a rate-of-change of magnetic flux density ${\displaystyle B}$ through a current loop: $${\displaystyle \mathcal{E}=-\frac{d{\mathrm{\Phi }}_B}{dt}.}$$

For an inductor consisting of a coil with ${\displaystyle N}$ loops this gives: $${\displaystyle \mathcal{E}=-N\frac{d{\mathrm{\Phi }}_B}{dt}.}$$

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.

Impedance

Both reactance ${\displaystyle X}$ and resistance ${\displaystyle R}$ are components of impedance ⁠${\displaystyle \mathbf{Z}}$⁠: $${\displaystyle \mathbf{Z}=R+\mathbf{j}X,}$$ where:

• ${\displaystyle \mathbf{Z}}$ is the complex impedance, measured in ohms;
• ${\displaystyle R}$ is the resistance, measured in ohms – it is the real part of the impedance ${\displaystyle R=\text{Re}(\mathbf{Z})}$;
• ${\displaystyle X}$ is the reactance, measured in ohms – it is the imaginary part of the impedance ${\displaystyle X=\text{Im}(\mathbf{Z})}$; and
• ${\displaystyle \mathbf{j}}$ is the square root of negative one, usually represented by ${\displaystyle \mathbf{i}}$ in non-electrical formulas. ${\displaystyle \mathbf{j}}$ is used so as not to confuse the imaginary unit with current, commonly represented by ⁠${\displaystyle \mathbf{i}}$⁠.

When both a capacitor and an inductor are placed in series in a circuit, their contributions to the total circuit impedance are opposite. Capacitive reactance ${\displaystyle X_C}$ and inductive reactance ${\displaystyle X_L}$ contribute to the total reactance ${\displaystyle X}$ as follows: $${\displaystyle X=X_L+X_C=\omega L-\frac{1}{\omega C},}$$ where:

• ${\displaystyle X_L}$ is the inductive reactance, measured in ohms;
• ${\displaystyle X_C}$ is the capacitive reactance, measured in ohms; and
• ${\displaystyle \omega }$ is the angular frequency, ${\displaystyle 2\pi }$ times the frequency in Hz.

Hence:

• if ⁠${\displaystyle X>0}$⁠, the total reactance is said to be inductive;
• if ⁠${\displaystyle X=0}$⁠, then the impedance is purely resistive;
• if ⁠${\displaystyle X<0}$⁠, the total reactance is said to be capacitive.

Note however that if ${\displaystyle X_L}$ and ${\displaystyle X_C}$ are assumed both positive by definition, then the intermediary formula changes to a difference: $${\displaystyle X=X_L-X_C=\omega L-\frac{1}{\omega C},}$$ but the ultimate value is the same.

Phase relationship

The phase of the voltage across a purely reactive device (i.e. with zero parasitic resistance) lags the current by ${\displaystyle \frac{\pi }{2}}$ radians for a capacitive reactance and leads the current by ${\displaystyle \frac{\pi }{2}}$ radians for an inductive reactance. 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 ${\displaystyle e^{\pm \mathbf{j}\frac{\pi }{2}}}$ in the impedance: $${\displaystyle \begin{array}{cc}{\mathbf{Z}}_C& =\frac{1}{\omega C}{e}^{-\mathbf{j}\frac{\pi }{2}}=\mathbf{j}\left(-\frac{1}{\omega C}\right)=\mathbf{j}{X}_C,\\ {\mathbf{Z}}_L& =\omega L{e}^{\mathbf{j}\frac{\pi }{2}}=\mathbf{j}\omega L=\mathbf{j}{X}_L.\end{array}}$$

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

General references

• Meade, R. (2002). Foundations of Electronics. Cengage Learning.
• Shamieh, C.; McComb, G. (2011). Electronics for Dummies. John Wiley & Sons.
• Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2004) [1949]. Sears and Zemansky's University Physics (11 ed.). San Francisco: Addison Wesley. ISBN 0-8053-9179-7.