reactance

What is reactance?

Reactance is a form of opposition generated by components in an electric circuit when alternating current (AC) passes through it. The term reactance applies only to AC circuits -- both serial and parallel -- not to direct current (DC) circuits. You can measure reactance in ohms (Ω) and symbolize it with X.

Inductance is the resistance that occurs when a component such as an inductor generates an electromagnetic field that impedes the current. Inductance is measured in henry (H) and symbolized with L. Capacitance, meanwhile, is resistance that occurs when a device such as a capacitor stores an electric charge that resists changes in voltage. Capacitance is measured in farad (F) and symbolized with C. A circuit's reactance can result from inductance, capacitance or a combination of both.

When AC passes through a component that generates reactance, a magnetic or electric field alternately stores it and releases that energy. If a circuit contains both inductance and capacitance, the two cancel each other out, and you can measure the circuit's total reactance by the difference between them.

Calculating inductance and capacitance

You calculate reactance using formulas based on angular frequency, a measure of angular displacement in a wave. Angular frequency is represented by the Greek letter omega (ω) and is equal to 2πf, where f is the circuit's frequency in hertz (Hz).

Angular frequency factors into both the inductance and capacitance formulas. The formula for inductance is the simpler of the two:

X_L = ωL = 2πfL

L represents inductance, as measured in henry. When reactance is specific to inductance, you symbolize it as X_L.

Capacitance is inversely proportional to a current's frequency. The formula used to calculate capacitance takes this inverse relationship into account:

X_C = 1 / ωC = 1 / 2πfC

This formula contains many of the same elements as the one for inductance but specifies an inverse structure. It uses C instead of L to represent capacitance, as measured in farad. When reactance is specific to capacitance, you symbolize it as X_C.

Reactance, resistance and impedance

Resistance refers to any component in a circuit that opposes the current's flow. Even the wire that carries the current can represent a source of resistance. Unlike reactance, electrical resistance acts as a type of friction that dissipates the current in the form of heat. In addition, resistance applies to both AC and DC circuits. Measure resistance in ohms, and symbolize it as R.

Resistance and reactance combine in a circuit to form a single property referred to as impedance, which you measure in ohms and symbolize with Z. The formula used to calculate impedance depends on whether you're working with a series AC circuit or parallel AC circuit. The formula for a series circuit is the simpler of the two:

Z = √(R² + (XL - XC)²)

This formula requires that you first calculate the circuit's inductance and capacitance before calculating the total impedance. Once you have these values, subtract the capacitance from the inductance. If a circuit includes one but not the other, you can use that measure alone. Everything within the inner parentheses is considered reactance, which you can represent with a simple X.

After you calculate the reactance, you should then square both the resistance and reactance, add them together and find the square root of their sum. This should give you the impedance.

These basic principles work similarly for a parallel AC circuit, except that you must also take into account that capacitance runs inversely proportional to a current's frequency. This means it requires a much more detailed formula:

Z = 1 / √((1 / R)² + ((1 / X_L) - (1 / X_C))²)

As with a series, you should start by plugging in your values: resistance, inductance and capacitance. Then, calculate the individual fractions within the denominator, following the basic algebraic rules of precedence. Next, subtract the capacitance from the impedance, as with the series formula. You can then square the resistance and reactance, add them together and determine the square root of their sum. Only then should you calculate the outer fraction, dividing 1 by the resultant denominator.

People sometimes treat reactance as the imaginary part of inductance. Because of this, many calculations incorporate an imaginary unit symbolized by the letter j, which represents the positive square root of -1. For example, you might see the impedance formula written as Z = R + jX when analyzing complex AC circuitry. Some might also use the letter i instead of j for this purpose.

When treated as imaginary numbers, you can assign inductive reactance positive imaginary number values. As the inductance of a component increases, its inductive reactance becomes larger in imaginary terms, assuming the frequency remains constant. As the frequency increases for a given value of inductance, the inductive reactance increases in imaginary terms.

Meanwhile, you can assign capacitive reactance negative imaginary number values. As the capacitance of a component increases, its capacitive reactance becomes negatively smaller -- i.e., closer to zero -- in imaginary terms, assuming the frequency holds constant. As the frequency increases for a given value of capacitance, the capacitive reactance becomes negatively smaller in imaginary terms.

Calculating the impedance of a series AC circuit

A circuit often includes one or more sources of resistance, impedance or capacitance mixed together in various ways. In this example, the circuit is a series AC with a 60 Hz frequency and includes the following components:

• One resistor: 20 Ω.
• One inductor 100 millihertz (mH).
• One capacitor: 100 microfarad (µF).

To calculate the impedance in this circuit, start by determining the inductor's reactance, represented by X_L:

X_L = 2πfL

X_L = 2 ⋅ π ⋅ 60 ⋅ (100 ⋅ 10^-3)

X_L = 37.70 Ω

The inductive reactance for this inductor is approximately 37.70 Ω -- rounded to two decimal places to simplify the equations. For the inductor's calculations, the equation uses 60 Hz for the frequency and 100 mH for the inductor. Because you rate the inductor in millihertz, multiply the 100 by 10^-3 to convert it to hertz.

Next, calculate the capacitive resistance, using 60 Hz for the frequency and 100 µF for the capacitance:

X_C = 1 / 2πfC

X_C = 1 / (2 ⋅ π ⋅ 60 ⋅ (100 ⋅ 10^-6)

X_C = 1 / 0.03769911184

X_C = 25.53 Ω

In this case, you would rate the capacitor in microfarad, so you must multiply it by 10^-6 to convert it to farad. The reactance capacitance for this capacitor comes out to approximately 25.53 Ω.

After calculating the inductance and capacitance, you can plug them into the impedance formula, along with the 20 Ω resistance:

Z = √(R² + (X_L - X_C)²)

Z = √(20² + (37.70 - 25.53)²)

Z = √(20² + 12.17²)

Z = √(548.1089)

Z = 23.41 Ω

The equation starts by replacing the values for R, X_L and X_C. The rest is just basic algebra, resulting in a rate of impedance of about 23.41 Ω.

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