Electrical Reactance

Electrical Reactance is a type of electrical resistance that impedes the flow of current in an AC circuit. It is denoted by the symbol “\(X\)” and measured in “\(Ω\)”. There are two types of reactance: capacitive reactance and inductive reactance. As the frequency increases, capacitive reactance decreases while inductive reactance increases. Reactance arises from the phase difference between the current and voltage and is based on the properties of inductance (inductors) and capacitance (capacitors). Understanding these characteristics is essential for designing and analyzing AC circuits, tuning high-frequency filters and resonant circuits, and improving power transmission efficiency.

This article will explain the two types of reactance, the methods for calculating them, and their relationship with impedance.

What is Electrical Reactance?

Electrical reactance opposes alternating current (AC) flow in an electrical circuit. It is a type of resistance that occurs only in AC circuits. When an alternating current passes through a coil (inductance) or a capacitor (capacitance), a phase difference between the current and voltage waveforms is created, resulting in reactance. It is denoted by the symbol “\(X\)” and measured in ohms (\(Ω\)). Reactance is essential in designing and analyzing AC circuits that use inductors and capacitors. It also constitutes the imaginary part of the complex impedance.

How to Calculate Reactance

Reactance (\(X\)) can be calculated from the ratio of voltage (\(V\)) to current (\(I\)) using the following formula:

\(X=\displaystyle \frac{V}{I}\ [\mathrm{\Omega}] \)

However, the behavior of reactance differs between coils and capacitors. The reactance of a coil is called “inductive reactance,” while the reactance of a capacitor is called “capacitive reactance.”

Difference Between Reactance and Resistance

Like reactance, there is “resistance,” which indicates the difficulty for electric current to flow. Resistance is represented by “\(R\)” and its unit is “\(Ω\)”. Since resistance also represents the “difficulty in current flow,” the formula for the relationship between current and voltage is the same as for reactance:

\(R=\displaystyle \frac{V}{I}\ [\mathrm{\Omega}]\)

Generally, “resistor” refers to a component that provides resistance.

There are two main differences between reactance and resistance. One is that reactance represents the “difficulty of current flow in AC circuits”, while resistance represents the “difficulty of current flow in both DC and AC circuits”.

The other difference is that reactance varies with frequency, whereas resistance does not change with frequency.

For example, connect a 5Ω resistor to an AC power source with an amplitude of 5V and a frequency of 100Hz and compare it to connecting it to an AC power source with an amplitude of 5V and a frequency of 200Hz. The magnitude of the current will be the same in both cases:

\(I = \displaystyle \frac{V}{R} = \displaystyle \frac{5 \, \text{[V]}}{5 \, \text{[Ω]}} = 1 \, \text{A}\)

Inductive Reactance

Inductive reactance is the resistance encountered by an alternating current (AC) as it passes through an inductor (coil). This resistance occurs because the inductor generates an opposing voltage in response to changes in the current, which acts to resist the change. Consequently, only inductive reactance slows the current flow in an inductive circuit and causes a phase difference between the current and voltage. Only inductive reactance is denoted by the symbol “\(X_L\)”.

Inductive Reactance Increases with Frequency

Inductive reactance arises from inductive elements such as coils and depends on the frequency of the AC and the inductance of the inductive element. Specifically, inductive reactance “\(X_L\)” is calculated using the following inductive reactance formula:

\(X_L\ =\ \omega L\ =\ 2\pi fL\ [\mathrm{\Omega}]\)

Where:

\(ω\): Angular frequency (\(2πf\)) \(rad/s\)
\(f\): Frequency
\(L\): Inductance in Henries (\(H\))

The unit of inductance “\(L\)” is Henry, a characteristic value determined by the coil’s shape and number of turns.

As the frequency “\(f\)” increases, inductive reactance increases proportionally. Therefore, inductive reactance is directly proportional to frequency. In a direct current (DC) circuit, where the frequency is 0 Hz, the inductive reactance is zero reactance. This means the inductive circuit behaves as if it has zero reactance to the current flow.

In Inductive Reactance, Electric Current Decreases with Frequency

In inductive reactance, current is inversely proportional to frequency. Inductive reactance is the ratio of voltage to current in an inductive circuit and is expressed by the following formula:

\(X_L=\displaystyle \frac{V}{I}\left[\mathrm{\Omega}\right]\)

Solve this formula for the current \(I\) give us:

\(I = \displaystyle \frac{V}{X_L} = \displaystyle \frac{V}{2\pi f L} \, \text{[A]}\)

This equation shows that the current I decreases as the frequency “\(f\)” increases, meaning they are inversely proportional.

Capacitive Reactance

Capacitive reactance is the opposition to the change in alternating current (AC) caused by a capacitor. This opposition arises because the capacitor stores and releases electrical energy as the current changes, leading to a voltage change across the capacitor. Consequently, capacitive reactance accelerates the flow of electric current in a capacitive circuit and introduces a phase difference between the current and voltage waveforms. The symbol for capacitive reactance is “\(X_C\)”.

Capacitive Reactance Decreases with Frequency

Only capacitive reactance decreases with frequency. Only capacitive reactance is determined by capacitive components like capacitors and depends on the frequency of the alternating current and the capacitance of the capacitor. Specifically, only capacitive reactance \(X_C\) is calculated using the following capacitive reactance formula:

\(X_C=\displaystyle \frac{1}{\omega C}=\displaystyle \frac{1}{2\pi fC}\ \left[\mathrm{\Omega}\right]\)

where:

• ・\(ω\): Angular frequency (\(2πf\)) [rad/s]
• ・\(f\): Frequency [\(Hz\)]
• ・\(C\): Capacitance [\(F\)]

The unit for capacitance “\(C\)” is “farad.” A capacitor’s capacitance value “\(C\)” is an inherent property determined by its physical characteristics.

As the frequency “\(f\)” increases, the capacitive reactance “\(X_C\)” decreases. This means capacitive reactance is inversely proportional to the frequency. In a direct current (DC) circuit, where the frequency is 0 Hz, the capacitive reactance becomes infinitely large.

In Capacitive Reactance, Electric Current Increases with Frequency

Capacitive reactance is the ratio of voltage to current in a capacitive circuit and is represented by the following formula:

\(X_C = \displaystyle \frac{V}{I} \, \text{[Ω]}\)

Solving this formula for current “I” gives:

\(I=\displaystyle \frac{V}{Xc}=V･2πfC[A]\)

This equation clearly shows that the current “I” increases as the frequency “f” increases, meaning they are directly proportional.

Total Reactance

Total reactance refers to the combined effect of inductive reactance (resistance caused by an inductor) and capacitive reactance (resistance caused by a capacitor) in an alternating current (AC) circuit. This combined resistance influences the relationship between the current and the voltage phases, thereby determining the behavior of the AC circuit. Inductive reactance opposes changes in current and causes the phase to lag, while capacitive reactance causes the current to lead to the voltage. Total reactance is calculated as the difference between these two types of reactance, determining whether the current lags behind or leads the voltage. The symbol for total reactance is typically “\(X\),” and its value is obtained by subtracting the capacitive reactance “\(X_C\)” from the inductive reactance “\(X_L\)”:

\(X = X_L – X_C = \omega L – \displaystyle \frac{1}{\omega C} = 2\pi f L – \displaystyle \frac{1}{2\pi f C} \, \text{[Ω]}\)

It’s important to note that both inductive reactance and capacitive reactance depend on the frequency. At a specific frequency, \(X_L = X_C\) meaning the total reactance is zero. This condition is called resonance, and the frequency at which this occurs is known as the resonant frequency

Reactance as a Component of Impedance

Understanding the difference between reactance and impedance is essential for efficient electric circuit design. This section delves into the characteristics of reactance and impedance, explaining how each affects an electric circuit differently.

Impedance quantifies the opposition faced by electric current in an AC circuit, incorporating both resistance and reactance. Represented by the symbol “\(Z\),” it is measured in ohms (\(Ω\)).

Impedance is commonly represented as complex impedance. Complex impedance uses complex numbers to describe the relationship between current and voltage in an electric circuit, where reactance (\(X\)) represents the imaginary part. The formula for complex impedance is expressed as:

\(Z = R + jX\)

Here, R is the component that consumes power, and X is the component that stores power. The symbol j represents the imaginary unit (where \(j^2 =−1\)).

Additionally, the magnitude (absolute value) of impedance can be calculated using the Pythagorean theorem as follows:

\(Z=\sqrt{\left(R^2+X^2\right)}\)

The Impact of Reactance on the Phase of Current

Reactance delays or advances the flow of current in an electric circuit. Inductive reactance delays the current flow due to the changing magnetic field in inductors, while capacitive reactance advances the current flow due to the charging and discharging of capacitors.

As a result, inductive reactance causes the current phase to lag behind the voltage, and capacitive reactance causes the current phase to lead the voltage. Specifically, in a circuit with inductive reactance, the current lags the voltage by 90 degrees, while in a circuit with capacitive reactance, the current leads the voltage by 90 degrees. This phase difference is critical in the design and analysis of AC circuits.

The phase difference in AC circuits directly affects the power factor, influencing the efficient transmission and consumption of power. A power factor close to 1 indicates efficient power usage, whereas a low power factor means significant power is wasted. Therefore, understanding and managing the effects of reactance is crucial for improving energy efficiency and reducing power costs.

Thus, reactance significantly impacts the current phase, and understanding this impact aids in the effective design and operation of electronic circuits and power systems.

The Relationship Between Reactance and AC Power

There are three types of AC power: apparent power, active power, and reactive power. Reactance is the main factor generating reactive power. Reactance causes a phase difference \(θ\) between voltage and current, resulting in reactive power.

1. Apparent Power (\(S\)): The total power represented by the product of voltage and current. Its unit is VA.

   \(S=V\times I\)

2. Active Power (\(P\)): The power that performs actual work and is consumed by resistance. Its unit is W.

   \(P=V\times I\times cos\theta\)

3. Reactive Power (\(Q\)): The component of energy that is stored and released, caused by inductance or capacitance. Its unit is VAR.

   \(Q=V\times I\times sin\theta\)

This phase difference increases as reactance increases, and the reactive power increases proportionally. Therefore, reactance is directly involved in the generation of reactive power.

Summary of Reactance

Reactance is a concept that represents the opposition to the flow of current in an AC circuit. It is denoted by the symbol “\(X\)” and measured in units of “\(Ω\).” Reactance is categorized into two types: inductive reactance (due to inductors) and capacitive reactance (due to capacitors). These types vary with frequency and change the current phase, making them essential for the design and analysis of AC circuits.

Inductive reactance increases with rising frequency, while capacitive reactance diminishes as frequency increases. These characteristics determine the current behavior within the AC circuit. The total reactance is also calculated as the difference between these two reactances. At the resonant frequency, the reactances cancel each other out, resulting in a total reactance of zero.

Understanding reactance leads to a better grasp of impedance. Impedance combines reactance and resistance and indicates the relationship between current and voltage. The following article will explain the details and calculation methods of impedance. We will then explore resonant circuits and examine how reactance and impedance affect resonance phenomena.