What is AC Power? Active Power, Reactive Power, Apparent Power

AC power is composed of active power (real power), reactive power, and apparent power. Active power is the electrical power consumed by household appliances and machinery for operation. The load does not consume reactive power but is necessary for power transmission. Apparent power is the combination of active and reactive power, indicating the total amount of power.

The balance of active, reactive, and apparent power affects the power supply’s efficiency and the power grid’s stability. This article explains the roles and functions of active, reactive, and apparent power, as well as their interrelationships.

What is Electrical Power?

AC power is electrical power that periodically changes current and voltage. Unlike DC power, AC power alternates, making it suitable for long-distance power transmission. AC power consists of three components: active, reactive, and apparent. Active power delivers energy, while reactive power is necessary to maintain the stability of power systems. Apparent power is the sum of these components, allowing for effective power management. This way, AC power is widely used in daily life and industry.

Definition of DC Power

The difference between alternating current AC power and direct current DC power lies in how the electric current flows. Alternating current AC power is characterized by the periodic change in the direction of the current and is widely used in homes and businesses. The unidirectional current flow characterizes direct current DC power and is mainly used in battery-powered and electronic devices. AC power is advantageous for long-distance power transmission and allows easy voltage conversion using transformers. DC power is highly energy-efficient and essential for many electronic devices. Let’s briefly review the differences between AC power and DC power.

In a DC circuit, the power consumed is the product of the DC voltage (\(V\)) and the direct current (\(I\)), usually represented as \(P\). The unit of measurement is watts. This is commonly referred to as power consumption. The constant voltage of the DC power supply and the direct current flowing to the load remain constant.

For example, the DC voltage and direct current applied to a load are represented by \(V\) and \(I\) respectively. In that case, the direct current DC power (\(P\)) can be expressed by the following formula:

DC power: \(P=I×V\)

Definition of AC Power

In an AC circuit, power is classified into three types: active power (\(P\)), reactive power (\(Q\)), and apparent power (\(S\)). This classification arises because the voltage supplied by the power source fluctuates over time, and the direction of the electric current periodically changes. The RMS voltage, current values, and the phase angle between them are used to calculate power.

The current and voltage have the following relationship for a typical sine wave.

\(V_{\text{rms}}=\displaystyle\frac{V_{\text{max}}}{\sqrt{2}}\)

\(I_{\text{rms}}=\displaystyle\frac{I_{\text{max}}}{\sqrt{2}}\)

Summarizing active power, reactive power, and apparent power is as follows:

The power factor indicates the efficiency with which AC power is utilized. It is expressed as the ratio of active power (watts) to apparent power (volt-amperes). A power factor closer to 1 indicates higher efficiency, meaning more power is effectively used.

Active Power (Real Power)

Active power, denoted by the symbol \(P\), is the power consumed by a load. Its unit is watts.
Active power (\(P\)) performs the “actual work” in an electrical circuit. Measured in watts, effective power defines the power consumed by the resistive part of the circuit (the load). While active power (\(P\)) is present in both AC and DC circuits, the calculation in an AC circuit accounts for the phase difference between voltage and current, which is not a factor in a DC circuit. However, in cases where the phase difference (\(\theta\)) is zero, the calculation of active power in an AC circuit is similar to that in a DC circuit.
Therefore, active power is calculated in an AC circuit by considering the phase difference between the voltage and current. In the absence of a phase difference, as in a purely resistive AC circuit, all useful power is directly supplied to the resistance and converted into heat, light, and work, similar to a DC circuit.

Formula for Active Power

Active power (\(P\)) is calculated as the product of the current (\(I\)) and the voltage (\(V\)). The formula is as follows:

\(P = V_{\text{rms}} × I_{\text{rms}} × cos\theta\)

Here, \(V_{\text{rms}}\) is the rms voltage applied to the load, \(I_{\text{rms}}\) is the rms current flowing through the load, \(\theta\) represents the phase angle between the current and voltage, and cos\(\theta\) signifies the power factor.

Note that this active power (\(P\)) is the power consumed by the load, also called power dissipated. Additionally, simply using the term ‘electric power’ usually refers to this active power (power dissipated).

Calculation Method and Example of Active Power

We will explain the specific calculation procedure for active power in detail and elucidate the method for accurately calculating active power considering voltage, current, and phase differences.

1. Measurement of Voltage and Current
Accurately measure the voltage and current. These values must be RMS values.

\(V_{\text{rms}}=\sqrt{\frac{1}{T} ∫_0^T [V(t)]^2 dt}\)

In the case of a general sine wave, the following relationship holds at the maximum value \(V_{\text{max}}\) :

\(V_{\text{rms}}=\displaystyle\frac{V_{\text{max}}}{\sqrt{2}}\)

For current, it also needs to be calculated in RMS values, with the following relationship holding:

\(I_{\text{rms}}=\displaystyle\frac{I_{\text{max}}}{\sqrt{2}}\)

2. Measurement of Phase Difference
We measure the phase difference (\(\theta\)) between current and voltage. This can be done by observing waveforms using an oscilloscope or similar device. The basic representation of sinusoidal waveforms in an AC circuit is as follows:

\(V_1(t)=V_{\text{max1}} \ sin⁡(ωt + \theta_1 )\)

\(V_2(t)=V_{\text{max2}} \ sin⁡(ωt + \theta_2 )\)

Here, \(V_{\text{max1}}\) and \(V_{\text{max2}}\) are the maximum values of the respective waveforms, \(ω\) is the angular frequency, and \(\theta\)₁ and \(\theta\)₂ are the phase angles.
We identify the two waveforms’ crossover points and use them as reference points. We then measure the time displacement to each waveform’s peak (or zero crossing) from the crossover points, which is the phase difference.

3. Calculation of Active Power
Based on the obtained values, active power (\(P\)) is calculated as the product of electric current (\(I\)) and voltage (\(V\)).

\(P=V_{\text{rms}}×I_{\text{rms}}×cos\theta\)

Understanding Active Power Through Real-World Examples

This demonstrates how active power calculations are applied through actual case studies.

Example: Household Electrical Devices

Let’s take the example of household electrical devices to understand the concept. Appliances like refrigerators typically use a sine wave current. Since the phase difference between current and voltage is almost zero in these devices, the active power is nearly equal to the product of the RMS voltage and RMS current.
Assume the voltage is 120V, the current is 5A, and there is no phase difference.

\(P=(120V)×(5A)×cos0°\)

\(P=(120V)×(5A)×1\)

\(P=600W\)

In this case, the household electrical device’s active power (power consumed) is 600W.

Example: Industrial equipment

A phase difference can occur in industrial equipment. For example, in equipment with varying loads, such as induction motors, a phase difference occurs and affects active power calculation.

Assuming the RMS voltage is 120V, the RMS current is 5A, and the phase angle is 30 degrees:

\(P=(120V)×(5A)×cos30°≒600×0.866≒519W\)

In this case, the industrial equipment’s active power (power dissipated) is approximately 519W.

Application and Importance of Active Power in the Real World

Accurately calculating active power is crucial in various contexts. For example, household electrical devices such as refrigerators and air conditioners help reduce unnecessary power consumption and enable efficient energy use. In industrial settings, it improves efficiency and optimizes production processes. Additionally, accurate active power calculation in renewable energy systems supports stable power supply management. The importance of these calculations lies in improving power quality, optimizing power flow, and ensuring the stability of power supply, which contributes to energy conservation and sustainable energy use.

In the industrial sector, precise active power calculation is essential for minimizing power losses and enhancing the system’s overall efficiency. This understanding and calculation also contribute to energy conservation and promoting sustainable energy, making it critical in modern energy management.

Reactive Power: The Reactive Component of Energy

The imaginary part of the complex number obtained from the product of current (\(I\)) and voltage (\(V\)) is called reactive power and is represented by the symbol \(Q\). The unit is volt-amperes reactive (\(VAR\)). Reactive power (\(Q\)) does not perform practical work but affects the phase difference between voltage and current waveforms. Reactive power does not exist in DC circuits because there is no phase difference between current and voltage in DC. Reactive power represents energy oscillation between the source and reactive components, such as inductors and capacitors, without consumption.

Reactive power mainly occurs when a phase difference between voltage and current exists. This phase difference (leading or lagging) is primarily influenced by non-resistive loads or inductive devices such as motors and transformers. Reactive power depends on the reactance generated by inductors and capacitors, which counteracts the effects of active power. The power stored in the magnetic field by inductors attempts to control the current, while the power stored in the electrostatic field by capacitors attempts to control the voltage.

The formula for Reactive Power

This section explains the specific calculations for reactive power. Reactive power (\(Q\)) can be calculated based on the phase difference (\(\theta\)) between the current (\(I\)) and the voltage (\(V\)) as follows:

\(Q=V_{\text{rms}}×I_{\text{rms}}×sin\theta\)

Here, \(Q\) represents the reactive power, \(V_{\text{rms}}\) is the rms value of the AC voltage, \(I_{\text{rms}}\) is the rms value of the alternating current, and \(\theta\) is the phase difference between the voltage and current.

The Role and Impact of Reactive Power in Power Systems

Reactive power (\(Q\)) oscillates between the AC power supply and the load and is not consumed by the load. The magnitude of reactive power mainly changes due to the phase difference (\(\theta\)) caused by the load’s inductance and capacitance components. The phase difference indicates the lag between voltage and current, which is the cause of reactive power generation.

Reactive power can lead to inefficiencies in power transmission and conversion processes as it requires handling additional power that does not contribute to practical work. Therefore, controlling reactive power can improve the stability and efficiency of power systems. Standard methods to manage reactive power include using capacitors or inductors for phase compensation, synchronous condensers, and other power correction devices to adjust and stabilize the phase angle between voltage and current.

Apparent Power: Total Amount of Energy

Apparent power is a measure of power in AC circuits, representing the total power combining active and reactive components, regardless of the phase difference between current and voltage. It indicates the total amount of electrical energy and has a mathematical relationship as a composite of active power (\(P\)) and reactive power (\(Q\)). The symbol \(S\) represents the apparent power, the product of the RMS voltage (\(V\)) applied to the AC circuit and the RMS current (\(I\)) flowing through the circuit. The unit is volt-ampere (VA). Apparent power (\(S\)) is the magnitude of the vector sum of active power (\(P\)) and reactive power (\(Q\)), typically represented in volt-amperes (\(VA\)).

Formula for Apparent Power

Apparent power (\(S\)) is expressed as the AC voltage and current product.

\(S=V_{\text{rms}}×I_{\text{rms}}\)

Here, \(S\) represents the apparent power, \(V_{\text{rms}}\) is the rms value of the AC voltage, \(I_{\text{rms}}\) is the rms value of the alternating current, and \(\theta\) is the phase difference between the voltage and current.

Practical Applications and Examples of Apparent Power

Understanding apparent power is crucial in the design and operation of power systems. For example, accurate calculation and comprehension of apparent power are necessary for capacity design of equipment and devices, load analysis of the power grid, and power quality improvement. In specific cases, proper management of apparent power is essential for ensuring efficient power transmission and effective device design. Optimizing apparent power enhances the efficiency of power systems and offers benefits such as extending the lifespan of electrical devices.

Apparent Power, Active Power, and Reactive Power

Apparent power (\(S\)) is a concept that includes both reactive power (\(Q\)) and active power (\(P\)). Understanding the relationship between apparent, active, and reactive power is essential for efficiently operating power systems. The power triangle visually explains these relationships. Here, we will explain how apparent power is composed.

Understanding and Drawing Power Vector Diagrams

Apparent power comprises two parts: active power, which is resistive power measured in watts, and reactive power, measured in volt-amperes, which represents what arises due to the phase difference between voltage and current. The vector addition of these two power components can be described as a power triangle. The power triangle includes three main components: \(P\) (active power), \(Q\) (reactive power), and |\(S\)| (apparent power). The angle \(\theta\) represents the phase difference between voltage and current. \(S\) (complex power) is the complex representation of apparent power, combining active and reactive power.

The three elements that constitute power in an AC circuit can be graphically represented by the three sides of a right triangle. The horizontal side represents the circuit’s active power (\(P\)), the vertical side represents the circuit’s reactive power (\(Q\)), and the hypotenuse represents the resulting complex power (\(S\)). As illustrated in this diagram, the magnitude of the apparent power (|\(S\)|) can be calculated as follows:

\(|S|=\sqrt{P^2+Q^2}\)

Furthermore, when using complex numbers, it can be expressed as follows:

\(S=P+jQ\)

Here, \(S\) is the complex power (the complex representation of apparent power), \(P\) is the active power, \(Q\) is the reactive power, and \(j\) is the imaginary unit.

In the vector diagram of electrical power, the current’s phase is considered concerning the voltage. This allows the active and reactive power components to be intuitively understood on the vector diagram.

Mathematically deriving the magnitude of apparent power

Each formula is as follows:
Active power \(P\)

\(P=V_{\text{rms}}×I_{\text{rms}}×cos\theta＝S×cos\theta[W]\)

Reactive power \(Q\)

\(Q=V_{\text{rms}}×I_{\text{rms}}×sin\theta＝S×sin\theta[VAR]\)

Apparent power \(S\)

\(S=V_{\text{rms}}×I_{\text{rms}}[VA]\)

Using the trigonometric identity in the above three formulas:

\(cos\theta^2+sin\theta^2=1\)

We get:

\(P^2+Q^2=S^2 (cos\theta^2+sin\theta^2)=S^2\)

The relationship between active power \(P\), reactive power \(Q\), and apparent power \(S\) is as follows:

\(S^2=P^2+Q^2\)

\(S=\sqrt{P^2+Q^2}\)

Apparent power (\(S\)) is the electrical power supplied by an AC power source. When reactive power (\(Q\)) increases, the power supplied by the AC power source also increases. In other words, as reactive power (\(Q\)) increases, the burden on the AC power source is expected to grow.

Reactive power does not directly reduce active power but increases the total power demand on the AC power source. Therefore, it is essential to ensure that the apparent power supplied by the power system is sufficient to meet the load requirements. Since the power source must supply the required volt-ampere (\(VA\)) power for any load, this is essential to understanding AC power sources.

What is the Power Factor?

The power factor (cos\(\theta\)) is one of the key indicators for evaluating electrical power quality. It shows how efficiently an electrical circuit uses the available power, with a higher power factor indicating less wasted energy in the form of reactive power. Power factor is defined as the ratio of active power (\(P\)) to apparent power (\(S\)), and it is usually expressed in a range from 0 to 1, with a value closer to 1 indicating more efficient use of power.

Accurate measurement of power factor is crucial for assessing the efficiency of power systems and minimizing unnecessary power losses. A low power factor can strain the power grid and lead to inefficient power use. Improving power factor leads to enhanced power efficiency and contributes to the effective utilization of energy.

Calculation and Measurement of Power Factor

Calculation of Power Factor

The power factor is related to the phase angle between the current and voltage waveforms, representing the cosine of this phase angle.

The following formula expresses the power factor (\(PF\)):

\(Power Factor (PF)=\displaystyle\frac{Active Power (P)}{Apparent Power (S)}\)

Here, \(P\) is the active power, and \(S\) is the apparent power, including both active and reactive.
The power factor is typically represented in the range from 0 to 1, with values closer to 1 indicating more efficient use of electrical power.

Calculation Example of the Power Triangle in Power Factor

The power triangle graph represents the active power (practical value), reactive power, and the specified phase difference between voltage and current.

Active power (\(P\)), reactive power (\(Q\)), and the phase difference (\(\theta\)) between voltage and current have the following relationships:

\(P=V_{\text{rms}}×I_{\text{rms}}×cos\theta\)

\(Q=V_{\text{rms}}×I_{\text{rms}}×sin\theta\)

Here, \(V\) is the RMS voltage, \(I\) is the RMS current, and \(\theta\) is the specified phase difference between the voltage and current. Suppose the voltage is 100V and the current is 10A, with the active power being 800W and the reactive power being 600VAR. The power factor from the active power can be expressed as follows.

The power factor cos\(\theta\) can be calculated as follows.

\(P=V_{\text{rms}} ×I_{\text{rms}}×cos\theta=100×10×cos\theta\)

\(cos\theta = \displaystyle\frac{P}{100×10} = \displaystyle\frac{800}{100×10} = 0.8\)

Similarly, from the reactive power, sin\(\theta\) can be represented as follows.

\(Q=V_{\text{rms}}×I_{\text{rms}}×sin\theta=100×10×sin\theta\)

\(sin\theta = \displaystyle\frac{Q}{100×10} = \displaystyle\frac{600}{100×10} = 0.6\)

The phase difference (\(\theta\)) between voltage and current is as follows.

\(\theta=Arcsin(0.6)≒36.87°\)

*Use frequency method

From the above, active power, reactive power, and phase difference can be calculated.

Example Calculation in an RL Circuit

Assume the following data is given:

Resistance (\(R\)): 50 Ω, Inductance (\(L\)): 0.1 H, AC supply: \(V\) = 100 V, Frequency: \(f\) = 50 Hz

Calculation of Impedance (\(Z\)):

\(X_L=2πfL=2×3.1416×50×0.1=31.416[Ω]\)

\(Z = \sqrt{R^2+(X_L)^2} \quad= \sqrt{\mathrm{50}^2+31.146^2} \quad= \sqrt{2500+986.96} \quad= \sqrt{3486.96} \quad≒ 59.05[Ω]\)

Calculation of Electric Current (\(I\)):

\(I=\displaystyle\frac{V}{Z}=\displaystyle\frac{100}{59.05}\quad≒1.694[A]\)

Calculation of Power Factor (PF) and Phase Angle (\(\theta\)):

\(PF=cos\theta=\displaystyle\frac{R}{Z}=\displaystyle\frac{50}{59.05} \quad≒ 0.847\)

Calculation of Apparent Power (\(S\)) Consumed:

\(S=VI=100×1.694=169.4[VA]\)

Measurement of Power Factor

Power factor can be measured in several ways. Below are the primary measurement methods.
Measurement Using a Power Meter
Power meters are commonly used to measure the practical value of power and the power factor. These meters measure the complex value of electrical power and calculate the effective power and power factor from the results.
Measurement Using a Clamp Meter
A clamp meter is a device that measures current nondestructively using a clamp attached to the power line. It acquires voltage and current information and uses it to calculate effective power and power factor.
Detailed Measurement Using a Power Analyzer
A power analyzer is a device capable of measuring various parameters of power systems with high precision. It can calculate the power factor directly from the phase difference between current and voltage and analyze multiple waveform parameters simultaneously.

Methods for Power Factor Improvement and Their Effects

Inappropriate power factors can lead to unnecessary consumption of electrical power. To improve the power factor, the following methods can be employed:
Addition of Capacitors: Capacitors can compensate for reactive power, thus enhancing the power factor. This is particularly effective when there are many inductive loads.
Use of Power Factor Correction Devices: Introducing power factor correction devices can automatically adjust the power factor, reducing power loss.
Load Optimization: Power factor improvement can be expected by properly managing loads and achieving efficient operation.
Applying these methods promotes efficient use of electrical power and enhances the overall efficiency of power systems.