Electric Power Single and Three Phase Power Active Reactive Apparent •

Table of Contents

Key Concepts in Electric Power:

Electric Power: The rate of electrical energy transfer in a circuit, measured in watts (W).
Single-Phase Power: Uses one alternating voltage and current wave, common in residential settings.
Three-Phase Power: Employs three alternating currents offset by 120 degrees, offering stable and efficient power ideal for industrial applications.
Active Power: The power that performs actual work in a circuit, measured in watts (W).
Reactive Power: Supports voltage levels for active power transfer but does no useful work. Measured in volt-amperes reactive (VAR).
Complex Power: A crucial concept. In a single-phase network, voltage (V) and current (I) can be represented as complex values: V * e^(jα) and I * e^(jβ), respectively, where α and β are the angles of the voltage and current vectors relative to a reference axis. Complex power is calculated by multiplying the voltage by the conjugate of the current. This product yields both the active and reactive power components.

Complex Power

Complex Power: A crucial concept. In a single-phase network, voltage (V) and current (I) can be represented as complex values: V * e^(jα) and I * e^(jβ), respectively, where α and β are the angles of the voltage and current vectors relative to a reference axis. Complex power is calculated by multiplying the voltage by the conjugate of the current. This product yields both the active and reactive power components.

The angle (α – β) represents the phase difference between voltage and current, commonly denoted as φ. The complex power (S) can be expressed as:
S = VI * e^(jφ)
Where:

A negative Q (when φ is negative, meaning current leads voltage) indicates a capacitive load.

P = VIcosφ (Active Power)

Q = VIsinφ (Reactive Power)
The magnitude of complex power, |S| = √(P² + Q²), is the apparent power, measured in volt-amperes (VA). Apparent power is the product of the absolute values of voltage and current. Since current is directly related to heating (Joule’s Law), electrical machines are rated based on their apparent power handling capacity within acceptable temperature limits.
In the complex power equation:

A positive Q (when φ is positive, meaning current lags voltage) indicates an inductive load.

Single Phase Power

Single-Phase Power Fundamentals
While single-phase transmission is rarely used today, understanding its principles is crucial before delving into modern three-phase systems. Key parameters in electrical systems are resistance, inductance, and capacitance.

Resistance

Resistance is an inherent property of materials that opposes current flow. This opposition arises from collisions between electrons and atoms, generating heat known as ohmic power loss. In a purely resistive circuit, voltage and current are in phase (no phase difference). The energy consumed by a resistor (R) carrying current (I) for time (t) is I²Rt. This energy is termed active energy, and the corresponding power is active power.

Inductance

Inductance: Inductance is the property of a circuit element (an inductor) to store energy in a magnetic field. During a portion of the AC cycle, the inductor stores energy; during another portion, it releases it back into the circuit. The energy stored in an inductor (L) carrying a current (I) is given by (1/2)LI². The power associated with inductance is reactive power.

Capacitance

Capacitance is the property of a circuit element (a capacitor) to store energy in an electric field. Like inductors, capacitors store energy during part of the AC cycle and release it during another. The energy stored in a capacitor (C) with a voltage difference (V) across its plates is (1/2)CV². This energy resides in the electric field between the plates. The power associated with a capacitor is also reactive power.

Active and Reactive Power in Single-Phase Circuits

Active Component and Reactive Component of Power

Consider a single-phase AC circuit where the current lags the voltage by an angle φ. The instantaneous voltage (v) and current (i) can be represented as:

i = Im * sin(ωt – φ)
Where Vm and Im are the maximum voltage and current, respectively. The instantaneous power (p) is given by:
p = vi

Active Power

Resistive Power

Active Power (Resistive Power):
First, consider a purely resistive circuit (φ = 0). In this case:
p = Vm * Im * sin(ωt) * sin(ωt) = (Vm * Im / 2) * (1 – cos2ωt)
Since cos2ωt cannot exceed 1, the instantaneous power (p) is always positive. This means energy always flows from the source to the load. This power, consumed due to the resistive component of the circuit, is called active power or resistive power.

v = Vm * sin(ωt)

Reactive Power

Inductive Power

In a purely inductive circuit, the current lags the voltage by 90 degrees (φ = +90°). Substituting this into the power equation shows that power alternates direction. During one quarter of the AC cycle, power is negative (energy flows from the load back to the source), and during the next quarter cycle, it’s positive (energy flows from the source to the load). This power oscillation occurs at twice the supply frequency. Because the energy is simply exchanged between the source and the inductor, with no net work done, this is called reactive power or inductive power. The average power over a full cycle is zero. The inductor stores energy in its magnetic field during one half-cycle and releases it during the next. The rate of this energy exchange is the reactive power.
(Capacitive Power follows in the next prompt)

Capacitive Power

In a purely capacitive single-phase power circuit, the current leads the voltage by 90 degrees (φ = -90°). Consequently, the power flow alternates directions. Over a full cycle, the power alternates between positive (from 0° to 90° and 180° to 270°) and negative (from 90° to 180° and 270° to 360°) half-cycles. This alternating power, with a frequency twice that of the supply, performs no useful work, just like inductive reactive power. Therefore, capacitive power is also classified as reactive power.

Active Component and Reactive Component of Power

Power Factor in AC Circuit - Your Electrical Guide

The power equation can be expressed as:
P = Vm * Im * cosφ * (1 – cos2ωt) + Vm * Im * sinφ * sin2ωt
The first term, Vm * Im * cosφ * (1 – cos2ωt), is always non-negative since (1 – cos2ωt) is always greater than or equal to zero. This component represents the active or real power. Its average value is non-zero, indicating that this power performs useful work.
The second term, Vm * Im * sinφ * sin2ωt, alternates between positive and negative values. Its average value is zero, meaning this component represents reactive power. It oscillates back and forth on the transmission line without doing any useful work.
While both active and reactive power are dimensionally equivalent to watts, reactive power is measured in volt-amperes reactive (VAR) to distinguish it as non-working power.
Single-phase power systems, where all voltages vary in unison, are typically generated by rotating a coil in a magnetic field or vice-versa. This produces single-phase alternating voltage and current. Different circuit elements (resistance, capacitance, and inductance, or combinations thereof) respond differently to sinusoidal inputs. We will analyze these responses individually to derive the single-phase power equation.

Single Phase Power Equation for Purely Resistive Circuit

Let’s analyze single-phase power calculations in a purely resistive circuit. Consider a circuit with a pure ohmic resistance, R, connected across a voltage source, V.
Where, V(t) = instantaneous voltage.
V_m = maximum value of voltage.
ω = angular velocity in radians/seconds.

According to Ohm’s law ,

Substituting value of V(t) in above equation we get,
Equations (1.1) and (1.5) demonstrate that V(t) and I_R are in phase. Therefore, in a purely resistive circuit, the voltage and current have no phase difference; they are in phase, as illustrated in Figure (b).

Instantaneous power,

From single phase power equation (1.8) it is clear that power consist of two terms, one constant part i.e.

and another a fluctuating part i.e.

That’s value is zero for the full cycle. Thus power through pure ohmic resistor is given as and is shown in fig(c).

Single Phase Power Equation for Purely Inductive Circuit

Let’s examine the single-phase power equation for a purely inductive circuit.
An inductor, a passive component, opposes the flow of AC current by generating a back EMF. The applied voltage must balance this back EMF. Consider a circuit with a pure inductor, L, connected across a sinusoidal voltage source, Vrms.
The voltage across the inductor is given by:
V = L * (dI/dt)
From this, it’s clear that the current (I) lags the voltage (V) by π/2 (90 degrees), or equivalently, the voltage leads the current by π/2. This phase difference is illustrated in Figure (e).
The instantaneous power is given by:
P = V * I = (Vm * sin(ωt)) * (Im * sin(ωt – π/2)) = -VmImsin(ωt)cos(ωt) = (-Vm*Im/2)sin(2ωt)
This single-phase power formula for a purely inductive circuit contains only a fluctuating term. The average power over a full cycle is zero. This signifies that no real power is consumed by the inductor; energy is stored in the inductor’s magnetic field during one half-cycle and released back to the source during the other half-cycle.

Single Phase Power Equation for Purely Capacitive Circuit

When AC passes through capacitor, it charges first to its maximum value and then it discharges. The voltage across capacitor is given as,

Thus it is clear from above single phase power calculation of I(t) and V(t) that in case of capacitor current leads voltage by angle of π/2.

Power through capacitor consists of only fluctuating term and the value of power for full cycle is zero.

Single Phase Power Equation for RL Circuit

A pure ohmic resistor and inductor are connected in series below as shown in fig (g) across a voltage source V. Then drop across R will be V_R = IR and across L will be V_L = IX_L.

These voltage drops are shown in form of a voltage triangle as shown in fig (i). Vector OA represents drop across R = IR, vector AD represents drop across L = IX_L and vector OD represents the resultant of V_R and V_L.

is the impedance of RL circuit.
From vector diagram it is clear that V leads I and phase angle φ is given by,

Thus power consist of two terms, one constant term 0.5 V_mI_mcosφ and other a fluctuating term 0.5 V_mI_mcos(ωt – φ) that’s value is zero for the whole cycle.
Thus its the only constant part that contributes to actual power consumption.
Thus power, p = VI cos Φ = ( rms voltage × rms current × cosφ) watts
Where cosφ is called power factor and given by,

I can be resolved in two rectangular components Icosφ along V and Isinφ perpendicular to V. Only Icosφ contributes to real power. Thus, only VIcosφ is called wattfull component or active component and VIsinφ is called wattless component or reactive component.

Single Phase Power Equation for RC Circuit

We know that current in pure capacitance, leads voltage and in pure ohmic resistance it is in phase. Thus, net current leads voltage by angle of φ in RC circuit. If V = V_msinωt and I will be I_msin(ωt + φ).

Power is same as in the case of R-L circuit. Unlike R-L circuit electrical power factor is leading in R-C circuit.

Three Phase Power Definition

Here’s a rephrased version:

Advantages of Three-Phase Power Generation
Generating three-phase power is more economical than producing single-phase power. This is due to the unique characteristics of three-phase power systems.

Characteristics of Three-Phase Power:
In a three-phase electric power system:

Phase Difference: The three voltage and current waveforms are offset by 120 degrees in each cycle of power.
Separate Power Circuits: Three individual single-phase powers are transmitted through three separate power circuits.
Time-Phase Difference: The voltages of the three powers are ideally 120 degrees apart from each other in time-phase.
Current Phase Difference: Similarly, the currents of the three powers are also ideally 120 degrees apart from each other.

Balanced Three-Phase Power System:
An ideal three-phase power system implies a balanced system, where:

The voltage and current waveforms are symmetrical.
The power delivered to each phase is equal.

This balance ensures efficient and reliable transmission of power.

A three phase system is said to be unbalanced when either at least one of the three phase voltage is not equal to other or the phase angle between these phases is not exactly equal to 120^o.

Advantages of Three Phase System

Why Three-Phase Power is Preferred Over Single-Phase Power
Three-phase power has several advantages over single-phase power, making it a more preferred choice for high-power applications.

Key Differences:

Constant Power: Three-phase power equation is time-independent, providing a constant flow of power. In contrast, single-phase power equation is time-dependent, resulting in a pulsating power supply.
Reduced Vibration: The constant power flow in three-phase systems reduces excessive vibration, making it ideal for high-rated motors.
Higher Rating: Three-phase machines have a 1.5 times greater rating than same-size single-phase machines.
Self-Starting: Three-phase induction motors are self-starting, eliminating the need for auxiliary starting means.
Improved Power Factor and Efficiency: Three-phase systems boast better power factor and efficiency compared to single-phase systems.

These advantages make three-phase power a more reliable and efficient choice for high-power applications.

Three Phase Power Equation

Understanding Three-Phase Power and Reactive Power
To calculate three-phase power, we consider a balanced system where the voltage and current in each phase differ by 120 degrees, with equal amplitude for each current and voltage wave.

Expressing Instantaneous Power in Each Phase:
The instantaneous power in each phase can be expressed as:

Red phase:
Yellow phase:
Blue phase:

Total Three-Phase Power:
The total three-phase power is the summation of individual power in each phase:

This expression shows that the total instantaneous power is constant and equal to three times the real power per phase.

Reactive Power in AC Circuits:
Reactive power is the magnetic energy flowing per unit time in an electric circuit, measured in VAR (Volt Ampere Reactive). It’s essential in AC circuits, particularly in inductive or capacitive loads.

Power Factor and Reactive Power:
The electrical power factor of equipment determines the amount of reactive power required. It’s the ratio of real power to total apparent power:

Where θ is the phase difference between voltage and current, and cosθ is the electrical power factor.

Reactive power is present in circuits with a phase difference between voltage and current. Inductive components take lagging reactive power (magnetic energy), while capacitive components absorb leading reactive power (electrostatic energy).

Reactive Power in AC Circuits:
In typical AC circuits (RL or RC), reactive power is taken from the supply for a half cycle and returned to the supply for the next half cycle.RL load is derived as:

V = V_msinωt , I = I_msin(ωt − θ)

Here, Q₁sin2ωt is reactive power that’s average value is zero, this shows that reactive power is never utilized.

Use of Reactive Power

Reactive power (VAR) is vital for the proper functioning of electrical machines and systems. Here’s how it impacts different components:

Motors and Transformers: The Role of Magnetizing Reactance

Electrical Motors: Motors rely on reactive power to generate the magnetic field necessary for operation. This field enables the motor to convert electrical energy into mechanical energy, resulting in a lagging power factor as the current phase trails behind the voltage phase.
Transformers: In transformers, reactive power is essential for establishing the magnetic field required for mutual induction in the primary winding. The lagging current in the primary winding reflects the consumption of reactive power needed to magnetize the core, facilitating the transfer of real power from the primary to the secondary winding.

The Importance of Reactive Power in Power Systems

Reactive power plays a crucial role in maintaining a stable and efficient power system.

Key Functions of Reactive Power:

Voltage Regulation: Reactive power helps maintain optimal voltage levels within the electrical system. This is essential for the reliable operation of electrical equipment. Insufficient reactive power can cause voltage drops, while excessive reactive power can lead to voltage rises.
Power Factor Improvement: Reactive power is necessary to achieve a higher power factor, which reduces energy losses and minimizes demand charges from utilities. To improve power factor, capacitor banks, synchronous condensers, and other power factor correction devices are used to supply the required reactive power, thereby reducing the lagging VARs drawn by inductive loads.

3. Impact of Reactive Power on System Stability

Voltage Stability: Proper management of reactive power is essential for voltage stability, especially during peak load conditions or disturbances (e.g., faults, sudden load changes). Techniques such as voltage control devices, load tap changers, and reactive power compensation systems help ensure that the reactive power supply (Q2) meets the reactive power demand (Q1).
Dynamic Response: Systems equipped with fast-acting reactive power compensation devices (like STATCOMs or SVCs) can respond quickly to changes in reactive power demand, thus enhancing the stability of the transmission network and improving overall system performance.

Summary

In conclusion, reactive power is indispensable in electrical machines, particularly motors and transformers, where it forms the backbone of magnetic field generation and energy conversion. Its management is critical for voltage stability, power factor correction, and ensuring reliable operation across the power system. Without adequate reactive power, the efficiency and reliability of electrical systems would be significantly compromised, leading to potential equipment damage and costly outages.

Reactive Power in Transmission Lines

In an electrical power transmission line, the flow of reactive power significantly influences the receiving end voltage. Proper management of voltage levels at the receiving end is crucial because excessive voltage can damage consumer equipment and lead to substantial losses. Frequent voltage fluctuations can occur due to external factors like lightning or faults in other phases, often resulting in equipment damage. To understand how voltage is affected by reactive power, we can express the receiving end reactive power as follows:

[
Q_r = \frac{V_s V_r \sin(\theta)}{X_l}
]

In this equation, ( \theta ) represents the power angle, which is typically maintained at a low value for stability, ( X_l ) is the reactance of the transmission line, ( V_s ) is the sending end voltage, and ( V_r ) is the receiving end voltage. Consequently, we can express ( Q_r ) as:

[
Q_r = Q_1 – Q_2
]

Here, ( Q_1 ) is the reactive power demanded by the load at the receiving end, while ( Q_2 ) is the reactive power supplied by the source at the sending end. It is important to note that we exclude the negative sign from our calculations; otherwise, if ( Q_r ) equals zero, ( V_r ) would also become zero, which is not a feasible scenario.

Case – 1

When the supply, Q2, matches the demand, Q1, the receiving end voltage (Vr) will equal the sending end voltage (Vs), which is the desired outcome.

Case – 2

it is essential to monitor both the excess and deficit of reactive power. Compensation for reactive power imbalances is achieved through various devices. In this context, a reactor absorbs surplus reactive power, while a capacitor provides reactive power to meet high demand.

For loads with a low electrical power factor, there is a significant reactive power demand. To address this, we can enhance the power factor by utilizing a capacitor bank, which lowers the var demand by supplying the necessary reactive power to the load. Other compensation methods include the use of shunt capacitors, synchronous phase modifiers, on-load tap changing transformers, and shunt reactors. An overexcited synchronous motor can also function in parallel with the load, acting as a capacitor, and is referred to as a synchronous condenser. A shunt reactor helps improve the electrical power factor. In on-load tap changing transformers, the turns ratio is adjusted as needed to uphold the desired voltage level, since the voltage difference between the sending and receiving ends influences the reactive power.

Mathematically, the reactive power (Q) required to improve the electrical power factor from cosθ1 to cosθ2 can be expressed as follows:

Where P represents the real power demand of the load (in watts). Conversely, if the electrical power factor needs to decrease from cosθ2 to cosθ1, the reactive power to be absorbed by the shunt reactor at the load end is defined by:

The required values of capacitance or inductance can then be calculated accordingly.

When the demand outweighs supply, Qr becomes negative, causing the receiving end voltage to drop below the sending end voltage. This situation is critical, as voltage management is fundamental for any electrical load and is deeply influenced by reactive power. During peak hours, the demand for reactive power rises, resulting in voltage dips, whereas in the morning, the lower demand leads to an increase in voltage levels. To stabilize the voltage levels, it is necessary to ensure Q1 = Q2.When demand is more and supply is less, Q_r becomes negative. And so the receiving end voltage becomes less than sending end voltage.

Case – 3

Monitoring both the excess and scarcity of reactive power is important, as previously mentioned. Various devices are used for compensation in this context. Specifically, a reactor absorbs excess reactive power, while a capacitor provides reactive power support during times of high demand.

When dealing with low electrical power factor loads, the demand for reactive power is significantly high. To address this, we increase the power factor by employing a capacitor bank, which reduces the demand for volt-amperes reactive (var) by supplying the necessary reactive power to the load. Other methods for improving power factor include using shunt capacitors, synchronous phase modifiers, on-load tap changing transformers, and shunt reactors. An overexcited synchronous motor connected in parallel with the load acts as a capacitor and is known as a synchronous condenser. To decrease the electrical power factor, a shunt reactor is utilized. In on-load tap changing transformers, the turns ratio is adjusted to maintain the desired voltage level, as the voltage difference between the sending and receiving ends influences the reactive power.

The reactive power (Q) required to increase the electrical power factor from cosθ1 to cosθ2 can be expressed mathematically, where P represents the real power demand of the load in watts. Conversely, if we aim to decrease the electrical power factor from cosθ2 to cosθ1, the reactive power that must be absorbed by the shunt reactor at the load end is defined by another expression. The necessary values of capacitance or inductance can be determined accordingly.

When demand is low and supply is high, the reactive power (Qr) becomes positive, leading to a situation where the receiving end voltage exceeds that of the sending end, which can be hazardous. This highlights the dependence of voltage management—an essential requirement for any electrical load—on reactive power. During daylight hours, the demand for reactive power rises, causing voltage dips, while in the early morning, reduced reactive power demand results in increased voltage levels. To maintain voltage stability, it is necessary to ensure that Q1 equals Q2.

Reactive Power Compensation

Monitoring the excess and scarcity of reactive power is essential, and various devices are used for compensation. In this context, reactors absorb excess reactive power, while capacitors supply reactive power during periods of high demand. Low electrical power factor loads exhibit a significant reactive power requirement; hence, it is necessary to enhance the power factor with capacitor banks, which reduce var demand by supplying the appropriate amount of reactive power to the load. Additional methods include utilizing shunt capacitors, synchronous phase modifiers, on-load tap-changing transformers, and shunt reactors. An overexcited synchronous motor, known as a synchronous condenser, can also be connected in shunt with the load to act as a capacitor. Meanwhile, shunt reactors are employed to decrease the electrical power factor. In on-load tap-changing transformers, the turns ratio is adjusted to maintain the desired voltage level, as the voltage difference between the sending and receiving ends influences reactive power. Mathematically, the reactive power (Q) required to improve the electrical power factor from cosθ1 to cosθ2 is expressed as follows:

Where P denotes the real power demand of the load (in watts). Conversely, if the goal is to lower the electrical power factor from cosθ2 to cosθ1, the reactive power that must be absorbed by the shunt reactor at the load end is calculated

Electric Power Single and Three Phase Power Active Reactive Apparent