Power Alternating Current

AC power, the kind we use at home and in industries, switches direction back and forth. Because of this, it works a bit differently than other types of electricity. In AC systems, power comes in three forms: real power, which does the actual work like running appliances; reactive power, which helps things like motors and transformers function properly; and apparent power, which is the total power being supplied. Knowing how these work helps us use electricity more efficiently.

1.0Alternating Current 

It is an electric current that periodically reverses direction, unlike direct current (DC), which flows only in one direction.

2.0Types of Power in AC Circuits

1. Active Power

• Active power, also known as real power or genuine power, is the portion of electrical power that is actually consumed or converted into useful work in a circuit such as running motors, lighting lamps, or producing heat.
• In general, it’s measured in watts (W), but usually expressed as kilowatts (kW) or megawatts (MW) for practical use.
• It is calculated as the average power using the formula: $P=VIcos\varphi$. Where, V is voltage, I is current, and ϕ is the phase angle between them.
• Active power represents the effective output of an electrical system and directly powers the load or device to perform its intended function.

2. Reactive Power

• Reactive power is the part of electrical power that doesn't do useful work but is needed to sustain electric and magnetic fields in AC systems, especially in devices like motors, transformers, and capacitors
• It is measured in volt-amperes reactive (VAR) and is denoted by Q.
• Reactive power is calculated using the formula: $Q=VISin\varphi$
• Reactive power doesn’t transfer real energy but is essential for voltage control and the efficient functioning of AC power systems.

3. Apparent Power

• It is the total power in an AC circuit, combining real and reactive power.
• It represents the combined effect of voltage and current, regardless of their phase difference.
• Calculated by multiplying voltage and current.
• Used to determine the proper size of electrical equipment like transformers and generators.
• Denoted by the symbol S.
• Measured in volt-amperes (VA), with larger units such as kilovolt-amperes (kVA) and megavolt-amperes (MVA).
• Mathematically, apparent power can be represented as: $S=P+jQ$

P is the real power (in watts), Q is the reactive power (in VARs), j is the imaginary unit

4. Instantaneous Power

• It is the power at any specific moment in time in an electrical circuit. Unlike average power, which is calculated over a period, instantaneous power varies continuously with time, especially in AC systems where voltage and current are sinusoidal.
• It is defined as the product of instantaneous voltage and instantaneous current. $p(t)=v(t).i(t)$

          p(t): instantaneous power in watts (W)

          v(t): instantaneous voltage

          i(t): instantaneous current

3.0Power Triangle

It is a graphical representation of the relationship between three types of power in an AC electrical circuit:

1. Real Power (P) – measured in watts (W)
2. Reactive Power (Q) – measured in volt-amperes reactive (VAR)
3. Apparent Power (S) – measured in volt-amperes (VA)

Note: These three components form a right-angled triangle, 

• Real Power (P) is the horizontal side (base)
• Reactive Power (Q) is the vertical side (perpendicular)
• Apparent Power (S) is the hypotenuse

$S^2=P^2+Q^2\Rightarrow S=\sqrt{P^2+Q^2}$

4.0Power Factor

It measures how efficiently electrical power is used, showing the ratio of real power to total (apparent) power.

$\text{Power Factor}=\frac{RealPower}{ApparentPower}=cos\varphi$

Types of Power Factor:

• Lagging Power Factor: Common with devices like motors and transformers. Here, the current comes a bit after the voltage.
• Leading Power Factor: Seen in setups with capacitor banks. In this case, the current moves ahead of the voltage.
• Unity Power Factor: The best case—voltage and current line up perfectly, so all the power is used efficiently.

Significance of Power Factor:

• High Power Factor (close to 1): Indicates efficient power usage, reduced energy losses, and lower electricity bills.
• Low Power Factor (much less than 1): Indicates poor efficiency, increased losses, and the need for larger capacity equipment to handle excess current.

5.0Power in AC Circuit

The rate of doing work or the amount of energy transferred by a circuit per unit time is known as power in AC circuits. It is used to calculate the total power required to supply a load.

Instantaneous Power

$P_{\text{inst}}=VI=\left(V_o\sin \omega t\right)\left(I_o\sin (\omega t+\varphi )\right)$

$=V_o{I}_o\sin \omega t\sin (\omega t+\varphi )=\frac{V_o{I}_o}{2}\left[2\sin \omega t\sin (\omega t+\varphi )\right]$

$\text{Hence }{P}_{\text{inst}}=\frac{V_o{I}_o}{2}\left[\cos \varphi -\cos (2\omega t+\varphi )\right]$

$(\therefore 2\sin A\sin B=\cos (A-B)-\cos (A+B))$

Note : Therefore frequency of power fluctuation is twice the frequency of applied ac-source.

Average Power :

$P_{av}=\frac{{\int }_0^T(V_{0}sin\omega t)(I_{0}sin(\omega t+\varphi ))dt}{{\int }_0^{T}dt}=\frac{V_o{I}_o}{T}\left[\cos \varphi {\int }_0^T{\sin }^2\omega tdt+\frac{\sin \varphi }{2}{\int }_0^T\sin \omega tdt\right]$

$P_{\text{av}}=V_0{I}_0[cos\varphi \frac{{\int }_0^{T}si{n}^2\omega tdt}{T}+\frac{sin\varphi }{2}\frac{{\int }_0^{T}Si{n}^2\omega tdt}{T}]=V_o{I}_o\left[\cos \varphi \times \frac{1}{2}+0\right]$

$P_{\text{av}}=\frac{1}{2}{V}_o{I}_o\cos \varphi \Rightarrow P_{\text{av}}=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$

Note: Hence $cos\varphi =\frac{R}{Z}$=Power factor of ac Circuit

6.0Power in Basic Circuit Elements

1. Average Power in Capacitive Circuit

$I=I_{0}cos\omega t$

$P_{av}=V_{rms}{I}_{rms}cos90^0$

$P_{av}=0$

2. Average Power in Inductive Circuit

$I=I_{o}cos\omega t$

$P_{av}=V_{rms}{I}_{rms}cos90°$

$P_{av}=0$

RMS Power: $P_{rms}=V_{rms}{I}_{rms}$

7.0Comparison of Active, Reactive & Apparent Power

8.0Measuring AC Power

9.0Applications of AC Power Analysis

1. Design and Optimization of Power Systems
2. Sizing of Electrical Equipment
3. Power Factor Correction
4. Energy Billing and Cost Analysis
5. Fault Detection and Protection System Design