Power in Single-Phase AC Circuits

Lesson 2: Real, Reactive, and Apparent Power

Learning Outcome: By the end of this lesson, you will be able to calculate real power, reactive power, and apparent power in single-phase AC circuits.

We will explore:

• Why power in AC circuits isn't a single, simple value.
• The definitions of Real Power (P), Reactive Power (Q), and Apparent Power (S).
• The units used for each type of power: Watts (W), Volt-Amps Reactive (VAR), and Volt-Amps (VA).
• The relationship between these three types of power, known as the Power Triangle.
• How to calculate each of these values in a practical circuit.

1. Power in AC Circuits: It's Not So Simple

In a simple DC circuit, power is just voltage times current ($P=V\times I$). In AC circuits, it's more complicated. The phase shift between voltage and current, caused by reactance, means that not all the power delivered by the source is actually consumed by the load to do useful work.

To understand this, let's start with a circuit with no phase shift and then see what happens when we introduce one.

As the video showed, when voltage and current are perfectly in phase ($\varphi =0^{\circ }$), the power is always positive. All the energy sent by the source is consumed by the load (e.g., turned into light and heat by the light bulb).

Now, let's see what happens when we add an inductor, which as we learned last lesson, causes the current to lag behind the voltage.

This "sloshing" of energy back and forth between the source and the load's reactive components (inductors and capacitors) is the key to understanding the different types of power in an AC system.

2. The Three Types of Power

Because some power does useful work and some is just stored and returned, we need three distinct terms to describe what's happening.

A very popular and intuitive way to understand these three types of power is the "beer analogy".

Now let's formally define these concepts, using the proper terminology and units.

To summarize:

• Real Power (P): This is the "useful" power that performs actual work, like creating light, heat, or rotating a motor. It is the beer in our analogy.

  • Unit: Watt (W) or kilowatt (kW).

• Reactive Power (Q): This is the power that oscillates back and forth, sustaining the electric and magnetic fields required by reactive components. It does no useful work but is necessary for many devices to operate. It is the foam in our analogy.

  • Unit: Volt-Ampere Reactive (VAR) or kilovar (kVAR).

• Apparent Power (S): This is the total power that the grid infrastructure (wires, transformers) must be able to carry. It's the vector sum of Real and Reactive power. It is the entire glass of beer and foam.

  • Unit: Volt-Ampere (VA) or kilovolt-ampere (kVA).

3. The Power Triangle and Calculation Formulas

Real power, reactive power, and apparent power are related to each other through a right-angled triangle, often called the Power Triangle. This is the most important tool for understanding and calculating power in AC circuits.

From this triangle, we can see the relationships using basic trigonometry and the Pythagorean theorem. Please read the following short text which lays out the mathematics clearly.

This gives us our fundamental set of formulas. For a circuit with RMS voltage $V$, RMS current $I$, and a phase angle $\varphi$ between them:

1. Apparent Power (S): This is the simplest calculation, as it ignores the phase angle.
   $S=V\times I$

2. Real Power (P): This is the component of Apparent Power that is in phase with the voltage.
   $P=S\times \cos (\varphi )=V\times I\times \cos (\varphi )$

3. Reactive Power (Q): This is the component that is 90 degrees out of phase.
   $Q=S\times \sin (\varphi )=V\times I\times \sin (\varphi )$

And the relationship between them is:
$S^2=P^2+Q^2$

4. Worked Example: Calculating Power for a Motor

Let's apply these formulas to a practical example. An electric motor is a common inductive load, meaning it has both resistance and inductance, and will therefore consume both real and reactive power.

The following video walks through a full calculation for a single-phase motor. It starts with the basic electrical properties (resistance and inductance) and calculates everything from there.

Let's recap the key results from that calculation:

• Given: Voltage = 120V, Frequency = 60Hz, Resistance (R) = 60 $\Omega$, Inductance (L) = 160 mH.
• Step 1: Calculate Reactance & Impedance. The video first calculates the inductive reactance ($X_L$) and then the total impedance ($Z$).
• Step 2: Calculate Current. Using Ohm's law for AC, $I=V/Z$.
• Step 3: Calculate Power.
  • Apparent Power (S): Calculated as $S=V\times I$.
  • Real Power (P): Calculated as $P=I^2\times R$.
  • The video found that for an apparent power of 338 VA, the real power was only 238.5 W. The rest is reactive power.

To see an extreme case, consider a load with only reactance and no resistance.

Lesson Summary

Key Takeaways:

• The phase shift between voltage and current in AC circuits leads to three distinct types of power.
• Real Power (P) is the energy consumed to do useful work, measured in Watts (W).
• Reactive Power (Q) is the energy stored and returned by reactive components, measured in Volt-Amps Reactive (VAR).
• Apparent Power (S) is the total power that must be supplied by the grid, measured in Volt-Amps (VA).
• These three are related by the Power Triangle, with the core formula being $S^2=P^2+Q^2$.
• The fundamental calculation formulas are:
  • $S=V\times I$
  • $P=V\times I\times \cos (\varphi )$
  • $Q=V\times I\times \sin (\varphi )$

Next Lesson Preview

You might have noticed that the grid has to supply the full Apparent Power (S), even though only the Real Power (P) does useful work. The ratio between these two ($P/S$) is a measure of how efficiently the power is being used. This ratio is called the Power Factor.

In our next lesson, we will explain the role of power factor in AC systems and its impact on grid efficiency. We will see why a low power factor is undesirable and what can be done to improve it.