Last time, we went over some basics concerning electric power. Things like power vs energy, the generalized power law (power at DC and resistive AC circuits), and an intro to reactive power appear in the previous post.

If you missed it, please read it before you read this one. Here it is: One Powerful Post: A Primer on Electric Power Part 1.

Let’s kick things off by going over some electric power related terms and concepts dealing with reactive circuits.

For now, we won’t really differentiate between reactive circuits that are primarily inductive (RL), capacitive (RC) or both (RLC). We’ll delve into more detail on that stuff in another post.

Electric Power Lingo

Important terms are in bold.

If we apply the generalized electric power law

P = I x V                (eq. 1)

to AC power, we can rewrite it as

p(t) = v(t)i(t)       (eq. 2).

This is the instantaneous power, where t represents a particular point in time. In AC circuits, this usually involves some nasty looking trig with cosines all over the place due to AC power’s sinusoidal nature.

This quantity is difficult to measure, but the average power is a bit easier to measure. In fact, the wattmeter, (you guessed it — it measures power), responds to average power.

To get the average power, the instantaneous power is averaged over one cycle. The equation for average power is given below.

P_avg = ½ V_p I_p cos (ϴ_v – ϴ_i)               (eq. 3).

The subscript p in equation 3 is the peak value of the sinusoid, which for a 120 V circuit is 170 V. The trig part is simply the cosine of the phase difference between the voltage and current. Reactive elements like inductors and capacitors cause this phase shift.

Another, even more useful measurement is the RMS value, or root mean square, also called the effective value. The RMS values of AC voltage and current are based upon equating the values of AC and DC power one would need to heat a resistor to the same temperature. In other words, a direct current whose values of I and V equal the RMS values of I and V for an alternating current will produce the same power.

The nice thing about RMS measurements is that they’re independent of time and frequency. The equation below is for 120 VAC circuits.

V_RMS = V_p x 0.707               (eq. 4).

Equation 4 gives us the RMS voltage where V_p is again the peak voltage. The formula for the RMS value of the current is similar – simply substitute peak voltage with peak current.

The actual formula for calculating the RMS value of other waveforms is kind of messy and involves calculus. We’ll save that for another post. Right now, you’ll have to trust me that equation 4 is right if it’s a 120 VAC circuit.

The formula for average power and power absorbed by a resistor can be rewritten in terms of the RMS voltage and currents.

P_avg = V_rmsI_rms cos(ϴ_v – ϴ_i)       (eq. 6)

P = I²_rms R                                      (eq. 7)

The power factor is the cosine of the phase difference between voltage and current. A lot can be said about power factor and power factor correction, but for now all we need are the basics. We can also regard the power factor as the ratio of the real power (resistive power) dissipated to the apparent power (more on that in a minute).

Purely resistive circuits have a power factor of 1, which is the best you can get. Purely reactive circuits have a power factor of 0. Most practical circuits fall somewhere between the two values.

Power factor can be leading or lagging. If it is leading, the current leads the voltage by some phase difference. If it’s lagging, the voltage leads the current by some phase difference. Power factor can affect your electric bill, so it’s important.

The real power is the only useful power and is the power the load actually dissipates. The reactive power (a.k.a. wattless power) is a measure of the energy exchanged between reactive elements and the load. The VAR (for volt-amp reactive) is the unit of measurement.

Purely reactive circuits dissipate no real power. They instead store energy either in a magnetic field or electric field and swap the energy back and forth with the rest of the circuit. Of course, in real life inductors have some resistance and capacitors aren’t perfect either.

The impedance these components show isn’t a real resistance and is instead a reactance. To calculate reactive power, we use a formula similar to equation 1 in that it is straight from Ohm’s Law and easy to apply.

P_X = I² x X_M           (eq. 8)

The subscript X in P_X means the power is reactive. The subscript M in X_M is either the capacitive reactance or inductive reactance and is measured in ohms just like resistance. The letter X itself is the reactance, be it capacitive or inductive.

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The formula for calculating inductive reactance is given in equation 9.

X_L = 2πfL               (eq. 9)

Where f is the frequency in Hertz and L is the inductance or value of the inductor in Henrys.

The formula for calculating capacitive reactance is given in equation 10.

X_C = 1/(2πfC)      (eq. 10)

Where f is again the frequency and C is the capacitance or value of the capacitor in Farads.

The apparent power is so called because it seems apparent that the power should be the voltage-current product, as compared to a DC circuit. It is measured in VAs (volt-amps) and not watts (unless it’s purely resistive). It is the product of the RMS values of voltage and current.

Think of it as the power that appears to be supplied by the load. It includes both real power and reactive power.

Electric Power and the Power Triangle

An electronics newbie may think that apparent power is simply the sum of the real power and the reactive power.

This is not true. The problem lies in the fact that simple arithmetic on reactive variables can’t be done without considering phase.

If you’re an electrical engineer (or EE student) you take this into account using phasors and imaginary/complex numbers. For the average hobbyist, this can seem daunting and complicated.

Lucky for those who haven’t taken any engineering level circuit analysis classes, there is an easier way to get the apparent power.

To illustrate this, we can use a simple diagram known as the power triangle, which we can see in the figure below.

Figure 1: the power triangle and how it relates to electric power

 

Understanding the following discussion and examples requires some (very) basic knowledge of trigonometry.

Using a right triangle, the relationship between real power (watts), reactive power (VARs), and apparent power (VAs) can be more easily understood.

You probably already know that in any right triangle the hypotenuse (apparent power in the above figure) is always the longest side. Here we can graphically see that the apparent power is greater than either the real power or reactive power and is the total power used by the circuit.

The cosine of 0 is 1; if we make the power factor angle small (ideally 0), then the triangle collapses on itself and the real power equals the apparent power. This gives us a power factor of 1 and is an ideal situation that only occurs in DC and purely resistive AC circuits.

Looking at the triangle, it is easy to see the amount of reactive power the circuit uses is what determines its efficiency. The more reactive power the greater the power factor angle, which lowers the power factor.

But is reactive power really all a big waste?

Sure, ideally, we should try to keep it to a minimum (especially if we pay for it!) but it turns out we need it. For example, inductive reactive power in electric motors forms the magnetic fields to spin the rotor. Without it the motor would not work.

For most households, employing power factor correction (we’ll talk more about this in a future post) might be a waste and too expensive to justify.

But, for large commercial and industrial facilities who pay tens of thousands per month for electricity, a significant savings can be realized.

Imagine figure 1 sitting in a Cartesian (X-Y) coordinate system with the left point of the triangle sitting at the origin. The real power runs along the x-axis, and the reactive power follows the y-axis. Figure 2 illustrates this.

Figure 2: the power triangle in Cartesian coordinates.

One important thing to note is that if the hypotenuse lies in the first quadrant of the coordinate system, the load is inductive with a lagging power factor.

If we were to flip the triangle along the x-axis so the hypotenuse now lies in the fourth quadrant, the load becomes capacitive with a leading power factor.

Believe it or not, it is possible for the hypotenuse to also lie in the second or third quadrant. This means that the load impedance has a negative resistance, which is possible in some circuits.

Calculating Electric Power with the Power Triangle

The power triangle is nice because it gives us a more intuitive understanding of electric power, but how do we actually calculate the different types of electric power?

You may have heard of the Pythagorean theorem. Using that and/or some simple trigonometry, we can easily calculate any value if we know any other two values.

The Pythagorean theorem states that in any right triangle:

A² + B² = C²          (eq. 11)

Where C is the hypotenuse or long side of the triangle and A and B are the other sides.

Using simple algebra, we can take the square root of both sides which gives us:

C = √A² + B²         (eq. 12)

Note that the square root sign covers the whole right side of the equation.

To put it another way:

VA = √P² + VAR²                (eq. 13)

Where VA is the apparent power, P is the real power, and VAR is the reactive power.

Here we don’t have to worry about phase angles – the Pythagorean theorem takes care of that for us.

From simple trigonometry, we can also note that:

cos (ϴ) = A/H     (eq. 14)

where ϴ is the power factor angle, A the side adjacent to it (the bottom of the triangle in figures 1 and 2), and H is the hypotenuse.

If we re-write this using power as the variables we get:

cos (ϴ) = real power / apparent power

or

cos (ϴ) = P/VA   (eq. 15)

This means that if we know the power factor or the power factor angle, we can get any of the three powers (sides of the triangle) as long as we know one of the sides.

Let’s do a few easy examples to illustrate the information given above.

Ex 1:

We know our circuit consumes 10 W or real power and 3 VAR of reactive power. How much apparent power does our contraption suck down?

Solution:

From the Pythagorean theorem we get equation 12 and use it here.

VA = √10² + 3² = √100 + 9 = √109

Apparent power = 10.44 VA

Ex 2:

My circuit consumes 5 W of real power, but the apparent power is found to be 6.3 VA. How much reactive power does my creation use?

Solution:

Here we know the reactive power and the real power. All we need to do is algebraically rearrange equation 12 a bit to obtain:

VAR = √VA² – P² or VAR = √6.3² – 5²

Which gives us VAR = √39.69 – 25   = √14.69

Giving us a VAR of about 3.83.

Ex 3:

A circuit has a power factor of 0.85 and draws 3 W of real power. How much apparent power does this thing consume?

Solution:

We know that power factor = cos (ϴ) and that the cosine is the adjacent side of the triangle over the hypotenuse. Using this information, we can write:

0.85 = 3 W / some amount of VAs

Rearranging algebraically, we can write:

0.85 x (some amount of VAs) = 3 W

Dividing both sides by 0.85, we see that the apparent power is equal to roughly 3.53 VA.

Note that if we want the phase angle for some reason, we can simply grab a calculator and take the inverse cosine of 0.85, which gives us about 31.79⁰. This means the voltage and current are 31.79⁰ out of phase with each other.

All the More Electric Power to You

By now we have a reasonable understanding of the difference between power and energy, purely resistive power, and calculating power in AC circuits, both resistive and reactive.

If some of the terms used in this post are unfamiliar, don’t worry. You should still be able to make power calculations using the information presented here.

In another post, we’ll delve into more detail about things we didn’t quite have room for in the two posts.

Things like the concept of impedance are important to understand, but right now you should be able to easily calculate the power your toaster uses or the power rating of the resistor needed in many of your circuits.

We’ll also take a closer look at power (and impedance) in RL, RC, and RLC circuits, since we now know that most AC circuits are not purely resistive.

Until then, drop us a comment and share any tips you have on saving power or choosing components for your circuits with the correct power rating.

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