C-bro

"Imagine" Their Usefulness


Remember all those boring high school lectures that concluded with an answer that didn’t even exist? Think hard, that little number i, the square root of -1. It’s a totally abstract and seemingly useless idea, but I intend to show that they are in fact very useful for electrical and computer applications. Now this is the absolute barebones information regarding the concept of imaginary numbers. There are tons of derivations required to bridge the gap between theory and application, but I’m just going to give some back ground and a brief synopsis of the final results. It’s partly because I’m lazy, but also because many of the steps involve insignificant tedious algebra.

Even with the abundance of material I left out, some of the concepts are still a little mind boggling. This is a reworked and reorganized approach to my FAQ since the original version proved somewhat confusing, and more sympathetic I could not be. From the very beginning, my primary concern was to simply inform people that imaginary numbers did have practical applications. I've decided to re-write the structure of this FAQ in hopes of creating a simplified explanation without having to understand the often-tricky mathematics. Here goes revision 2 and hopefully you guys can take a bit more away from it. However, even with the simplified explanations, one must still understand basic electrical properties like Ohm's law and be aware of things like resistors, capacitors, inductors, and the difference between voltage and current. I still kept all my background justification, but it's no longer the focal point of my explanation. It is there for those of you who wonder "why" and "how" after reading the applications of imaginary numbers. However, the time for small talk is over and I present you with the applications of imaginary numbers.


Power Factor Correction
The easiest form of power factor correction deals with a motor. Since a motor is really just a big coil of wire, it behaves like an inductor. And since the world isn't perfect, that big coil of wire has an inherent resistance to current flow. Very simply, our motor is the same as a resistor and an inductor in series. When you have a sinusoidal signal, inductive or capacative loads (like a motor or power supply) produce a very distinct effect. You can see in the diagram below that with an inductive load the voltage and current get shifted out of alignment (see details). Think of it like jumping rope. If you're not jumping at the same time the rope comes around, your success is quite pitiful. The same goes for power in a motor. If the voltage and current aren't lined up (or out of phase) then the motor can't perform at it's maximum efficiency.


The idea of power factor correction is to reverse that "phase shift" caused by the inductive load by adding a capacitor. This pushes the current and voltage back into alignment, and allows the motor to perform at maximum efficiency. You can see the "correction" happening in the diagram above. To illustrate the merits of PFC, again think of jumping rope. If you're in phase with the rope, you're jumping just as the rope goes beneath you and you can go for hours. Imaginary numbers come into play when you figure out the voltage, current, and inductors/capacitors involved in the circuits. When you find the right combination of values, it brings the power factor up to 1, and the circuit is once again efficient (loosely speaking).

This is used extensively in factories and workplaces in an attempt to increase efficiency and lower energy cost. In some cases, having a low power factor will not only require you to purchase more electricity (since only a fraction is being used for real work), the power company will also charge more for that electricity since it’s being used so inefficiently. Computer power supplies also use forms of active and passive power factor correction to increase efficiency.


Filters

If you’ve ever listened to a speaker that contained more than a single driver, or received more than 1 channel on your television, you can thank filters and imaginary numbers. Filters are often made up of a combination of inductors and capacitors to do just want they say: filter an AC signal. In a speaker, the input signal is filtered to different drivers, not only for sound quality, but because playing a 20Hz tone on a tweeter would result in physical damage. In televisions, you only get 1 input signal, but it can carry several hundred channels. Each channel has its own frequency, and it’s the TV’s responsibility to filter out all the other channels, and display only the frequency you desire.

The design of these complex systems is often done by computer, but at the heart of all the computations lies, you guessed it, imaginary numbers. In the case of filters especially, with any more than a few inductors/capacitors, the only humanly possible way to calculate the output is with imaginary numbers. Otherwise you’re stuck trying to solve something like a 5th or 10th order differential. VERY difficult.

While there are many other applications of imaginary numbers, I found these two the most exciting and practical. If you found any of this interesting, I would definitely recommend reading up on AC signals and systems. You may pull your hair out trying to compute L’s, C’s, bandwidth, and a host of other properties, but please remember, imaginary numbers are your friend.

I pulled this all out of my head from what I remember from circuit analysis, so if you find any errors or inconsistencies please let me know. Also, please be kind to my GIF images. I either did them in paint or screen capped them from Word, so they dont' come out entirely clearly.

Additional Details


WARNING: If you don’t have a background in math or science, even after reading this, it most likely won’t make a whole lot of sense.


Now that we have everything converted into their complex forms, the analysis becomes much simpler. Think of two resistors in series. It’s only naturally that you add their values together to give a total resistance. The same goes for complex impedances. In this example I’ll use a motor. Ideally, a motor can be modeled as an inductor and a resistor. Quite simply, the total impedance of the motor would be:


We’ll also assume that the input voltage has a magnitude of A, with a phase angle of 0°. With an angle of 0°, a phasor is simply its magnitude (shown as vinput above), so it makes the example calculations much simpler.

Now with any circuit, you want to know the voltage and current. Ohm’s Law for impedances is the same as resistances: V= IZ. Knowing the voltage and impedance, we’ll use I = V/Z to solve for current. We find the expression for current to be:


It looks more like an alphabet than a number, but it shows one thing. Using formula 4) we can see that the real part of I, is equal to A cos Φ. Since the R, ω, and L are all positive, then cos Φ and therefore Φ must also be positive. Since the current has a non-zero angle associated with it, the current has been phase shifted by a degree of Φ. Since we said the voltage in this case has a phase of zero, the current and voltage waveforms are no longer in phase. In other words, they are still the same frequency, but the maximums and minimums don’t line up (see diagram below). This is a revelation when it comes to the efficiency of a circuit (a PSU for example) and I’ll explain why.


Since power is the product of voltage and current (think dot product), as the phase angle between them increases, their product decreases. In our example, the angle between the current and voltage was Φ. A perfectly efficient circuit will operate with a power factor of 1. Since our circuit operates at a power factor of cos Φ, and Φ is positive, then the power factor is less than 1, and is therefore not at maximum efficiency. Power factor correction, as mentioned above, is the solution to this unfavourable situation.

C-bro

Hey! A special thanks goes out to cowboyz who actually read this FAQ. The insterest will soon pick up. I can feel it...

13 3 @ 7 l 3 13 0 y

whoa! somthing just went way over my head!
good FAQ.......probably

__________________
My cynical ranting: pet peeves on OCN
-people who say merely "do want" and reduce the reply box to a checklist of your emotions.
-people who misuse "a" and "an"
-people who think posting inane pictures makes them clever or funny
-people who will inevitably try to reply to me with all three of the above.

RonindeBeatrice

Good luck with that....

I'm a programmer not an engineer so my brain now hurts.... I'm gonna lie down.

Krunk_Kracker

I....uh.....kinda followed that...so I guess it was good, lol.

Good FAQ! Informative and you need to really pay attention when reading, lol.

__________________

C-bro

Thanks for the patience and interest guys. I made some major changes to the format and complexity of some of the parts. I've worded the applications in much simpler terms, and provided the mathematics simply as proof behind what I was saying.

PGT96AJT

Very nice explaination, I feel like I'm taking circuits class again.

Imaginary number are very important when dealing with AC power.

I belive the correct expression would be "My brain hertz".