Complex Numbers In Engineering?

Kareir

Hey all,

In another thread I mentioned that I had bought a book (50 Mathematical Ideas You Really Need To Know - Tony Crilly) that had a chapter on complex numbers. In this chapter, the author says that complex numbers are used in engineering, though fails to explain how.
If anyone could explain the idea behind how complex numbers have a real-world application to engineering, I would be very grateful.

Thanks,
_Kar.

mathew

used a lot in Elec to represent phase and magnitude..

IDCI

Also mech - to represent phases in dynamics. I'm sure there's more potential uses for the basic concept but none come to mind right now.

That sounds like an interesting book btw.

Kareir

It is a pretty good book:

http://www.dubraybooks.ie/display.asp?K=9781847240088&st_01=50+mathematical&sf_01=kword_index&m=1&dc=6


_Kar.

Stoner

As discussed WRT "Elec to represent phase and magnitude"

when electricity is generated it is typically presented in 3 phases, each phase being exactly 120 degrees out of phase of the other, 3 phases x 120 degrees = a 360 degree circle/cycle.

When generators are connected up to supply a national grid with electricity they all have to generate at the same phase exactly, if you have 4 generators powering a grid and one goes out of phase, it will actually appear in part as an additional load, <><> the more out of phase it is the less generation it does

This additional load can cause strain on the other generators and one by one they can become over loaded and go out of phase themselves and appear as more load. the voltage a generator supplies has to match the phase of the voltage of the other generators.

That's why in films sometimes to see aerial shots of cities with large sections of lighting going down block or section after section, here the generators are becoming over loaded, pole hop, go out of phase and a chain reaction has been set off, all because one generator was out of phase and someone made an error.

http://en.wikipedia.org/wiki/Northeast_Blackout_of_1965
(there are ways to avoid this type of thing now , but I was looking for something that would aid you visually, this is just a quick example and not complete or perfect and is full of engineering holes)
Therefore there is a requirement for phase sequences etc and engineers have to know how to add additional generators to grid.

I hope this is a suitable example, I'm sure there are better ones out there.

Michael Collins

Complex numbers are used widely in engineering - particularly electronic but also mechanical and probably a bit in most other diciplines, and they are used a lot in quantum mechanics too.

Their main use is when trying to describe oscillating phenomena, in which case we use objects called 'phasors'. A phasor converts a trigonometric function into a complex number, for example, a sinusoidal (i.e. sine or cosine wave) time varying wave can be described by:

Acos(wt+phase)

and this can be converted into a phasor using Euler's Formula to get:

Ae^(j*phase)

note that this is just a complex number (in polar form - remember De Moivre's theorem?).

So before I get carried away here, for electronic engineers the basic idea is in linear circuits you'll only have sinusoids of the same frequency, and in solving the circuit you'll often have to add two sinusoids with different phases together:

Acos(wt + phase1) + Bcos(wt + phase2)

and you'd like to combine this into a single sinusoid, which is a bit of a challenge. Try it if you don't belive me:

3cos(wt+pi) + 4cos(wt+pi/2) = Xcos(wt+phase) - you have to find 'X' and 'phase'

but the key is if you use phasors, it just comes down to adding two complex numbers together - which is easy! I did this for the above and I got:

X = 5 and phase = 2.2 radians, which took me about 30 seconds! So in summary complex numbers turn difficult trigonometric equations into simple algebraic ones.

So a valid question now might be "Why all the interest in sinusoidal signals, what about all the other types of signals?", well by use of the Fourier Series, any periodic waveform can be made up of a (possibly infinite) sum of sinewaves, and hence if you can solve a circuit for one sinewave, you can solve it for a whole range of different signals.

There's a good deal of deeper theory about the use of complex numbers in communications, such as the Hilbert Transform (of which phasors are a special case).

Hopefully this gives you an idea of how all that complex number stuff is actually useful! Feel free to ask questions on anything that I haven't made clear.

Turbulent Bill

Complex numbers are used in engineering because many parameters are vectors - they can't be described by a single scalar number. In mechanics, for example, position, velocity and acceleration are all vector quantities - in 2D these can be represented as complex numbers.

Most mathematical operations for scalar numbers also have a complex equivalent, so you can usually treat complex numbers in broadly the same way as scalars.

Michael Collins

Turbulent Bill wrote: »

Complex numbers are used in engineering because many parameters are vectors - they can't be described by a single scalar number. In mechanics, for example, position, velocity and acceleration are all vector quantities - in 2D these can be represented as complex numbers.

Most mathematical operations for scalar numbers also have a complex equivalent, so you can usually treat complex numbers in broadly the same way as scalars.

What's the advantange in using complex numbers instead of vectors do you know?

Darren1o1

Michael Collins wrote: »

What's the advantange in using complex numbers instead of vectors do you know?

Big thing in Mech is to determine the response of a system after a disturbance. i.e. is a structure stable.

Michael Collins

Darren1o1 wrote: »

Big thing in Mech is to determine the response of a system after a disturbance. i.e. is a structure stable.

As in linear systems analysis or control systems? Since complex numbers are vectors I can't see any obvious advantage in using one over the other. There are definite advantages in using complex numbers instead of trigonometric functions alright, which is the case in control systems.

Turbulent Bill

Michael Collins wrote: »

As in linear systems analysis or control systems? Since complex numbers are vectors I can't see any obvious advantage in using one over the other. There are definite advantages in using complex numbers instead of trigonometric functions alright, which is the case in control systems.

Complex numbers and vectors aren't always equivalent. Vector quantities (usually) can only be described by complex numbers when there are just two independent parameters for the quantity - magnitude and phase of a voltage, for example. It's easy then to map these to the real and imaginary parts of a complex number. Often a vector quantity can have more than two parameters, though. The position of a point in 3D space, for example, is given by the position vector's magnitude and at least two angular dimensions to reference axes, all of which are indepedent. You could decompose the overall position vector into two or more complex component vectors with different reference planes, but obviously this gets a bit messy.

I agree about the representation - it's much easier to write R*e^(i*theta) than R*(cos(theta) + i*sin(theta)) all the time!

Sea Sharp

Also capacitors and inductors are like complex versions of resistors.

For example a wire could have a resistance of 40 +j10 ohms. This means there's resistance and inductance on the wire.

If you put a sinusoid into this wire, the output will be out of phase (as discussed above) with teh input.

Using complex numbers as opposed to trigonometry cuts problem solving from 4xA4 pages to a half an A4 page

Bruthal

Stoner wrote: »

As discussed WRT "Elec to represent phase and magnitude"

when electricity is generated it is typically presented in 3 phases, each phase being exactly 120 degrees out of phase of the other, 3 phases x 120 degrees = a 360 degree circle/cycle.

When generators are connected up to supply a national grid with electricity they all have to generate at the same phase exactly, if you have 4 generators powering a grid and one goes out of phase, it will actually appear in part as an additional load, <><> the more out of phase it is the less generation it does

This additional load can cause strain on the other generators and one by one they can become over loaded and go out of phase themselves and appear as more load. the voltage a generator supplies has to match the phase of the voltage of the other generators.

That's why in films sometimes to see aerial shots of cities with large sections of lighting going down block or section after section, here the generators are becoming over loaded, pole hop, go out of phase and a chain reaction has been set off, all because one generator was out of phase and someone made an error.

http://en.wikipedia.org/wiki/Northeast_Blackout_of_1965
(there are ways to avoid this type of thing now , but I was looking for something that would aid you visually, this is just a quick example and not complete or perfect and is full of engineering holes)
Therefore there is a requirement for phase sequences etc and engineers have to know how to add additional generators to grid.

I hope this is a suitable example, I'm sure there are better ones out there.

The generator once synchronized will stay in phase i`d say, held in sync by the grid, but it will appear as a load if its not being driven adequately by the power source driving it, it will be like a motor instead of a generator.

Stoner

robbie7730 wrote: »

The generator once synchronized will stay in phase i`d say, held in sync by the grid, but it will appear as a load if its not being driven adequately by the power source driving it, it will be like a motor instead of a generator.

em I'd say the electrical load on a generator can be considered to be like a mechanical load from a calculation point of view (I'm just making a statement ). The more electrical load placed on the generator (from the grid it feeds) the more work it has to do adding more and more load will eventually tip it over, but it will pole hop first and go part out of sync, so it will be part generating and part load, it will continue to pole hop as the load increases eventually enough pole hops and it will be completly out of phase and not contributing to generation and straining the other generators supplying the gird.
+1 to GaNjaHaN's post , it was a better example that mine.

Bruthal

Stoner wrote: »

em I'd say the electrical load on a generator can be considered to be like a mechanical load from a calculation point of view (I'm just making a statement ). The more electrical load placed on the generator (from the grid it feeds) the more work it has to do adding more and more load will eventually tip it over, but it will pole hop first and go part out of sync, so it will be part generating and part load, it will continue to pole hop as the load increases eventually enough pole hops and it will be completly out of phase and not contributing to generation and straining the other generators supplying the gird.
+1 to GaNjaHaN's post , it was a better example that mine.

yes it was a good statement, i did`t disagree with it

A few pole hops and the generator probably be in the skip

years ago they used to use light bulbs to synch them, imagine that

John_C

Generally, anything which moves in circles will be easier to describe in complex numbers.