Complex Valued Electric Field

[john1989]

Hi

I am new to EM and i have a question.

When using Maxwell equations to compute time harmonic electric fields we use phasors for an easier solution.

The result is a complex valued electic field E(z) where E(z) is a vector. To get the time harmonic field we use E=Re{E(z)*exp(-i*2*pi*f*t)}

My question is: in the complex valued field E(z) what is the physical meaning of the real and the imaginary part the may come as a result of the equations.

For example if we get E(z)=(4+5i)x where x is the vector of the x-axis what is the physical meaning of 4 and 5 of its amplitude

Thanks in advance

 

[jsgruszynski]

The complex coefficient, exp(j omega t), is simply a time/phase delay added to the real solution.

 

[john1989]

i'm not asking about the time harmonic coefficent

i'm asking if the amplitude of exp(i*w*t) is a complex number

thanks though

 

[yungman]

We always use $\cos(\omega t -kz)$ to represent harmonic wave.

$$e^{j(\omega t-kz)}= \cos (\omega t -kz)+ j\sin (\omega t -kz)\;\Rightarrow \; \Re e [e^{j\omega t-kz}]=\cos (\omega t -kz)$$

Nothing more than that.

 

[john1989]

We always use $\cos(\omega t -kz)$ to represent harmonic wave.

$$e^{j(\omega t-kz)}= \cos (\omega t -kz)+ j\sin (\omega t -kz)\;\Rightarrow \; \Re e [e^{j\omega t-kz}]=\cos (\omega t -kz)$$

Nothing more than that.

ok
but what if the $$e^{j(\omega t-kz)}$$ has a complex amplitude
thats what I'm asking
what is the physical meaning of this amplitude
for example does it mean that it has to components of different phase?

 

[yungman]

ok
but what if the $$e^{j(\omega t-kz)}$$ has a complex amplitude
thats what I'm asking
what is the physical meaning of this amplitude
for example does it mean that it has to components of different phase?

That is a very good question that I wonder about. I hope someone can answer. All I know is the imaginary part plays a role, for example if the function is

$$je^{j\omega t-kz}\;\Rightarrow\; \Re e [je^{j\omega t-kz}]=-\sin (\omega t -kz)$$

The sine term comes out in this case. I hope someone can explain this.

 

[Floid]

Guessing here without really thinking through:

cos(x+90) = -sin(x)

So your complex amplitude is still just a phase shift...

 

[yungman]

I understand the phase shift part. The question is even more fundamental, why not just use phase shift and eliminate all the complex stuff? This is where I get loss. I just use the complex notation because everyone use it.

As as practising engineer, I like to deal with "real" things. Seem like this is more driven by math than real life.

 

[dlgoff]

...why not just use phase shift and eliminate all the complex stuff?

The handling of the impedance of an AC circuit with multiple components quickly becomes unmanageable if sines and cosines are used to represent the voltages and currents. A mathematical construct which eases the difficulty is the use of complex exponential functions.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html"

 

[yungman]

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html"

I have no issue using all the formulas in the link, in fact as I said, we are forced to learn using it. Books pretty much give out the formulas as is and live with it. I understand the complex number are just a way to provide of phase shift calculation and all that.

The only thing I can think of is if we use phase angle in trig function to do calculation, it can get more complicate and long. by using:

$$e^{jx}=\cos x +j\sin x$$

I think you make calculation easier in complex form. If so, I sure wish the books will explain this. The best explanation I ever saw was what I posted in #4. You go read most of the physics, engineering books, they pretty much say "use this"...in a nice way!

To be honest, I yet to find an advanced level EE book explain things in English yet. You get into electromagnetics, Fourier transform, laplace transform, all books start out in Calculus equation from the get go. In fact, after studying PDE, I really wonder how undergrad student really appreciate electromagnetics or electrodynamics without a semester of PDE that most of the colleges don't require for undergrad. All the explanation are in calculus, ODE and PDE. Why can't they write in English so people can understand it. I question most of the EE student gone through EM really know what it is.

On this note, I have been into heavy duty circuit and system design for almost 30 years, RF, data acq, HV etc, all sort of analog, transistors and IC design. I can tell you I never even use any calculus beyond the first semester or maybe a little of the integration. Usually never exceed basic stuff like super position, thevenin?? etc. All the math that I learn is nothing more than to be able to understand the textbooks. Once I understand the book, I almost never have to look back at the math again. there is something really wrong with this picture. In fact I never have calculus knowledge beyond the second semester until I stop working! I only study all the calculus in the last few years.

This is the reason you see people graduated with good grades never able to transition into real life career. I have seen so many lousy engineer with advanced degree...even from MIT and UC Berkley!

 

[Floid]

The question is even more fundamental, why not just use phase shift and eliminate all the complex stuff? This is where I get loss. I just use the complex notation because everyone use it.

Because sometimes it is easier to work with the complex notation versus the phase shift. But in the end it is two ways of saying the same thing with phase shift being much easier for most people to conceptualize.

It is sort of like having cartesian, spherical, cylinderical coordinate systems. You can solve any problem using just one, but you can transition between them to make solving some problems and conceptualizing the situation easier.

 

[yungman]

Because sometimes it is easier to work with the complex notation versus the phase shift. But in the end it is two ways of saying the same thing with phase shift being much easier for most people to conceptualize.

It is sort of like having cartesian, spherical, cylinderical coordinate systems. You can solve any problem using just one, but you can transition between them to make solving some problems and conceptualizing the situation easier.

Problem is it takes me a while to realize that. Books don't usually explain that, they just start out using complex notation.

That is the reason of my long ranting post, books don't explain the reason, they start out with all the math instead of explaining in English. Look at EM books, starting out with Divergence and Curl right about on the first page! I had to study 3 times over on three different books to try to understand in English. You can really go through the whole class, getting an A in the class without understanding it. I first started out with Ulaby and did ALL the problems, then I went on to Cheng and also did a lot of the problems. It was not until I study the third time with Griffiths that is really for Physics major that I feel I finally get the feel of it. I don't even dare to say I understand EM, just a feel of it. Griffiths is about the best book NOT used in EE major, I have 8 books on EM, if EE student don't use Griffiths, I really really question EE student really know much about EM. Yes you can get A's, I am sure I can get an A in college after the first time studying Ulaby ( which is an easy book). You can learn how to plug in the formulas and get the right answer! That's why I feel there is such a big disconnect and that is the reason why so many good student never transition into a good engineer.

 

[fishguy]

thanks for explaining such a wonderful notation, especially introducing Griffiths's book here

 

[meBigGuy]

ok
but what if the $$e^{j(\omega t-kz)}$$ has a complex amplitude
thats what I'm asking
what is the physical meaning of this amplitude
for example does it mean that it has to components of different phase?

$$e^{j(\omega t-kz)}$$ has an amplitude of 1 and a phase. The amplitude is real. The quantity being expressed can also be represented by two numbers, a "real" part and an "imaginary" part which are actually just the sin and cos of the magnitude/phase that was originally expressed. If you want to give it a complex amplitude, you have to multiply it by a complex number.

Complex numbers are simply a "rectangular" way to represent a magnitude and phase. Sometimes you can relate easily to what the parts represent, sometime it's pretty abstract. Take Impedance, for example, which is resistance and reactance. But other times the "magnitude in the X direction" is just that.

 

[Jeffrey Yang]

Hi

I am new to EM and i have a question.

When using Maxwell equations to compute time harmonic electric fields we use phasors for an easier solution.

The result is a complex valued electic field E(z) where E(z) is a vector. To get the time harmonic field we use E=Re{E(z)*exp(-i*2*pi*f*t)}

My question is: in the complex valued field E(z) what is the physical meaning of the real and the imaginary part the may come as a result of the equations.

For example if we get E(z)=(4+5i)x where x is the vector of the x-axis what is the physical meaning of 4 and 5 of its amplitude

Thanks in advance

Hi John:

I hope you can see this "so late" reply. :-)

Yes, I'm also thinking this "fundamental" question. When we have used the time-harmonic approximation, why we still get a complex value amplitude. Some books I've read also "emphasise" that spatial part E(r) is complex. In my opinion, such a "complex valued" amplitude means "it can be complex valued" if the fields contains an extra phase. Exp(i*w*t) has an initial phase equals zero, therefore if the fields has an extra phase it must be wrote into the spatial part. Of course, if there has no extra phase, the spatial amplitude would be real. The so-called extra phase here can be determined by the source of the field, or ourselves, for physical and mathematical reasons.

This is my understanding. Overall, the complex valued amplitude is just a complete statement.

How do you think about?

 

[sophiecentaur]

This is also a late entry but what do you actually mean by a Complex Amplitude? The complex notation for periodic functions always implies "the real part of" when you get to the end of your calculation. So what situation will generate the need for a 'complex' amplitude? What would you be trying to represent in that mathematical model? If your interest is 'just' mathematical then just follow the rules and you will obtain an answer but it could be a case of GIGO, scientifically.

 

[Jeffrey Yang]

This is also a late entry but what do you actually mean by a Complex Amplitude? The complex notation for periodic functions always implies "the real part of" when you get to the end of your calculation. So what situation will generate the need for a 'complex' amplitude? What would you be trying to represent in that mathematical model? If your interest is 'just' mathematical then just follow the rules and you will obtain an answer but it could be a case of GIGO, scientifically.

To my own experience, the "complex amplitude" will be a little bit "physically" meaningful when we discuss the resonance system. In a well-confined system the resonance mode can only be real valued (even if we use complex notation) with real resonance frequency. However, for a real open resonance system, mode is partially leaking away. In such a case, both the amplitude and resonant frequency will be complex. In such a case, the complex amplitude meaning the system is non-Hermitian, or free propagating in some degree.

But anyway, the complex amplitude just means a spatial phase information is contained here.

 

[sophiecentaur]

But anyway, the complex amplitude just means a spatial phase information is contained here.

All that can go 'inside' the exponential bit, surely?? ( as in ωt - kx ) When there are multiple traveling waves (a standing wave pattern), that will just be a sum of such exponential terms. Are you saying that you can represent that in terms of a complex amplitude function? I guess I shouldn't be surprised.

 

[Jeffrey Yang]

All that can go 'inside' the exponential bit, surely?? ( as in ωt - kx ) When there are multiple traveling waves (a standing wave pattern), that will just be a sum of such exponential terms. Are you saying that you can represent that in terms of a complex amplitude function? I guess I shouldn't be surprised.

I mean even you using the exponential form exp[i*(wt-kx)], which is complex due to the phase information, to construct a standing wave pattern, the imaginary part "of the spatial component", or "of amplitude" we discussed here, will finally disappear. Such as

sin( πx/nL) *exp(i*wt)= 1/2i * {exp[i*(πx/nL)] + exp[-i*(πx/nL)]} * exp(i*wt)

This is just because the standing wave will only has in phase part or anti-phase part in the space, and therefore the spatial mode pattern will always be real. But once the boundary condition becomes weaker, which means the mode is leaky, the spatial pattern will be complex again. It also corresponds to whether the system is Hermitain or not, although they are not directly shown.

 

[sophiecentaur]

That makes sense now.
Cheers.

 

[anorlunda]

There has been much resistance to complex arithmetic over the ages, mostly because people felt it to be overtly complex (pun intended). In reality, once you learn it you will find it to be a great simplification and it will open new horizons for you.

May I recommend the book "An Imaginary Tale: The Story of i [the square root of minus one" By Paul J. Nahin.
It is a great read about the historic evolution of these ideas. It explains very well why it is worth your while to learn it, and by the way it teaches you as you read.

Find it https://www.amazon.com/dp/B004EV36WK/?tag=pfamazon01-20.