Why does current lag behind voltage in inductor?

[Metaleer]

hi haleycomet2!


(this thread is sooo long that I'm not sure what we're talking about , but …) if this is a circuit with only a battery an inductor and no resistance, then KVL means that, at any instant, the "back emf" must equal the voltage

(hmm … having said that, i have vague memories of seeing comments that where there's an inductor, the electric field isn't conservative, so I'm not sure where that fits in )​

You're probably thinking of how Walter Lewin (MIT Physics professor) argues (rather convincingly, I must say) that in the presence of inductors, Kirchhoff's voltage law does not apply, since this law applies for conservative fields, and changing B-fields (inductors) mean that the field is no longer conservative (as the curl of the E-field no longer vanishes, by the Faraday-Lenz law). However, you can still use the Faraday-Lenz law to see if this statement about the back emf and the voltage is true.

 

[sophiecentaur]

It still bothers me that some people see the emf as a 'problem'. Perhaps the definition of volts as "joules per coulomb" and just accept what you get; it is just the energy situation at different places in the circuit.

 

[haleycomet2]

Could anyone suggest any link or books which have a deeper introduction of inductor(i have read“all about circuit” ,"hyperphysics" website,and A-level physics by Nelkon and Parker)?I am sorry for troubling u all and maybe i need time to study more and think more to understand this.
Thank you.

 

[cabraham]

Ok then, you are talking about me now.

How does one get a current flowing if one neither pushes nor pulls?

Sorry if I've misinterpreted the meat of your message. This thread reminds me somewhat of my foray into "How the hell do diodes work?" a few years back. I drilled down to the quantum level, and got totally lost, so I won't go there.(thread hijack preempted)

But on the most fundamental level, we have 4 copper atoms. Two pairs of dual copper atoms separated by space. To have a current flow, we have to move at least one electron from one atom to the other. This transfer of an electron will have an effect on an electron in the other copper atom pair(induction).

Of course by now, you are seeing where I'm coming from. If one neither pushes nor pulls, how do you get the electron to flow from atom A to atom B?

But this also introduces the fact that there is no such thing as a perfect anything. Getting an electron to move from atom A to atom B required a force, and knowing that electrons are not massless, this implies that there was a resistance. So we can remove the dreadful infinities that plague many a college textbook when discussing this topic on the macroscopic conceptual scale.

e^--->
A-B
(magical pixie magnetic force)
C-D
<--e^-​

dear FSM, please let me have one more get out of banned card for using the magical pixie magnetic force again...

I responded to this post a day or 2 ago, but I hit "preview post" instead of "submit post". It was never entered. So here it is.

Initially some type of force is needed to commence conduction. Electrons, in the case of metals, & holes and/or ions for other media, must begin motion, in order to transfer their energy to other charges, enabling them to conduct.

The initial energy needed to do this comes from a redox reaction in a battery, or from mechanical input enrgy for a generator. You've probably seen those desktop gadgets w/ 5 or so steel balls suspended on strings. They touch each other, so that if 1 ball is moved outward, it strikes the 2nd ball, & the 5th ball at the other end rises. It eturns to its starting point, & the 1st ball rises, etc. The principle here is transfer of energy & momentum. Of course somebody had to provide energy to get the process started.

Current conduction is similar to the suspended balls gadget. Once charges are given energy to put them in the conduction band, that energy is transferred to other particles. The electrons conducting in a metal are not being "pushed". They are energized initially, then transfer said energy to other particles.

As far as resistance goes, that involves atomic collisions. Electrons collide w/ the lattice, energy is dissipated as photons are released. The photon emission is in the IR wavelength around 5 micron, & is felt as heat.

So if an inductor is energized, it tends to maintain a constant current. If the source is disconnected & the inductor shorted, the current will remain constant forever if no resistance were present. But, for a resistive inductor, electrons conduct & bang into the lattice losing energy. This energy is radiated as heat. Hence the energy in the inductor is decreasing, as is the current, since W = L*I^2/2.

Did this help?

Claude

 

[martix]

I guess that's a common issue with a lot of people, that and capacitors.

Got one for you.(though I don't know if it's been mentioned already since I won't bother reading through the whole thread)

It is just my intuitive interpretation of the concepts.
Think of an inductor as the electrical equivalent of a flywheel. It's a great analogy for me and I think it works on many levels.
Just as a capacitor's mode of operation can be compared to that of a spring.

Where one deals with electrical energy, so does the other deal with its mechanical counterpart(here's hoping that those are concepts more intuitively familiar to you).

 

[Bassalisk]

I responded to this post a day or 2 ago, but I hit "preview post" instead of "submit post". It was never entered. So here it is.

Initially some type of force is needed to commence conduction. Electrons, in the case of metals, & holes and/or ions for other media, must begin motion, in order to transfer their energy to other charges, enabling them to conduct.

The initial energy needed to do this comes from a redox reaction in a battery, or from mechanical input enrgy for a generator. You've probably seen those desktop gadgets w/ 5 or so steel balls suspended on strings. They touch each other, so that if 1 ball is moved outward, it strikes the 2nd ball, & the 5th ball at the other end rises. It eturns to its starting point, & the 1st ball rises, etc. The principle here is transfer of energy & momentum. Of course somebody had to provide energy to get the process started.

Current conduction is similar to the suspended balls gadget. Once charges are given energy to put them in the conduction band, that energy is transferred to other particles. The electrons conducting in a metal are not being "pushed". They are energized initially, then transfer said energy to other particles.

As far as resistance goes, that involves atomic collisions. Electrons collide w/ the lattice, energy is dissipated as photons are released. The photon emission is in the IR wavelength around 5 micron, & is felt as heat.

So if an inductor is energized, it tends to maintain a constant current. If the source is disconnected & the inductor shorted, the current will remain constant forever if no resistance were present. But, for a resistive inductor, electrons conduct & bang into the lattice losing energy. This energy is radiated as heat. Hence the energy in the inductor is decreasing, as is the current, since W = L*I^2/2.

Did this help?

Claude

Can you explain what field of electrical engineering are you working on? I study electrical engineering, first year, telecommunications department and they don't teach me that stuff. They only went so far when explaining electricity, I almost failed my exams because I didn't focus on studying, I focused on understanding things that I've learned.

And yes, this makes things very clear. Can you recommend me some book(s) that you have studied from? I will buy them online if i have to, just give me the names.

Thanks

 

[cabraham]

I am a broad EE generalist, w/ primary focus on analog & power electronics. I develop motor drives, smps (switch mode power supplies), microcontroller & dsp based systems, analog filters, to name a few.

Understanding quantum mechanics, "qm", is something nobody has yet achieved. To fully understand inductance, conduction, energy bands, thermo, etc., involves deep level of qm. The late great Richard Feynmann, one of the giants of the 20th century science community, flatly stated that he does not fully comprehend qm, & that anyone who claims otherwise is fooling nobody but themself. I frankly admit before the world that qm is beyond me.

When I took modern physics as an undergrad, as well as e/m fields, I just accepted what was taught w/o questioning or attempting to comprehend it fully. If the phy prof says this is how the science community sees it, I nod my head & say "yes sir." We can explain things down to a level where we cannot go further. Axioms are developed based on verifiable repeatable observation/testing, & all else is derived from those axioms. One cannot "prove" or explain an axiom in terms of something more fundamental because the most fundamental quantities we can detect give birth to the axiom itself.

Books I'd recommend are any uni-approved EE fields texts as well as physics fields texts. Also, circuit theory texts give a great mAcroscopic explanation of inductors, but fields texts give a better mIcroscopic view. Also, good texts on energy conversion, dealing w/ motors & generators, transformers, solenoids, etc. are great references. Fitzgerald, Kingley, & Umans text "Electric Machinery" is quite good. I just had Dr. Umans for a power course spring 2010. It was great.

Anyway, I hope I've helped. BR.

Claude

 

[Bassalisk]

I am a broad EE generalist, w/ primary focus on analog & power electronics. I develop motor drives, smps (switch mode power supplies), microcontroller & dsp based systems, analog filters, to name a few.

Understanding quantum mechanics, "qm", is something nobody has yet achieved. To fully understand inductance, conduction, energy bands, thermo, etc., involves deep level of qm. The late great Richard Feynmann, one of the giants of the 20th century science community, flatly stated that he does not fully comprehend qm, & that anyone who claims otherwise is fooling nobody but themself. I frankly admit before the world that qm is beyond me.

When I took modern physics as an undergrad, as well as e/m fields, I just accepted what was taught w/o questioning or attempting to comprehend it fully. If the phy prof says this is how the science community sees it, I nod my head & say "yes sir." We can explain things down to a level where we cannot go further. Axioms are developed based on verifiable repeatable observation/testing, & all else is derived from those axioms. One cannot "prove" or explain an axiom in terms of something more fundamental because the most fundamental quantities we can detect give birth to the axiom itself.

Books I'd recommend are any uni-approved EE fields texts as well as physics fields texts. Also, circuit theory texts give a great mAcroscopic explanation of inductors, but fields texts give a better mIcroscopic view. Also, good texts on energy conversion, dealing w/ motors & generators, transformers, solenoids, etc. are great references. Fitzgerald, Kingley, & Umans text "Electric Machinery" is quite good. I just had Dr. Umans for a power course spring 2010. It was great.

Anyway, I hope I've helped. BR.

Claude

U were more than helpful. But I myself am very passionate about learning. My college department offers 5% of QM that I want to know. I am self-taught at home QM to get at least a glimpse how something works. My dream is to study QM, but there is no such college here. Only parts of some departments go so far.

Well I understand that you don't have to know QM to actually understand something. U can accept it by heart. But here is the thing. I want to understand. I think I am the only one amongst 300 students that really go down to quantum level to understand some processes.

My assistants told me: kid you are going to go crazy, you cannot understand everything we are trying to teach you. I said: So be it.Now you see why I bug this thread so much...

Thanks for the book, I will check it out, you were very helpful. High level opinions like yours here should be highlighted.

 

[sophiecentaur]

If you really think you can square the effects of Inductance with QM then you are welcome to try. Frankly, anyone who could do that should have a pretty thorough understanding of the Classical approach- Maxwell etc..
The good thing about the Classical approach is that all it demands is the ability to understand the wonderful world of Integral Calculus and also some ability with vectors. There are many basic EM theory books to degree level which will get you a fair way to the way fields and structures interact. I look at the ones I bought at Uni (mid '60s) and it's amazing just how much the classical approach can do for you. Snag is, you can't wave yer arms about with EM theory. It's far too businesslike and not quite so much open to 'interpretation', which is why QM appeals to so many people as a conversation piece.

 

[cabraham]

If you really think you can square the effects of Inductance with QM then you are welcome to try. Frankly, anyone who could do that should have a pretty thorough understanding of the Classical approach- Maxwell etc..
The good thing about the Classical approach is that all it demands is the ability to understand the wonderful world of Integral Calculus and also some ability with vectors. There are many basic EM theory books to degree level which will get you a fair way to the way fields and structures interact. I look at the ones I bought at Uni (mid '60s) and it's amazing just how much the classical approach can do for you. Snag is, you can't wave yer arms about with EM theory. It's far too businesslike and not quite so much open to 'interpretation', which is why QM appeals to so many people as a conversation piece.

Yes I agree w/ you that the classical approach per Maxwell et al offers a wealth of insight. For practical engineering applications, classical more than suffices. I guess we all got off the deep end delving into quantum theory. It's interesting to ponder, but as you mentioned, the study, analysis, & design of transformers, solenoids, filters, machines, etc., can be done full justice using classical methods.

Maxwell's axioms stand like a Gibraltar, today, yesterday, & tomorrow. Thanks to you & all for a most interesting discussion.

Claude

 

[RANA WAJID]

as we know that the ohmic resistance of the inductor is very low.
when we apply some potential or voltages to the coil the coil energies and a flux is produced in it.
There are three types of effects when we apply voltages to the coil.
1) As according to the self induction when due to the change in direction or magnitude of flux then emf induces in that coil.
2) we know that when electricity is given to any motor than back emf induces in that motor so here back emf also produced and due to which a current also flows through the inductor which is opposite in direction to the supply current.this back current provides some resistance in the flow of current that's why current lags from voltage in inductors.
3) third and last one effect relates with the LENZ's law according to which current always opposes its own generating process.
so by the above given reasons we say that the current lags from the voltage in the inductor so the back current is partially stopped that current.
Due to that partial stop of current voltages become leading.
THIS ANS IS GIVEN TO ME BY OUR GOOD TEACHER
MUSSADIQ ALI RAZA

 

[jim hardy]

I've seen that this question has been asked but never answered because guy was silent on formulas.

I can say that i am familiar with formulas, and i know how to derive the equation for current, and that II/2 lag, but here is the thing... Why? i know that formulas say so... But can someone please try to explain this to me in more touchable and intuitive way ?Thanks

touchable and intuitive, eh?
All you need is to believe Lenz's law.

here's how i came to believe it at the primal level. I felt it.
You need a decent sized 115 volt transformer, maybe a pound or two, and an analog multimeter with RX1 scale such as Simpson 260.
Or, lacking a meter, a D cell battery and a smallish transformer.

Set the multimeter on RX1, zero it and connect to the transformer's primary.
You will see the meter move to indicate the winding resistance of just a few ohms.
Repeat until you get a feel for how fast the needle moves.
Then reverse the leads, and you should notice a short hesitation before needle begins its travel.
Note difference in delay when you reverse or don't reverse leads between readings .

That delay is the transformer's inductance opposing the (increasing) current flow from the meter. A Simpson 260 on RX1 scale will push about 100 milliamps into low ohms. Lenz's law says inductance will oppose change in flux. That's the delay you observe on the meter.
When you don't reverse leads, the transformer is less able to resist because core was left mildly magnetized, so there's less change in flux than when you do reverse leads.

Now pinch the two transformer leads beneath fingers of one hand.
Apply multimeter again and note the shock you feel when removing the test lead.
That is the inductance opposing the reduction of flux - it will literally "Bite the hand..."

>>>>>NOTE --use one hand, don't ever intentionally pass current through your chest.<<<<<

if you don't have a multimeter , use a smaller transformer and a D cell...
>>> Repeat:: smaller transformer <<<

feeling that shock should help you believe that inductance vigorously opposes change in flux... quite vigorously.

Now - back to definition of inductance

inductance (L) = flux linkages per ampere
L = n * Phi / I ;; n=turns, I = amps, Phi = flux
so flux = I * L/ n ;; which says ( L and n being constants ) current and flux are in proportion no time delay
so the inductor tries to maintain flux by maintaining current;
which it does by producing voltage that you can feel

and it's symmetric, inductance will oppose an increase or a decrease in current.An ideal inductor could hold out forever but a real one reaches limit on current through its copper or flux through its core.
Recall that a sinewave is a mathematical special case, where time delay becomes phase...

That simple experiment should help you believe the formulas.
But - use a transformer you can pick up with one hand. A big one would be dangerous.

or just take my word for it...
old jim

 

[Bassalisk]

touchable and intuitive, eh?
All you need is to believe Lenz's law.

here's how i came to believe it at the primal level. I felt it.
You need a decent sized 115 volt transformer, maybe a pound or two, and an analog multimeter with RX1 scale such as Simpson 260.
Or, lacking a meter, a D cell battery and a smallish transformer.

Set the multimeter on RX1, zero it and connect to the transformer's primary.
You will see the meter move to indicate the winding resistance of just a few ohms.
Repeat until you get a feel for how fast the needle moves.
Then reverse the leads, and you should notice a short hesitation before needle begins its travel.
Note difference in delay when you reverse or don't reverse leads between readings .

That delay is the transformer's inductance opposing the (increasing) current flow from the meter. A Simpson 260 on RX1 scale will push about 100 milliamps into low ohms. Lenz's law says inductance will oppose change in flux. That's the delay you observe on the meter.
When you don't reverse leads, the transformer is less able to resist because core was left mildly magnetized, so there's less change in flux than when you do reverse leads.

Now pinch the two transformer leads beneath fingers of one hand.
Apply multimeter again and note the shock you feel when removing the test lead.
That is the inductance opposing the reduction of flux - it will literally "Bite the hand..."

>>>>>NOTE --use one hand, don't ever intentionally pass current through your chest.<<<<<

if you don't have a multimeter , use a smaller transformer and a D cell...
>>> Repeat:: smaller transformer <<<

feeling that shock should help you believe that inductance vigorously opposes change in flux... quite vigorously.

Now - back to definition of inductance

inductance (L) = flux linkages per ampere
L = n * Phi / I ;; n=turns, I = amps, Phi = flux
so flux = I * L/ n ;; which says ( L and n being constants ) current and flux are in proportion no time delay
so the inductor tries to maintain flux by maintaining current;
which it does by producing voltage that you can feel

and it's symmetric, inductance will oppose an increase or a decrease in current.


An ideal inductor could hold out forever but a real one reaches limit on current through its copper or flux through its core.
Recall that a sinewave is a mathematical special case, where time delay becomes phase...

That simple experiment should help you believe the formulas.
But - use a transformer you can pick up with one hand. A big one would be dangerous.

or just take my word for it...



old jim

You got my attention. I understood mostly of it. I will definitely try this. :D

 

[jim hardy]

I understood mostly of it...

thanks.

i sort of swapped terms on you there, using current and flux interchangeably before showing that they are indeed one another's image...
but i think that is a key detail to grasp when figuring out inductance.
grasping that little insignificant(?) point will help you understand how real inductors depart from ideal ones.

Keep it simple. (That helps things "go well".)


old jim

 

[psparky]

I didn't get to read the whole thread...but will throw in my two cents nonetheless.

Reactance of an inductor is JWL...or WL <90 degrees.

V=IR...In this case R = JWL

When you divide the voltage by 90 degrees...you get a minus 90 degrees.

Therefore current is lagging voltage.

A capacitor does just the opposite 1/(JWC).

You get a current vector in the plus 90 degree direction. This is how power factor correction is achieved.

Any questions?

 

[sophiecentaur]

So many of you guys don't want to get your feet wet in Calculus. However, if you consider that the voltage that appears across an inductor is proportional to the rate of change of the current, then the current will be the integral of the voltage. When you apply this to a sinusoidal signal, you get exactly the lead and lag effects you would expect.
Why is that not a perfectly good explanation?

 

[Bassalisk]

This thread is almost 8 moths old. In these 8 months my calculus became very good. So this lagging physically I now understand, but with now good calculus its even simpler.But I do not regret for asking this to be explained on physical level.

 

[sophiecentaur]

Can you claim to be merely 'physical' once you have introduced sines and phases?

 

[psparky]

To sophiecentar...your explanation is fine...

I just explained it in a different way that proves it mathematically as well...and it also spurs thoughts toward power factor correction.

 

[Bassalisk]

Can you claim to be merely 'physical' once you have introduced sines and phases?

Well, all I'm saying is that I had troubles with math. I had to get the exam ready. I couldn't learn whole calculus in a week, so i tried to find a common sense and use my current knowledge of physics to get a feeling for that lag.

I did write that I knew about the formula for voltage across the inductor, but did I had a true feeling and knowledge of what derivative and integral means? no.

All I knew was, when I do derivative of a sine, i get a cosine. That is as far as my mind went. Later, when I finished calculus course, and I had to study for my final exam, I had to learn like 250 pages of pure calculus to pass. This final exam is not written, you talk with professor and he can ask you anything.

I had to learn all those theorems, fundamental theorems of derivatives, integrals, derivatives and calculus in general(Fermat's La grange's Cauchy's Cantor's need I go on?) . Did that help me understand a, I may say now simple problem like lag in inductor ? Yes ! Of course it did. But at the point I was asking this question, 8 months ago, I didn't have as good math as I do now.

And yes, I can say that I can "feel" the sine function now. Its not just another function for me.

 

[sophiecentaur]

OK but is "j" 'physical'? The idea of complex numbers comes way along the line in Maths, I should have thought.

Glad to hear you have laid the ghost of hard Maths. One step at a time and it will yield!

 

[Bassalisk]

OK but is "j" 'physical'? The idea of complex numbers comes way along the line in Maths, I should have thought.

Glad to hear you have laid the ghost of hard Maths. One step at a time and it will yield!

Yea j isn't physical. I got that. I understood that its only a good tool that somehow works very well when going back and forth between frequency and time domain.

I am very happy that I truly understand how j is incorporated.

Not once I found myself asking: "There must be SOME kind of relation between reactive resistance and imaginary numbers, other than mathematical. This square root of number one, the whole idea of imaginary numbers, and the way they are set up, there must be something there. I know that nobody can answer this question for me. I do know that currently it is just said that it works, and that there is no good reason for why is reactive part imaginary and I know great minds have thought this through and it works, but still you have that little tingling feeling in the back of your head, every time you think of it"

But one time I did that, I went deeper and deeper and deeper, then I stopped. Where did I stop? At the axioms of math. And there are no answers there. They are axioms. If people want to question this relation, then ask a question: What is number? Why are numbers defined like this, why -2 smaller than 2 etc. This just math, pure and purest science that was, is, and will ever exist.

 

[Bassalisk]

But then again, this science and all this math I learned, It affected me as a person. I am questioning the existence of God, luck and all that stuff that makes life easier. I am beginning to think deterministically. All this is very hard on me.

 

[sophiecentaur]

Aren't we just differentiating between what's very familiar, what's not so familiar and what we jus can't grasp at all? It's all very personal at this level.
If you 'feel' the result of some maths because you've done it so often that it's second d nature then I guess it becomes like the sums you do in your head when you're running downstairs. You just 'do it'. I suppose something like that could be described as Physical.
But is it really?
To my mind, truly 'physical' models are more Concrete Operational (Piagetian Cognitive Levels - google him) and don't support useful hypothesising. Mostly, people on these fora are using more (possibly a private, personal) maths than they are aware of. For Physical, read 'Familiar'.
Formal Maths is useful / essential for seriously progressing but it's also a useful common language because it is less likely to cause misunderstanding. I can't think, for instance, of a more concise term than 'Integral' to describe basically what's going on in many situations.

I think that, to have made the comments you did, implies a comfortable level of Formal Operational thought. The same goes for a lot of other contributors. (Ye Gods - was that a compliment? I'd better watch myself.)
Of course, Formal Operational thought may not actually imply 'reasonableness'. haha

 

[Bassalisk]

Aren't we just differentiating between what's very familiar, what's not so familiar and what we jus can't grasp at all? It's all very personal at this level.
If you 'feel' the result of some maths because you've done it so often that it's second d nature then I guess it becomes like the sums you do in your head when you're running downstairs. You just 'do it'. I suppose something like that could be described as Physical.
But is it really?
To my mind, truly 'physical' models are more Concrete Operational (Piagetian Cognitive Levels - google him) and don't support useful hypothesising. Mostly, people on these fora are using more (possibly a private, personal) maths than they are aware of. For Physical, read 'Familiar'.
Formal Maths is useful / essential for seriously progressing but it's also a useful common language because it is less likely to cause misunderstanding. I can't think, for instance, of a more concise term than 'Integral' to describe basically what's going on in many situations.

I think that, to have made the comments you did, implies a comfortable level of Formal Operational thought. The same goes for a lot of other contributors. (Ye Gods - was that a compliment? I'd better watch myself.)
Of course, Formal Operational thought may not actually imply 'reasonableness'. haha

This forum helped me in many ways, other than EE. These posts I read, like yours, are very well written. And I really stress my brain to truly understand what you are trying to say, as you may assume English is not my first language.

I will of course progress as I go to upper years. I will of course be more familiar with math and I will start to think mathematically more, and less physically.

Thank you for your suggestions, I will definitely research them out.

 

[sophiecentaur]

Why thank you kind sir. Your own English is not arf bad at all. You make sense and don't make me smile. What more can one ask for?

 

[Bassalisk]

Why thank you kind sir. Your own English is not arf bad at all. You make sense and don't make me smile. What more can one ask for?

No problem at all. Thats why you are Science Advisor and I am student with a lot of questions. Knowledge is my reward sir. :)

 

[Pablo Verdugo]

I'll try to use the perspective of power. As far as ii know, a conductive predominant circuit has positive reactive power, so S=P+jQ. Now, in per unit, we have S=VI*
By substituting terms I=S*/V* which means I=(P-jQ)/V*
S=P-jQ gives you a negative angle so current lags voltage, if you want to assume an infinite bar which would make the voltage angle equal to zero.

This may not be the explanation you were looking for, but it's just another perspective.

 

[sophiecentaur]

No problem at all. Thats why you are Science Advisor and I am student with a lot of questions. Knowledge is my reward sir. :)

omg - a mutual appreciation society!

 

[jim hardy]

what an interesting couple pages.

i'm not anti-math but am sympathetic toward those who struggle with it.

being mildly autistic, my awkwardness made math more difficult for me than it should have been. simple arithmetic mistakes scuttled many a calculus problem that i'd set up correctly but blew the evaluating part from a dropped sign or something.

result is i am real skeptical of formulas unless i can "feel" them.

when you're as dumb as me you have to work twice as hard as normal people and that's why i try to explain things simply - if it keeps somebody from giving up it's worthwhile.

for anybody stuggling with concept behind operator j ;
here's what i decided felt right:
multiplying by operator j shifts phase 90 degrees
multiplying twice shifts you 180 degrees,
which is exactly same as multiplying by -1
so obviously j is sqrt(-1) , for when you multiply it by itself you get -1 and that's a square root.
And of course it's imaginary because everybody knows negative numbers don't have square roots.

works for me. I'm no Euler.
but i learn from most everybody i meet.
great discussion guys , thanks.



old jim