complex numbers and vectors

[PG1995]

Hi

What does complex conjugate, a-ib, stand for compared to its pair a+ib? I had thought that complex conjugate of a complex number represents 180° rotation on the plane. For example, 2+3i has an angle of 56° and so I had thought that its complex conjugate 2-3i would have angle of 236°. But I was wrong because 2-3i has an angle of -56° or 304°.

I think I build this concept during solving problems of circuit analysis. For instance, in this problem you can see the direction of current I1 and direction of current source are 180° out of phase, therefore I took complex conjugate of 2<90° which represents current source and reached correct answer.

In short, in terms of rotation what does complex conjugate mean? I think a+ib means clockwise rotation and a-ib counterclockwise rotation. Kindly guide me. Thank you.

Regards
PG

 

[steveB]

A rotation of 180 degrees is equivalent to muliplication by -1. If you have a complex number z=x+jy, rotation by 180 degrees gives -z=-x-jy. Obviously, this is different than complex conjugation which gives z*=x-jy.

The operation of complex conjugation can be thought of as a mirror image with the axis of symmetry being the real axis. As you say, instead of rotating one way, the complex cojugate rotates the other way. This is easier to see in polar notation. z=r exp(jθ) has complex conjugate z*=r exp(-jθ). Clearly, the angle is in the opposite direction.

 

[PG1995]

Thank you, Steve.

Q1:
But I think I was right in this problem that the direction of current I1 and direction of current source are 180° out of phase. But it was coincidence that both 180° rotation which is represented by -z=-x-jy and complex conjugation which can be thought of as mirror image along real axis summed up to the same thing in this particular case; 180° rotation would mean 270°, and mirror image of 90° would also mean 270°. Could you please confirm it if I have it right?

Q2:
Could you please help me with this query too?

Q3:
Could you kindly guide me with this query too?

Regards
PG

 

[steveB]

For Q1, of course in the special case where a complex number is imaginary, multiplying by -1 (180 deg rotation) and taking the complex conjugate (mirror image) amount to the exact same result. This is obvious in rectangular representations z=jy results in -z=-jy and z*=-jy, hence -z=z* in this special case.

For Q2, I'm confused on what you are asking. You point out that phasors should not be applied to multiplication of sine waves, even if the frequency is the same. Your reference gives an example and show the conflict. Yet, you apply phasors to the same example and seem to also show the conflict. I can only assume that maybe you are not understanding the impediment to using phasors in this case. The phasor notation assumes a particular frequency, which is left out of the notation. However, mathematical multiplication of two sign waves of the same frequency typically generates a DC component plus a sine wave at twice the frequency. This means the frequencies subtracted to zero and added to twice the value, yet the phasor notation knows nothing about any change in frequency. So a true answer of 0 and 2w for frequency components is in conflict with the phasor description which says frequency is w only.

 

[PG1995]

Thanks a lot.

For Q2, I'm confused on what you are asking. You point out that phasors should not be applied to multiplication of sine waves, even if the frequency is the same. Your reference gives an example and show the conflict. Yet, you apply phasors to the same example and seem to also show the conflict.

Where is the conflict in my solution?! Actually I was trying to demonstrate that there was no conflict and I had thought my end result was correct.

Regards
PG

 

[steveB]

Thanks a lot.



Where is the conflict in my solution?! Actually I was trying to demonstrate that there was no conflict and I had thought my end result was correct.

Regards
PG

The conflict is that you get the wrong answer because you are applying a technique, which has underlying assumptions, to a problem that does not meet those required assumptions. Why did you think your end result was correct?

 

[steveB]

For Q3, you got to the point where you could say

[latex]\frac{{\bf A}\cdot {\bf u}}{A}=1[/latex]

I would then write this as follows ...

[latex]\frac{{\bf A}}{A}\cdot {\bf u}=1[/latex]

Then I would note the key property of a unit vector that ...

[latex]{\bf u}\cdot {\bf u}=1[/latex]

from which I would conclude that ...

[latex]\frac{{\bf A}}{A}= {\bf u}[/latex] provided that the magnitude of A/A equals one.

However, this condition is true because the magnitude of a ratio equals the magnitude of the numerator divided by the magnitude of the denominator, and the total magnitude is clearly 1.

This is a bit of an awkward proof, and I expect there is a more elegant formulation, but it's an obvious fact anyway, so we don't need to go crazy finding the best method.

 

[PG1995]

Thank you very much, Steve.

Re: Q2 from previous post

The conflict is that you get the wrong answer because you are applying a technique, which has underlying assumptions, to a problem that does not meet those required assumptions. Why did you think your end result was correct?

I thought my answer was correct because it seemed I didn't do any usual silliness! Where did I go wrong in any of the math here? If we can't apply phasor notation to product of two sinusoids then where is this rule used?

Could you please help me with this query too? Thanks.

Regards
PG

 

[steveB]

Well, you keep asking me questions and my answers aren't working here, so let's reverse the process. You can answer the following.

What does a phasor represent?

What frequency is represented by a phasor?

What assumptions (such as linearity) are specified for the use of phasors?

The example in question, is it a linear or nonlinear operation?

You quoted an example, did you notice the answer they gave without the use of phasors? That is, did you see the DC component and sine wave with twice the frequency?

What frequency is implied in your answer, and does it match the correct answer?

Once you answer these questions, we can identify what the stumbling block is to understanding this issue. Note that one can always apply a method, even when the required assumptions are not true. Then you will get an incorrect answer. So, how do you know when an answer is correct? In this case, the answer is easy. Phasors give a fast way to get answers; so, do it the hard way and do it the phasor way and see if they match. If they don't match, why don't they match? The usual answer is "nonlinearity". Linear systems generally produce output signalss that are at the same frequency as the input signals. Nonlinear systems, generate additional frequencies.

 

[PG1995]

Thanks a lot, Steve.

Seriously I'm not evading your questions. Whenever you ask a counter-question I do try to find purpose of your asking that question and then try to correct my confusion without actually answering your question.

I think I understand it now.

If you have time please also give this query a look. Thank you.

Regards
PG

 

[steveB]

Than
I think I understand it now.

I think you are getting closer to understanding, but I'm not sure I agree with your viewpoint entirely. You apply phasors to the LHS and see no problem doing that for a nonlinear operation of multiplication, and then you say phasors can't be applied to the right hand side. If anything, the RHS is in better position to apply phasor notation, if you add the idea of superposition. You can add two phasors - one at zero frequency and one at 2w frequency (the idea of adding phasors of different frequencies is dubious however, and requires special rules). The phasor notation on the LHS is a bit nonsensical because superposition can't be applied directly and phasors should not be multiplied.

My view is that phasors can't be applied to either side because doing so requires disobeying established rules for their application.

However, with that said, the real problem is that if you blindly apply phasor operations/notation to both sides without regards to rules and assumptions, you get different answers on each side. This clearly shows that an assumption was not valid. If you have two things that are equal, and apply an operation to both sides, you expect both sides to still be equal. When this doesn't work, you need to question the assumptions required for applying the operation.

 

[steveB]

If we can't apply phasor notation to product of two sinusoids then where is this rule used?

That is a rule for complex numbers, not for phasors. Phasors are special objects that have an implied exp(jwt) that is ignored in the analysis. They represent vectors that rotate in the complex plane at a particular frequency. The magnitude is the radius of the circular rotation and the angle is the starting angle at t=0.

When all phasors are at the same frequency, you can factor the exp(jwt) out of all complex numbers and apply complex math to the remaining phasor. The problem with multiplying phasors is that it changes the frequency, and the assumption that exp(jwt) is present in phasors is violated.

Simple answer? - Phasors are not exacty (just) complex numbers and certain rules must be obeyed when you use them. If you do try to multiply, then new rules are needed. In fact, when we calculate power, we need to multiply, so people have figured out the special rules for describing power with phasors. However, those rules assume that the system is linear also.

 

[PG1995]

Many thanks. I understand it now.

When you get time please also have a look on this query?

Best regards
PG

 

[steveB]

When you get time please also have a look on this query?

I can't follow what you are asking or saying at all. At the botom you show A=x+jy, and then say there is a phasor for x and a phasor for y, or something like that. I don't know what this means, and the entire bottom part seems wrong (EDIT: I shouldn't say wrong, but rather unusual and unessessarliy complicated) , unless I'm misunderstanding what you are doing. Just because something is a complex number, doesn't mean it's a phasor (EDIT: although it is often possible to use math that supports that viewpoint, which might be the case here, but agian - unusual and complicated). A phasor is a complex number that has been assigned to another complex number that rotates in the complex plane at a fixed frequency. They seem to call that a sinor (which I've never heard of before). So a phasor represents the sinor, but ignores the exp(jwt) part. The sinor is a rotating complex vector that represents a real signal. The conversion of the sinor to the real signal is accomplished by taking the real part of the sinor.

In the end, the final signal is f(t)=A cos(wt+θ), where A is the magnitude of the phasor and θ is the angle of the phasor. I don't know why we need all this detail about sinors and A(t) etc. It's making a simple situation overly complicated.

That's really all you need to worry about. Then, we learn rules for dealing with phasors that give us correct answers in solving problems with linear circuits. It's important to understand the foundation and the rules so that we don't apply phasors to the wrong problems and so we don't make mistakes when we apply them to the right problems.

EDIT: In light of what you said in another thread about using phasor methods for EM field problems and considering 3D vectors with phasor like (complex) values, I would tone down some of what I said above. However, I'm still confused on some of the details of what you did. In particular, I'm confused by the idea that we have separate phasors for x and y.