Conjugate of angles and phasors

[mivey]

A brief note on the correct relationship between conjugating phasors and inverting phasors.

There are three common ways we use "conjugate" in relation to algebraic expressions, phasors, or angles:

1) Changing the sign of the second half of an algebraic expression

2) Changing the sign of the imaginary part of a complex expression (or changing the sign of the phasor's angle). This would be the most common use in a discussion on electric system phasors.

3) Conjugate angles are two angles that sum to 360?


In another thread, there were some incorrect uses of "conjugate" and "conjugation". The incorrect uses were:

1) Calling the conjugate of a phasor the same as the inverse of a phasor.

2) Calling the conjugate of an angle the same as the inverse of a phasor angle.

3) Saying that conjugating an angle is the same as inverting an angle.


Notes on error #1:
When we speak of inverting a phasor, we do so by taking the negative of the phasor, or by taking the negative of the complex number representation, or by shifting the phasor angle by 180?. When we conjugate a phasor, we do so by changing the sign of the phasor's angle, or by changing the sign of the imaginary portion of its complex number representation. So, taking the conjugate of a phasor and inverting a phasor are not the same. For example:

Invert{C@Θ?} = -1 * C@Θ? = C@(Θ? ? 180?)
Invert{A + jB} = -1 * (A + jB) = -A - jB

Conjugate{C@Θ?} = C@(-1 * Θ?) = C@(-Θ?)
Conjugate{A + jB} = A + j(-1 * B) = A - jB


Notes on error #2:
When inverting a phasor, we can do so by taking the negative of the phasor, or by shifting the phasor angle by 180?. If by calling the conjugate of an angle the angle found when we find the missing angle of a conjugate pair, we subtract the known angle from 360?. So, the conjugate angle is not the same as the inverse phasor angle. For example:

Invert{C@Θ?} = -1 * C@Θ? = C@(Θ? ? 180?)

Conjugate pairing angle{Θ?} = (360? - Θ?) => C@(360? - Θ?)


Notes on error #3:
Inverting the angle was referring to the angle change when inverting the phasor, or changing to the opposite direction. Not so sure what was meant by conjugating the angle as it is not common terminology. If conjugating an angle means finding the conjugate pair, see note on error#2. If conjugating an angle means the angle from conjugating a phasor, see note on error #1.

 

[gar]

120412-0739 EDT

A useful discussion on the complex conjugate with a graphical illustration can be found at http://en.wikipedia.org/wiki/Complex_conjugate .

When you use capacitors for power factor correction of an inductive load a complex conjugate operation is performed.

.

 

[Besoeker]

When you use capacitors for power factor correction of an inductive load a complex conjugate operation is performed.

I don't generally do that. Just a simple bit of mathematics.
An example.
I have a 150kW 0.86 pf load that I want to correct to 0.95 pf.
The 150kW and 0.86 pf give me load kVA and kVAr as 174kVA and 89kVAr.
Just Pythagoras.
With basic trig I can work out what kVAr is needed for 0.95 pf as 150*tan(acos 0.95)
That's 49kVAr. Now a bit of simple arithmetic gives me 89-49 = 40.
So I need a 40kVAr PFC cap.

That's how I do it.

 

[mivey]

120412-0739 EDT

A useful discussion on the complex conjugate with a graphical illustration can be found at http://en.wikipedia.org/wiki/Complex_conjugate .

Thanks.

As part of the other thread, double subscipts and angles were used together in the examples of inversions and conjugations (like Vxy@180?). Have you ever seen notation like that or any documentation to describe how it is supposed to be used? I have not found any and can't make sense of it using the results from the other thread.

 

[mivey]

I don't generally do that. Just a simple bit of mathematics.
An example.
I have a 150kW 0.86 pf load that I want to correct to 0.95 pf.
The 150kW and 0.86 pf give me load kVA and kVAr as 174kVA and 89kVAr.
Just Pythagoras.
With basic trig I can work out what kVAr is needed for 0.95 pf as 150*tan(acos 0.95)
That's 49kVAr. Now a bit of simple arithmetic gives me 89-49 = 40.
So I need a 40kVAr PFC cap.

That's how I do it.

Of course. But I think gar was just referring to the mirroring about the real axis that takes place.

Add: in other words, the j49 portion of your work is the negating action of complex conjugation

 

[Besoeker]

Of course. But I think gar was just referring to the mirroring about the real axis that takes place.

Add: in other words, the j49 portion of your work is the negating action of complex conjugation

I just like simple.

 

[mivey]

I just like simple.

Get 'er done!

 

[rattus]

Well no:

Well no:

Thanks.

As part of the other thread, double subscipts and angles were used together in the examples of inversions and conjugations (like Vxy@180?). Have you ever seen notation like that or any documentation to describe how it is supposed to be used? I have not found any and can't make sense of it using the results from the other thread.

There is nothing in the preceding expression which tells us the magnitude which is a constant, always positive.

In algebraic terms, Vxy is the unknown; it is not a constant. For example, the proper expression is:

Vxy = 120Vrms @ 180

Which means Vxy is the RMS voltage on node 'x' relative to the RMS voltage on node 'y' with a phase angle of 180 degrees. That is, it is the potential DIFFERENCE between the two nodes, and we must SUBTRACT to obtain the potential DIFFERENCE.

 

[mivey]

In algebraic terms, Vxy is the unknown; it is not a constant. For example, the proper expression is:

Vxy = 120Vrms @ 180

Thanks. That's what I thought. I guess the improper use was trying to convey something, but it did not makes sense and was producing two different results.