How Is Polar to Rectangular Conversion Used in Complex Number Calculations?

[Dcmagni]

I'm having trouble trying to calculate how the answer below was achieved from an example i have seen, see below:

208L0 - 2.5L90 x 27.42L36.9 which is then calculated to 255.12L-12.4.

I have tried converting everything to rectangular form, subtract where required and the convert back to polar form to multiply. To be honest I have tried every way and can not figure this out. Any help is appreciated.

 

[berkeman]

Welcome to PF.

208L0 - 2.5L90 x 27.42L36.9 which is then calculated to 255.12L-12.4.

I'm having a hard time parsing what you wrote. Are your "L" characters supposed to be indicating angles? ( ∠ )

And is the "x" character meant to represent some kind of multiplication (cross product?)

 

[Dcmagni]

Welcome to PF.I'm having a hard time parsing what you wrote. Are your "L" characters supposed to be indicating angles? ( ∠ )

And is the "x" character meant to represent some kind of multiplication (cross product?)

Yes apologies, it would not let me enter the angle sign for some reason. Also the 'x' is multiplication
Thank you

 

[PeroK]

Yes apologies, it would not let me enter the angle sign for some reason. Also the 'x' is multiplication
Thank you

Are these complex numbers?

 

[The Electrician]

I'm having trouble trying to calculate how the answer below was achieved from an example i have seen, see below:

208L0 - 2.5L90 x 27.42L36.9 which is then calculated to 255.12L-12.4.

I have tried converting everything to rectangular form, subtract where required and the convert back to polar form to multiply. To be honest I have tried every way and can not figure this out. Any help is appreciated.

If you want help, you must show all the steps of your calculations in detail. Using my HP50 calculator which can do complex arithmetic I get a result of 255.11795 < -12.408174

 

[berkeman]

I have tried converting everything to rectangular form, subtract where required and the convert back to polar form to multiply. To be honest I have tried every way and can not figure this out.

Can you show us some of your work so we can check it over for mistakes? You should be able to copy/paste the angle symbol that I posted to make the notation clearer.

 

[Dcmagni]

So far I have the understanding that subtraction is easier in polar form and multiplication in rectangular form, I need to transform into rectangular form.

I have first multiplied 2.5∠90 x 27.42∠36.9.
= 68.55∠126.9 (I have multiplied the polar magnitude and addition for the angle)

Then I have converted 208∠0 - 68.55∠126.9 into rectangular form.
208∠0 =
(208)(cos0) = 208
(208)(sin0) = 0
Therefore = 208+j0

-68.55∠126.9 =
(68.55)(cos126.9) = -41.16.
(68.55)(sin126.9) = j54.82
Therefore = -41.16+j54.82

Resulting in 208+j0 - 41.16+j54.82.
Subtracting to me gives a result of: 166.84 - j54.82 (Subtracting the magnitudes and the angles)

I have then converted 166.84 - j54.82 back to polar form.

Magnitude = √166.84^2 - 54.82^2
= 157.57
Angle = arctan -54.82/166.84
= -18.19

Therefore my final result is 157.57∠-18.19. Which is not the same as the result from the example that was given.

 

[PeroK]

Therefore = -41.16+j54.82

Resulting in 208+j0 - 41.16+j54.82.
Subtracting to me gives a result of: 166.84 - j54.82 (Subtracting the magnitudes and the angles)

You've added these two complex numbers. Whereas, you were supposed to subtract one:

208L0 - 2.5L90 x 27.42L36.9 which is then calculated to 255.12L-12.4.

 

[Dcmagni]

You've added these two complex numbers. Whereas, you were supposed to subtract one:

Sorry I don't understand what you mean, where have I added?

I have subtracted as follows:
208 - 41.16 = 166.84
j0 - j54.82 = -j54.82.
Therefore resulting in 166.84 - j54.82

 

[PeroK]

Sorry I don't understand what you mean, where have I added?

I have subtracted as follows:
208 - 41.16 = 166.84
j0 - j54.82 = -j54.82.
Therefore resulting in 166.84 - j54.82

What's $208 - x$ when $x = -41$?

 

[Dcmagni]

What's $208 - x$ when $x = -41$?

249? If so then this answer is still incorrect as the answer gives 255.12

 

[PeroK]

249? If so then this answer is still incorrect as the answer gives 255.12

That's just the real part.

 

[The Electrician]

So far I have the understanding that subtraction is easier in polar form and multiplication in rectangular form, I need to transform into rectangular form.

I have first multiplied 2.5∠90 x 27.42∠36.9.
= 68.55∠126.9 (I have multiplied the polar magnitude and addition for the angle)

Then I have converted 208∠0 - 68.55∠126.9 into rectangular form.
208∠0 =
(208)(cos0) = 208
(208)(sin0) = 0
Therefore = 208+j0

-68.55∠126.9 =
(68.55)(cos126.9) = -41.16.
(68.55)(sin126.9) = j54.82
Therefore = -41.16+j54.82

Resulting in 208+j0 - 41.16+j54.82.
Subtracting to me gives a result of: 166.84 - j54.82 (Subtracting the magnitudes and the angles)

I have then converted 166.84 - j54.82 back to polar form.

Magnitude = √166.84^2 - 54.82^2
= 157.57
Angle = arctan -54.82/166.84
= -18.19

Therefore my final result is 157.57∠-18.19. Which is not the same as the result from the example that was given.

When you did this:

"-68.55∠126.9 =
(68.55)(cos126.9) = -41.16.
(68.55)(sin126.9) = j54.82
Therefore = -41.16+j54.82"

You forgot to apply the leading minus sign which was at the very beginning.
The correct result is: +41.16-j54.82

You need to include parentheses where appropriate. Next you can either subtract (-41.16+j54.82), or you can add (+41.16-j54.82), which would give:

"Resulting in 208+j0 -(-41.16+j54.82).
Subtracting to me gives a result of: 249.16 - j54.82 (Subtracting the magnitudes and the angles)"

You are not subtracting magnitudes and angles. You are subtracting real parts and imaginary parts.

Converting to polar gives: 255.12 ∠-12.41

 

[PeroK]

Note that if $z = z_1 - z_2z_3$, then:
$$|z|^2 = r_1^2 + r_2^2r_3^2 - 2r_1r_2r_3\cos(\theta_1 - \theta_2 - \theta_3)$$

 

[Dcmagni]

When you did this:

"-68.55∠126.9 =
(68.55)(cos126.9) = -41.16.
(68.55)(sin126.9) = j54.82
Therefore = -41.16+j54.82"

You forgot to apply the leading minus sign which was at the very beginning.
The correct result is: +41.16-j54.82

You need to include parentheses where appropriate. Next you can either subtract (-41.16+j54.82), or you can add (+41.16-j54.82), which would give:

"Resulting in 208+j0 -(-41.16+j54.82).
Subtracting to me gives a result of: 249.16 - j54.82 (Subtracting the magnitudes and the angles)"

You are not subtracting magnitudes and angles. You are subtracting real parts and imaginary parts.

Converting to polar gives: 255.12 ∠-12.41

Ok thank you for your reply and feedback.

So the process I am using is correct which is a good start. Looking at what you have put and inputting the correct figures I have gotten.

208 + j0 + 41.16 - j54.82 gives 241.16 - j54.82 (As you have suggested above)

Converting to polar:
√241.16^2 + (-54.82^2)
= √58158.15 + (-3005.23)
= √55152.92
=234.85

arctan 54.82/241.16 = -12.81

= 234.85 ∠ -12.81

I just do not understand where I am going wrong.

 

[PeroK]

I just do not understand where I am going wrong.

Negative numbers square to positive numbers!

 

[PeroK]

I just do not understand where I am going wrong.

Alternatively, with $z = a+bj$, we have:
$$|z|^2 = a^2 + b^2 \ne a^2 - b^2$$

 

[Dcmagni]

Negative numbers square to positive numbers!

Even still.

Converting to polar:
√241.16^2 + (-54.82^2)
= √58158.15 + (+3005.23)
= √61163.38
=247.31

arctan 54.82/247.31 = -12.49

= 247.31 ∠ -12.49

I have just tried this after reading another article but it still doesn't seem to come up with the right answer.

 

[PeroK]

Even still.

Converting to polar:
√241.16^2 + (-54.82^2)
= √58158.15 + (+3005.23)
= √61163.38
=247.31

arctan 54.82/247.31 = -12.49

= 247.31 ∠ -12.49

I have just tried this after reading another article but it still doesn't seem to come up with the right answer.

You're just making basic mistakes like typing $241$ instead of $249$.

Why not put the calculation on a spreadsheet?

 

[PeroK]

r_1	theta_1	r_2	theta_2	r_3	theta_3		x	y	r	theta
208​	0​	2.5​	1.570796​	27.42​	0.644026​		249.1588​	-54.8184​	255.1179​	-12.4082​

 

[Dcmagni]

You are right and thank you for all your help. Heads jammed juggling this and work at the same time. Again, appreciate your help.

 

[The Electrician]

Ok thank you for your reply and feedback.

So the process I am using is correct which is a good start. Looking at what you have put and inputting the correct figures I have gotten.

208 + j0 + 41.16 - j54.82 gives 241.16 - j54.82 (As you have suggested above)

Converting to polar:
√241.16^2 + (-54.82^2)
= √58158.15 + (-3005.23)
= √55152.92
=234.85

arctan 54.82/241.16 = -12.81

= 234.85 ∠ -12.81

I just do not understand where I am going wrong.

You have a simple arithmetic error. Are you doing your arithmetic by hand? Don't you have a calculator or a calculator app for your phone?

You have:
"208 + j0 + 41.16 - j54.82 gives 241.16 - j54.82 (As you have suggested above)"

but it should be:

"208 + j0 + 41.16 - j54.82 gives 249.16 - j54.82 (As you have suggested above)"

Edit: Sorry, I just noticed that PeroK already pointed this out.

 

[JimJCW]

So far I have the understanding that subtraction is easier in polar form and multiplication in rectangular form, I need to transform into rectangular form.

I can see that the original question has already been addressed. Here I would like to comment on the statement mentioned in the above quote. I think, as demonstrated below, it should be,

The multiplication of two complex numbers is easier in polar form and addition/subtraction of two complex numbers is easier in rectangular form.​

Let’s first represent a complex number in polar form with (r, θ). We have

z2 = (2.5, 90)​

z3 = (27.42, 36.9)​

z2×z3 = (2.5*27.42, 90+36.9) = (68.55, 126.9)​

[See Eq. (3.11) of Complex variables.]​

Let’s next represent a complex number in rectangular form with (x, yi), where i = sqrt(-1).
Then

z1 = (208, 0i)​

z2×z3 = [68.55*cos(126.9), 68.55*sin(126.9) i] = (-41.16, 54.82i)​

z1 - z2×z3 = (208+41.16, 0i-54.82i) = (249.16, -54.82i)​

Convert it into polar form we have:

z1 - z2×z3 = (255.12, -12.41)​