Complex numbers rectangular form

[LDC1972]

Homework Statement

Given the equivalent impedance of a circuit can be calculated by the expression:

Z = Z1 X Z2 / Z1 + Z2

If Z1 = 4 + j10 and Z2 = 12 - j3, calculate the impedance Z in both rectangular and polar form.

Homework Equations

Multiplication and division of complex numbers.

The Attempt at a Solution

I want to solve the rectangular first. As I want to totally understand how this is done. My attempt so far:

Z1 + Z2 = 4 + j10 + 12 + j3
= 4 + 12 + J10 - J3
Z1 + Z2 = 16 + j7

Z1 x Z2 = 4 + j10 x 12 - j3
= (4 x 12) + (12 x j10) + (4 x -j3) + (j10 x j3)
= 48 + j120 - j12 - j^2 30

Since j^2 = -1
Then
Z1 x Z2 = 48 + j108 - (-1) 30
= 48 + j108 +30
Z1 x Z2 = 78 + j108

Z = 78 + j108 multiply by conjugate 16 - j7
----------- -------
16 + j7 16 - j7

Z = -1248 - j1728 - j546 - j^2 756
----------------------------------
16^2 + 7^2

-1248 - 756 = 2004
j1728 - j546 = j1182

Z = -2004 - j1182
---------------
305

-2004 / 305 = -6.570491803
j1182 / 305 = 3.875409836

Z = 6.5705 ± j3.8754

 

[LDC1972]

Looking at this again I think the end result should be:

Z = -6.5705 + j3.8754

??

 

[Mark44]

Homework Statement

Given the equivalent impedance of a circuit can be calculated by the expression:

Z = Z1 X Z2 / Z1 + Z2

I'm sure you really mean this:
Z = (Z₁Z₂)/(Z₁ + Z₂)

Knowledgeable folks would read the right side you wrote as
$$Z_1 X (\frac{Z_2}{Z_1}) + Z_2$$

When you write fractions with a sum in the numerator or denominator, USE PARENTHESES!

If Z1 = 4 + j10 and Z2 = 12 - j3, calculate the impedance Z in both rectangular and polar form.

Homework Equations

Multiplication and division of complex numbers.

The Attempt at a Solution

I want to solve the rectangular first. As I want to totally understand how this is done. My attempt so far:

Z1 + Z2 = 4 + j10 + 12 + j3
= 4 + 12 + J10 - J3
Z1 + Z2 = 16 + j7

So far, so good.

Z1 x Z2 = 4 + j10 x 12 - j3
= (4 x 12) + (12 x j10) + (4 x -j3) + (j10 x j3)
= 48 + j120 - j12 - j^2 30

Since j^2 = -1
Then
Z1 x Z2 = 48 + j108 - (-1) 30
= 48 + j108 +30
Z1 x Z2 = 78 + j108

Correct here as well.

Z = 78 + j108 multiply by conjugate 16 - j7
----------- -------
16 + j7 16 - j7

This is the right approach.

Z = -1248 - j1728 - j546 - j^2 756
----------------------------------
16^2 + 7^2

Your arithmetic is off here. 78 * 16 should be a positive number and 16 * 108j should be a positive number times j.

-1248 - 756 = 2004
j1728 - j546 = j1182

Z = -2004 - j1182
---------------
305

-2004 / 305 = -6.570491803
j1182 / 305 = 3.875409836

Z = 6.5705 ± j3.8754

There shouldn't be ±. It's one or the other.

Looking at this again I think the end result should be:

Z = -6.5705 + j3.8754

??

 

[LDC1972]

Thanks, I am very close then!
Need to go through the signs again and see what I come out with.
Then do polar and see if they correspond I guess?

 

[LDC1972]

Thanks again, nights sleep and went through it again quickly.
Now have Z = 6.5705 + j3.8754

I think this is right?

Will confirm in polar form today!

 

[LDC1972]

Just did the much simpler polar calculation and got the exact result as above :-)

Thanks Mentor for your pointing out my sign errors. I must of been tired!

 

[LDC1972]

Just did the much simpler polar calculation and got the exact result as above ( after converting to rectangular) :-)

Thanks Mentor for your pointing out my sign errors. I must of been tired!

Just did the much simpler polar calculation and got the exact result as above ( after converting to rectangular)

 

[KEEPitSIMPLES]

great thread, thought the signs were off too. were filling it with negatives and it was confusing me as I had a positive result. some other examples I have seen are finding 78*(-j7) as a positive and was throwing all my basic ideas off. glad you got to it in the end and put in the work. nice!

 

[JT1996]

I have gone through the rectangular form but I am not entirely sure how to find the polar form. Do you start the question again but transfer the initial complex numbers to polar form and work through it again or do you just transform the answer to polar form please?

 

[gneill]

I have gone through the rectangular form but I am not entirely sure how to find the polar form. Do you start the question again but transfer the initial complex numbers to polar form and work through it again or do you just transform the answer to polar form please?

You could do it either way. However, if you've already got the solution in one form (rectangular) then it's much easier to simply convert that to the other form (polar) rather than redo the same calculations from start.

 

[JT1996]

You could do it either way. However, if you've already got the solution in one form (rectangular) then it's much easier to simply convert that to the other form (polar) rather than redo the same calculations from start.

Thank you very much. I got 7.828/_30.532degrees. Is this right please?

 

[gneill]

Thank you very much. I got 7.828/_30.532degrees. Is this right please?

You'll have to show details of your calculations. We don't simply confirm homework answers here without seeing the work behind it.