# Complex numbers rectangular form

1. Jul 9, 2013

### LDC1972

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

Multiplication and division of complex numbers.

3. 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

2. Jul 9, 2013

### LDC1972

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

Z = -6.5705 + j3.8754

??

3. Jul 9, 2013

### Staff: Mentor

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!!!

So far, so good.
Correct here as well.
This is the right approach.
Your arithmetic is off here. 78 * 16 should be a positive number and 16 * 108j should be a positive number times j.
There shouldn't be ±. It's one or the other.

4. Jul 9, 2013

### 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?

5. Jul 9, 2013

### 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!

6. Jul 10, 2013

### 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!!

7. Jul 10, 2013

### LDC1972

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

8. Apr 1, 2017

### 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!