# Complex numbers in ac circuit

#### Blackfriars

Joined Nov 3, 2020
9
Hi can anyone tell me how to find missing resistance in a parallel ac circuit using complex numbers
I have found the resistor using the current triangle but i have to use complex to find it i was told

#### Papabravo

Joined Feb 24, 2006
16,785
If you have complex impedances in parallel then
$\frac{1}{Z_T}=\frac{1}{Z_1}+\frac{1}{Z_2}+\frac{1}{Z_3}$

#### Blackfriars

Joined Nov 3, 2020
9
So i have the voltage current and inductor value but no value for resistor so i am finding it hard to find resistor value using complex numbers

#### Papabravo

Joined Feb 24, 2006
16,785
So the voltage divided by the current will be the complex impedance. That's Ohms Law.
The complex impedance will consist of the resistance which is the real part and an inductive reactance which is the imaginary part.

#### MrAl

Joined Jun 17, 2014
8,475
Hi can anyone tell me how to find missing resistance in a parallel ac circuit using complex numbers
I have found the resistor using the current triangle but i have to use complex to find it i was told
Hi,

You really should show the circuit otherwise we are just guessing as to what configuration you are working with. We dont even know yet what the goal is. What is the resistor for.

I can give an example of an LCR circuit.
First you calculate the total impedance with known L and known C and unknown R. That gives you something compelx like:
(a+b*j)/(aa+bb*j)

Now we know we need a form like:
c+d*j
so we sort of equate those two:
(a+b*j)/(aa+bb*j)=c+d*j

We also know that we can convert that left hand side into a single complex number:
A+B*j
and so we end up with:
A+B*j=c=d*j

and now we just equate the real and imaginary parts separately:
c=A
d=B

So really it is just a matter of knowing how to calculate impedances in parallel, and then reducing that result into a form with just one real part and one imaginary part like c+d*j.

So in this case we do this:
1. Compute Z's in parallel with unknown R.
2. Reduce into c+d*j

then after that you can solve for R if you know what the total impedance is supposed to be.

You should elaborate your question though so we know exactly what you want to do.
A schematic helps the most.

#### Blackfriars

Joined Nov 3, 2020
9
Hi this is the circuit i am trying to solve for resistance but it has to be done by using complex numbers i do not know where to start
Thanks

#### Blackfriars

Joined Nov 3, 2020
9
This is a calculation i have done so far but is not acceptable as it does not use complex numbers i have been told
Thanks

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#### The Electrician

Joined Oct 9, 2007
2,848
This is a calculation i have done so far but is not acceptable as it does not use complex numbers i have been told
Thanks
This is another problem with Vs =500 V and L =0.318 H
The problem you're asking for help on has Vs = 310 V and L = 0.141 H

Can you express the impedance of the unknown R in parallel with 0.141 H as a complex number?

#### Blackfriars

Joined Nov 3, 2020
9
Hi the numbers do not really matter
It is how do i get the resistance value for this type of circuit using complex numbers
Thanks

#### Papabravo

Joined Feb 24, 2006
16,785
The key to this problem is under standing how to compute the impedance of R || L
$Z_R= R+j0\; \text and\; Z_L=0+ j\omega L$
Now refer back to post #2 and compute the total impedance of those two components

#### Blackfriars

Joined Nov 3, 2020
9
Sorry how do you compute Zr when you do not have a value for the resistor
Thanks

#### Papabravo

Joined Feb 24, 2006
16,785
Sorry how do you compute Zr when you do not have a value for the resistor
Thanks
The value of R is the unknown. You must use Ohms Law!
You know the voltage and you know the current so you compute the total impedance.
Then you set the numerical value of the total impedance equal to the expression you get from the parallel combination of R and L

I do not think you need to worry about complex numbers because the voltage and current do not have a specified phase relationship to each other. I thin this mean you can consider them to be in phase.
My spreadsheet result is R= 43.3940705207381 the same to about 4 significant figures

Last edited:

#### Blackfriars

Joined Nov 3, 2020
9
Can you have a look at this calculation this is for 500v with 10amps with inductor 0.318 H
Thanks

#### The Electrician

Joined Oct 9, 2007
2,848
This looks like a calculation of the parallel combination of the 0.318 H inductor with a 50 ohm resistor. Where does the 50 come from?

But wherever it comes from the calculation is not correct. 39.98 is the real part of the computation but there is an imaginary part of 20i.

For the current you need to use the magnitude of the impedance, not just the complex value of the impedance. The magnitude is the square root of the sum of the squares of the real and imaginary parts.

#### The Electrician

Joined Oct 9, 2007
2,848
If the impedance of the unknown resistor is Zr = R + j0 and the impedance of the inductor is ZL = 0 + j w L, what would be the result of plugging those expressions into the product over the sum formula for the parallel combination of impedances? After you get the result as a complex number, you must take the magnitude of that result to use in the computation of the current from Vs.

#### Blackfriars

Joined Nov 3, 2020
9
Hi but i have been goven total current it is 10amps it is resistance i need
I hope you get what i mean
Thanks

#### The Electrician

Joined Oct 9, 2007
2,848
I am aware of what your goal is. If you want to do this with complex numbers, you must derive an expression for the parallel impedance of the unknown resistor R and the impedance of the inductor, j w L. Once you have the expression for the impedance of the parallel combination of R and L, call it Zp. Next you must calculate the magnitude of Zp, call it |Zp|.

Then you can write an expression for the current from Vs. That expression will be Vs / |Zp|. That expression will be somewhat complicated. Set the expression equal to the current from Vs which is 10 amps and you will have an equation which you can solve for R.

But, first, use the product over the sum formula and derive a complex number for the impedances R + j0 and 0 + j w L in parallel.

#### The Electrician

Joined Oct 9, 2007
2,848
I do not think you need to worry about complex numbers
He says in post #1: " how to find missing resistance in a parallel ac circuit using complex numbers " and " i have to use complex to find it i was told "

#### Blackfriars

Joined Nov 3, 2020
9
Hi i solved the circuit with 500 v 0.318 H and 10 A and i got 57 ohms
Using the current triangle
But i have been told by my lecturer to solve it with using complex numbers not the current triangle
Thanks

#### The Electrician

Joined Oct 9, 2007
2,848
Hi i solved the circuit with 500 v 0.318 H and 10 A and i got 57 ohms
Using the current triangle
But i have been told by my lecturer to solve it with using complex numbers not the current triangle
Thanks
Have you read post #17? You need to actually do what I've explained if you want to solve the problem with complex numbers.