Homework Help: Determine currents Ia, Ib and Ic

1. Oct 16, 2013

agata78

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

Determine currents Ia, Ib & Ic in the network?

2. Relevant equations

FIRST EMF

RT = R1 + (R2 x R3) / (R2 + R3)

RT = 47 + (j100 x ( -j75)) / (j100 + (-j75))

RT = 47+ 300j (ohms)

Ohms Law then states that:

I1 = E1 / RT

I1 = 12∠0º / 83-84.48j

I1 = (996+1013.76j) / (14025.87)

Using the Current Division Laws leaves:

I2 = I1 x ( R3 ) / (R2 + R3)

I2 = ( ((996+ 1013.76j) / 14025.87) x ( -j75 )) / (j100 + -j75)

I2 = ( (2988-3041.28j) / 14025.87) ∠0º

I3 = I1 x ( R2 ) / (R2 + R3)

I3 = I1 x ( j100 ) / (j100 + -j75)

I3 = ((3984+4055.04j)/ 14025) ∠0º

SECOND EMF

RT = R3 + (R1 x R2) / (R1 + R2)

RT = - j75 + ( 47 x j100 ) / ( 47 + j100 )

RT = ? Ω

Ohms Law then states that:

I4 = E2 / RT

I4 = 10∠0º / ?

I4 = ?

Using the Current Division Laws leaves:

I5 = I4 x ( R2 ) / (R1 + R2)

I5 = ? x ( j100 ) / ( 47 + j100 )

I5 = ? x ?

I5 = ? Ω

I6 = I4 - I5

I6 = ? _ ?

I6 = ? Ω

So:

IA = I1 - I5

IA = ? - ?

IA = ?∠ ?º Ω

IB = I3 - I4

IB = ? - ?

IB = ?∠ ?º Ω

IC = I2 + I6

IC = ? + ?

IC = ?∠ ?º Ω

Can someone please let me know if the information provided is answering the question correctly? Thanks!

In the diagram R3 is j75. However, it is suppose to read -j75.

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Last edited: Oct 16, 2013
2. Oct 16, 2013

Staff: Mentor

Wow, looks like a lot of work taking the superposition approach. Why not write a pair of mesh equations? Two of the currents will pop out as the mesh currents, and the third is just the sum of the other two.

3. Oct 16, 2013

agata78

I spent all day calculating all theses, and everytime i have a different figure and they dont look good. I have a simillar example but with much easier numbers ( no j).
A pair of mesh equations?

4. Oct 16, 2013

agata78

Ok i will have a go!

-12 +100j (I1 + I2) + 47I1= 0 and -100j (I1 + I2) +10- (-75j)I2 = 0

Am i right?

5. Oct 16, 2013

agata78

The answer for I2 = 0.4- 4jI1 and for I1= 147jI1+400I1=12-40j.

Which looks rubish and very not correct!
I need to finish this asap my time is running out!

6. Oct 16, 2013

Staff: Mentor

I'd choose my mesh currents to correspond to the currents Ia and Ib . Then "walk" both loops in the directions of the mesh currents and sum the potential changes:

Loop1:
+12 - Ia47 - (Ia + Ib)j100 = 0

Loop2:
+10 - Ib(-j75) - (Ia + Ib)j100 = 0

arriving at what appears to be the same equations that you found. When I solve for Ia and Ib I get reasonable values.

If you think carrying through all the complex manipulations while you solve the equations might be tripping you up, try leaving the component values as symbols and solve the equations symbolically. Plug in complex values in the final expressions for Ia and Ib and then reduce.

7. Oct 16, 2013

agata78

12- Ia47 - (Iaj100+Ibj100) =0
12- Ia47- Iaj100- Ibj100=0
Ia47 - Iaj100= Ibj100-12
Ia(47-j100)=Ibj100-12

Ia= (Ibj100-12) / (47-j100)

Is it right so far?

8. Oct 16, 2013

Staff: Mentor

9. Oct 16, 2013

agata78

And for loop 2:

10-Ib (-75j)-(Ia+Ib)j100=0
10+Ibj75-Ia100j-Ibj100=0
Ibj75-Ibj100-Ia100j=-10
Ibj(-25)- Ia100j=-10
Ibj(-25)-100j((Ibj100-12)/ (-47-j100)) =-10

Ibj(-25) -((10000Ibj2 - 1200j)/ (-47-j100))= -10

Ibj(-25) - ( 10000(-1) Ib -1200j ) / (-47-j100) =-10

Ibj(-25) +(( 10000 Ib + 1200j) / (-47-j100)) = -10

And what next?

10. Oct 16, 2013

agata78

ooops, thank you!

11. Oct 16, 2013

agata78

235bj +1500b= (-11280j + 24000-2 ) / (47-100j)

b= ( (-11280j + 24000-2 ) / (47-100j) ) / (235j +1500).

Its after midnight will jump to bed and do some checking on my algebra.
thank you

12. Oct 16, 2013

Staff: Mentor

Continue banging away until you've solved for Ib. Then use that Ib to go back to a previous expression for Ia and solve for Ia.

It's a lot of work carrying out all these complex operations for every manipulation of the equations. I would suggest solving symbolically first, then plug in the values for the impedances in the final expressions. Then it's just two expressions to evaluate with complex numbers, rather than a whole series of complex expansions and shuffles over and over.

I'll think about whether there's another approach that would be less time consuming.

13. Oct 16, 2013

Staff: Mentor

Okay, I'm thinking that a nodal analysis approach might be a bit more streamlined. If you take the top middle node and call its potential Vx, and if you leave the complex impedances as R2 and R3, then the node equation becomes:

$\frac{Vx - 12}{47} + \frac{Vx}{R2} + \frac{Vx - 10}{R3} = 0~~~~~$ whence:

$Vx = \frac{R2(12 R3 + 470)}{R2 R3 + 47 (R2 + R3)}$

Plug in values for R2 and R3 and solve for Vx (the only really messy expression to deal with), then:

$Ia = \frac{12 - Vx}{47}$

$Ib = \frac{10 - Vx}{R3}$

$Ic = Ia + Ib$

14. Oct 16, 2013

The Electrician

You are torturing yourself!

Once you have learned how to do calculations with complex arithmetic, and have practiced for a while, it's time to move on to modern methods using a computer or calculator. As you can see, it's easy to make mistakes when doing such an involved series of complex calculations by hand.

Your school should have Matlab or Mathcad or something similar on the school network, which can do these sorts of calculations.

Also, you can find applications for free on the web that can do complex arithmetic. You can even find complex matrix solvers.

If you plan to continue your studies in a field that requires these sorts of calculations, you might consider buying a handheld calculator that can do complex arithmetic.

Hewlett-Packard has recently introduced a new calculator, the HP Prime, and this introduction has led to a discounting in price for the older calculators. The older HP50G can do complex arithmetic, including complex matrix calculations, and can be had for under \$100. This is a very powerful calculator and you would be able to use it for your entire academic career.

Edit: Here's an online calculator for complex numbers using the RPN format of HP calculators:

http://www.alpertron.com.ar/CALC.HTM

Last edited: Oct 16, 2013
15. Oct 17, 2013

agata78

Ok,

Vx= 12.67 + 4.28j

Last edited: Oct 17, 2013
16. Oct 17, 2013

agata78

Thanks, i checked it out, looks ok but quite complicated for me. but for sure i will try to purchase software to help me with calculation asap!

17. Oct 17, 2013

agata78

Ia = -0.014-0.09j
Ib=( -2.67-4.28j) / -75j

I didnt calculate Ic yet cause a amd be doesnt look very good!
Can you check it out for me?

Last edited: Oct 17, 2013
18. Oct 17, 2013

Staff: Mentor

Doesn't look right. Calculate the numerator and denominator of the expression separately and show your values. Maybe we can pin down where the slip occurred.

19. Oct 17, 2013

agata78

Vx= 12.67 + 4.28j
Ia = -0.014-0.09j
Ib=( -2.67-4.28j) / -75j

I found a mistake, i changed it. Is it better now? if not i will post my calculations :-(

20. Oct 17, 2013

Staff: Mentor

The numbers look good. Might as well express Ib in decimal form too.

21. Oct 17, 2013

agata78

ok,

100j(12 (-75) +470 ) / (100j(-75j) +47 (100j+(-75j))
100j(-900j +470) / -7500j2 +47 (100j-75j)
-90000j2 + 47000j / 7500 +(47 x 25j)
90000+47000j / 7500 + 1175j
(90000+47000j)(7500-1175j) / (7500+1175j ) (7500-1175j)
(675000000-105750000j+352500000j-55225000j2) / ( 56250000-1380625j2)
(675000000+55225000-246750000j ) / 56250000+1380625
730225000+246750000j / 57630625
12.67+ 4.28j

22. Oct 17, 2013

agata78

Ib= -0.17-0.237j
Ic= Ia+ IB
Ic= 4.3156j+ 12.727

23. Oct 17, 2013

agata78

Is there a way to check the answer is correct?

24. Oct 17, 2013

Staff: Mentor

Something went wrong when you finished off Ib. Check the math. Try out the online calculator!

25. Oct 17, 2013

agata78

I checked wolframalpha. it says:

0.570667+ (0.0356 / j)

Its even worse!