# Thread: More Complex Numbers

1. ## More Complex Numbers

I am still working on these complex numbers. This one is giving me trouble:

Given the following:

$\displaystyle z_{1}=R_{1}+R+jwL$

$\displaystyle z_{2}=R_{2}$

$\displaystyle z_{3}=\frac{1}{jwC_{3}}$

$\displaystyle z_{4}=R_{4}+\frac{1}{jwC_{4}}$

Express R and L in terms of the constants R1, R2, C3 and C4.

We are also told that $\displaystyle z_{1}z_{3}=z_{2}z_{4}$

So this last bit is obviously the clue to solving the whole thing. I worked it out:

$\displaystyle \frac{R_{1}+R+jwL}{jwC_{3}}=R_{2}(R_{4}+\frac{1}{j wC_{4}})$

I did lots of algebraic manipulation after that, but nothing came of it. Would somebody please give me a hand?

(I may be crap at maths, but I'm getting good at LaTeX!)

Regards,

Evanator

2. Originally Posted by evanator

$\displaystyle \frac{R_{1}+R+jwL}{jwC_{3}}=R_{2}(R_{4}+\frac{1}{j wC_{4}})$
Given this is correct, I haven't actually checked!

To solve for $\displaystyle R$:

1. Multiply both sides by $\displaystyle jwC_3$

2. Take $\displaystyle jwL$ from both sides

3. Take $\displaystyle R_1$ from both sides

To solve for $\displaystyle L$:

1. Multiply both sides by $\displaystyle jwC_3$

2. Take $\displaystyle R_1$ from both sides

3. Take $\displaystyle R$ from both sides

d. Divide both sides by $\displaystyle jw$

Post what you get...

3. Thanks, man. It doesn't look right, but here is what I got:

$\displaystyle R_{1}+R+jwL=jwC_{3}(R_{2}R_{4}+\frac{R_{2}}{jwC_{4 }})$

$\displaystyle =jwC_{3}R_{2}R_{4}+\frac{jwC_{3}R_{2}}{jwC_{4}}$

$\displaystyle =jwC_{3}R_{2}R_{4}+\frac{C_{3}R_{2}}{C_{4}}$

$\displaystyle R=jwC_{3}R_{2}R_{4}+\frac{C_{3}R_{2}}{C_{4}}-jwL-R_{1}$

And now L:

$\displaystyle R_{1}+R+jwL=jwC_{3}R_{2}R_{4}+\frac{C_{3}R_{2}}{C_ {4}}$

$\displaystyle jwL=jwC_{3}R_{2}R_{4}+\frac{C_{3}R_{2}}{C_{4}}-R_{1}-R$

$\displaystyle L=C_{3}R_{2}R_{4}+\frac{C_{3}R_{2}}{jwC_{4}}+\frac {R_{1}}{jw}+\frac{R}{jw}$

It doesn't feel like that's what I'm supposed to do. I still have the imaginary parts hanging around for one thing.

4. Originally Posted by evanator
I still have the imaginary parts hanging around for one thing.
Sometimes these guys will cancel out, have you tried this?

It is also preferred that you have your answers in the form $\displaystyle z_i = a+bj$

Therefore you may have to pull those imaginary numbers from the demoninator by multiplying some complex conjugates.

5. Set the imaginary and real parts of the equation equal to each other and see what you get. What kind of variable is w? So take your resulting equation from $\displaystyle Z_1Z_3=Z_2Z_4$ multiply both sides by $\displaystyle jwC_3C_4$ and equate coefficients of j and then equate coefficients of the real part.

6. I have to admit to being stumped by this. I don't normally like doing this, but I looked at the answers in the back of the book and we are tantalizingly close.

It says that

$\displaystyle R=\frac{R_{2}C_{3}-R_{1}C_{4}}{C_{4}}$

and

$\displaystyle L=R_{2}R_{4}C_{3}$

So the real parts are right (pretty much).

7. So j and w are imaginary.

8. Nearly there. Just typing it up.

9. $\displaystyle jwC_{3}C_{4}(\frac{R_{1}+R+jwL}{jwC_{3}})=jwC_{3}C _{4}(R_{2}R_{4}+\frac{R_{2}}{jwC_{4}})$

$\displaystyle RC_{4}+R_{1}C_{4}+jw(LC_{4})=R_{2}C_{3}+jw(C_{3}C_ {4}R_{2}R_{4})$

Taking the real parts:

$\displaystyle RC_{4}+R_{1}C_{4}=R_{2}C_{3}$

$\displaystyle RC_{4}=R_{2}C_{3}-R_{1}C_{4}$

$\displaystyle R=\frac{R_{2}C_{3}-R_{1}C_{4}}{C_{4}}$

and the imaginary parts (coefficients of jw):

$\displaystyle LC_{4}=C_{3}C_{4}R_{2}R_{4}$

$\displaystyle L=C_{3}R_{2}R_{4}$

It felt good to get it in the end, even though I had lots of help. Thanks a lot, Krahl and Pickslides.