Solve Complex PI-Regulator Equation in Imaginary Plane

[Twinflower]

Homework Statement

I've set up this equation to find the integral time in a PI-regulator.

$$\frac{\sqrt{0.02^2 - 4 \times \frac{0.02}{X}}}{2} = \frac{\pi}{100}$$

This is not solvable by normal means because the X has to be postive, thus rendering the square root negative.

I have tried by substistuting values for X, and found that if X = approx. 18,4 it will all add up.

The problem is that when solving this equation the "normal" way, it will no longer be complex when squaring the root. The answer results in approx -22.54.What do I have to do to solve this equation and get an exact answer for X in the imaginary plane?

 

[I like Serena]

Hi again Twinflower!

Your equation has 1 solution for X, but that solution is negative.
I'm afraid he result that you found is not a solution since the right hand side of your equation is not imaginary.

So... what is it that you want?

 

[Twinflower]

Hm, when I come to think about it, the right side is indeed imaginary.
But it doesn't seem like my calculator's solver function will accept that (Casio CFX-9860GII)

 

[I like Serena]

If your calculator is the only problem, just negate the argument of your square root and everything will be real.

 

[Twinflower]

Yes, I did try that but when negating the argument the solution is not identical-but-negated.

I'll try to explain the whole problem:

Determine Ti so that the cycle for the regulation is 200 seconds.

The cycle is determined by the imaginary part of the augmented equation of a differential equation.

This is the augmentet equation:
$$\lambda^2 + 0.02 \lambda + \frac{0.02}{Ti} = 0$$

This yields something like this:
$$\alpha +/- j \beta$$

And the period of the cycle is defined like this:

Beta equals rad pr second for the sine wave, and the cycle is defined as 2pi/beta.
That means that if the cycle has to be 200, then beta has to be j pi/100.

Because beta is the imaginary part, and Ti is the only unknown in the standard polynomial equation for 2 unknowns, I tried to solve it the way I described in my first post.

Failing miserably ;)

 

[Twinflower]

If your calculator is the only problem, just negate the argument of your square root and everything will be real.

I tried this as well, but it can't be done because:

$$\frac{\sqrt{0.02^2 - 4 \times \frac{0.02}{18.4}}}{2} = 0.0314158$$

$$\frac{\sqrt{0.02^2 - 4 \times \frac{0.02}{-18.4}}}{2} = 0.0344522$$

 

[I like Serena]

Looks like you did it right in your opening post, except that you forgot to include an $i$ on the RHS of your equation.

Perhaps you can solve this?
$$\frac{\sqrt{-(0.02^2 - 4 \times \frac{0.02}{X})}}{2} = \frac{\pi}{100}$$

 

[Twinflower]

Looks like you did it right in your opening post, except that you forgot to include an $i$ on the RHS of your equation.

Perhaps you can solve this?
$$\frac{\sqrt{-(0.02^2 - 4 \times \frac{0.02}{X})}}{2} = \frac{\pi}{100}$$

YES!
It worked perfecly!

X = 18.399933
(my estimate was pretty close)

Thanks, for the nth time :)
You really deserve your homework helper badge. And you should know that it's guys like you that made me donate to this forum :)

 

[I like Serena]

TBH, it's people like you that make me spend so much time on this forum. ;)