unity feedback system

[logistic_guy]

here is the question

For the unity feedback system shown, where \(\displaystyle G(s) = \frac{K(s + 10)(s + 20)}{(s + 30)(s^2 - 10s + 100)}\), do the following:

(a) Sketch the root locus.
(b) Find the range of gain, \(\displaystyle K\), that makes the system stable.
(c) Find the value of \(\displaystyle K\) that yields a damping ratio of \(\displaystyle 0.707\) for the system's closed-loop dominant poles.
(d) Find the value of \(\displaystyle K\) that yields closed-loop critically damped dominant poles.




my attemb
\(\displaystyle (s + 30)(s^2 - 10s + 100) = 0\)
\(\displaystyle s^3 - 10s^2 + 100s + 30s^2 - 300s + 3000 = 0\)
\(\displaystyle s^3 + 20s^2 - 200s + 3000 = 0\)
i know this basic algebra. i forgothow to solve \(\displaystyle 3\) degree equation

 

[khansaheb]

here is the question

For the unity feedback system shown, where \(\displaystyle G(s) = \frac{K(s + 10)(s + 20)}{(s + 30)(s^2 - 10s + 100)}\), do the following:

(a) Sketch the root locus.
(b) Find the range of gain, \(\displaystyle K\), that makes the system stable.
(c) Find the value of \(\displaystyle K\) that yields a damping ratio of \(\displaystyle 0.707\) for the system's closed-loop dominant poles.
(d) Find the value of \(\displaystyle K\) that yields closed-loop critically damped dominant poles.

View attachment 38880


my attemb
\(\displaystyle (s + 30)(s^2 - 10s + 100) = 0\)
\(\displaystyle s^3 - 10s^2 + 100s + 30s^2 - 300s + 3000 = 0\)
\(\displaystyle s^3 + 20s^2 - 200s + 3000 = 0\)
i know this basic algebra. i forgothow to solve \(\displaystyle 3\) degree equation

How many roots does a cubic equation have?
Hint:

For this problem you do NOT need to solve the cubic equation - explicitly. Solving quadratic will provide you the "sought" after roots.

 

[logistic_guy]

thank

How many roots does a cubic equation have?

three

Solving quadratic will provide you the "sought" after roots.

i know how to solve quad i need cube. i can plag in this equation in wolframalfa and get the solution. the problem there's no wolframalfa in the test and calculater is forbiden. i need to practice to solve cubic quickly.

i remember there's theorem but it's waste time it use trial and error. if \(\displaystyle n\) is a root \(\displaystyle (x - n)\) is a factor
checking roots manually take time. it's not effecient method.

so what's the quick way to solve cubic equation by hand?

 

[Dr.Peterson]

so what's the quick way to solve cubic equation by hand?

Seriously? You expect everything to have a "quick way"? Cubics, in general, don't.

But in this case there is. You can do it quickly if you can factor it. And in this case, it was already partially factored! (They were kind enough to make this one that can be solved easily.)

my attempt
\(\displaystyle (s + 30)(s^2 - 10s + 100) = 0\)
\(\displaystyle s^3 - 10s^2 + 100s + 30s^2 - 300s + 3000 = 0\)
\(\displaystyle s^3 + 20s^2 - 200s + 3000 = 0\)
i know this basic algebra. i forgothow to solve \(\displaystyle 3\) degree equation

You skipped right past the step you needed. (And you've been told that you need to solve only a quadratic. Did you look for one?)

 

[logistic_guy]

Seriously? You expect everything to have a "quick way"? Cubics, in general, don't.

(And you've been told that you need to solve only a quadratic. Did you look for one?)

khan tell me to solve quadratic. that's i'm told you mean?

You skipped right past the step you needed.

i can't think of a forth step. it's just simplification. i just calculated the brackets multiplication in one step to save time

 

[khansaheb]

......calculated the brackets multiplication......

You do not need to do that - think!! and think hard!!

 

[khansaheb]

(s+30)(s² −10s+100)=0

One solution is s = -30

Then we have left

(s² −10s+100)=0...................Quadratic