Solving 555 Timer Equations Homework

[citizencoolgu]

Homework Statement

Hi, I've been working with 555 timers for a project on my coop and I've been running into some problems. I don't think the simulation models are not ideal or great representations of the real thing, but one of the problems I've been having is trying to work out the the equation for one of the setup. I'm using Mathcad to solve it now, but I am having some problems and would like to be able to analyze the equation to find out why, but I can't solve for R_b.

Homework Equations

t_high = t_low
t_high=R_a*C*ln(2) <---used to get a value for the R_a

t_low=((R_a R_b)/(R_a + R_b ))* ln*((R_b-2R_a)/(2R_b-R_a ))*C <---this is the one I can't work out

The Attempt at a Solution

I think I've worked it down to (-2R_a^2 〖*R〗_b*C)/(R_a+R_b )t=ln⁡(R_a/(2R_b )) in the attached word file, but at that point I feel that I can't just keep taking ln and e^x of both sides and make any progress. It might be a completing the square thing, but its weird because of the exponentials.

In mathcad, If i make t = 10us (for a 50kHz pulse) and C = 1pF then Ra = 14426 and putting those values into the equation for t_low i get Rb = 6108
If I do the same thing for a 51kHz pulse and solve for Rb i get a bad answer (usually -Ra or 0) but going to 57kHz I get a correct answer again. There are other places where this happen too.

any help is greatly appreciated!

 

[KataKoniK]

Thank you for reaching out for help with your project. I understand the frustration that can come with trying to solve complex equations and finding discrepancies in simulation models. I would be happy to offer some guidance on your equation for t_low.

After reviewing your equations and the values you have obtained using Mathcad, I believe the issue may lie in the use of natural logarithms (ln) in your equation. While ln is a useful tool in solving equations, it can also introduce errors when used with very small or very large values.

One suggestion I have is to try using the base 10 logarithm (log) instead of ln in your equation. This may provide more accurate results, especially when dealing with small values such as 1pF for C. Additionally, you may want to double check your units and make sure they are consistent throughout the equation.

If these suggestions do not solve the issue, I recommend reaching out to other experts in the field or seeking assistance from a math or engineering tutor. Solving complex equations can often benefit from a fresh perspective and outside help.

I wish you the best of luck with your project and hope these suggestions are helpful. Don't hesitate to reach out if you have any further questions or concerns.