# RC circuit with pulse input voltage; finding the time constants

I've the following circuit:

Question: What are the time constants of the voltage at node $$\\text{Y}_2\$$ when the up part of the puls comes and when the down part of the puls comes?

MY WORK:

The voltage at node $$\\text{Y}_2\$$ can by using the following analysis:

1. The input voltage (starts at $$\t=0\$$ at $$\8\$$ V and is high for $$\\frac{1}{20}\$$ seconds and turn than back to $$\0\$$) is given by (in the s-domain using the Lapalce transform): $$\text{u}_\text{in}\left(\text{s}\right)=\int_0^\frac{1}{20}8e^{-\text{s}t}\space\text{d}t=8\cdot\frac{1-\exp\left(-\frac{\text{s}}{20}\right)}{\text{s}}\tag1$$
2. The relation between the voltage at node $$\\text{Y}_2\$$ and the input voltage is given by: $$\frac{\text{u}_{\text{Y}_2}\left(\text{s}\right)}{\text{u}_\text{in}\left(\text{s}\right)}=\frac{600+\frac{1}{10^{-6}\text{s}}}{600+600+\frac{1}{10^{-6}\text{s}}}=\frac{5000+3\text{s}}{5000+6\text{s}}\tag2$$

So, the voltage at node $$\\text{Y}_2\$$ is given by:

$$\text{u}_{\text{Y}_2}\left(\text{s}\right)=8\cdot\frac{1-\exp\left(-\frac{\text{s}}{20}\right)}{\text{s}}\cdot\frac{5000+3\text{s}}{5000+6\text{s}}\tag3$$

Plotting the function (in the time domain gives):

So, for finding the time constant ($$\\approx63\text{%}\$$ of the maximum value) we need to solve:

$$8\cdot\left(1-\frac{1}{e}\right)=\mathcal{L}_\text{s}^{-1}\left[8\cdot\frac{1-\exp\left(-\frac{\text{s}}{20}\right)}{\text{s}}\cdot\frac{5000+3\text{s}}{5000+6\text{s}}\right]_{\left(t\right)}\space\Longleftrightarrow\space$$ $$t\approx0.368223\space\text{ms}\tag4$$

Am I correct in my analysis or not? Because I also thought about using $$\\tau=\text{R}\cdot\text{C}\$$ so then we get $$\\tau=(600+600)\cdot10^{-6}=1.2\$$ miliseconds, is that right or is my previous analysis right? Because I got two different answers

• Just a note. The two R values aren't summed for your cross-check $\tau$.
– jonk
Nov 29 '18 at 19:25
• @jonk Okay, but what is my mistake there? Nov 29 '18 at 19:26
• What does $Y_2$ see looking back in? (Assuming you weren't doing all your math and were just trying to take a first-guess at it.)
– jonk
Nov 29 '18 at 19:27
• @jonk just one resistor of $600\space\Omega$, I think :) Nov 29 '18 at 19:28
• What about your ideal voltage source (pulse) as well as your capacitor? Would $Y_2$ only see one of those? Or would it see both? I haven't checked your math analysis, but I think you may have a reasonable answer given my own guess at about where it should be close to without doing the math. (Of course, I could be wrong. I'm just pointing out that my "off the cuff" computation came close to your result.)
– jonk
Nov 29 '18 at 19:31