Hi Michael, I'm back in this notebook for teaching purpose, and I really want to thank you for the notebook you shared.
Thank you Federico Il giorno sabato 4 marzo 2023 alle 08:01:38 UTC+1 m...@duke.edu ha scritto: > I know this is almost a year old, but is this notebook helpful to you: > > https://colab.research.google.com/drive/1p5WP-twoxD5HKq9gziNWKqSi8Xo_cotg?usp=sharing > -Michael > > On Tuesday, June 28, 2022 at 4:52:12 PM UTC-4 federic...@gmail.com wrote: > >> Thank you Oscar, >> onestly I don't get the same resoult of yours, because I get 3 result , >> not just one like you: >> one solution for for τ/R =−i/(100πRi) , one solution for τ/R =i/(100πRi) >> and another is otherwise >> Anyway, the otherwise seems to be possible to be simplified and become >> like your solution, so thank you very much for the help. >> It should be nice to avoid the 3 solution and keep just the otherwise one >> and get it simplified, but I just want to thank you for your help. >> >> >> Il giorno martedì 28 giugno 2022 alle 12:05:54 UTC+2 Oscar ha scritto: >> >>> On Tuesday, 28 June 2022 at 08:03:05 UTC+1 federic...@gmail.com wrote: >>> >>>> >>>> Hi Aaron, thank you for the reply. >>>> >>>> The code I have in my notebook (I'm using google colab) is this now (at >>>> the end of the post), but using w is not working anymore. >>>> Anyway, if I move back to use the line w = 2 * sp.pi * 50 I still get >>>> a unreadble result. >>>> The result should be something like the nice wikipedia result I found >>>> there: >>>> >>>> https://it.wikipedia.org/wiki/Circuito_RC#Risposta_in_frequenza_del_circuito_RC >>>> Is there a way to get the a readble result? (something like the >>>> following latex) >>>> {\displaystyle v_{C}(t)=v_{C}(0)e^{-{\frac {t}{\tau }}}+K\cos(\omega >>>> t+\theta )} >>>> The wikipedia result has tau=R*C, theta is an angle displacement, and >>>> K don't know what is. >>>> >>> >>> The result that I see is similar: >>> >>> In [13]: diffeq >>> Out[13]: >>> d vo⋅cos(100⋅π⋅t) vc(t) >>> ──(vc(t)) - ─────────────── + ───── >>> dt C⋅R C⋅R >>> >>> In [12]: res >>> Out[12]: >>> -t >>> >>> ─── >>> >>> C⋅R >>> >>> 100⋅π⋅C⋅R⋅vo⋅sin(100⋅π⋅t) vo⋅cos(100⋅π⋅t) vo⋅ℯ >>> >>> vc(t) = ───────────────────────── + ────────────────── - >>> ────────────────── >>> 2 2 2 2 2 2 2 2 2 >>> >>> 10000⋅π ⋅C ⋅R + 1 10000⋅π ⋅C ⋅R + 1 10000⋅π ⋅C ⋅R >>> + 1 >>> >>> The main difference is that the constant K is given explicitly here in >>> terms of C, R and v0. If you want to substitute for tau and K you can do >>> something like: >>> >>> In [19]: tau, K = symbols('tau, K') >>> >>> In [20]: res.subs(C, tau/R).subs(vo, K*(1 + 10000*pi**2*tau**2)) >>> Out[20]: >>> -t >>> ─── >>> τ >>> vc(t) = 100⋅π⋅K⋅τ⋅sin(100⋅π⋅t) + K⋅cos(100⋅π⋅t) - K⋅ℯ >>> >>> You can see now something more similar. The difference is that you have >>> sin and cos rather than cos with a phase shift theta but those are >>> equivalent under trigonometric identities. Another difference is the >>> constant K on the exponential term but the wikipedia solution is incorrect >>> there: >>> >>> >>>> In the italian wikipedia there's this piece (I try to translate) that I >>>> would like to reproduce in a notebook: >>>> >>>> Let's see how does the RC circuit works with a sine wave. We can use >>>> voltage Kirchhoff law: >>>> {\displaystyle V_{0}\cos(\omega t)=R\cdot i(t)+v_{C}(t)} >>>> >>>> we can rewrite the equation like: >>>> {\displaystyle V_{0}\cos(\omega t)=RC{\frac {dv_{C}(t)}{dt}}+v_{C}(t)} >>>> >>>> and then solve the differential equation with constant coefficients >>>> with a known therm: >>>> {\displaystyle {\frac {dv_{C}(t)}{dt}}+{\frac {1}{\tau >>>> }}v_{C}(t)={\frac {V_{0}\cos(\omega t)}{\tau }}} >>>> >>>> where \tau =RC is still the time constant of the circuit. >>>> The general solution come from the sum of the associated homogeneous >>>> solution: >>>> >>>> >>>> v_{C}(t)=v_{C}(0)e^{-{\frac {t}{\tau }}} >>>> >>> It is incorrect to use v_C(0) here. There should be a constant there but >>> it is not generally equal to the initial condition. >>> >>>> and a particolar solution: >>>> {\displaystyle K\cos(\omega t+\theta )\ } >>>> >>>> where K is a constant. So: >>>> >>>> >>>> {\displaystyle v_{C}(t)=v_{C}(0)e^{-{\frac {t}{\tau }}}+K\cos(\omega >>>> t+\theta )} >>>> >>> You can see that this is incorrect if you just substitute t=0. >>> >>> -- >>> Oscar >>> >> -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/6e3dd9ff-d945-418f-8a56-9b004108027an%40googlegroups.com.
