Question about this Capacitor voltage integral equation

AI Thread Summary
The discussion focuses on deriving the capacitor voltage equation from the integral of current. The original equation, v(t) = 1/C ∫t-∞ i(τ) dτ, is transformed by recognizing that the initial voltage v(t0) represents the history of current flow. By splitting the integral into two parts, the relationship v(t) = v(t0) + 1/C ∫tt0 i(τ) dτ is established. This shows how the voltage at time t depends on both the initial condition and the current flowing from t0 to t. The explanation clarifies the connection between past current and present voltage in circuit analysis.
JMFernandez
Messages
2
Reaction score
2
Hi.

I don´t know if this question should be in the maths forum, but as it´s related with circuit analysis, I will post it here. I just would like to know how you get:

v(t) = 1/C ∫tt0 i(τ) dτ + v(t0)

From:

v(t)=1/C ∫t-∞ i(τ) dτ

I just know the basics of calculus and I don´t know how to operate the second equation to get the first one.

Thank you in advance.
 
Engineering news on Phys.org
JMFernandez said:
Hi.

I don´t know if this question should be in the maths forum, but as it´s related with circuit analysis, I will post it here. I just would like to know how you get:

v(t) = 1/C ∫tt0 i(τ) dτ + v(t0)

From:

v(t)=1/C ∫t-∞ i(τ) dτ

I just know the basics of calculus and I don´t know how to operate the second equation to get the first one.

Thank you in advance.
The equation describes how the cap voltage changes as current flows through it. Fortunately, all of the history of past current flow(s) is represented by the voltage at any time. That is what the initial voltage ##v(t_o)## is. Since that doesn't depend on the variable ##t##, we can just call it a constant value, the "initial condition" of the capacitor. So,

$$v(t) = \frac{1}{C} \int_{-∞}^{t} i(\tau) \, d\tau = \frac{1}{C} \int_{-∞}^{t_o} i(\tau) \, d\tau + \frac{1}{C} \int_{t_o}^{t} i(\tau) \, d\tau \equiv v(t_o) + \frac{1}{C} \int_{t_o}^{t} i(\tau) \, d\tau$$
 
Last edited:
Thank you. Very clear and concise explanation!!
:smile::smile:
 
  • Like
Likes berkeman and DaveE
I have recently moved into a new (rather ancient) house and had a few trips of my Residual Current breaker. I dug out my old Socket tester which tell me the three pins are correct. But then the Red warning light tells me my socket(s) fail the loop test. I never had this before but my last house had an overhead supply with no Earth from the company. The tester said "get this checked" and the man said the (high but not ridiculous) earth resistance was acceptable. I stuck a new copper earth...
Thread 'Electromagnet magnetic field issue'
Hi Guys We are a bunch a mechanical engineers trying to build a simple electromagnet. Our design is based on a very similar magnet. However, our version is about 10 times less magnetic and we are wondering why. Our coil has exactly same length, same number of layers and turns. What is possibly wrong? PIN and bracket are made of iron and are in electrical contact, exactly like the reference design. Any help will be appreciated. Thanks. edit: even same wire diameter and coil was wounded by a...
Thread 'Beauty of old electrical and measuring things, etc.'
Even as a kid, I saw beauty in old devices. That made me want to understand how they worked. I had lots of old things that I keep and now reviving. Old things need to work to see the beauty. Here's what I've done so far. Two views of the gadgets shelves and my small work space: Here's a close up look at the meters, gauges and other measuring things: This is what I think of as surface-mount electrical components and wiring. The components are very old and shows how...
Back
Top