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29th January 2018, 16:47 #1
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capacitor charge with current source + resistor
Hi all,
I know that I should be able to do this within 5 minutes, but I am not.
I simply do not remember my mathematics well enough anymore! :_(
I have a capacitor C.
It is charged using a constant current source (Ik) and
parallel to that source is a resistor R from supply Vcc, charging it with current Ir.
Total charge current through the capacitor is hence Ic = Ik + Ir.
I want to deduct an equation telling me what the voltage Vc across a capacitor is, at any specific time t from time = 0.
Charge is 0 at time = 0.
I can easily simulate it, but now I want to have that equation, because I want to calculate it.
If you can help, please do. :)
/c0x

29th January 2018, 18:56 #2
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Re: capacitor charge with current source + resistor
I'm very confused. Can you show a schematic?
But if what you say "Ic=Ik+Ir" is true, then your equation is simply:
Vc=1/c*Integral(Ic dt)
But Ir is going to vary as the capacitor charges, right?
We need a schematic.

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29th January 2018, 20:23 #3
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Re: capacitor charge with current source + resistor

29th January 2018, 21:32 #4
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29th January 2018, 22:22 #5
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29th January 2018, 22:38 #6
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Re: capacitor charge with current source + resistor
Superposition is the easiest method for this specific circuit. The circuit implements 2 simple first order LPF for both sources, .
Here is your solution. Just apply superposition and see page 4: http://hamblen.ece.gatech.edu/3710/ch5.pdfLast edited by CataM; 29th January 2018 at 22:48.

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30th January 2018, 00:04 #7
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Re: capacitor charge with current source + resistor
Hi CataM,
Sounds simple, but not so easy for me. Maybe I should go back to school.
Am I on the right track if I think that one way to go is:
* generating expressions for Ic in s domain for both sources separately,
* then add the expressions to get combined expression for Ic,
* reduce and
* finally do inverse Laplace to get a time domain expression?
/c0x

30th January 2018, 03:36 #8
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Re: capacitor charge with current source + resistor
You are correct in acknowledging that you need a good review of linear differential equations. I think the best way to do this problem is by node analysis, because you need only one equation with only one unknown, specifically the voltage across the capacitor. Call that voltage "v".
The node equation is:
Solving for v, we get:
Substituting c = 1 nF :
Find the inverse Laplace:
Plot the equation for 50 microsecconds:
Any questions?
RatchHopelessly Pedantic
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30th January 2018, 07:02 #9
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Re: capacitor charge with current source + resistor
Ik=100 uA =0.1 mA; given.
Ir=5/10k =0.5 mA; calculated (valid at t=0)
Const current source has infinite impedance. IrIk must flow through the capacitor. (t=0)
The capacitor charges exponentially to final potential of 4V with a time const of 10k * 10nF
How do you get 6V at t = infinity?

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30th January 2018, 07:17 #10
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Re: capacitor charge with current source + resistor
How do you get 6V at t = infinity?

30th January 2018, 07:43 #11
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Re: capacitor charge with current source + resistor
Yes, that is true.
Const current source has infinite impedance. IrIk must flow through the capacitor. (t=0)
The capacitor charges exponentially to final potential of 4V with a time const of 10k * 10nF
How do you get 6V at t = infinity?Hopelessly Pedantic

30th January 2018, 08:05 #12
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Re: capacitor charge with current source + resistor
5V + 10k*100µA according to the source polarities in the post #3 schematic ...

30th January 2018, 09:33 #13
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Re: capacitor charge with current source + resistor
Seeing the solution I can accept the procedures. Some I recognize, some is forgotten long time ago. Maybe some of it I never actually learned.
I acknowledge that I should practice this sort of circuit analysis more, and I will try and do so, because this is so simple and I want (and need) to be able to do it.
For now, I can move on. Thank you Ratch, and everyone else who contributed.
Yay, my LTSpice and Console Calculator finally agree. :)
/c0x

30th January 2018, 14:38 #14
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Re: capacitor charge with current source + resistor
Yes but it is to much time consuming.
An easier method is like this:
1) Transform the voltage source and resistor to current source in parallel with resistor. https://en.wikipedia.org/wiki/Source_transformation
2) Sum up both current sources.
3) Convert the summed current sources in parallel with the resistor to voltage source in series with resistor
4) The remaining circuit is already solved in the link of post #6.

2nd February 2018, 18:15 #15
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Re: capacitor charge with current source + resistor
No current exists through the capacitor. If it did, the capacitor would be defective. Charge accumulates on one plate and depletes on the other plate for a net charge gain/loss of zero. The capacitor separates the charges and thereby becomes energized. That means the capacitor stores energy, not charge.
Flow of charge is current. The current through the dielectric is called "displacement current" see for details: https://en.wikipedia.org/wiki/Displacement_current
Alternating current flows through a capacitor just like a resistor: except for the phase.
   Updated   
Some I recognize, some is forgotten long time ago. Maybe some of it I never actually learned...
The initial condition is not defined: it will be tempting to assume that you want a solution of the steady state (which is trivial).
To give a possible idea (there are several) consider a switch on the left of the top node connecting the const current source to the rest. The time starts when you close the switch and you wish the voltage and currents as a function of t.
If you consider the switch on the capacitor side (top central node; connecting the capacitor) then you may get a different result.
But if you consider the switch connected to the resistor side (right of the node) then you get another result (that can be rather interesting)...
And there are other possibilities.
But the problem can be more complex in real life.

2nd February 2018, 18:42 #16
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Re: capacitor charge with current source + resistor
Capacitance is a proportional measure of the amount of energy a capacitor can store at a specified voltage. A 1 farad capacitor can store a half joule of energy at one volt.
Charge must flow. The capacitor stores charge.
The energy is actually stored in the dielectric (it can be a vacuum) in the electric field.
Flow of charge is current.
The current through the dielectric is called "displacement current" see for details: https://en.wikipedia.org/wiki/Displacement_current
Alternating current flows through a capacitor just like a resistor: except for the phase.
   Updated   
c0x,
Thanks for your thanks. The conditions I assumed was that the capacitor was initially unenergized, and both the current and voltage sources were switched on at the same time.
RatchHopelessly Pedantic

2nd February 2018, 19:12 #17
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Re: capacitor charge with current source + resistor
The net charge of a capacitor is the same at 0,100 or 1000 volts, specifically zero...
Capacitance of earth is about 100 uF

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2nd February 2018, 19:34 #18
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7th May 2018, 05:42 #19
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Re: capacitor charge with current source + resistor
Can you show a schematic?
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