# What prevents abrupt voltage change in a circuit with a capacitor?

Let's say we have a simple circuit consisting of a power supply and a resistor, and currently the input voltage is 0V. We now apply a voltage of 5V to the circuit (like a step increase - instantaneously). The voltage across the resistor changes instantaneously to 5V.

If a capacitor is introduced into this circuit, it will gradually charge until the the voltage across it is also approximately 5V, and the current in this circuit will become zero.

My question:
What is now preventing us from suddenly changing the voltage from 5V to let's say 10V (again like a step increase - instantaneously)? We could do it before the capacitor was introduced, but why not now?

The answer I have thus far always gotten is that for that to happen, according to i=C*(dv/dt), the current flowing in the circuit must be infinite, and since that cannot happen, the voltage cannot be changed instantaneously.

The problem is that we ARE changing the voltage of the power source instantaneously, just like before the capacitor was introduced. The introduction of the capacitor has not somehow taken away our ability to change the voltage of the power source (instantaneously), has it???

So, in this circuit with the capacitor included:

1. Can we change the input voltage instantaneously or not? (theoretically)
2. If we can, what will happen to the current? Will it try to become infinity as i=C*(dv/dt) states? Or is there something else that I'm missing?
3. If we can't, is the capacitor somehow magically preventing this change? How can any component actually affect the input voltage of the circuit?
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The current can't become infinite, otherwise the voltage would become 0. Power supplies aren't some magic electricity-making thing. –  Ignacio Vazquez-Abrams Aug 31 '14 at 13:57
Then what will happen? –  M. Hassaan Aug 31 '14 at 14:31
It appears that when you're adding the capacitor to the circuit you're connecting it in parallel with the resistor. True? –  EM Fields Aug 31 '14 at 18:21
No, it's in series. –  M. Hassaan Sep 2 '14 at 15:28

It's like the "paradox" of the immovable object meeting the irresistible force. In reality, neither can exist.

A real power source will have some impedance. A real capacitor will have some impedance. Real wires connecting them have resistance and inductance.

So in reality when you slap a fairly 'stiff' power source across a fairly good real capacitor there's a spark and the capacitor charges through those series resistances with some ringing and stuff due to the inductances. Ignoring the inductances, the voltage difference would simply divide in ratio to the internal resistance of the capacitor and the internal resistance of the power supply and the wire resistance.

To answer your specific question: If the capacitor and voltage source and wires are ideal, you have a mathematical problem, like division by zero. It's of no consequence in the real world- it just illustrates that the ideal models of the power source and the capacitor and wires are insufficiently accurate to describe their real-world behavior. Modelling any one of those as a real part with real resistance (and inductance) will make the mathematical problem go away, but it won't likely give you an accurate indication of what is actually happening.

For example, if the wire (or the capacitor or the power supply) had 10m$\Omega$ resistance, and there is a 10V difference, you could predict you'd see 1000A (which is very high, but not infinity) and the capacitor would charge very quickly. In reality that isn't likely going to happen because of other non-ideal factors. If the 10m$\Omega$ was modeled as in the power supply, the power supply voltage would drop. If the 10m$\Omega$ was modeled as in the capacitor, the voltage would suddenly appear across the capacitor terminals. If the 10m$\Omega$ was modeled as in the wire, the voltage would appear across the wire. But none of those is very realistic.

If you modeled as a circuit with no resistance at all and a tiny bit of inductance (even superconducting wires of any length have inductance) then a simple mathematical model would predict ringing that would persist forever, energy sloshing back and forth between the inductance and capacitance at an angular frequency of $\omega_0 = {1 \over \sqrt{LC}}$.

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If we continue with this simplistic model, with everything ideal, is there no way to explain what will happen if we change the voltage instantaneously? Do we absolutely HAVE to use a more accurate model? –  M. Hassaan Aug 31 '14 at 14:48
Yes, you have to use a more accurate model or it blows up mathematically. You could use a very small resistance and look at what happens as the resistance -> 0, but it won't be remotely close to what really happens at the instant of connection (the prediction for some time later should be fairly accurate- but that's not what you're asking about). Alfred's use of the Dirac delta function shows one way that it can be treated mathematically. –  Spehro Pefhany Aug 31 '14 at 15:15
Yes, you need a more accurate model. That model is based on Maxwell's equations, which take into account the movement of charges and the electrical and magnetic fields they create. All other statements, such as "you can't instantaneously change the voltage on a capacitor" are simply verbal shorthand for commonly-encountered situations, but don't necessarily apply in the corner cases such as what you are proposing. –  Dave Tweed Aug 31 '14 at 15:17

I like Spehro's answer, but I think he might be over-reading your question. In this 5V to 10V step is the resistor still in the circuit. If so then the step from 0V to 5V, will look pretty much like the step from 5V to 10. There will first be all the (extra) voltage across the resistor (5 Volts), and the C will start to charge up to the final 10V.

(Or maybe I'm making your question too simple?)

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Basically I've read that a capacitor resists abrupt changes in voltage, and I'm trying to create a scenario in which we put it against such a change, and saying that "Hey, we CAN change it abruptly." But I don't know what will happen afterwards. –  M. Hassaan Aug 31 '14 at 14:54
@GeorgeHerold I bought a Sphero the other day. It's a battery powered ball that rolls around under smart phone or tablet control. ;-) –  Spehro Pefhany Aug 31 '14 at 15:18
OK, Then +1 for Spehro's answer. In any circuit there are all sorts of different time scales, infinitely fast exists only in our imagination. But it's a useful approximation sometimes. –  George Herold Aug 31 '14 at 15:20
Grin, I say that. Spher... just trips so lightly off the finger tips. –  George Herold Aug 31 '14 at 15:23

Can we change the input voltage instantaneously or not? (theoretically)

The answer is a qualified yes. Formally, the voltage across the capacitor can be of the form

$$v_C(t) = 5u(t)$$

where $u(t)$ is the unit step function. In that case, the capacitor current is

$$i_C(t) = C\cdot 5\delta(t)$$

where $\delta (t)$ is the unit impulse 'function' (distribution) which is roughly defined as being zero for all $t$ except when $t=0$ where the area is equal to one.

However, this solution is maximally non-physical since, in ideal circuit theory, we assume that voltage and current rates of change are small enough that electromagnetic radiation can be ignored.

When we get a solution that is not consistent with this assumption, the solution is non-physical and, thus, of 'academic' interest only.

Even if the voltage source internal impedance could be made arbitrarily small, one cannot avoid the inherent self-inductance of a circuit that encloses a non-zero area.

Thus, to find a physically relevant solution, we must model the self-inductance and radiation resistance of the circuit as is done here for the "two-capacitor paradox".

When this done, we find that the solution for the voltage across the capacitor does not instantaneously change and the circuit current is finite.

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Could you please elaborate on what you mean by "maximally non-physical science"? –  M. Hassaan Aug 31 '14 at 15:00
@MuhammadHassaanAyyub, to instantaneously change the voltage across a capacitor by a finite amount requires that one instantaneously change the charge on each plate by a finite amount. This would require a current impulse. But, as you many know, a current impulse requires infinite bandwidth, i.e., a current impulse contains all frequencies with equal weight. But there is no physical circuit or capacitor that even remotely approximates this characteristic. –  Alfred Centauri Aug 31 '14 at 21:44