# How can an AC circuit have positive current and negative voltage?

While trying to fully understand capacitors, I have run into a concept I can't understand.

I understand that voltage lags behind current by 90 degrees when connected to an AC circuit, and I understand why it happens, but how can you have positive current and negative voltage for those 90 degrees? How can voltage flow one way and current flow the other? Maybe I'm thinking about it too hard.

• What happens with a battery? When there is an energy source the current flows in the opposite direction to with an energy sink. Apr 9, 2022 at 1:15
• Note that voltage does not flow. Apr 9, 2022 at 1:27
• Lots of answers for you, no responses at all from you. Not a good omen, my opinion. Are you still there? Able to expand your question with a specific, concrete example from somewhere you are reading that has led to your question here? It would help to see what you are looking at... a lot.
– jonk
Apr 9, 2022 at 5:11

How can an AC Circuit Have Positive Current and Negative Voltage?

Current and voltage in a circuit are completely independent, and only depend on what sources/loads are connected together.

You don't need AC for it. You don't need capacitors. It works at DC. And since you can control it independently at DC, you can also do so at AC. You can have the voltage be a sine wave, and the current be a triangular wave. I'm not kidding. Independent means independent.

Below, on the left, is a circuit that does just that at DC: a voltage source connected to a current source. Both can be configured for whatever voltage and current value you want. You can have non-zero current with zero voltage, non-zero voltage with zero current, positive current with negative voltage, and vice versa.

simulate this circuit – Schematic created using CircuitLab

Voltmeter VM1 is optional. You can remove it. it's just a single circuit, with two active components: a voltage source and a current source. A single loop. We connect a current meter in series with it, and a voltmeter in parallel with the voltage source.

On the right above you can see what happens when you replace the current source with a 1 Ohm resistor. The current then flows the "normal" way. Or, rather, the way you consider "normal".

This may seem counterintuitive because it's easy to only think of resistors connected to ground as valid loads. Resistors are useful, but they are "just" voltage-controlled current sources, whose control equation is Ohm's law: $$I(t)={1 \over R}\cdot V(t),$$ where $$\t\$$ is time.

But there's a wide variety of control equations you can use for a current source, including one where the current is independent of voltage but changing, or even just constant. A capacitor also acts as a voltage-controlled current source, it just uses a different control equation - one that's a differential equation, specific to ideal capacitors: $$I(t)=C \cdot {{\rm d}V(t) \over {\rm d}t}.$$

Now you may say: hey, no fair, that's just some ideal elements in a simulator. No way you can build one, right? Ahem. You can. Not only that, you can make an OK one from about 10 bipolar or mosfet transistors, or two op-amps. And no, the op-amps don't need to be unobtainium or ideal. Just a basic LM358 or two LM741s will do.

simulate this circuit

But wait! There's more. You don't need anything as "weird" as "current sources" or "fancy" as op-amps. Two batteries and a resistor is all you need. And yeah, with a resistor you can still get negative current: you need to connect other end to something other than 0V:

simulate this circuit

It gets even fancier: you can build virtual capacitors without using any capacitors at all, just op-amps and inductors. Such circuits are called gyrators. They invert impedances, so a complex impedance of an inductor is made to "appear" to the connected circuit as a complex impedance of opposite sign: a capacitor.'

And it gets fancier than that still: you can implement any V-I relationship you want in code, using an ADC, a DAC, a voltage-controlled current source circuit, and a CPU. At low frequencies - say <100Hz, this works using some super-rudimentary devices like an Arduino Uno, its internal ADC, a PWM digital output pin to do the job of a DAC. For small currents (single mA) and voltages between 1-2V, you only need a capacitor to convert the PWM digital output into a variable bidirectional current, controlled by some equation implemented in code. Within those basic limits, you can make the thing act like a capacitor, an inductor, a tank circuit, a filter, and so on.

I think this may be useful to you: https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/pages/readings/

This will walk you through electromagnetism in a more focused way than is possible here.

Localised current through a resistor, wire or other device always flows from high to low voltage; if there’s no voltage then current won’t flow. However, localised voltage doesn’t necessarily correlate with supply voltage - if the supply is at its peak and connected to a capacitor that’s charged to the same voltage then no current will flow. You’ll often hear the term ‘potential difference’ when referring to voltage between two points.

• That’s not correct. Current can flow without a difference in voltage - if it is induced by a magnetic flux change. Apr 9, 2022 at 3:08
• @hacktastical - there must still be a voltage present, otherwise transformers would deliver current but no voltage.
– Frog
Apr 9, 2022 at 3:11
• @Frog Next time you have some superconducting wire to play with, you may want to try it out. Keep the field strength within limits so as not to lose superconductivity. Apr 9, 2022 at 3:15
• Does "other device" include inductors, capacitors, generators, ...? Apr 9, 2022 at 3:56
• @The Photon - touché - but although current flows in and out of a capacitor it doesn’t flow through it. Its’s borderline physics/semantics/philosophy though.
– Frog
Apr 9, 2022 at 9:03

Here is a better tutorial on "Series RLC Circuit Analysis" with good visuals, all the math, and detailed explanations for the leading/lagging voltages on the components.
https://www.electronics-tutorials.ws/accircuits/series-circuit.html

We have an AC source, a capacitor and a resistor.

Let's say the AC source is at 0.5 V. If the capacitor is currently at -0.7 V, current will flow against the voltage direction of the AC source (negative you could call it).

Conversely, if the AC source is at -0.5 V, while the capacitor is at 0.7 V, current will flow in the positive direction while the voltage source is negative.

So there could be situation where the capacitor overpowers the voltage source and current flows in a direction opposite to the voltage source's direction.

Small note: There must be a resistor in the circuit - if all you have is an AC source and a capacitor, it wouldn't converge as the AC source and capacitor's voltages won't necessarily cancel out (in real life there's some resistance in the wires even if you don't put a resistor in a series).

I highly recommend the following circuit simulator: