# How does a negative current start in the inductor in ac circuit?

As shown in this image, how does a negative current appear at the beginning, when the voltage starts to increase ?

I understand the current lead lag mathematics behind it. I just can't figure out, what is going on with the electronics inside the wire. How can an increasing voltage, cause a reverse current ?

Update

Thanks to Andy aka for pointing out the right solution with correct references. I realised that the answer to the anomaly lies in the equation of the inductor.

Since for an inductor v = L * di/dt

Therefore current = Integral (v/L * dt)

Integral of Vsin(wt)/L = -Vcos(wt)/L

And the TRICKY part is to take integrals between 2 points A and B

So the current at a time t = -Vcos(wt)/L - (-Vcos(0))/L = -Vcos(wt) + V/L

The cos(0) is the DC component that "lifts" the entire current wave above the 0-line so the current keeps flowing only in positive direction (assuming a 0 resistance path). This is mentioned here -

Observation

This is probably why the SMPS of my CPU cabinet makes a loud noise right when I turn it on. The sound persists for about a minute and then fades away. Because the voltage supply starts with a 0V at t=0 and the ac current flows only on the positive side of the graph above the 0 level, giving a higher dc value.

Notes

I tried reading a few more online tutorials on circuits to understand the very basics of RL and RC AC circuits.

1. Every RL and RC ac circuit has 2 currents. One is the steady-state current I(ss) and another is the transient current I(tr).

2. The phasor method calculates only the steady-state current of it. Which many believe to be the "real" behavior of the circuit, while it is not.

3. This link shows the calculation of the transient current in a RL ac circuit - http://www.ee.nthu.edu.tw/~sdyang/Courses/Circuits/Ch09_Std.pdf

4. I have created a partsim simulation of the RL ac circuit, which shows the whole thing nicely - https://www.partsim.com/simulator/#69215

• Simple way to think of an inductor is an element that fights against changes in current, the same way that a capacitor is an element that fights against changes in voltage. – lucas92 Jan 20 '17 at 14:14

When you apply a sinewave voltage to an inductor, current has no option other than to begin at zero - it cannot suddenly adopt a negative value as would be implied in the steady-state waveforms.

In fact it has a name - it's called inrush current and particularly in motors, inductors and transformers it can cause core saturation because, the current rises above the peak of the normal steady-state value: -

This is why some inductive loads are connected to the AC supply when the voltage waveform is at its maximum peak (negative or positive) because that is when the current would naturally flow through a zero-crossing point.

Without core saturation (or other losses) the current would remain having a positive average DC value. Here's a picture that shows what I mean: -

Picture taken from here.

When voltage is applied at a zero crossing the current will remain having a dc value for some time and this dc offset will ebb away with losses due to R1.

• I was just about to add another answer here, but you almost have it in its entirety, with the exception of one alteration that would make it perfect. 'Without core saturation (or other losses)' is not the condition. If you replace that with 'without series resistance in the inductor' then it would be right. Shunt losses don't have that effect, and neither does saturation (though saturation increases current magnifying the effect of series resistance). Core losses equivalent to series R do work, but that can easily be omitted at this level of discussion. – Neil_UK Jan 21 '17 at 6:21
• amazing, finally someone demystified the solution !! – Silver Moon Jan 21 '17 at 6:33
• @Andyaka Can the dc component be neutralised by adding a low impedance capacitor in series with the inductor ? – Silver Moon Jan 22 '17 at 13:27
• You have to choose a capacitor that is big enough in value to avoid series resonance and therefore you will still get the problem for several cycles. Why not simulate using LTSpice or similar. – Andy aka Jan 22 '17 at 13:42
• @Andyaka Is there a way to completely suppress the transient dc right at t=0, no matter how the voltage starts ? – Silver Moon Jan 23 '17 at 5:56

What is shown in the diagram is about one period in steady state.
It does not show the conditions when the sinusoidal soucre is switched on at t=0 (transient state).

I.e. you have to imagine that many more cycles have happened before. So at t=0 the coil is already energized and current and voltage phases are stabilized.

"Start of the alternating current cycle" does not mean that alternating current is switched on at that moment. It is just the start of the repeating cycle.

• what would happen in a real circuit, if everything was 0 at t=0. how would the graph of current look compared to the voltage ? – Silver Moon Jan 20 '17 at 11:38
• You need more information: especially you need to know the series resistance of the circuit (which is ignored in this very basic picture and doesn't matter in steady state as long as it is small). – Curd Jan 20 '17 at 11:44
• is it really that complicated to understand. the capacitor circuit with ac source was much more intuitive and easier to visualize. – Silver Moon Jan 20 '17 at 12:02
• Actually capacitor with voltage source works in principle exactly like inductor with current source. That principle is called duality. – Curd Jan 20 '17 at 14:39
• Points up for paragraph one..!! – soosai steven Jan 20 '17 at 18:07

An inductor is a energy storage device but it stores energy as magnetic energy. As opposed to a capacitor which stores energy as electrical energy (electrons).

If you start with thinking about current through an inductor what you have to think about is the rate of change of current through the inductor resulting in certain amplitude of voltage across it that opposes the rate of change. So it generates a negative voltage across it. When I say negative I mean the opposite polarity of a voltage that would be generated across a resistor say.

This is essentially Lenz's Law that is important to understand to also understand Faraday's Law.

If you force a voltage across a capacitor its the rate of change of voltage that gives rise to a current that will oppose the rate of change.

At a basic physical level its all about momentum and the fact that if you have mass you cannot move through a medium without something else acting against you. Natural systems do not like change.

In the case of inductors and coils this is really great for us as it allows us to generate power and convert power. $$emf = -L\frac{di}{dt} = -\frac{d\Phi}{dt}$$ $\Phi$ is the magnetic flux

• "If you force a voltage across an inductor its the rate of change of voltage that gives rise to a current that will oppose the rate of change" - methink you talk about capacitors not inductors. In a capacitor, the rate of change of voltage determines the current. In an inductor the rate of change of current determines the voltage. – Andy aka Jan 20 '17 at 17:54