# Why does inductor current lag the applied voltage at its terminal by 90 degrees?

I studied electromagnetism and I am currently working with Inductors.

I could not figure out the physical reason based on electromagnetics on why inductor current lags the applied voltage by 90 degrees?

Everywhere I find the Formula as the starting point of explanation which is partially satisfying.

Induced EMF =L di(t)/dt

From the above equation, I understood that when there is a change in current there will EMF induced in the coil terminals. But How this induced EMF makes the current to lag the supply voltage V(t) = Vm*sin(wt)? (Please don't just differentiate and say there is a lag, I am looking for a physical sense on why this happens)

Lets us consider the AC voltage is applied across the terminal at time t=0; At this time there is almost no current. From this transient state how the steady-state is reached where the current is delayed with respect to the voltage?

Can someone help me to figure out why the current lags the Supplied voltage by using electromagnetism, the motion of Charges or any other basic concepts?

• What does “microscopic reason” mean here? I am very confused by your question. You say “I understood that when there is a change in current there will EMF induced in the coil terminals” but then reject $v=L \ di/dt$ but they are literally the same thing – Dale Aug 10 '19 at 13:14
• @Dale Everywhere I see the steady-state of the inductor current and voltage waveforms in AC circuit are considered? What happens initially for the first time when an alternating voltage is applied across the terminals? – VKJ Aug 10 '19 at 13:55
• @Dale I have edited the question. – VKJ Aug 10 '19 at 15:04
• If you consider a turn on transient, you don't have a single frequency signal, so you have to think carefully about how you will define the "phase" of the voltage and current signals in that scenario. – The Photon Aug 10 '19 at 15:40
• Most circuit theory texts teach this using the Laplace transform, but I'm not sure if that's what would give you the best qualitative understanding. – The Photon Aug 10 '19 at 15:42

The EMF will be induced only after the current starts flowing.

This is not correct. The EMF in an inductor has nothing to do with the magnitude of the current that is flowing (for example, whether it is zero or non-zero).

It only depends on whether the current is changing.

From a physics point of view, this comes from Faraday's Law of Induction:

$$\mathcal{E} = -\frac{d\Phi}{dt}$$

In the inductor, the magnetic flux $$\Phi$$ is proportional to the current, so we can express this as the constitutive relation of the inductor,

$$V=L\frac{dI}{dt}$$

You can see it's entirely possible for there to be an EMF produced, even if the inductor current is zero, so long as the rate of change of the current ($$\frac{dI}{dt}$$) is non-zero.

If you connect, at $$t=0$$, an ideal voltage source to an ideal inductor, the inductor EMF is immediately produced to counter the applied voltage (satisfying Kirchhoff's Voltage Law), and the inductor current immediately starts changing.

The current signal lags the voltage signal if the applied voltage is sinusoidal because the current continues to increase ($$\frac{dI}{dt}>0$$) for as long as the applied voltage is positive, thus it reaches its peak when the voltage just returns to 0 at the end of the positive half-cycle of the voltage waveform.

• Comments are not for extended discussion; this conversation has been moved to chat. – Chris Aug 11 '19 at 17:23

Another answer has explained that the current lags behind the voltage because the induced EMF in the inductor "cancels out" some or all of the applied voltage.

One way to see why it lags by a phase of 90 degrees, and not some other value, is the fact that unlike resistors, ideal inductors and capacitors do not dissipate any energy. They store the energy in a magnetic field (inductor) or electric field (capacitor) as the applied current (inductor) or voltage (capacitor) increases, and release it again as it decreases. The only phase angles between the current and voltage which don't dissipate any energy over a complete cycle are +90 and -90 degrees. The current lags the voltage by 90 degrees in an inductor, and leads it by 90 degrees in a capacitor.

For real inductors and capacitors, their internal resistance does dissipate some electrical energy as heat, and the phase angle is not exactly 90 degrees. The phase angle in a resistor is of course zero.

• Can I say that the energy transfer is done at a slow rate to the magnetic domains from electrons that overcome induced emf? So the flow of charges is also slowed and hence the current slow down/lag the voltage? – VKJ Aug 11 '19 at 15:57

I think the key "physical" reason here is that the induced EMF opposes the change in the supplied voltage. The total voltage across the inductor, that consists of the supplied voltage + the induced EMF, tends to lag behind the supplied voltage as it builds up because of the opposition of the induced EMF. The lag in current is a direct consequence of that.

• In general It's said that induced EMF will be of same magnitude and opposite in direction in case of varying supply voltage.So the sum of supplied voltage and induced EMF won't be zero??? – VKJ Aug 10 '19 at 16:47
• For an ideal inductor with zero resistance, the supplied voltage and induced EMF are equal and opposite. If you applied a constant voltage, the current would increase linearly for ever, to maintain a constant induced EMF. Of course this doesn't happen for a real power supply which can't produce an infinite current, or for a real inductor where the winding has some resistance. In that case the induced EMF is less than the supplied voltage, and the difference is given by Ohm's law using the internal resistance of the inductor. – alephzero Aug 11 '19 at 1:32
• @alephzero Can I say that the EMF induced across the terminals of the inductor will cause current between the terminals of the inductor and thus this current continues to flow out of the inductor - the circuit?? – VKJ Aug 11 '19 at 2:52

It is easy to show this using the terminal relation $$v = L di/dt$$, but you are looking for another explanation.

It is easier to see that happens to the current when you hook up a voltage source across an inductor if you use the integral form of the terminal relation: $$i(t) = i(0) + \frac{1}{L}\int\limits_0^tv(t')dt'.$$

Intuitively, without ever referring to this equation, you know that inductors resist the change in the current through them, in accordance with Faraday's law. Since the magnetic field within the inductor cannot change abruptly with time, neither can the current, which is proportional to the magnetic flux. When you apply time-varying voltage, the current cannot immediately follow this variation, a consequence of which is the phase lag when the voltage is sinusoidal. The mathematical way of saying this is that up to a constant, the current is proportional to the time integral of the voltage.

If you just start applying a constant voltage across the inductor at $$t=0$$, the current will increase continuously for as long as the voltage is applied. This illustrates the inability of the current to catch up with sudden changes in voltage, and the "delay" in the current response to a voltage input.

• Can I say that the energy transfer is done at a slow rate to the magnetic domains from electrons that overcome induced emf. So the flow of charges is also slowed and hence the current slow down/lag? – VKJ Aug 11 '19 at 15:42
• No, there don't need to be magnetic domains at all. These domains exist in ferromagnetic materials, which magnetic induction doesn't require. Work is done by pushing charges against the induced emf: this stores energy in the magnetic field through the inductor. I don't see a direct connection between this and the "charge flow slowing down" though. At the end it comes down to Faraday's law, for which I suspect there may not be an explanation as simple as you would like there to be. – Puk Aug 11 '19 at 17:17
• Then Why is that magnetic field can't change abruptly?? – VKJ Aug 11 '19 at 20:07
• Because of Faraday's law. An abrupt change in magnetix flux would be accompanied by infinite emf, or infinite E-field. This is not physical. – Puk Aug 12 '19 at 1:19