# Why does current lag 90 degrees behind the voltage when an inductor is present?

When we supply a voltage to an inductor then the phase difference between voltage and current is 90 degrees (ideally.)

Mathematically, I have seen how it becomes 90 degrees but can someone help me to understand it in an intuitive way?

• Lenz's Law, in opposing the change of currents in inductors, causes that lag. May 1, 2018 at 3:58
• why 90 deg?? why not any other deg? May 1, 2018 at 3:59
• Because d/dt sin(t) = cos(t) = sin(t + 90°) May 1, 2018 at 4:17
• Or maybe it was sin(t - 90°). Too sleepy to remember right now. May 1, 2018 at 4:24
• @SredniVashtar “If you accept the fact that sinusoidal signals are the building blocks of all other signals” ... not necessarily. We just like to say so because it makes the math easier. May 1, 2018 at 4:51

Because inductors are made to react against the change in current, it causes it to lag behind the voltage. When you apply a voltage to an inductor, you make a magnetic field. Henceforth, if the magnetic field varies with respect to time, there is an electric field that opposes the magnetic field inside the inductor. In other words, the electric field generated by the voltage behaves like a wall to the magnetic field generated by the current.

This behavior is described by Len'z Law.

$\displaystyle EMF=-\frac{\partial \Phi_B}{\partial t}$, which happens to be the negative of Faraday's Law of Induction. I didn't draw this, I just found it on Wikipedia. But as you can see, the magnetic field $B$ is being pushed back.

The 90 degrees that you're thinking of comes from the sine function... The sine function represents the reactive current. When you plot the cosine and sine functions, the graphs are 90 degrees of a phase difference.

If you're asking why it's 90 degrees rather than something else like 45 degrees, then again, it's because of the sine and cosine functions.

So why is there a sine function? Because when you talk about the voltage and current, there's a derivative relationship.

$\displaystyle V(t)_L = L\frac{di(t)_L}{dt} \rightarrow \cos(t)=\frac{d \sin(t)}{dt}=\sin(t-90°)$

Typical AC voltage is described by a cosine sinusoidal function.

The voltage across an inductor is the derivative of the current. So when the current is 0, its derivative is at a maximum. When the current is at the peak of a sinusoid, the derivative is 0.

• Well he said he understand the mathematics May 1, 2018 at 3:50

If you have a volt meter, a coil of wire and a magnet and statically place the magnet close to the coil you do not get any DC voltage shown on the voltmeter. However, if you move the magnet around you get a voltage induced in the coil whose magnitude is proportional to the change in magnetic flux experienced by the coil. In other words: -

$$V = N\dfrac{d\phi}{dt}$$

But you can replace the magnet with a current source. If you apply a ramping waveform of current to the coil it produces a ramping magnetic flux and across the terminals of the inductor the voltage will be constant. That constant voltage is proportional to the rate of change of flux.

You might be able to see that this is mathematical differentiation so, applying a sinusoidal current through the coil (from the current source) produces a voltage that is the differential of the sine wave across the terminals. That means the voltage is a cosine and thus, in AC analysis the voltage and current are displaced by 90 degrees.

The magnetising current (B field) reacts to the rate of change in charge in the source voltage, and the point at which the source voltage rate of change is fastest, is where the gradient of the sinusoidal function is the steepest (which is through the ‘neutral’ voltage point of the source).

An inductor can be visualized as an object having inertia, with force being applied through a spring. The force (voltage) compresses the spring, but the object moves with some delay (phase shift). If the force is then reversed, the object will slow down and then reverse direction as the spring alternately stretches and compresses. The action is something like a yo-yo, or a paddle ball on an elastic cord. When the paddle or the hand move up and down in a smooth motion, the object will follow, but with a delay that is analogous to a phase shift.